Annales Henri Poincaré - Volume 4

Ann. Henri Poincar´e 4 (2003) 1 – 34 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/0101-34 DOI 10.1007/s00023-003-01...

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Ann. Henri Poincar´e 4 (2003) 1 – 34 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/0101-34 DOI 10.1007/s00023-003-0120-1

Annales Henri Poincar´ e

Elliptic-Hyperbolic Systems and the Einstein Equations Lars Andersson and Vincent Moncrief Abstract. The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasi-linear elliptic-hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum space-times.

1 Introduction In order to construct solutions to classical ﬁeld equations with constraints, such as the Yang-Mills and Einstein equations, it is often necessary to rewrite the system, either by extracting a hyperbolic system, or by performing a gauge ﬁxing. The gauge ﬁxing may result in a hyperbolic system, as is the case for example using Lorentz gauge for the Yang-Mills equations, or space-time harmonic coordinates for the Einstein equations. For discussions of hyperbolicity and gauge choices for the Einstein equations, see [11, 6]. On the other hand, there are interesting gauge choices which lead to a coupled elliptic-hyperbolic system, such as the Coulomb gauge for Yang-Mills, which was used in the global existence proof of Klainerman and Machedon [10]. For the Einstein equations, the constant mean curvature gauge leads to an elliptic equation for the Lapse function, and to an elliptic-hyperbolic system for the second fundamental form kij , cf. [4]. In this paper we introduce and study a gauge condition for the Einstein equations, which is a combination of constant mean curvature gauge and a spatial harmonic coordinate condition. This leads to an elliptic-hyperbolic system where the hyperbolic part is a modiﬁed version of the Einstein evolution equations, and where the elliptic part consists of the deﬁning equations for Lapse and Shift.

1.1

The gauge fixed vacuum Einstein evolution equations

Let M be a compact, connected, orientable C ∞ manifold of dimension n ≥ 2 and ¯ = R × M . We deﬁne t : M ¯ → R by projection on the ﬁrst component. We let M ¯ so that the level sets of t, Mt = {t} × M are will consider Lorentz metrics g¯ on M Cauchy surfaces. When there is no room for confusion we write simply M instead of Mt .

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¯ , g¯), let T be a time-like normal to Mt , let the Lapse function N Given (M and Shift vector ﬁeld X be deﬁned by ∂t = N T + X, and assume N > 0 so that T is future directed. Let {ei }ni=1 , be a time-independent frame on M and let {ei }ni=1 be its dual frame. Let e0 = T and let {eα }nα=0 = {e0 , e1 , . . . , en } so that {eα }nα=0 is an ¯ (which is neither global nor time-independent), with dual adapted frame on M frame {eα }nα=0 . In some cases we will use local coordinates xi , and use the notation ∂i = ∂/∂xi , for the coordinate frame and the corresponding ﬁrst derivative operators. In the following we will use frame indices, unless otherwise speciﬁed, and let greek indices take values in 0, 1, . . . , n while lower case latin indices take values in 1, . . . , n. The Lorentz metric g¯ is of the form g¯ = −N 2 dt ⊗ dt + gij (ei + X i dt) ⊗ (ej + X j dt),

(1.1)

where g = gij ei ⊗ ej is the induced metric on M . The second fundamental form k ¯ is given by of M in M 1 1 kij = − LT g¯ij = − N −1 (∂t gij − LX gij ), 2 2 where L denotes the Lie derivative operator. The vacuum Einstein equations ¯ αβ = 0 R

(1.2)

can be written as a system of evolution and constraint equations for (g, k). The vacuum Einstein evolution equations are ∂t gij = −2N kij + LX gij , ∂t kij = −∇i ∇j N + N (Rij + trkkij − 2kim k mj ) + LX kij ,

(1.3a) (1.3b)

and the vacuum constraint equations are R − |k|2 + (trk)2 = 0,

(1.4a)

∇i trk − ∇ kij = 0.

(1.4b)

j

A solution to the Einstein evolution and constraint equations is a curve t → (g, k, N, X) which satisﬁes (1.3,1.4). Assuming suﬃcient regularity, the space-time metric g¯ given in terms of (g, N, X) by (1.1) solves the vacuum Einstein equations (1.2) if and only if the corresponding curve (g, k, N, X) solves (1.3,1.4). The system (1.3,1.4) is not hyperbolic, and to get a well-posed evolution problem we must modify the system. We will do this by ﬁxing the gauge.

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Let gˆ be a ﬁxed C ∞ Riemann metric on M with Levi-Civita covariant derivaˆ ˆ k . Deﬁne the vector ﬁeld V k by tive ∇ and Christoﬀel symbol Γ ij ˆ i ej ), V k = g ij ek (∇i ej − ∇

(1.5a)

ˆ kij ). V k = g ij (Γkij − Γ

(1.5b)

or in local coordinates,

Then −V k is the tension ﬁeld of the identity map Id : (M, g) → (M, gˆ), so that Id is harmonic exactly when V k = 0, see [5] for background on harmonic maps. The constant mean curvature and spatial harmonic coordinates (CMCSH) gauge condition is given by the equations trg k = t k

V =0

(constant mean curvature)

(1.6a)

(spatial harmonic coordinates).

(1.6b)

Remark 1.1. In this paper we will restrict our attention to the homogeneous gauge conditions given above. However, it seems likely that the ideas presented here can be generalized to include gauge source functions, with the gauge conditions (1.6) replaced by for example trg k = t + f 0 , k

k

V =f ,

(1.7a) (1.7b)

where f 0 , f k are a function and a space-like vector ﬁeld on M , independent of the data. ˆ g be deﬁned on symmetric 2-tensors by Let the second order operator ∆ ˆ g hij = 1 ∇ ˆ m (g mn µg ∇ ˆ n hij ), ∆ µg

(1.8)

√ ˆ m (g mn µg ) where µg = det g is the volume element on (M, g). Using the identity ∇ n ˆ g hij may be written in the form = −V µg , ∆ ˆ g hij = g mn ∇ ˆ m∇ ˆ m hij − V m ∇ ˆ m hij . ∆ ˆ m∇ ˆ n hij . A ˆ g hij = g mn ∇ In particular, if the gauge condition V = 0 is satisﬁed, ∆ computation, cf. Section 3, shows that 1ˆ Rij = − ∆ g gij + Sij [g, ∂g] + δij , 2 where the symmetric tensor δij is deﬁned by δij =

1 (∇i Vj + ∇j Vi ), 2

(1.9)

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and Sij [g, ∂g] is at most of quadratic order in the ﬁrst derivatives of gij . Thus the system gij → Rij − δij is quasi-linear elliptic. In order to construct solutions to the Cauchy problem for the system consisting of the Einstein evolution and constraint equations (1.3, 1.4) together with the gauge conditions (1.6), we will consider the following modiﬁed form of the Einstein evolution equations, ∂t gij = −2N kij + LX gij ,

(1.10a)

∂t kij = −∇i ∇j N + N (Rij + trkkij − 2kim k mj − δij ) + LX kij ,

(1.10b)

coupled to the elliptic deﬁning equations for N, X, needed to preserve the imposed gauge conditions, −∆N + |k|2 N = 1, ∆X + R f X − LX V = (−2N k i

i

f

i

(1.11a) mn

ˆ m en ) + 2∇ X )e (∇m en − ∇ m

n

i

i + 2∇m N km − ∇i N kmm .

(1.11b)

If δij = 0, in particular if V k = 0, then (1.10) coincides with the Einstein vacuum evolution equations (1.3). The vacuum Einstein evolution equations in CMCSH gauge is the coupled system (1.10–1.11). In view of the fact that gij → Rij − δij is elliptic, the system (1.10) is hyperbolic, and the coupled system (1.10–1.11) is elliptic-hyperbolic.

2 The Cauchy problem for quasi-linear hyperbolic systems In this section we prove that the Cauchy problem for a class of quasi-linear hyperbolic evolution equations, which includes coupled elliptic-hyperbolic systems of the form (1.10–1.11), is strongly well posed. The techniques used are not new, cf. [12, 17, 3, 2] for treatments of various aspects of the problem for classical quasi-linear hyperbolic systems. The methods of Kato [8] for general quasi-linear evolution equations can presumably be used to prove the results stated here. However, in view of the abstract nature of the techniques involved in the approach of Kato, we have decided to give a reasonably complete treatment of the Cauchy problem for the class of equations that is of interest from the point of view of applications in this paper and its sequel [1]. Let Λ[g] be the ellipticity constant of g, deﬁned as the least Λ ≥ 1 so that Λ−1 g(Y, Y ) ≤ gˆ(Y, Y ) ≤ Λg(Y, Y ),

∀Y ∈ T M.

(2.1)

In the following, all norms and function spaces will be deﬁned with respect to gˆ. Let g¯ be deﬁned in terms of g, N, X by (1.1). Let Λ[¯ g] = Λ[g] + ||N ||L∞ + ||N −1 ||L∞ + ||X||L∞ .

(2.2)

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For a curve of metrics t → g¯(t), it is convenient to deﬁne Λ(T ) = sup Λ[¯ g(t)].

(2.3)

t∈[0,T ]

We write D for ﬁrst order spatial derivatives. The action of D on tensors is ˆ w.r.t. gˆ. Let D¯ deﬁned using the covariant derivative ∇ g be the ﬁrst order spatial derivatives of g¯. Then |Dg| + |DN | + |DX| ≤ C(Λ[¯ g ])|D¯ g |, |D¯ g | ≤ C(Λ[¯ g ])(|Dg| + |DN | + |DX|), where | · | denotes the pointwise norm. We recall some deﬁnitions and facts from analysis which will be needed in ˆ i∇ ˆ j be the Laplace operator deﬁned the proof of local existence. Let ∆gˆ = gˆij ∇ with respect to the background metric gˆ, acting on functions or tensors, and let

D = (1 − ∆gˆ )1/2 . Let W s,p denote the Sobolev spaces and let H s = W s,2 . Then W s,p = D−s Lp , for s ∈ R, 1 < p < ∞, with norm ||u||W s,p = || Ds u||Lp . In case s is a non-negative integer, W s,p , 1 ≤ p ≤ ∞ is the closure of C ∞ (M ) w.r.t. the equivalent norm |k|≤s ||Dk u||Lp . We will without further notice use the same notation for spaces of tensor ﬁelds as for spaces of functions on M . For I ⊂ R an interval, we use the notation F (I; W s,p ) for the space of curves of class F with values in W s,p . Spaces which will be used are F = C, C 0,1 , C j , L∞ , L1 , W j,1 , where C 0,1 denotes the space of continuous functions with one (time) derivative in L∞ . We use the notation OP s for pseudo-diﬀerential operators with symbol in s the H¨ ormander class S1,0 , see Taylor [17] for details. For P ∈ OP s , s ∈ R, ||P u||W r,p ≤ C||u||W r+s,p ,

for r ∈ R.

(2.4)

In particular, Ds ∈ OP s and D ∈ OP 1 . The following basic inequalities will be used. Assume 1 0, W s,p ∩ L∞ is an algebra, and the inequality ||uv||W s,p ≤ C(||u||L∞ ||v||W s,p + ||u||W s,p ||v||L∞ )

(2.5)

holds. In particular if s > n/p, then ||uv||W s,p ≤ C||u||W s,p ||v||W s,p . 2. Product estimate II (special case of [13, Theorem 9.5 3], see also [17, §3.5]). Assume ti ≥ 0, i = 1, 2, some ti > 0. Then for s ≤ min(t1 , t2 , t1 + t2 − n/p), (where the inequality must be strict if some ti = n/p), ||uv||W s,p ≤ C||u||W t1 ,p ||v||W t2 ,p .

(2.6)

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3. Composition estimate ([17, §3.1], see also [15, Theorem 1]) Let 1 ≤ s < µ, and suppose F ∈ C µ (R) with F (0) = 0, where C µ denotes the H¨older space. Then for u ∈ W s,p ∩ L∞ , ||F (u)||W s,p ≤ C||u||W s,p (1 + ||u||µ−1 L∞ ). 4. Commutator estimate I ([17, Prop. 3.6.A]) Assume P ∈ OP s , s > 0, σ ≥ 0, then ||[P, u]v||W σ,p ≤ C(||Du||L∞ ||v||W s−1+σ,p + ||u||W s+σ,p ||v||L∞ ).

(2.7)

5. Commutator estimate II ([17, (3.6.2)]) Assume P ∈ OP 1 . Then ||[P, u]v||Lp ≤ C||Du||L∞ ||v||Lp . Restricting to Sobolev spaces of integer order, the above inequalities can be proved using the classical methods of calculus. We next introduce the class of nonlinear evolution equations which will be considered. As our application is to the Einstein evolution equations, we will consider symmetric 2-tensors on M as the unknowns, but it should be stressed, that the proof generalizes essentially without change to sections of general vector bundles over M . We will think of the symmetric 2-tensors uij , vij as sections of the vector bundle Q of symmetric 2-tensors over (M, g) with ﬁber inner product u, v given by

u, v = uij vkl gˆik gˆjl and the corresponding norm |u| deﬁned by |u| = u, u1/2 . The natural ﬁber inner product on derivatives is ˆ m u, ∇ ˆ n vg mn , ˆ ∇v ˆ g = ∇

∇u, ˆ g . The covariant derivative is metric, with corresponding norm |∇u| ˆ Y v, ˆ Y u, v + u, ∇ Y u, v = ∇ ˆ m uij , and the rough Laplacian ∆ ˆ g on Q, deﬁned by ˆ Y u)ij = Y m ∇ where (∇ ˆ m (g mn µg ∇ ˆ n hij ), ˆ g hij = 1 ∇ ∆ µg cf. (1.8), is self-adjoint with respect to the natural L2 inner product on Q, ˆ ˆ ˆ g u, vµg .

∇u, ∇vg µg = −

∆ M

M

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ˆ i of ∇, ˆ The curvature on Q can be computed in terms of the Riemann tensor R jkl ˆ m∇ ˆ n hij − ∇ ˆ m hij = −R ˆ n∇ ˆl ˆl ∇ imn hlj − R jmn hil . Let the operator L with coeﬃcients given by (g, N, X) be deﬁned by ˆ Xu u ∂t u − N v − ∇ L[g, N, X] = ˆ gu − ∇ ˆ Xv . v ∂t v − N ∆

(2.8)

We will use the notation U = (u, v), Hs = H s × H s−1 , W 1,∞ = W 1,∞ × L∞ . Write L[U] for L given by (2.8), with (g, N, X) = (g, N, X)[U], and consider Cauchy problems of the form (2.9) L[U]U = F [U], U t=0 = U 0 . Deﬁne the space CTk (Hs ), 1 ≤ k ≤ s, to be CTk (Hs ) = ∩0≤j≤k−1 C j ([0, T ]; Hs−j ). Definition 2.1. A number T > 0 is called a time of existence in Hs for the Cauchy problem (2.9) if there is a unique solution U ∈ C([0, T ]; Hs ) ∩ C 1 ([0, T ]; Hs−1 ) to (2.9). The maximal time of existence in Hs for 2.9 is T+ = sup{T : T is a time of existence for (2.9)}. The Cauchy problem (2.9) is called strongly locally well posed in C k (Hs ) if the solution map U 0 → U is continuous as a map Hs → CTk (Hs ) for a time of existence T = T (U 0 ) > 0, which depends continuously on U 0 ∈ Hs . The continuity of U 0 → U is called Cauchy stability. The following deﬁnition states the regularity properties of L, F which will imply that the Cauchy problem (2.9) is strongly locally well posed. Definition 2.2. Let V ⊂ Hs be connected and open. The system L[U]U = F [U] is called quasi-linear hyperbolic in V, if the maps U→ h[U] = (g[U], N [U], X[U]), U→ F [U],

V → Hs V → Hs

(2.10a) (2.10b)

are deﬁned and continuous, g¯ = g¯[U] deﬁned in terms of (g, N, X)[U] satisﬁes Λ(¯ g ) < ∞ for U ∈ V, and there is a continuous function CL = CL (U 0 ) such that for each U 0 ∈ V the following holds. s s 1. B1/C (U 0 ) ⊂ V, where B1/C (U 0 ) is the ball in Hs of radius 1/CL , centered L L at U 0 .

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s 2. The maps (2.10) are Lipschitz on B1/C (U 0 ) with Lipschitz constant CL L r w.r.t. H , 1 ≤ r ≤ s, explicitly

||h[U1 ] − h[U2 ]||H r ≤ CL ||U1 − U2 ||Hr ||F[U1 ] − F[U2 ]||Hr ≤ CL ||U1 − U2 ||Hr s for all U1 , U2 ∈ B1/C (U 0 ). L

3. The maps h, F have Fr´echet derivatives Dh, DF satisfying ||Dh[U]U ||H s−1 ≤ CL ||U ||Hs−1

||DF [U]U ||Hs−1 ≤ CL ||U ||Hs−1 s for all U ∈ B1/C (U 0 ). L

4. Let m be an integer, 1 ≤ m ≤ s − 1. For all integers j, 1 ≤ j ≤ m, {i }ji=1 , j i ≥ 1, i=1 i = m, the Fr´echet derivatives Dj h[U], Dj F [U] of order j are s Lipschitz functions from B1/C (U 0 ) to the spaces of multilinear maps L j

Hs−i → H s−m ,

i=1 j

Hs−i → Hs−m ,

i=1

respectively. We call m the order of regularity of (2.9). Remark 2.1. The order of regularity determines the regularity of the solution w.r.t. time. For strong local well-posedness, it is suﬃcient to have order of regularity m = 1. The following is the main result of this section. Theorem 2.3. Let s > n/2 + 1. Assume that (2.9) is quasi-linear hyperbolic, regular of order 1 ≤ m ≤ s − 1, in V ⊂ Hs . Then the following holds. 1. (Strong local well-posedness) The Cauchy problem (2.9) is strongly locally well posed in C m+1 (Hs ) with time of existence which can be chosen depending only on M, Λ[¯ g[U 0 ]], ||D¯ g [U 0 ]||L∞ , ||U 0 ||Hs , ||¯ g [U 0 ]||H s , and the constant 0 CL = CL (U ) in Deﬁnition 2.2. 2. (Continuation) Let T+ be a maximal time of existence for (2.9). Then either the solution leaves V at T+ , or T+ = ∞ or g ], ||D¯ g ||L∞ , ||∂t g||L∞ , CL ) = ∞. lim sup max(Λ[¯

(2.11)

tT+

The rest of this section is devoted to the proof of Theorem 2.3. In Subsection 2.1, the basic energy estimate, Lemma 2.7 is proved. Local existence and uniqueness is proved in Subsection 2.2, Cauchy stability is proved in Subsection 2.3 and the continuation principle, point 2.3 is proved in Subsection 2.4.

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Energy estimates

¯ = [0, T ] × M , and assume that In this subsection, ﬁx T > 0, s ≥ 1, let M ∞ 1,∞ 0,1 ∞ g ∈ L ([0, T ]; W )∩C ([0, T ]; L ) and N, N −1 , X ∈ L∞ ([0, T ]; W 1,∞ ). Unless otherwise stated, all constants in this subsection depend only on T and Λ(T ). For the applications in this paper, there is no loss of generality in assuming, in this ¯ . We will consider the linear system subsection, that all ﬁelds are C ∞ on M (2.12) L[g, N, X]U = F , U t=0 = U 0 , where L is given by (2.8) and u U= , v

0 u U = , v0 0

Fu F= . Fv

Given g, X, let ρ be deﬁned by 1 ρ = − (∂t g − LX g), 2

(2.13)

¯ , g¯). Then under the so that N −1 ρ is the second fundamental form of M in (M ∞ ¯ present assumptions, ρ ∈ L (M ). Deﬁne the energy E = E(t, U) by 1 ˆ 2 + |v|2 )µg . E(t, U) = (|u|2 + |∇u| g 2 Mt Lemma 2.4. Assume that U ∈ L∞ ([0, T ]; H2 ) ∩ C 0,1 ([0, T ]; H1 ) is a solution to (2.12). Then (2.14) |∂t E| ≤ C(E 1/2 ||F ||H1 + (1 + ||ρ||L∞ )E), where C = C(Λ[¯ g ]). Proof. A computation shows ˆ m u, ∇ ˆ n Fu g mn ∂t E =

u, N v + Fu + ∇ Mt ˆ m u, R ˆ rn uX r g mn + v, Fv µg + ∇ 1 ij 2 2 2 ˆ ˆ ˆ +

∇i u, ∇j uρ − (|u| + |∇u|g + |v| )trρ µg , 2 Mt ˆ rn u is given by where the curvature term R ˆ rn uij = +R ˆ l ulj + R ˆ l uil . R irn jrn An application of the Schwartz inequality gives the result.

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Using (2.14) to estimate ∂t E 1/2 and integrating the resulting inequality gives

|E 1/2 (T ) − E 1/2 (0)| ≤ C

T

||F ||L1 ([0,T ];H1 ) +

0

(1 + ||ρ(t)||L∞ )E 1/2 (t)dt .

(2.15) An application of the Gronwall inequality gives E 1/2 (T ) ≤ CeC

ÊT 0

||ρ(t)||L∞ dt

(E 1/2 (0) + ||F ||L1 ([0,T ];H1 ) ).

(2.16)

Higher order regularity is proved by estimating Es (T ) = Es (T ; U) deﬁned by Es (T ) = ||U||L∞ ([0,T ];Hs ) ,

s ≥ 1.

(2.17)

We have Es (T ; U) ≤ CE1 (T ; Ds−1 U),

s ≥ 1.

(2.18)

The inequality (2.16) gives Lemma 2.5. Assume U ∈ L∞ ([0, T ]; H2 ) ∩ C 0,1 ([0, T ]; H1 ) solves (2.12). Then with ρ given by (2.13), E1 (T ; U) ≤ CeC

ÊT 0

||ρ(t)||L∞ dt

(E1 (0; U) + ||F ||L1 ([0,T ];H1 ) ).

(2.19)

In order to derive higher order energy estimates, we will apply (2.19) to

Ds−1 U using the identity L Ds−1 U = [L, Ds−1 ]U + Ds−1 F . In order to do this we must estimate the commutator [L, Ds−1 ]U. This is done in the following Lemma. Lemma 2.6. Let s > 1. Assume (g, N, X) ∈ H s ∩ W 1,∞ , U ∈ Hs ∩ W 1,∞ . Then ||[L, Ds−1 ]U||H1 ≤ C(Λ[¯ g ])(||¯ g ||W 1,∞ ||U||Hs + ||¯ g ||H s ||U||W 1,∞ ).

(2.20)

Proof. By construction, [∂t , D] = 0, and hence ˆ X ]u [ Ds−1 , N ]v + [ Ds−1 , ∇ s−1 [L, D ]U = ˆ g ]u + [ Ds−1 , ∇ ˆ X ]v . [ Ds−1 , N ∆ In the following we will write D for a ﬁrst order operator with smooth coeﬃcients, ˆ X = BD with B of order zero. ˆ ∈ OP 1 , so that ∇ such as given by for example ∇ ˆ s−1 may be used Recall that [D, Ds−1 ] ∈ OP s−1 . In case s is an integer, then ∇ instead of Ds−1 in this proof. We need to estimate the following quantities: ||[ Ds−1 , N ]v||H 1 , ˆ g ]u||L2 , ||[ Ds−1 , N ∆

ˆ X ]u||H 1 , ||[ Ds−1 , ∇ ˆ X ]v||L2 . ||[ Ds−1 , ∇

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We will treat each term separately. For the ﬁrst term, the commutator estimate gives ||[ Ds−1 , N ]v||H 1 ≤ C||DN ||L∞ ||v||H s−1 + ||N ||H s ||v||L∞ . The identity [BD, Ds−1 ]u = B[D, Ds−1 ]u + [B, Ds−1 ]Du,

(2.21)

gives using the product and commutator estimates, ˆ X ]u||H 1 ≤ C(||X||W 1,∞ ||u||H s + ||X||H s ||Du||L∞ ). ||[ Ds−1 , ∇ This takes care of the second term. Next, the identity [ Ds−1 , BD]v = Ds−1 [B, D]v + [ Ds−1 , D]Bv + [D Ds−1 , B]v gives ˆ X ]v||L2 ≤ C(||X||W 1,∞ ||v||H s−1 + ||X||H s ||v||L∞ ), ||[ Ds−1 , ∇ which takes care of the fourth term. Finally, for the third term, we may in view of ˆ g in the form (1.8) write the second order operator P = N ∆ P u = AD2 u + BDu. ˆ we have With D = ∇, ˆ m∇ ˆ n u, AD2 u = N g mn ∇ ˆ l u = −N g mn (Γl BDu = −N V l ∇

mn

ˆ l )∇ ˆ l u, −Γ mn

with A ∈ H s ∩ W 1,∞ , B ∈ H s−1 ∩ L∞ . The term [ Ds−1 , BD]u can be estimated in L2 by expanding the commutator and using the product estimates to get ||[ Ds−1 , BD]u||L2 ≤ C(||B||L∞ ||u||H s + ||B||H s−1 ||Du||L∞ ). The identity [AD2 , Ds−1 ]u = A[D2 , Ds−1 ]u + [A, Ds−1 D]Du + ( Ds−1 DA)(Du), (2.22) together with the commutator and product estimates gives ||[AD2 , Ds−1 ]u||L2 ≤ C(||A||W 1,∞ ||u||H s + ||A||H s ||Du||L∞ ). We now have ||[P, Ds−1 ]u||L2 ≤ C ((||A||W 1,∞ + ||B||L∞ )||u||H s +(||A||H s + ||B||H s−1 )||Du||L∞ ) .

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This gives ˆ g ]u||L2 ≤ C(Λ[¯ g]) ((||g||W 1,∞ + ||N ||W 1,∞ )||u||H s ||[ Ds−1 , N ∆ +(||g||H s + ||N ||H s )||Du||L∞ ) . Collecting the above and using Λ[¯ g], ||¯ g ||, ||D¯ g || for the terms involving norms of g, N, X, gives the result. Using the Gronwall inequality, Lemmas 2.4, 2.5 and 2.6, gives the following higher order energy estimate for the linear system (2.12). Lemma 2.7. Assume g¯ ∈ L∞ ([0, T ]; H r ∩ W 1,∞ ), ∂t g ∈ L∞ ([0, T ]; H r−1 ∩ L∞ ), r ≥ 2. Let U = (u, v) be a solution to (2.12) satisfying u ∈ L∞ ([0, T ]; H r ∩W 1,∞ )∩ C 0,1 ([0, T ]; H r−1 ), v ∈ L∞ ([0, T ]; H r−1 ∩ L∞ ) ∩ C 0,1 ([0, T ]; H r−2), r ≥ 2. Then, for 1 ≤ s ≤ r, there is a constant C = C(T, Λ(T )) so that Es (T ; U) ≤ CeC

ÊT 0

(||ρ||L∞ +||D¯ g||L∞ )dt

(Es (0; U) + ||F ||L1 ([0,T ];Hs ) T + ||¯ g ||H s ||U||W 1,∞ dt). (2.23a) 0

In particular, if s > n/2 + 1, Es (T ; U) ≤ CeC

ÊT 0

(||ρ||H s−1 +||¯ g||H s )dt

(Es (0; U) + ||F ||L1 ([0,T ];Hs ) ).

(2.23b)

To prove (2.23b), note that if s > n/2 + 1, the quantities ||ρ||L∞ , ||D¯ g ||L∞ , and ||U||W 1,∞ , are dominated by ||ρ||H s−1 , ||¯ g ||H s , and Es (T ; U).

2.2

Local existence and uniqueness

This subsection is devoted to the proof of local existence and uniqueness for (2.9). The proof is an iteration argument, following [12]. Note that by Deﬁnition 2.2, CL = CL (U 0 ) is continuous in U 0 . Further, Λ[¯ g[U]], ||D¯ g [U]||L∞ can be estimated in terms of Λ[¯ g [U 0 ]], ||D¯ g [U 0 ]||L∞ and CL . In the rest of this subsection, unless otherwise stated, constants will depend only g [U 0 ]||L∞ , ||U 0 ||Hs , ||¯ g [U 0 ]||H s , and the constant CL = CL (U 0 ). on M, Λ[¯ g[U 0 ]], ||D¯ 0 ∞ s Let R = 1/CL . Let {Um }m=0 ⊂ C ∞ ∩ BR/4 (U 0 ) be a sequence of approxi0 mations of U given by smoothing. We will construct a sequence of approximate solutions 1 s 0 {Um }∞ m=0 ⊂ C ([0, T∗ ]; BR (U )), 0 for some T∗ > 0, to be chosen, with initial data for Um given by Um . 0 Set U0 (t) ≡ U0 , F0 ≡ 0, and let L0 = L[g0 , N0 , X0 ] be the time-frozen version of L deﬁned by setting (g0 (t), N0 (t), X0 (t)) ≡ (g[U00 ], N [U00 ], X[U00 ]).

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For m > 0, let (gm , Nm , Xm ) = (g[Um ], N [Um ], X[Um ]), Lm = L[Um ], Fm = F [Um ], and for m ≥ 0 deﬁne Um+1 to be the solution of the linear Cauchy problem 0 Um+1 t=0 = Um+1 . (2.24) Lm Um+1 = Fm , The existence of solutions for the Cauchy problem for linear hyperbolic problems with smooth coeﬃcients is standard, the proof given in [16, Theorem 3.3] is easily adapted to the present situation. Suppose {Um }m m =0 is a sequence of solutions to s (2.24) with Um taking values in BR (U 0 ). By point 2.2 of Deﬁnition 2.2, we get ||∂t g[Um (t)]||H s−1 ≤ CL ||∂t Um ||Hs−1 .

(2.25)

Let ρm = − 21 (∂t gm − LXm gm ). From (2.24) it follows, using the energy estimate, that ||∂t Um ||Hs−1 is bounded by a constant depending on CL and hence in view of (2.25) we get a bound on ||ρm ||H s−1 + ||¯ gm ||H s . Given this estimate, it follows from Lemma 2.7, that there is a constant CR < ∞, so that as long as {Um }m m=0 s takes values in BR (U 0 ), the energy estimate Es (T ; Um+1 ) ≤ CR (Es (0; Um+1 ) + ||Fm ||L1 ([0,T ];Hs ) )

(2.26)

holds for Um+1 , for T ≤ 1. The restriction T ≤ 1 is made so that CR does not 0 ∞ }m=0 such that depend on T . We can choose the sequence {Um 0 s ∈ BR/4 (U 0 ) for m ≥ 0 Um

CR ||Um (0, ·) − Um (0, ·)||Hs < R/4 for m, m ≥ 0.

(2.27) (2.28)

We will prove, for a T∗ ≤ 1 to be chosen, convergence for this sequence in L∞ ([0, T∗ ]; Hs ) ∩ C 0,1 ([0, T∗ ]; Hs−1 ) to a limit U ∈ C([0, T∗ ]; Hs ) ∩ C 1 ([0, T∗ ]; Hs−1 ), by the following steps: ∞ s 0 1. {Um }∞ m=0 ⊂ L ([0, T∗ ]; BR (U )), by Lemma 2.8 ∞ 1 0,1 2. {Um }∞ ([0, T∗ ]; H0 ), by m=0 is a Cauchy sequence in L ([0, T∗ ]; H ) ∩ C Lemma 2.9, with limit U satisfying

(U, ∂t U) ∈ Cw ([0, T∗ ]; Hs × Hs−1 ) ∩ L∞ ([0, T∗ ]; Hs × Hs−1 ), by Lemma 2.10. See (2.30) below for the deﬁnition of Cw .

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3. U is a solution to (2.9). 4. U ∈ C([0, T∗ ]; Hs ) ∩ C 1 ([0, T∗ ]; Hs−1 ), Lemma 2.11. Lemma 2.8 (Boundedness in high norm). There is a time T∗ > 0 such that {Um } ⊂ s L∞ ([0, T∗ ]; BR (U 0 )). s (U0 )), Proof. We must prove that there is a T∗ > 0 so that if Um ∈ L∞ ([0, T∗ ]; BR ∞ s 0 then Um+1 ∈ L ([0, T∗ ]; BR (U )). In order to do this we consider

Lm (Um+1 − U0 ) = Fm − Lm U0 It follows from (2.26) that

0 ||Um+1 − U0 ||L∞ ([0,T ];Hs ) ≤ CR ||Um+1 − U00 ||Hs + ||Fm ||L1 ([0,T ];Hs ) + ||Lm U0 ||L1 ([0,T ];Hs )

and we see using (2.28) and the fact that ||f ||L1 ([0,t]) ≤ t||f ||L∞ ([0,t]) , that there is s (U 0 )), then a T∗ ≤ 1 so that if Um ∈ L∞ ([0, T∗ ]; BR ||Um+1 − U0 ||L∞ ([0,T∗ ];Hs ) < R/2.

(2.29)

U00

By construction U0 (t) ≡ and hence it follows from (2.27) that Um+1 ∈ s L∞ ([0, T∗ ]; BR (U 0 )). This completes the proof of Lemma 2.8. Lemma 2.9 (Convergence in low norm). There is a time T∗ > 0, so that {Um } is Cauchy in L∞ ([0, T∗ ]; H1 ) ∩ C 0,1 ([0, T∗ ]; H0 ). Remark 2.2. In order to handle the cases with s < 3, we show in Lemma 2.9 that {Um } is Cauchy in L∞ ([0, T∗ ]; H1 )∩C 0,1 ([0, T∗ ]; H0 ), in particular we have control of ∂t vm only in H −1 . In case s ≥ 3, this can be avoided and H1 , H0 can in the rest of this section be replaced by H2 , H1 . In particular if s ≥ 3, ∂t vm is Cauchy in L∞ ([0, T∗ ]; L2 ). Proof. Let T∗ be as in Lemma 2.8. Let T ≤ T∗ . We compute Lm (Um+1 − Um +1 ) = Fm − Fm − (Lm − Lm )Um +1 and hence

0 0 ||Um+1 − Um +1 ||L∞ ([0,T ];H1 ) ≤ CR ||Um+1 − Um +1 ||H1 + ||Fm − Fm ||L1 ([0,T ];H1 )

+||(Lm − Lm )Um +1 ||L1 ([0,T ];H1 ) .

Using the Lipschitz property of the map U → (g, N, X, F ) we see that by possibly decreasing T∗ we get for all m, m ≥ m0 , 0 0 ||Um+1 − Um +1 ||L∞ ([0,T∗ ];H1 ) ≤ CR ||Um+1 − Um +1 )||H1 1 + ||Um − Um ||L∞ ([0,T∗ ];H1 ) . 2

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0 0 As Um is Cauchy in H1 , we can by thinning out the sequence {Um } get

0 0 CR ||Um+2 − Um+1 ||H1 < R/2. m

Let βm = Then,

0 0 CR ||Um+1 − Um ||H1 ,

m ≥ 0 and am = ||Um − Um−1 ||L∞ ([0,T∗ ];H1 ) , m ≥ 1.

1 am+1 ≤ βm + am , 2 Solving the diﬀerence equation

m ≥ 1.

1 am+1 − am = βm 2 gives

∞

m=1

am = 2

∞

βm + 2a1

m=1 ∞

and which shows {Um } is Cauchy in L ([0, T∗ ]; H1 ). In the equation of motion, which determines ∂t Um , there will occur terms of the form g ij ∂i ∂j um and X j ∂j vm . In order to show that ∂t Um is Cauchy in L∞ ([0, T∗ ]; H0 ), we need to show that these terms are Cauchy in H −1 . To see this, recall that multiplication is continuous H s × H 1 → H 1 for s > n/2 + 1, cf. product estimate II. In view of the fact that H −1 is the dual to H 1 , this implies that multiplication deﬁnes a continuous map H s × H −1 → H −1 for s > n/2 + 1. It is now clear from the mapping properties of (g, N, X, F ) and the equation of motion, that {Um } is Cauchy in L∞ ([0, T∗ ]; H1 ) ∩ C 0,1 ([0, T∗ ]; H0 ). The dual space to H s is H −s . Let (φ, u)−s,s denote the duality pairing of H with H s . Then |(φ, u)−s,s | ≤ ||φ||H −s ||u||H s . Deﬁne the space Cw ([0, T ]; H s ) of weakly continuous functions on [0, T ] with values in H s , i.e., −s

Cw ([0, T ]; H s ) = {u : (φ, u)−s,s ∈ C([0, T ]),

for all φ ∈ H −s }

(2.30)

s Lemma 2.10 (Weak convergence). Let {um }∞ m=1 ⊂ C([0, T ]; H ), s > 0 be a ∞ s bounded sequence and assume {um } is Cauchy in L ([0, T ]; H ) for some s , 0 ≤ s < s. Then there is a u ∈ Cw ([0, T ]; H s ) ∩ L∞ ([0, T ]; H s ) so that for all φ ∈ H −s , (φ, um )−s,s → (φ, u)−s,s uniformly in L∞ [0, T ] as m → ∞.

Proof. Let φ ∈ H −s be arbitrary and ﬁx > 0. Let C be a constant so that ||um ||H s ≤ C for all m. Let u be the limit of {um } in L∞ ([0, T ]; H s ). By the uniform bound on ||um ||L∞ ([0,T ];H s ) , we ﬁnd u ∈ L∞ ([0, T ]; H s ). Recall H −s is dense in H −s , hence there is a φ ∈ H −s so that ||φ − φ ||H −s < /3C. Since um → u in H s , we may choose m large enough so that ||φ ||H −s ||um − u||H s < /3. Then at t ∈ [0, T ], |(φ, um )−s,s − (φ, u)−s,s | ≤ |(φ − φ , um )−s,s | + |(φ , um − u)−s ,s | + |(φ − φ, u)−s,s | < .

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As was arbitrary, we ﬁnd that the sequence (φ, um )−s,s ∈ C([0, T ]) converges uniformly to (φ, u)−s,s ∈ C([0, T ]). Hence u ∈ Cw ([0, T ]; H s ) as φ ∈ H −s was arbitrary. By Lemma 2.10, Um → U ∈ Cw ([0, T∗ ]; Hs ). Next we prove that U solves (2.9). It is clear from the construction that U(0) = U 0 , so we need to show that L[U]U − F[U] = 0. Consider the sequence {Um } deﬁned by (2.24). We compute L[U]U − F[U] = (L[U] − L[Um ])U + L[Um ](U − Um+1 ) + (F [Um ] − F[U]). By Lemma 2.9, {Um } converges to U in L∞ ([0, T∗ ]; H1 ) ∩ C 0,1 ([0, T∗ ]; H0 ). By the Lipschitz property of U → (g, N, X, F ), the ﬁrst and third terms in the right-hand side tends to zero in L∞ ([0, T∗ ]; H1 ) as m → ∞. It follows from the deﬁnition of L and the discussion in the proof of Lemma 2.9 that the second term tends to zero in L∞ ([0, T∗ ]; H0 ). However, the left-hand side is independent of m and hence equals zero. This proves that U is a solution to (2.9). At this stage we know that t → (U, ∂t U) is weakly continuous and that U is a solution to (2.9). In order to prove U ∈ C([0, T∗ ]; Hs ) ∩ C 1 ([0, T∗ ]; Hs−1 ), we need the following Lemma. Lemma 2.11 (Continuity). Assume U is a solution to (2.9) with F ∈ Hs , and assume (U, ∂t U) satisﬁes (U, ∂t U) ∈ Cw ([0, T ]; Hs ×Hs−1 )∩L∞ ([0, T ]; Hs ×Hs−1 ), s > n/2 + 1. Then (U, ∂t U) ∈ C([0, T ]; Hs × Hs−1 ). Proof. First recall that continuity on [0, T ] is equivalent to right and left continuity at each t ∈ [0, T ]. Changing the direction of time gives an equation of the same type, so it is suﬃcient to prove strong right continuity at t ∈ [0, T ]. By a reparametrization, there is no loss of generality in assuming t = 0. We know from Lemma 2.10 that t → (U, ∂t U) is weakly continuous. In order to prove strong continuity, we will use the fact that if wm → w weakly, then wm → w strongly if ||wm || → ||w||, cf. [14, §12, exercise 3]. By ﬁxing (g, N, X)[U(t)] in the deﬁnition of E, we may deﬁne a norm |||U |||2s;t = E(t; Ds−1 U )

(2.31)

which for ﬁxed t, is an equivalent norm to || · ||Hs . By (2.18), (2.15) and Lemma 2.6, |||U(t)|||s;t → |||U(0)|||s;0 as t 0.

By compact embedding, U ∈ C([0, T ]; Hs ) for s < s, and hence we have g ∈ C([0, T ]; H s ). From this fact it follows easily that |||U(t)|||s;0 → |||U(0)|||s;0

as t 0,

(2.32)

and hence U is right continuous in Hs at t = 0. The corresponding property for ∂t U follows from equation (2.9).

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By Lemma 2.11, we have U ∈ C([0, T∗ ]; Hs ) ∩ C 1 ([0, T∗ ]; Hs−1 ). It follows from (2.9) and the assumptions on the map U → (g, N, X, F ) that U ∈ CTm+1 (Hs ), ∗ where m is the order of regularity of (2.9). The contraction property used in the proof of Lemma 2.9 shows that U is the unique solution to (2.9).

2.3

Cauchy stability

In this section we will give a proof of Cauchy stability following [2, 3]. We will give the proof only for the case s = k, k integer, k > n/2 + 1, and assuming that (2.9) is regular of order m = k − 1. The proof is easily adapted to noninteger s > n/2 + 1 and general m. Let || · || denote || · ||H for integer. Introduce the norm

|||U(t)|||k = ||∂tj U(t)||Hk−j . 0≤j≤k−1

Deﬁne the spaces CT (Hk ) = ∩0≤j≤k−1 C j ([0, T ]; Hk−j ), L1T (Hk ) = ∩0≤j≤k−1 W j,1 ([0, T ]; Hk−j ), with norms

|||U|||k,T =

||U||C j ([0,T ];Hk−j )

0≤j≤k−1

Uk,T =

||U||W j,1 ([0,T ];Hk−j ) .

0≤j≤k−1

Consider the linear problem L[h]U = F ,

U t=0 = U 0

where we use h to denote the coeﬃcients g, N, X of L. We will use the convention that |||h|||,T and

h,T denotes the norm deﬁned analogously to the above but using H k instead of Hk . From the energy estimate, Lemma 2.7 we get for 2 ≤ ≤ k, integer, |||U|||,T ≤ C(|||h|||k,T )(||U 0 || + |||F(0)|||−1 +

F ,T ).

(2.33)

∞

Using the density of C in Sobolev spaces, the proof of the following approximation Lemma is straightforward. Lemma 2.12 (Approximation). Given any U 0 ∈ H , F ∈ L1T (H ), ≥ 2, integer, there are for any > 0, U0 ∈ C ∞ , F ∈ C ∞ such that ||U 0 − U0 || < |||F(0) − F (0)|||−1 <

F − F ,T < .

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2.3.1 Perturbation estimate for ≤ k − 1 Consider the linear problems L[h]U = F , L[h ]U = F , L[h]U = F ,

U t=0 = U 0 U t=0 = U 0 U = U0 . t=0

In the following write L = L[h], L = L[h ] and assume h, h ∈ CT (Hk ). In applying the energy estimates in the following we will let C be a constant depending on T as well as on Λ[¯ g], |||h|||k,T , |||h |||k,T . All of these quantities are under our control, using the a priori estimates. We calculate L (U − U ) = F − F + (L − L )U L(U − U ) = F − F . Let Ω () = ||U0 || + |||F (0)|||−1 +

F ,T , so that by (2.33), for ≤ k, |||U |||,T ≤ CΩ (). Then, for ≤ k − 1,

(L − L )U ,T ≤ C|||h − h |||k−1,T Ω+1 (). Now we get from (2.33), for ≤ k − 1, |||U − U |||,T ≤ C ||U 0 − U0 || + |||F (0) − F (0)|||−1 +

F − F ,T + |||h − h |||k−1,T Ω+1 ()} . It is important to note that this works only for ≤ k − 1. The energy estimate (2.33), gives when applied to U − U for ≤ k − 1, |||U − U |||,T ≤ C. Putting this together gives for ≤ k − 1, |||U − U |||,T ≤ C + ||U 0 − U 0 || + |||F(0) − F (0)|||−1 +

F − F ,T + |||h − h |||k−1,T Ω+1 ()} .

(2.34)

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2.3.2 Perturbation estimate for = k Let B = ∂t − X, so that LU = BU − JU,

where J =N

0 ∆

I . 0

Let δ = BU. Then δ solves Lδ = F¯ ,

δ t=0 = U¯0 ,

where F¯ = [B, J]U + BF U¯0 = (JU + F )t=0 . We have the corresponding primed identities. Remark 2.3. For general s we have F¯ under control in Hs−1 , and this is also true for U¯0 . This observation allows one to generalize the proof to general s, m. From (2.34) applied to δ − δ , we get |||δ − δ |||k−1,T ≤ C + ||U¯0 − U¯0 ||k−1 + |||F¯ (0) − F¯ (0)|||k−2 +

F¯ − F¯ k−1,T + |||h − h |||k−1,T Ωk () . The terms ||U¯ 0 − U¯ 0 ||k−1 , |||F¯ (0) − F¯ (0)|||k−2 ,

F¯ − F¯ k−1,T can be estimated in terms of ||U 0 − U 0 ||k , |||F(0) − F (0)|||k−1 ,

F − F k,T . This gives |||δ − δ |||k−1,T ≤ C + ||U 0 − U 0 ||k + |||F(0) − F (0)|||k−1 +

F − F k,T + |||h − h |||k−1,T Ωk ()} , where now C also depends on ||U 0 ||k , ||U 0 ||k . We now use an elliptic estimate for U − U . A computation shows J(U − U ) = δ − δ + F − F + (J − J)U .

(2.35)

From the deﬁnition of J and standard elliptic theory we get the estimate ||U||k ≤ C(||JU||k−1 + ||U||k−1 ). Hence (2.35) implies the estimate ||U − U ||k ≤ C(||δ − δ ||k−1 + ||F − F ||k−1 + ||h − h ||k−1 ||U ||k + ||U − U ||k−1 ).

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Now calculate ∂t (U − U ) = BU − B U + XU − X U = δ − δ + X(U − U ) + (X − X )U . Iterating this estimate gives together with the above |||U − U |||k,T ≤ C + ||U 0 − U 0 ||k + |||F(0) − F (0)|||k−1 +

F − F k,T + |||h − h |||k−1,T Ωk ()} .

(2.36)

2.3.3 Application to the nonlinear system Consider the Cauchy problems L[U]U = F [U], L[U ]U = F [U ],

U t=0 = U 0 , U t=0 = U 0 .

We wish to estimate |||U − U |||k,T in terms of U 0 − U 0 . By the assumptions, if U, U are close, there is a constant CLip (not to be confused with CL ), such that |||F[U](0) − F[U ](0)|||k−1 ≤ CLip ||U 0 − U 0 ||k−1 ,

F[U] − F[U ]k,T ≤ CLip

U − U k,T , |||h − h |||k−1,T ≤ CLip |||U − U |||k−1,T . By (2.36) we now get an estimate of the form |||U − U |||k,T ≤ C + ||U 0 − U 0 ||k +

U − U k,T + |||U − U |||k−1,T Ωk ()} . The term

U −U k,T can be eliminated from the right-hand side by an application of the Gronwall inequality. This gives (2.37) |||U − U |||k,T ≤ C + ||U 0 − U 0 ||k + |||U − U |||k−1,T Ωk () . Now consider a sequence {Uα0 } ⊂ Hk , such that Uα0 → U 0 in Hk as α → ∞. Let Uα solve L[Uα ]Uα = F [Uα ], Uα t=0 = Uα0 . By compact embedding we can choose a subsequence so that |||Uα − U|||k−1,T → 0. It follows from (2.37), using the fact that was arbitrary, that |||Uα − U|||k,T → 0,

as α → ∞.

This completes the proof of Cauchy stability and point 2.3 of Theorem 2.3 is proved.

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Continuation

It remains to prove the continuation principle, point 2.3 of Theorem 2.3. Suppose for a contradiction, (2.11) does not hold and that T+ < ∞ is the maximal time of existence. Then by the proof of local existence, there is a uniform lower bound for the time of existence with initial data U(t), t < T+ . This contradicts T+ < ∞. The fact that U 0 → T+ is continuous follows from Cauchy stability. This completes the proof of Theorem 2.3.

3 The Cauchy problem for the modified Einstein evolution equations Let n ≥ 2 and ﬁx s > n/2 + 1. The vacuum Einstein equations have special structure in dimension 2 + 1 which we do not make use of, but we allow n = 2 here and in the following sections for completeness. Let P be the operator deﬁned by ˆ m en ), P Y i = ∆X i + Ri f X f − LX V i − 2∇m X n ei (∇m en − ∇ where V is given by (1.5). Let the operators B, E be deﬁned by Bf = −∆f + |k|2 f, i ˆ i ). + ∇i f kmm + 2f k mn (Γimn − Γ Ef = −2∇m f km mn

With A given by

B A= E

0 P

,

the deﬁning equation (1.11) can be written in the form N 1 A = . X 0

(3.1)

(3.2)

The operator A is second order elliptic. Deﬁne V to be the set of symmetric covariant tensors (g, k) ∈ H s × H s−1 such that g is a Riemann metric

(3.3a) 2

2

The operators B, P : H → L , are isomorphisms at (g, k).

(3.3b)

Suppose V is nonempty. For (g, k) ∈ V, let CN X = CN X (g, k) be a constant so that for functions u and vector ﬁelds Y , ||u||L2 ≤ CN X ||Bu||L2 ,

(3.4a)

||Y ||L2 ≤ CN X ||P Y ||L2 .

(3.4b)

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Let (g 0 , k 0 ) ∈ V be given. In this section we prove that the Cauchy problem for the system (1.10,1.4,1.6), with initial data = (g 0 , k 0 ), (3.5) (g, k) t=0

is strongly locally well posed. We will refer to this problem as the CMCSH Cauchy problem with data (g 0 , k 0 ). Theorem 3.1. The CMCSH Cauchy problem with initial data (g 0 , k 0 ) ∈ V is strongly locally well posed in C k (Hs ), k = s. In particular, there is a time of existence T∗ > 0 so that the solution map (g 0 , k 0 ) → (g, k, N, X) is continuous H s × H s−1 → CTk∗ (H s × H s−1 × H s+1 × H s+1 ). g 0 ], ||¯ g 0 ||H s and Here T∗ can be chosen to depend only on CN X (g 0 , k 0 ), Λ[¯ 0 ||k ||H s−1 . In particular, T∗ can be chosen so that it depends continuously on (g 0 , k 0 ) ∈ H s × H s−1 . Let T+ be the maximal time of existence of the solution to the CMCSH Cauchy problem with data (g 0 , k 0 ). Let CN X = CN X ((g(t), k(t)) be deﬁned by (3.4). Then either T+ = ∞ or g ], ||D¯ g ||L∞ , ||k||L∞ , CN X ) = ∞. lim sup max(Λ[¯ tT+

To prove Theorem 3.1 we will show that the CMCSH Cauchy problem is quasi-linear hyperbolic in V, regular of order s − 1. The result then follows from Theorem 2.3. The following Lemma gives the basic estimates used in the proof of Theorem 3.1. Lemma 3.2. The set V deﬁned by (3.3) is an open subset of H s × H s−1 . Let N, X be the solution to (1.11) w.r.t. (g, k) and let g¯ be deﬁned by (1.1) in terms of (g, N, X). Let the map σ : (g, k) → (N, X) be deﬁned by solving (1.11). g], ||D¯ g ||L∞ such that There is a constant CL > 0 depending only on CN X , Λ[¯ the following holds. s s (g 0 , k 0 ) ⊂ V, where B1/C (g 0 , k 0 ) is the ball in Hs of radius 1/CL 1. B1/C L L centered at (g 0 , k 0 ).

2. The Fr´echet derivative Dσ satisﬁes ||Dσ(g, k)(g , k )||H s ≤ CL ||(g , k )||Hs−1 s for (g, k) ∈ B1/C (g 0 , k 0 ). L

3. σ has the Lipschitz property ||σ(g, k) − σ(g , k )||H r+1 ≤ CL ||(g, k) − (g , k )||Hr , s for all (g, k), (g , k ) ∈ B1/C (g 0 , k 0 ), 1 ≤ r ≤ s. L

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4. For integers m, j, i , such that m = s − 1, 1 ≤ j ≤ m, 1 ≤ i , i i = m, s the Fr´echet derivative Dj σ is a Lipschitz map from B1/C (g 0 , k 0 ) to the L space of multilinear maps j

Hs−i → H s+1−m .

i=1

Remark 3.1. Note that the Lemma gives an estimate for N, X ∈ H s+1 . This extra regularity for N, X will be important later in the proof of Theorem 3.1. Proof. Consider the second order elliptic system for (N, X) given by (1.11). In this system the zeroth order coeﬃcients depend on the Ricci tensor Ri j and derivatives of the vector ﬁeld V i given by (1.5), and thus, at ﬁrst glance, appear to contain second derivatives of gij , i.e., terms in H s−2 . In order to prove that (N, X) ∈ H s+1 × H s+1 for (g, k) ∈ V, we must prove that these second derivative terms cancel. It is clear from (1.11) that it is suﬃcient to restrict our attention to the system Y i → ∆Y i + Ri f Y f − LY V i , where V i is given by (1.5). We have ∆Yc + Rcf Yf = g ab ∇a (∇b Yc + ∇c Yb − ∇m Y m gbc ). We will compute in local coordinates. Writing ∇b Yc + ∇b Yc = LY gbc in terms of ∂i gives LY gbc = Y f ∂f gbc + gbf ∂c Y f + gcf ∂b Y f . Using this together with ∇m Y m = 12 g mn LY gmn shows after some manipulations that the terms in ∆Yc + Rcf Yf containing second derivatives of gab can be written in the form 1 ab m g Y ∂m (∂a gbc + ∂b gac − ∂c gab ). (3.6) 2 On the other hand, using the explicit form of the Christoﬀel symbol, Γijk =

1 il g (∂j gkl + ∂k gjl − ∂l gkj ) 2

and the form of V d in local coordinates (1.5b) one sees that (3.6) exactly cancels the terms in gcd LY V d containing second order derivatives in gij . Let the operators B, P, A be as in (3.1). It follows from the fact that A is lower triangular that if B, P : H 2 → L2 are isomorphisms, then also A : H 2 → L2 is an isomorphism. With u = (N, X), (3.2) and hence (1.11) is of the form A(g, Dg, k)ul = F l (g, Dg, k)

(3.7)

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where F is a smooth function of its arguments and A is a second order elliptic system of the form A(g, Dg, k)ul = g mn ∂m ∂n ul + bl mi (g, Dg, k)∂i um + clm (g, Dg, k)um ,

(3.8)

where b, c are smooth functions of their arguments. It now follows from standard elliptic theory, cf. [17] that an inequality of the form ||u||H r+2 ≤ C(||Au||H r + ||u||H r ),

0 ≤ r ≤ s − 1,

(3.9)

holds. Uniqueness together with compact embedding implies that the lower order term can be eliminated from this inequality. We sketch the standard argument for this. Suppose that an inequality of the form ||u||H r+2 ≤ C||Au||H r does not hold. Then there is a sequence {ui }∞ i=1 with ||ui ||H r+2 = 1, and ||Aui ||H r → 0 as i → ∞. Since M is compact, H r+2 is compactly embedded in H r , and therefore there is a subsequence {uj } which converges to some u∗ which by construction satisﬁes Au∗ = 0. By (3.9), in fact u∗ ∈ H r+2 . The existence of a solution u∗ to Au = 0 contradicts the assumption that equation (3.7) has unique solutions. Thus, an inequality of the form ||u||H r+2 ≤ C||Au||H r ,

0 ≤ r ≤ s − 1,

(3.10)

holds. It follows from the commutator estimates as in Lemma 2.7, that the constant C in (3.10) depends only on CL . It is convenient to use the notation h = (g, k), Hs = H s × H s−1 . We write A[h] = A[g, Dg, k], F [h] = F (g, Dg, k) etc. To prove that C depends continuously on h ∈ Hs , let h be close to h in Hs and calculate ||A[h ]u||H r ≥ ||A[h]u||H r − ||(A[h ] − A[h])u||H r ≥ C −1 ||u||H r+2 − C1 ||h − h||Hs ||u||H r+2 which by choosing h suﬃciently close to h shows that the estimate ||u||H r+2 ≤ C ||Au||H r holds in a neighborhood of g with a uniform constant C arbitrarily close to C. In particular, for h close to h, A[h ] is an isomorphism. From this follows easily the ﬁrst part of the Lemma and point 3.2. Point 3.2 follows from elliptic estimates for (3.7). The Lipschitz property, point 3.2 for the map (g, k) → (N, X) is a consequence of 3.2 and the mean value inequality. The proof of point 3.2 is straightforward, the basic tool being the following product estimate which is a consequence of the product estimate II. Let r > n/2, i ≥ 1, i = 1, . . . , j, i i = m, m ≤ r. Then multiplication is a bounded multilinear map j H r−i → H r−m . i=1

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Proof of Theorem 3.1. We will show that Theorem 2.3 applies to the modiﬁed Einstein evolution equations (1.10,1.11). In order to do this we must ﬁrst write the system (1.10,1.11) with initial data (g, k)t=t0 = (g 0 , k 0 ), as a quasi-linear hyperbolic system (in the sense of Deﬁnition 2.2) of the form L[U] = F [U], U t=t0 = U 0 . Let uij = gij ,

(3.11a)

vij = −2kij .

(3.11b)

Then uij , vij are symmetric 2-tensors. We expand the Lie derivatives LX uij , LX vij ˆ in terms of ∇, ˆ m uij + ulj ∇ ˆ i X l + uil ∇ ˆ jXl (LX u)ij = X m ∇ ˆ as and similarly for LX vij . Similarly ∇i ∇i N can be written in terms of ∇ ˆ j N − 1 g lm (∇ ˆ i∇ ˆ i gjm + ∇ ˆ j gim − ∇ ˆ m gji )∇ ˆ l N. ∇i ∇j N = ∇ 2 The Ricci tensor Rij of gji is quasi-linear elliptic, up to a gauge term. This is seen from the identity 1ˆ Rij = − ∆ g gij + δij + Sij 2 where Sij [g, ∂g] is given by Sij =

1 ˆ l mjn + glj g mn R ˆ l min ) (gli g mn R 2 1 mn ls ˆ ˆ l gim + ∇ ˆ s gjn − 1 ∇ ˆ i gml ˆ j gns ∇ ˆ i gml ∇ ∇j gns ∇ + g g 2 2 ˆ n gjs − ∇ ˆ s gjn ˆ m gil ∇ ˆ m gil ∇ +∇ 1 ˆ − V l∇ l gij , 2

where ˆ l ). V l = g mn (Γlmn − Γ mn In the computation giving Sij , is it convenient to make use of the fact that Sij ˆ i = 0 at the center of is a tensor, and work in a local coordinate system with Γ jk coordinates. Let Sij [u, ∂u] be the expression corresponding to Sij [g, ∂g]. Now we may write the system that shows up in the proof of local existence in the form ∂t uij = N vij + LX uij ˆ g uij + LX vij + Fij ∂t vij = N ∆

(3.12a) (3.12b)

26

with

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1 −2Sij − umn vmn vij + vim vnj umn 2 ˆ i∇ ˆ j N − 1 ulm (∇ ˆ i ujm + ∇ ˆ j uim − ∇ ˆ m uji )∇ ˆ lN +2 ∇ 2

Fij = N

(3.13)

where umn = (u−1 )mn . In order to apply the energy estimates as presented in Section 2, we further ˆ This gives the system in the form expand the Lie derivative in terms of ∇. ˆ X uij + F1ij ∂t uij = N vij + ∇ ˆ g uij + ∇ ˆ X vji + F2ij ∂t vij = N ∆

(3.14a) (3.14b)

where ˆ i X l + uil ∇ ˆ jXl F1ij = ulj ∇ ˆ i X l + vil ∇ ˆ jXl F2ij = Fij + vlj ∇ with Fij given by (3.13). Finally, assuming that the deﬁning equations for N, X have unique solutions, let N = N [u, v], X = X[u, v] be given by (1.11). Then (1.10) takes the form ˆ X u + F1 [u, v] ∂t u = N v + ∇ ˆ gu + ∇ ˆ X v + F2 [u, v]. ∂t v = N ∆

(3.15)

Introduce the notation U = (u, v), then g = g[U], k = k[U] and F [U] = (F1 [U], F2 [U]). Further, via (1.11) we have N = N [U], X = X[U]. Then we can write equation (3.15) in the form L[U]U = F [U], U t=t0 = U 0 (3.16) with L given by (2.8), and U 0 given in terms of (g 0 , k 0 ) by (3.11). Clearly (3.16) is a system of the form considered in Deﬁnition 2.2. From the deﬁnition of F and Lemma 3.2, it follows that the CMCSH Cauchy problem with data in V is quasilinear hyperbolic, regular of order m = s − 1, and hence Theorem 3.1 follows from Theorem 2.3. Remark 3.2. The proof that F : Hs → Hs is the only place in the proof of Theorem 3.1 where the “additional regularity” N, X ∈ H s+1 × H s+1 is used, cf. Remark 3.1.

4 Evolution of gauges and constraints Let n ≥ 2 and ﬁx s > n/2 + 1. Let gˆ be a ﬁxed C ∞ metric on M with Levi-Civita ˆ Introduce the constraint and gauge quantities (A, F, V i , Di ) covariant derivative ∇.

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by A = trk − t,

(4.1a)

ˆ i ej ) V = g e (∇i ej − ∇ k

ij k

2

2

(4.1b)

F = R + (trk) − |k| − ∇i V , Di = ∇i trk − 2∇m kmi . i

(4.1c) (4.1d)

Let also

1 (∇i Vj + ∇j Vi ). (4.2) 2 A calculation shows the constraint and gauge quantities (A, F, V i , Di ) satisfy a hyperbolic system when (g, k, N, X) solve the modiﬁed evolution equation (1.10– 1.11). Using an energy estimate for this hyperbolic system, we will prove that if (A, F, V i , Di ) = 0 for the initial data (g 0 , k 0 ), then (A, F, V i , Di ) ≡ 0 along the solution curve (g, k, N, X) with δij =

(g, k) ∈ ∩0≤≤m ≤ C ((T0 , T1 ); H s− × H s−1− ), (N, X) ∈ ∩0≤≤m C ((T0 , T1 ); H

s+1−

×H

s+1−

),

(4.3a) (4.3b)

where m = s − 1, constructed in Theorem 3.1. In computing the time derivatives of the constraint quantities A, F, V, D, we note the fact that the eﬀect of Lie dragging gij , kij by X, is that the quantities A, F, V, D are also Lie dragged. We have using (1.10–1.11) and (4.1), ∂t V i = LX V i + N Di

(4.4a)

∂t A = LX A + N F

(4.4b)

∂t F = LX F + ∇i N Di + 2N F trk + 2N k ij δij + V f ∇f (N trk) + N ∆A + 2∇i N ∇i A

(4.4c)

∂t Di = LX Di + ∇i N F + N trkDi + N (∆Vi + Rif V ) + N trk∇i A + 2∇m N δmi . f

(4.4d)

In doing these computations we have used the expressions for the Fr´echet derivatives of of Γ and R, 1 im g (∇j hkm + ∇k hjm − ∇m hjk ) 2 DR.h = −∆hi i + ∇i ∇j hij − Rij hij

DΓijk .h =

(4.5) (4.6)

and the identities −2∆∇j X j + ∇i ∇j (∇i Xj + ∇j Xi ) = ∇m RX m + 2Rim ∇i Xm LY hij = Y ∇m hij + hmj ∇i Y m

m

(4.7) + him ∇j Y . (4.8) m

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The leading order terms in (4.4) are ∂t A ∼ = NF ∂t F ∼ = N ∆A i ∼ ∂t V = N Di ∂t Di ∼ = N (∆Vi + Rif V f ). Using the product and composition estimates stated in Section 2 and the deﬁnition of Rij in terms of gij , one ﬁnds that at a Riemann metric gij , the map gij → Rij is smooth and satisﬁes ||Rij ||H s−2 ≤ C(Λ[g])||g||H s (1 + ||g||H s ), for s > n/2 + 1. Deﬁne the energy E = E1 + E2 by 1 E1 = (|A|2 + |∇A|2 + |F |2 )µg 2 1 (|V |2 + |∇V |2 + |D|2 )µg . 2 The following Lemma gives the energy estimate required for proving that the Einstein vacuum constraints and the CMCSH gauge is conserved by the modiﬁed Einstein evolution equations.

and

E2 =

Lemma 4.1. Let (g, k, N, X) be a solution to (1.10, 1.11), satisfying the regularity condition (4.3). There is a constant C = C(g, k, N ) so that |∂t E| ≤ CE. Proof. We compute ∂t E using (4.4). Due to covariance, the LX terms in (4.4) can be dropped. From the assumptions, it follows that ∂t gij ∈ H s−1 ⊂ L∞ . Therefore, we only need to consider the terms in ∂t E involving ∂t A, ∂t F , ∂t V i , and ∂t Di . Using (4.4) and performing a partial integration, it is easy to check that |∂t E1 | ≤ CE. It remains to consider ∂t E2 . It is straightforward to show ∂t remains to consider 1 ∂t (∇i V j ∇k V l g ik gjl + Di Dj g ij )µg . 2

|V |2 µg ≤ CE. It

The only terms which need to be considered in detail are those where ∂t hits Γjim , V j or Di . The term involving ∂t Γjim is of the form (∂t Γjim )V m ∇k V l g ik gjl µg . (4.9)

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The terms involving ∂t V j , ∂t Di yield, after a partial integration, the expression (4.10) N Rif V f Di µg . From the assumptions, we have ∂t Γjim and Rif bounded in H s−2 . Further, V and D are bounded in H 1 and L2 , respectively, by E. The expressions (4.9) and (4.10) are of the form uvw with u ∈ H s−2 , v ∈ H 1 , w ∈ L2 . By the product rule (2.6), with t1 = s − 2, t2 = 1, p = 2, we have since s > n/2 + 1, ||uv||L2 ≤ C||u||H s−2 ||v||H 1 . An application of the Cauchy inequality gives the estimate | uvwµg | ≤ C||u||H s−2 ||v||H 1 ||w||L2 . Together with the above this gives |∂t E2 | ≤ CE. This completes the proof of Lemma 4.1. Lemma 4.1 and the Gronwall inequality now shows that if E = 0 initially, then E = 0 along the solution curve. We now have Theorem 4.2. Let (g 0 , k 0 ) ∈ V, and assume (M, g 0 , k 0 ) satisfy the Einstein vacuum constraint equations (1.4) and the gauge conditions (1.6), i.e., (A, F, V, D) = 0. Then the space-time metric g¯ deﬁned in terms of the solution (g, k, N, X) of the CMCSH Cauchy problem, is a solution of the Einstein vacuum equations.

5 Isomorphism property Let n ≥ 2 and ﬁx s > n/2 + 1. In this section, we use local coordinates unless otherwise stated. Deﬁne G to be the set of (g, k) ∈ H s × H s−1 such that g is a Riemann metric

(5.1a)

(A, F, V, D) = 0 at (g, k) .

(5.1b)

Thus (g, k) ∈ G precisely when (g, k) satisﬁes the constraint equations (1.4) and the gauge conditions (1.6). In this section we will work in the CMC time t = trg k. For initial data (g 0 , k 0 ) with t0 = trg0 k 0 , let (T− , T+ ), T− < t0 < T+ be a maximal existence interval, deﬁned by analogy with the notion of maximal existence time. Theorem 5.1. Assume that (g 0 , k 0 ) ∈ G and that gˆ has negative sectional curvature. Then the CMCSH Cauchy problem is strongly locally well posed in C k (Hs ), k = s and the Lorentz metric g¯ constructed from the solution (g, N, X) is a vacuum solution of the Einstein equations. Further, the following continuation principle holds. Let t0 = trg0 k 0 < 0 and let (T− , T+ ), T− < t0 < T+ , be a maximal existence

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interval for the CMCSH Cauchy problem in CMC time t = trg k. Then either (T− , T+ ) = (−∞, 0) or lim sup(Λ[¯ g ] + ||D¯ g ||L∞ + ||k||L∞ ) = ∞ as t T+ or as t T− . We begin by considering the operator P deﬁned by ˆ i ). P Y i = ∆Y i + Ri f Y f − LY V i − 2∇m Y n (Γimn − Γ mn We will need some material concerning vector and tensor ﬁelds along maps. Consider a map φ : (M, gij ) → (N, hαβ ). We use latin indices for coordinates on M and greek indices for coordinates on N . The bundles of tensor ﬁelds along φ, ⊗k T ∗ M ⊗ φ−1 T N have natural connections which we denote by D. Here φ−1 T N is the pullback of T N to M along φ. We work this out in local coordinates for k = 0, 1, 2. Di v γ = ∂i v γ + N Γγαβ v α ∂i φβ

(5.2)

Di ξjγ = ∂i ξjγ − M Γkij ξkγ + N Γγαβ ξjα ∂i φβ γ Di ξjk

=

γ ∂i ξjk

−

M

γ Γij ξk

−

M

γ Γik ξj

+

N

(5.3) α Γγαβ ξjk ∂i φβ

(5.4)

where if the left-hand side is evaluated at x ∈ M , all objects on N are evaluated at φ(x). On the bundles ⊗k T ∗ M ⊗φ−1 T N , there is a natural inner product ·, · which in local coordinates is given by

v, w = v α wβ hαβ

ξ, η = ξiα ηjβ hαβ g ij and similarly for higher order tensors. It is straightforward to check that D is metric w.r.t. ·, ·. Further, letting ∆D v α = g ij Di Dj v α the identity

∆D v, wµg = −

Dv, Dwµg M

M

holds, and thus the operator ∆D is self-adjoint with respect to the L2 pairing

v, wµg . M

Lemma 5.2. Assume gˆ has negative sectional curvature and V = 0. Then P : H s+1 → H s−1 is an isomorphism. Proof. Let φ : M → M be a diﬀeomorphism. Then ∗ ∗ (φ∗ V )i = (φ∗ g)mn (φ g) Γimn − (φ gˆ) Γimn .

(5.5)

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This follows from the fact that the diﬀerence of Christoﬀel symbols transforms as a tensor, or by a direct computation using the identities (φ∗ g) k r s Γij = ∂m (φ−1 )k ∂i ∂j φm + (g) Γm rs ∂i φ ∂j φ (φ∗ V )i (x) = ∂m (φ−1 )i V m (φ(x)). Let Y be a vector ﬁeld on M and let φs deﬁned by ∂s φs = Y ◦ φs , φ0 = Id, be the ﬂow of Y . A computation shows ∗ 1 ∂s s=0 (φs g) Γkij = g kl (Riljm Y m + Rjlim Y m + ∇i ∇j Yl + ∇j ∇i Yl ) 2 and similarly for

(φ∗ ˆ) k sg Γij .

Recall that ∂s s=0 φ∗s V i = [Y, V ]i = −LV Y i .

Now consider the identity (5.5) with φ replaced by φs . Diﬀerentiating with respect to s and evaluating at s = 0, gives the identity j ˆ nY i + R ˆ m∇ ˆi P Y i = g mn (∇ mjn Y )

(5.6)

ˆ of gˆ are raised and lowered with gˆ. where the indices on the Riemann tensor R ˆ in terms of Γ ˆ we have Using the deﬁnition of ∇ j ˆ m∇ ˆ n Y i = g mn (∂m ∇ ˆ nY i − Γ ˆ i ˆi ˆ ˆr ∇ g mn ∇ mn r Y + Γjm ∇n Y ) ˆ n Y i − Γr ∇ ˆ rY i + Γ ˆ nY j ) + V r ∇ ˆ rY i ˆi ∇ = g mn (∂m ∇ mn

jm

ˆ by Γ. where we used the deﬁnition of V to replace Γ Now we will think of Y as a vector ﬁeld along the map Id : (M, g) → (M, gˆ). For clarity we will use lower case greek indices for objects associated with gˆ. Recalling the deﬁnition of the connection D and the operator ∆D , we see that ˆ m∇ ˆ n Y α = ∆D Y α + V r Dr Y α . g mn ∇ Hence equation (5.6) takes the form ˆ α Y β + V r Dr Y α . P Y α = ∆D Y α + g mn R mβn Now assume as in the statement of the Lemma that V = 0. Then ˆ αmβn Y α Y β µg .

P Y, Y µg = −

DY, DY µg + g mn R M

(5.7)

M

ˆ αmβn Y α W β is symmetric and when gˆ has The quadratic form (Y, W ) → g mn R negative sectional curvatures then there is a λ > 0 so that ˆ αmβn Y α Y β ≤ −λ2 |Y |2 . g mn R

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It follows that an estimate of the form ||Y ||L2 ≤ C(Λ[g])||P Y ||L2 holds, and the Lemma now follows from the fact that P is second order elliptic of the form (3.8), cf. the proof of Lemma 3.2. Proof of Theorem 5.1. Let P be as above, let the operators B, E, A be deﬁned as in section 3. By the proof of Lemma 3.2, A is an elliptic second order operator of the form (3.8). Therefore, in order to prove that A is an isomorphism, it is enough to prove that ker A = ker A∗ = {0}. It follows from the maximum principle that B is an isomorphism as long as trg k = 0. By Lemma 5.2, P is an isomorphism. In view of the fact that A is lower triangular, the isomorphism property now follows from the isomorphism property for B and P . We have now proved that G ⊂ V and hence the conclusion of Theorems 3.1 and 4.2 apply. It remains to prove the continuation principle. In view of Theorem 3.1, it is enough to estimate CN X in terms of Λ[¯ g] and (trg k)−1 . By (5.7) it follows that ||Y ||L2 ≤ C(Λ(g))||P Y ||L2 . Similarly, we have (trg k)2 (Bf, f )L2 ≥ |∇f |2 + |k|2 f 2 ≥ C(Λ(g)) ||f ||2L2 3 M which gives the corresponding estimate for B. Now we have bounded CN X in terms of Λ[¯ g] and (trg k)−1 . Finally, to see that T+ ≤ 0, note that as M admits a metric gˆ with negative sectional curvature, it admits no metric with nonnegative scalar curvature, cf. [7, Corollary A, p. 94]. In case n = 2, this is a consequence of the uniformization theorem. Now suppose that T+ > 0. Then trg k = 0 must occur during the evolution, at which instant the scalar curvature of g is nonnegative by the Hamiltonian constraint (1.4a), which gives a contradiction. Therefore we have T+ ≤ 0. This completes the proof of Theorem 5.1.

Acknowledgments Part of this work was done during a visit to the Institute of Theoretical Physics, Santa Barbara, Universit´e Paris VI, the Albert Einstein Institute, Golm, and ´ L’Institut des Hautes Etudes Scientiﬁques, Paris. The paper was ﬁnished while the authors were enjoying the hospitality and support of the Erwin Schr¨ odinger Institute, Vienna.

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References [1] L. Andersson and V. Moncrief, On the global evolution problem in 3+1 dimensional general relativity, in preparation. [2] H. Beir˜ao da Veiga, Structural stability and data dependence for fully nonlinear hyperbolic mixed problems, Arch. Rational Mech. Anal. 120, no. 1, 51–60 (1992). [3]

, Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible Euler equations, Comm. Pure Appl. Math. 46, no. 2, 221–259 (1993).

[4] Yvonne Choquet-Bruhat and James W. York, Geometrical well-posed systems for the Einstein equations, C. R. Acad. Sci. Paris S´er. I Math. 321, no. 8, 1089–1095 (1995). [5] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10, no. 1, 1–68 (1978). [6] Helmut Friedrich and Alan Rendall, The Cauchy problem for the Einstein equations, Einstein’s ﬁeld equations and their physical implications, Springer, Berlin, 2000, pp. 127–223. [7] Mikhael Gromov and H. Blaine Lawson, Jr., Positive scalar curvature and the ´ Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. (1983), no. 58, 83–196 (1984). [8] Tosio Kato, Abstract evolution equations, linear and quasi-linear, revisited, Functional analysis and related topics, 1991 (Kyoto), Springer, Berlin, 1993, pp. 103–125. [9] Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41, no. 7, 891–907 (1988). [10] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in R3+1 , Ann. of Math. (2) 142, no. 1, 39–119 (1995). [11] Sergiu Klainerman and Francesco Nicol` o, On local and global aspects of the Cauchy problem in general relativity, Classical Quantum Gravity 16, R73– R157 (1999). [12] A. Majda, Compressible ﬂuid ﬂow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. [13] Richard S. Palais, Foundations of global non-linear analysis, W.A. Benjamin, Inc., New York-Amsterdam, 1968.

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Ann. Henri Poincar´e

[14] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York, 1973, McGraw-Hill Series in Higher Mathematics. [15] Winfried Sickel, Composition operators acting on Sobolev spaces of fractional order – a survey on suﬃcient and necessary conditions, Function spaces, diﬀerential operators and nonlinear analysis (Paseky nad Jizerou, 1995), Prometheus, Prague, 1996, pp. 159–182. [16] Christopher D. Sogge, Lectures on nonlinear wave equations, International Press, Boston, MA, 1995. [17] Michael E. Taylor, Pseudodiﬀerential operators and nonlinear PDE, Birkh¨ auser Boston Inc., Boston, MA, 1991. Lars Andersson1 Department of Mathematics University of Miami Coral Gables, FL 33124 USA email: [email protected] Vincent Moncrief 2 Department of Physics Yale University P.O. Box 208120 New Haven, CT 06520 USA email: [email protected] Communicated by Sergiu Klainerman submitted 18/03/02, accepted 04/12/02

To access this journal online: http://www.birkhauser.ch

1 Supported in part by the Swedish Natural Sciences Research Council (SNSRC), contract no. R-RA 4873-307 and the NSF, contract no. DMS-0104402 2 Supported in part by the NSF, contracts no. PHY-9732629 and PHY-0098084

Ann. Henri Poincar´e 4 (2003) 35 – 62 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/01035-28 DOI 10.1007/s00023-003-0121-0

Annales Henri Poincar´ e

Global Existence of Solutions to the Coupled Einstein and Maxwell-Higgs System in the Spherical Symmetry Dongho Chae

Abstract. We prove the global unique existence of classical solutions to the Einstein equations coupled with Maxwell-Higgs system for small initial data under the spherical symmetry. We also obtain the decay estimates of the solutions, and ﬁnd that the corresponding space-time is time-like and null geodesically complete toward the future. For the proof we reduce the system to a single ﬁrst order integrodiﬀerential equation, and use the contraction mapping theorem in the appropriate function spaces. We also obtain the completeness of space-time along the future directed time-like lines exterior to a region which resembles the even horizon of the Reissner-Nordstr¨ om black hole.

Introduction The Cauchy problem of the Einstein equations is initiated by Choquet-Bruhat[3], and the general type of local unique existence of solution is established using the harmonic coordinates (see [13], [16], [23] for updated surveys). Compared to the local existence results following [3], most of global existence theorems are based on the assumptions on the size of initial data and on symmetries of the Lorentzian geometries to construct. The most remarkable one of them is the global existence result for the vacuum Einstein equations for initial data close to the trivial one due to Christodoulou and Klainerman[11], the argument of which is being simpliﬁed in [12], [22]. We also refer the corresponding results for the spatially closed spacetime due to Andersson and Moncrief[1]. For the Einstein-matter system there is a semi-global result for the Einstein-Maxwell-Yang-Mills system for small data due to Friedrich[14]. For the small data global existence of the Einstein equations coupled with the other matters we note the self-graviting scalar system by Christodoulou[6], and the self-graviting Vlasov-Poisson system by Rein and Rendall[24] both under the assumption of the spherical symmetry of the space-time. In this paper we are concerned on the global existence problem for the Einstein equations coupled with the Maxwell-Higgs ﬁelds in the spherical symmetry. We note that our matter ﬁeld is quite general in the sense described below. Our system reduces to the system of (massless) self-graviting charged scalar ﬁelds assuming the zero Higgs potential term, V (|φ|) = 0, for which under the spherical symmetry there are numerical studies due to Hod and Piran([19], [20]) regarding to the critical behaviours during the collapse. Our result would be the ﬁrst rigorous mathematical analysis incorporating their model as a special case. For the Maxwell-Higgs

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equations on the background of Minkowski space we have global existence of solutions with arbitrary size of energy norm due to Ginbre and Velo[17] (two space dimension), and Erdley and Moncrief[14] (three space dimension). (See also previous small data global existence results due to Choquet-Bruhat[4], Choquet-Bruhat and Christodoulou[5].) Also, if we assume φ = φ∗ (real scalar ﬁeld), Aµ = 0 (no Maxwell ﬁeld coupled), 1 |φ|p+1 , then our system reduces to the coupled Einstein and the and V (|φ|) = p+1 nonlinear Klein-Gordon equation, which, setting V (|φ|) = 0, further reduces to the spherically symmetric self-graviting (neutral) scalar system previously studied by Christodoulou in a series of papers ([6], [7], [8], [9], [10]). We note that the nonlinear Klein-Gordon equation on the background of Minkowski space was studied by Struwe[25] in the spherical symmetry, and by Grillakis[18] without assumption of the symmetry. Our main purpose in this paper is that there exists a unique global classical solution to the coupled Einstein Maxwell-Higgs system for small initial data under the assumption of spherical symmetry. In the case we do not have assumption on the size of initial data it is expected that a black hole forms during evolution for large data. In this case we will show that the space-time is complete along the time-like lines exterior to a region determined by the ﬁnal Bondi mass and the charge. The region resembles the Reissner-Nordstr¨ om black hole. Our study is motivated mainly by the work of Christodoulou in [6]. The organization of this paper is the following. In Section 1 we introduce the coupled Einstein and Maxwell-Higgs equations, and reduce them to the single ﬁrst order integro-diﬀerential equation. In Section 2 we state and prove the main theorem on the global unique existence of solution. In Section 3 we study the completeness of the space-time along the time-like lines. We close this section by describing our metric under the assumption of spherically symmetric space-time. We consider the space and time oriented Lorentzian manifold diﬀeomorphic to R4 , on which the group SO(3) acts as an isometry, and the group orbits are the metric spacelike 2-spheres. The invariants of the group form a time-like curve in the space-time, which is the world line of the center of the spheres. In this spherically symmetric space-time it is convenient to introduce the function r, deﬁned by A , r= 4π where A is the area of the 2-sphere. Thus the metric on the 2-sphere is given by ds2 = r2 dΣ2 = r2 (dθ2 + sin2 θdϕ2 ). The quotient of the space-time by SO(3) is the Lorentzian 2-manifold with signature 0. We will deﬁne a coordinate u, which is a constant on every future null cones with vertices on the centers of the 2-spheres. With this coordinate system we can represent the metric in the form g(u, r)du2 − 2g(u, r)dudr + r2 dΣ2 , ds2 = −g(u, r)˜

(0.1)

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37

where g and g˜ tend to 1 as r goes to inﬁnity. We will write ds2 = gµν dxµ dxν in the following sections.

1 The coupled Einstein and Maxwell-Higgs system The Lagrangian of the Maxwell-Higgs ﬁelds in a given metric gµν is 1 Fµα Fνβ g µν g αβ − g µν Dµ φ(Dν φ)∗ − V (|φ|), (1.1) 8π √ where φ = φ1 + iφ2 , i = −1 is a complex scalar ﬁeld, and Dµ φ = ∂µ φ + iAµ is the (gauge) covariant derivative, Fµν = ∂µ Aν − ∂ν Aµ . V (·) is a continuously diﬀerentiable function, further conditions on which will be imposed in the later sections. The corresponding energy-momentum-stress tensor Tµν is 1 1 Tµν = Fµα Fνβ g αβ − gµν Fρα Fσβ g ρσ g αβ 4π 4 1 (1.2) +Re{Dµ φ(Dν φ)∗ } − gµν g αβ Dα φ(Dβ φ)∗ − gµν V (|φ|). 2 LMH = −

The coupled Einstein and Maxwell-Higgs equation is 1 Rµν − gµν R = 8πTµν . 2

(1.3)

This system is coupled with the matter ﬁeld equations ∇µ T µν = 0,

(1.4)

which corresponds to g µν Dµ Dν φ =

∂V (|φ|) ∂φ∗

(1.5)

with its complex conjugate, and ∇ν F µν = 4πJ µ ,

J µ = Im{φ(Dµ φ)∗ }.

Following [6], we introduce the new function h=

∂(rφ) . ∂r

Then,

1 r ¯ φ=h= h(s)ds. r 0 We introduce the local charge function r √ Q(u, r) = J 0 dv = 4π J 0 −γr2 dr, B(0,r)

0

(1.6)

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which, physically, represents the total charge inside B(0, r), the ball of radius r, at the retarded time u. Here we use the notation γ = det(gµν ). In our choice of metric in (0.1) we can represent the charge function in terms of h as r ¯ ∗ h − hh ¯ ∗ )ds. Q(u, r) = 4πi s(h (1.7) 0

In the spherical symmetry we assume Aθ = Aϕ = 0 as usual. Then, we can choose our gauge so that Ar = 0. Thus, among the four gauge ﬁeld components we are left only with Au = Au (u, r) as a nontrivial unknown. Then, the radial component of (1.6) can be integrated to give r Q Au = gds. (1.8) 2 s 0 The {rr} component of (1.3) is

2 ∂φ 2 1 ∂g = 8π . r g ∂r ∂r

(1.9)

This, combined with the formula ∂φ h − ¯h = ∂r r gives

g = exp −4π

r

∞

¯2 |h − h| ds . s

(1.10)

The {ur} component of (1.3) is 2 1 Q2 g 1 g˜ ∂ g 1 ∂φ ¯ , g˜ + + gV (|h|) +f − 1 = 8π g˜ g r2 g˜ g ∂r g˜ 2 ∂r 8π r4 which can be integrated with respect to g˜ as 1 r Q2 8π r 2 ¯ g˜ = s V (|h|)gds 1 − 2 gds − r 0 s r 0 1 r Q2 8π r 2 ¯ = g¯ − gds − s V (|h|)gds, r 0 s2 r 0

r where we set g¯ = 1r 0 gds. Using (1.9), (1.10) and (1.11), we can rewrite the equation (1.5) as Dh =

2 ¯ 1 ¯ − Q (h − h)g ¯ − ihA0 − 4πgr ∂V (|h|) , ¯ − iQ g h (g − g˜)(h − h) 2r 2r3 2r ∂ ¯h∗

where D=

g˜ ∂ ∂ − . ∂u 2 ∂r

(1.11)

(1.12)

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2 Global existence of solutions In this section we consider the initial value problem of (1.10)–(1.12) with the initial data h0 (r) = h(0, r). Let us introduce the function space X deﬁned by X = { h(·, ·) ∈ C 1 ([0, ∞) × [0, ∞)) | hX < ∞ }, where hX

2 3 ∂h := sup sup (1 + r + u) |h(u, r)| + (1 + r + u) (u, r) . ∂r u≥0 r≥0

We also introduce X0 = { h(·) ∈ C 1 ([0, ∞)) | hX0 < ∞ }, where hX0

2 3 ∂h := sup (1 + r) |h(r)| + (1 + r) (r) . ∂r r≥0

and denote h0 X0 = d. Below we will also use the space Y containing X, and deﬁned by Y = {h(·, ·) ∈ C 1 ([0, ∞) × [0, ∞)) | h(0, r) = h0 (r), where

hY < ∞},

2 hY = sup sup (1 + r + u) |h(u, r)| . u≥0 r≥0

The purpose of this section is the proof of the following theorem. Theorem 2.1 Let us assume that the function V (·) in (1.1) is a twice continuous diﬀerentiable function, and there exists a constant K0 ≥ 0 such that 2 ∂ V (|φ|) 2 ∂V (|φ|) |φ| ≤ K0 |φ|p+1 ∀φ ∈ C |φ| + (2.1) |V (|φ|)| + ∂φ∗ ∂φ∂φ∗ for some p ∈ [3, ∞). Suppose we have an initial data h(0, r) ∈ C 1 ([0, ∞)) such −3 ). Let us put d = h(0, ·)X0 . Then, that h(0, r) = O(r−2 ) and ∂h ∂r (0, r) = O(r there exists δ > 0 such that if d < δ, then there exists a unique global classical solution h ∈ C 1 ([0, ∞) × [0, ∞)) of (1.10)–(1.12) with h(0, r) as the initial data. This solution has the decay property: ∂h −2 −3 (2.2) |h(u, r)| ≤ C(1 + u + r) , ∂r (u, r) ≤ C(1 + u + r) , and the corresponding space-time is time-like and null geodesically complete toward the future.

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Remark 2.1. In the case of self-graviting scalar ﬁelds as studied by Christodoulou in [6], the order of decays for initial data and solutions is higher than ours obtained ∂h(u,r) in Theorem 1, namely he obtained |h(u, r)| + r ∂r = O(r−3 ) for the solution −3 ). In h(u, r) under the assumption on the initial data |h(0, r)| + r ∂h(0,r) ∂r = O(r our case we ﬁnd diﬃculties in obtaining similar decay rates. Currently we do not know if this lower decay estimates is just from technical diﬃculty in the method of the proof of the theorem, or an essential obstacle. Remark 2.2. If we set V (|φ|) = 0 in (1.2), then the energy momentum tensor of Maxwell-Higgs ﬁelds reduces to that of (massless) self-graviting charged scalar ﬁelds, and the matter ﬁeld equations become g µν Dµ Dν φ = 0,

∇ν F µν = 4πIm{φ(Dµ φ)∗ }.

In this case we ﬁnd that the reduced equation (1.12) becomes Dh =

2 1 ¯ − Q (h − h)g ¯ − ihA0 , ¯ − iQ g h (g − g˜)(h − h) 3 2r 2r 2r

where 1 g˜ = g¯ − r

r

0

(2.3)

Q2 gds. s2

Thus Theorem 2.1 implies that we have the global existence of classical solution for the equations of (massless) self-graviting charged scalar ﬁelds with the similar decays as in (2.2 ) under the similar hypothesis on the initial data as in the theorem. Remark 2.3. In the case φ = φ∗ (real scalar ﬁeld) we have Q = 0 (see (1.7)), and p+1 thus Aµ = 0 (see (1.8)), and Fµν = 0, and thus choosing V (|φ|) = |φ| p+1 , the matter ﬁeld equations become g φ = φ|φ|p−1 , which is the nonlinear Klein-Gordon equation, and can be reduced to Dh =

1 ¯ − 4πgrh| ¯ h| ¯ p−1 , (g − g˜)(h − h) 2r

where 8π g˜ = g¯ − (p + 1)r

r 0

(2.4)

¯ p+1 gds. s2 |h|

In this case Theorem 2.1 implies the global existence of classical solutions of the coupled Einstein and nonlinear Klein-Gordon system, which is obtained previously in [2].

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41

Proof of Theorem 2.1. We consider the mapping h → F (h), which is deﬁned as the solution of the ﬁrst order linear partial diﬀerential equation DF =

¯ 1 Q2 ¯ − iF A0 − 4πgr ∂V (|h|) , (2.5) ¯ − iQ g h (g − g˜)(F − ¯ h) − 3 (F − h)g 2r 2r 2r ∂ ¯h∗

with the initial condition F (h)(0, r) = h(0, r)

(2.6)

Let us denote B(0, x) = {f ∈ X | f X ≤ x}. We will prove the theorem by verifying that for suitable x and d = d(x) the mapping F (·) satisﬁes: (i) F : B(0, x) → B(0, x), and (ii) there exists λ = λ(x) ∈ (0, 1) such that F (h1 ) − F(h2 )Y ≤ λh1 − h2 Y , i.e., F contracts in Y . Then, the standard contraction mapping theorem provides us the unique ﬁxed point h ∈ X such that F (h) = h, which is the solution of the nonlinear problem (1.10)–(1.12). We set hX = x. Let r(u) = χ(u; r0 ) be the solution of the ordinary diﬀerential equation 1 dr = − g˜(u, r), du 2

r(0) = r0 .

We denote r1 = χ(u1 ; r0 ), then (2.7) gives 1 u1 r1 = r0 − g˜(u, χ(u; r0 ))du. 2 0 Using this characteristic, we can represent F as the integral as follows. u1 g − g˜ Q2 − 3 g − iA0 du F (u1 , r1 ) = h(0, r0 ) exp 2r 2r 0 χ u1 u1 2 g − g˜ Q + − 3 g − iA0 du [f ]χ du, exp 2r 2r 0 u χ where we denoted

(2.7)

(2.8)

(2.9)

¯ Q2 iQ ¯ ∂V (|h|) 1 f= (g − g ˜ ) − g h − 4πgr g − . 2r3 2r 2r ∂ ¯h∗

We estimate ¯ ≤ |h|

1 r

0

r

hX

2 ds

(1 + s + u)

=

x . (1 + u) (1 + r + u)

(2.10)

(2.11)

42

Dongho Chae

And,

r ∂h ds (u, s) ds ≤ x ∂s (1 + s + u)3 r r 1 1 1 − 2 . 2 (1 + r + u)2 (1 + r + u)

|h(u, r) − h(u, r )| ≤ = From this we have ¯ |(h − h)(u, r)|

≤ ≤ =

1 r

Ann. Henri Poincar´e

r

(2.12)

r

|h(u, r) − h(u, r )|dr x r 1 1 − 2 dr 2r 0 (1 + r + u)2 (1 + r + u) xr . 2 2 (1 + r + u) (1 + u) 0

(2.13)

Thus,

∞

0

¯2 x2 |h − h| dr ≤ r 4 (1 + u)2

0

∞

x2 ds = . (1 + s + u)4 24 (1 + u)4 s

(2.14)

This, combined with (1.10), implies g(u, 0) ≥ exp −

πx2 6 (1 + u)4

.

Using this inequality, we estimate r r ¯2 ∂g |h − h| (u, s) ≤ 4π ds s r ∂s r r πx2 s ≤ 2 4 ds (1 + u) r (1 + s + u) 1 πx2 1 1 = − 2 2 2 (1 + r + u)2 (1 + u) (1 + r + u) 1 1 1 − (1 + u) − , 3 (1 + r + u)3 (1 + r + u)3 and obtain |(g − g¯)(u, r)|

≤ ≤

(2.15)

(2.16)

1 r r ∂g (u, s) dsdr |g(u, r) − g(u, r )|dr ≤ r 0 r ∂s 0 r 2 1 πx 1 1 − 2 (1 + r + u)2 (1 + u)2 r 0 (1 + r + u)2

1 r

r

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43

1 1 1 dr − (1 + u) 3 − 3 3 (1 + r + u) (1 + r + u) =

πx2 r2

3.

3

(2.17)

3 (1 + u) (1 + r + u)

The charge Q is estimated as r r ∗ ∗ ¯ ¯ ¯ |Q(u, r)| = 4πi s(h h − hh )dr ≤ 8π s|h||h|ds 0 0 r 8πx2 s 4πx2 r2 ≤ ds = 2 2. (1 + u) 0 (1 + s + u)3 (1 + u) (1 + r + u)

(2.18)

Thus, 1 r

0

r

|Q|2 ds s2

≤ =

16π 2 x4 4

r

s2

4 ds

(1 + u) r 0 (1 + s + u) 16π 2 x4 r2 6 3. 3 (1 + u) (1 + r + u)

We also estimate the potential term as r r 1 2 ¯ ¯ p+1 ds ≤ K0 s V (| h|)ds s2 |h| r r 0 0 r K0 xp+1 s2 ≤ p+1 p+1 ds (1 + u) r 0 (1 + s + u) r s2 K0 xp+1 ≤ p+1 4 ds (1 + u) r 0 (1 + s + u) K0 xp+1 r2 = p+2 3. 3 (1 + u) (1 + r + u)

(2.19)

(2.20)

From (2.17) and (2.19) we have |g − g˜| ≤ ≤

1 r |Q|2 8π r 2 ¯ |g − g¯| + ds + s V (|h|)ds r 0 s2 r 0 C0 (x2 + x4 + xp+1 )r2 . (1 + u)3 (1 + r + u)3

Combination of (2.19) and (2.20) with (2.15) yields 8π r 2 1 r |Q|2 ¯ ds − s V (|h|)ds g˜ ≥ g¯(u, 0) − r 0 s2 r 0 πx2 8πK0 p+1 16 ≥ exp − x . − π 2 x4 − 6 3 3

(2.21)

(2.22)

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Ann. Henri Poincar´e

Let x1 be the smallest positive root of the equation 8πK0 p+1 πx2 16 x = 0. exp − − π 2 x4 − 6 3 3

(2.23)

Then, the function

πx2 8πK0 p+1 16 x k = k(x) = exp − − π 2 x4 − 6 3 3

(2.24)

satisﬁes 0 < k ≤ 1 for all x ∈ [0, x1 ). From (2.10) we estimate 2 ¯ Q ¯ + Q gh ¯ + 4π gr ∂V (|h|) ¯ + 1 (g − g˜)h |f | ≤ 3 g h 2r 2r 2r ∂ ¯h∗ = {1} + {2} + {3} + {4}.

(2.25)

Combining (2.18) with (2.11), we have {1} ≤

8π 2 x5 r 5

5

(1 + u) (1 + r + u)

≤

8π 2 x5 5

4.

(1 + u) (1 + r + u)

(2.26)

From (2.21) and (2.11) we obtain {2} ≤

3 5 p+2 1 ¯ ≤ C1 (x + x + x ) . |g − g˜||h| 4 3 2r (1 + u) (1 + r + u)

Similarly {3} ≤

2πx3 (1 + u)3 (1 + r + u)2

,

(2.27)

(2.28)

and ¯p≤ {4} ≤ 4πK0 r|h|

4πK0 rxp 4πK0 xp p p ≤ 3 2. (1 + u) (1 + r + u) (1 + u) (1 + r + u)

(2.29)

Adding (2.26)–(2.29), we have C2 (x3 + x5 + xp + xp+2 )

|f | ≤ For r(u) = χ(u; r0 ) we have r(u) = r1 +

1 2

3

2

(1 + u) (1 + r + u)

u1

u

.

1 g˜(u , r(u ))du ≥ r1 + k(u1 − u), 2

(2.30)

(2.31)

and, since k ∈ (0, 1] for x ∈ (0, x1 ], 1 + u + r1 +

k u1 k (u1 − u) ≥ k(1 + + r1 ) ≥ (1 + u1 + r1 ). 2 2 2

(2.32)

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45

Thus, 0

u1

3

|[f ]χ |du

5

p

p+2

≤ C2 (x + x + x + x ≤ ≤

u1

)

3

2

(1 + r1 + u1 ) k 2 C3 (x3 + x5 + xp + xp+2 ) 2

(1 + r1 + u1 ) k 2

∞

du

2

(1 + u) (1 + r + u)

0

4C2 (x3 + x5 + xp + xp+2 )

1

χ

du 3

(1 + u)

0

.

(2.33)

Now we estimate u1 g − g˜ Q2 − 3 g − iA0 du 0 2r 2r χ u1 u1 2 u1 g − g˜ Q ≤ + g du du + [A0 ]χ du 3 2r 2r 0 0 0 χ χ ≤

I1 + I2 + I3 .

(2.34)

Combining the estimate (2.32) with (2.21), we have C0 (x2 + x4 + xp+1 ) u1 r I1 ≤ du 3 3 2 (1 + u) (1 + r + u) χ 0 C0 (x2 + x4 + xp+1 ) ∞ 1 2 4 p+1 ≤ ). 5 ≤ C5 (x + x + x 2 (1 + u) 0 From (2.18) we obtain immediately I2

u1

≤

≤

8π 2 x4 r 4

∞

8π 2 x4

7 du

(1 + u)

0

du

4

(1 + u) (1 + r + u)

0

=

(2.35)

χ

8 2 4 π x , 3

(2.36)

and from the estimate |A0 | ≤ =

r

0

|Q| 4πx2 ds ≤ 2 s2 (1 + u) 4πx2

r 0

ds

(2.37)

3

(1 + u) (1 + r + u)

we have I3 ≤

∞ 0

4πx2

4 du

(1 + u)

=

2

(1 + s + u)

8 2 πx . 3

(2.38)

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Dongho Chae

Ann. Henri Poincar´e

Adding (2.3), (2.36) and (2.38), we obtain u1 Q2 g − g˜ − 3 g − iA0 du ≤ C6 (x2 + x4 + xp+1 ). 2r 2r 0 χ

(2.39)

From (3.26) we have 1 k 1 + r0 ≥ 1 + r1 + ku1 ≥ (1 + r1 + u1 ), 2 2

(2.40)

and |h(0, r0 )| ≤

d d 4d ≤ ≤ 1 2. 2 2 (1 + r0 )2 (1 + r1 + 2 ku1 ) k (1 + r1 + u1 )

(2.41)

Combining this with (2.33) and (2.39), we obtain u1 Q2 g − g˜ − 3 g − iA0 du |F(u1 , r1 )| ≤ |h(0, r0 )| exp 2r 2r 0 χ u1 u1 Q2 g − g˜ − 3 g − iA0 du |[f ]χ |du exp + 2r 2r 0 u χ ≤

4 exp[C6 (x2 + x4 + xp+1 )] 2

(1 + r1 + u1 ) k 2

d + C3 (x3 + x5 + xp + xp+1 ) , (2.42)

from which we have C7 (d + x3 + x5 + xp + xp+1 ) exp[C6 (x2 + x4 + xp+1 )], k2 (2.43) where we set C7 = 4 max{C3 , 1}. Now let us denote sup {(1 + r + u)2 |F(u, r)|} ≤

r,u≥0

G(u, r) = with

∂F (u, r) ∂r

∂h (0, r0 ). ∂r Diﬀerentiating (2.5) with respect to r we obtain 1 ∂˜ g g − g˜ Q2 g + − 3 − iA0 G DG(u1 , r1 ) = 2 ∂r 2r 2r 2 1 ∂ 3 1 ∂A0 Q ∂Q Q2 ∂g (g − g˜) − 2 (g − g˜) + − i + − g − F 2r ∂r 2r 2r4 r3 ∂r 2r3 ∂r ∂r G(0, r0 ) =

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

47

¯ ∂¯ ¯ ∂ ¯h∗ ∂ 2 V (|h|) ∂ 2 V (|h|) h∗ + 2 ∂r ∂¯ h∗ ∂ ¯ h ∂r ∂h∗ 2 ¯ ∂h Q g − g˜ iQ − g g− + 2r3 2r 2r ∂r 2 3Q 1 ∂ Q ∂Q i ∂Q (g − g˜) + g− 3 + g+ 2r ∂r 2r4 r ∂r 2r ∂r 2 Q iQ iQ ∂g 1 ¯ + 2 g − 2 (g − g˜) h. − + 2r3 2r ∂r 2r 2r +

(2.44)

As previously we can solve the linear equation (2.44) with respect to G by using the characteristic introduced in (2.7) and (2.8) as follows: u1 1 ∂˜ g g − g˜ Q2 g ∂h G(u1 , r1 ) = (0, r0 ) exp + − 3 − iA0 du ∂r 2 ∂r 2r 2r 0 χ u1 u1 2 1 ∂˜ g g − g˜ Q g + + − 3 − iA0 du [f1 ]χ du, exp 2 ∂r 2r 2r 0 u χ (2.45) where we set 2 1 ∂ 3Q 1 ∂A0 Q ∂Q Q2 ∂g (g − g˜) − 2 (g − g˜) + − i f1 = − g − F 2r ∂r 2r 2r4 r3 ∂r 2r3 ∂r ∂r ¯ 2 2 ¯ ∂ ¯h ∂ 2 V (|h|) ¯ ∂ ¯h∗ ∂h ∂ V (|h|) Q g − g˜ iQ − g + + g − + 2 2r3 2r 2r ∂r ∂r ∂ ¯h∗ ∂ ¯h ∂r ∂h∗ 2 3Q 1 ∂ Q ∂Q i ∂Q (g − g˜) + g g− 3 + g+ 2r ∂r 2r4 r ∂r 2r ∂r 2 Q iQ iQ ∂g 1 ¯ + + 2 g − 2 (g − g˜) h. − (2.46) 2r3 2r ∂r 2r 2r We write |f1 | ≤ 2 2 2Q 1 ∂ Q ∂g ∂A0 1 Q ∂Q 2r ∂r (g − g˜) − 2r2 (g − g˜) + 3r4 − r3 ∂r g + 2r3 ∂r + ∂r |F | 2 ¯ Q g − g˜ iQ p−1 ∂ h ¯ + 3g − − g + 2K0 |h| ∂r 2r 2r 2r 2 1 ∂ 2Q Q ∂Q i ∂Q (g − g˜) + g g− 3 g+ + 2r ∂r 3r4 r ∂r 2r ∂r 2 Q iQ iQ ∂g 1 ¯ |h| + + − g − (g − g ˜ ) 3 2 2 2r 2r ∂r 2r 2r ¯ = (A1 + A2 + A3 + A4 )|F | + A5 + (A6 + A7 )|h|. (2.47)

48

Dongho Chae

Ann. Henri Poincar´e

From (1.11) we compute ∂˜ g ∂¯ g 1 r Q2 Q2 8π r 2 ¯ ¯ = + 2 gds − 3 g + 2 s V (|h|)gds − 8πrV (|h|)g. ∂r ∂r r 0 s2 r r 0

(2.48)

g g−¯ g Thus, using the formula, ∂¯ ∂r = r , and the previous estimates (2.19) and (2.20) we have r ∂˜ |Q|2 |Q|2 8π r 2 g ≤ |g − g¯| + 1 gds + 3 + 2 s |V (|φ|)|ds + 8πr|V (|φ|)| ∂r r r 2 0 s2 r r 0 πx2 r 16π 2 x4 r 16π 2 x4 r ≤ + + 3 3 6 3 4 4 3 (1 + u) (1 + r + u) 3 (1 + u) (1 + r + u) (1 + u) (1 + r + u) 8πK0 rxp+1 8πK0 rxp+1 + p+3 4 + 3 (1 + u) (1 + r + u) 3 (1 + u)p+1 (1 + r + u)p+1 C8 (x2 + x4 + xp+1 )r ≤ (2.49) 3 3. (1 + u) (1 + r + u)

As in (2.13), we have ¯2 ∂g 4π|h − h| πx2 r ≤ ≤ . ∂r 2 r (1 + u) (1 + r + u)4

(2.50)

The estimate (2.21) implies immediately |g − g˜| C0 (x2 + x4 + xp+1 ) ≤ 3 3. 2 2r 2 (1 + u) (1 + r + u)

(2.51)

Combining (2.49), (2.50) and (2.51), we immediately have A1 ≤ We calculate and estimate ∂Q ∂r ≤

C9 (x2 + x4 + xp+1 )

3.

2

(2.52)

(1 + u) (1 + r + u)

¯ ∗ h − hh ¯ ∗ | ≤ 8πr|h||h| ¯ 4πr|h 8πrx2

≤

3.

(2.53)

(1 + u) (1 + r + u)

This, combined with (2.18) provides us with A2

≤ ≤

32π 2 x4 4

4

9 (1 + u) (1 + r + u) C10 x4 . 3 (1 + u) (1 + r + u)4

+

32π 2 x4 3

5

(1 + u) (1 + r + u)

(2.54)

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

49

As previously, (2.50) and (2.18) imply A3 ≤

8π 3 r2 x6 6

8

(1 + u) (1 + r + u)

≤

8π 3 x6

6,

6

(2.55)

(1 + u) (1 + r + u)

and

|Q|g 4πx2 ≤ 2 2. 2 r (1 + u) (1 + r + u) Similarly, from previous estimates, we deduce 2 ¯ Q g − g˜ |Q| |h − h| p−1 ¯ A5 ≤ 2r3 + 2r + 2r + 2K0 |h| r 3 5 p p+2 C11 (x + x + x + x ) . ≤ 3 4 (1 + u) (1 + r + u) A4 ≤

(2.56)

(2.57)

By (2.53)

i ∂Q 8πx2 r ∂r ≤ (1 + u) (1 + r + u)3 . This, and the previous estimates (2.49), (2.50) and (2.54) yield A6

≤ ≤

C9 (x2 + x4 + xp+1 ) 2

3

(1 + u) (1 + r + u) C12 (x2 + x4 + xp+1 ) 3

(1 + u) (1 + r + u)

+

C10 x4

3

4

(1 + u) (1 + r + u)

+

(2.58)

4πx2 (1 + u) (1 + r + u)3

.

(2.59)

By estimates (2.18) and (2.50) we have 2 |Q| |Q| ∂g |Q| + A7 ≤ + 2r3 2r ∂r 2r2 8π 3 r2 x6 4π 2 x3 2πx2 ≤ + + 6 6 3 4 2 2 (1 + u) (1 + r + u) (1 + u) (1 + r + u) (1 + u) (1 + r + u) C13 (x2 + x3 + x6 ) ≤ (2.60) 2 2. (1 + u) (1 + r + u) Summing (2.52)–(2.60) up, we obtain |f1 | ≤

C14 (x2 + x4 + x6 + xp )|F(u, r)|

≤

C10 (x3 + x5 + xp + xp+2 )

(1 + u)4 (1 + r + u)4 C11 (x2 + x4 + xp+1 ) C13 (x2 + x3 + x6 ) ¯ + 3 + 2 2 |h| (1 + u) (1 + r + u) (1 + u) (1 + r + u)

(1 + u)2 (1 + r + u)2

+

C14 (x2 + x4 + x6 + xp ) 2

4

sup {(1 + r + u)2 |F(u, r)|}

(1 + u) (1 + r + u) u,r≥0 C15 (x3 + x4 + x5 + x6 + +x7 + xp + xp+2 ) . + (1 + u)2 (1 + r + u)3

(2.61)

50

Dongho Chae

Ann. Henri Poincar´e

Now, using the inequality along the characteristics, as in (3.26) and (3.27), we know k u1 u (2.62) 1 + r(u) + ≥ k(1 + r1 + ) ≥ (1 + r1 + u1 ), 2 2 2 and thus we obtain in general u1 u1 rs 1 du ≤ du p q p q−s (1 + u) (1 + r + u) χ (1 + u) (1 + r + u) 0 0 χ ∞ 2m du ≤ du k m (1 + r1 + u1 )m 0 (1 + u)q−s+p−m 2m = (2.63) m (q − s + p − m − 1)k m (1 + r1 + u1 ) for q −s+p−m > 1, where q, s, p, m are positive integers. Applying this inequality, we ﬁnd that u1 C16 (x2 + x4 + x6 + xp ) 2 sup {(1 + r + u) |F(u, r)|} |[f1 ]χ |du ≤ k (1 + r + u ) 1 1 u,r≥0 0 + ≤

≤

8C15 (x3 + x4 + x5 + x6 + +x7 + xp + xp+2 ) k 3 (1 + r1 + u1 )3

C17 k 3 (1

2 3 {(x

+ r1 + u1 )

+ x4 + x6 + xp )(d + x3 + xp + xp+1 )

+ exp[C6 (x2 + x4 + xp+1 )] + x3 + x4 + x5 + x6 + +x7 + xp + xp+2 } C17 exp[C6 (x2 + x4 + xp+1 )] k 3 (1 + r1 + u1 )

3

×{(x2 + x4 + x6 + xp )(d + x3 + xp + xp+1 ) + +x3 + x4 + x5 + x6 + +x7 + xp + xp+2 }.

(2.64)

Using the estimate (2.49), we have u1 u1 g 1 ∂˜ r 2 4 p+1 du du ≤ C7 (x + x + x ) 2 ∂r χ (1 + u)3 (1 + r + u)3 χ 0 0 ∞ 1 ≤ C7 (x2 + x4 + xp+1 ) 5 du (1 + u) 0 ≤

C18 (x2 + x4 + xp+1 ).

(2.65)

Combining this with (2.39), we obtain u1 1 ∂˜ g g − g˜ Q2 g + − − iA du 0 3 2 ∂r 2r 2r 0 χ u1 u1 g − g˜ Q2 g g 1 ∂˜ du ≤ 2r − 2r3 − iA0 du + 2 ∂r χ 0 0 χ ≤ C19 (x2 + x4 + xp+1 ).

(2.66)

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

51

Similarly to (2.41) we estimate ∂h 8d (0, r0 ) ≤ h0 X0 ≤ 3 ∂r u0 3 3 k (1 + r1 + u1 ) 1 + r0 + 2

(2.67)

for all x ∈ [0, x1 ). Thus from (2.67) and (2.64) u1 ∂h |[f1 ]χ |du exp[C19 (x2 + x4 + xp+1 )] |G(u1 , r1 )| ≤ (0, r0 ) + ∂r 0 C21 exp[C20 (x2 + x4 + xp+1 )] {d + (x2 + x4 + x6 + xp ) ≤ 3 k 3 (1 + r1 + u1 ) (d + x3 + x5 + xp+1 ) + x3 + x4 + x5 + x6 + x7 + xp + xp+2 }, (2.68) where we set C20 = C6 + C19 . From this we have C21 exp[C20 (x2 + x4 + xp+1 )] 3 ∂F (u, r) ≤ sup (1 + r + u) ] ∂r k3 u,r≥0 ×{d + (x2 + x4 + x6 + xp )(d + x3 + x5 + xp+1 ) +x3 + x4 + x5 + x6 + x7 + xp + xp+2 }.

(2.69)

Combining (2.69) with (2.43), we thus have F (h)X

C7 exp[C6 (x2 + x4 + xp+1 )] (d + x3 + x5 + xp + xp+1 ) k2 C21 exp[C20 (x2 + x4 + xp+1 )] + {d + (x2 + x4 + x6 + xp ) k3 (d + x3 + x5 + xp+1 ) + x3 + x4 + x5 + x6 + x7 + xp + xp+2 } C22 ≤ exp[C20 (x2 + x4 + xp+1 )](2 + x2 + x4 + x6 + xp ) k3 ×(d + x3 + x4 + x5 + x6 + x7 + xp + xp+1 + xp+2 ). (2.70)

≤

Let us set Λ1 (x)

=

xk 3 exp −C20 (x2 + x4 + xp+1 ) C22 (2 + x2 + x4 + x6 + xp ) −(x3 + x4 + x5 + x6 + x7 + xp + xp+1 + xp+2 ).

(2.71)

Then, we ﬁnd that Λ1 (0) = 0, Λ1 (0) > 0, and Λ1 (x) → −∞ as x → ∞. Thus there exists x0 ∈ (0, x1 ) such that Λ1 (x) is monotone increasing function on [0, x0 ]. For every x ∈ (0, x0 ) we deduce that if d < Λ1 (x), then F X ≤ x, and F : B(0, x) → B(0, x).

52

Dongho Chae

Ann. Henri Poincar´e

We now show that the mapping h → F (h) contracts in Y . In order to prove this we consider equations (2.1) for h1 and h2 in X with h1 (0, r) = h2 (0, r), and take diﬀerence between them. We put Θ = F (h1 ) − F(h2 ), and use the obvious notations gj = g(hj ), Qj = Q(hj ), Fj = F (hj ), Gj = G(hj ), and A0j = A0 (hj ) for j = 1, 2. We assume max{h1 X , h2 X } < x. Then, we have ∂Θ 1 ∂Θ − g˜1 ∂u 2 ∂r

=

1 1 ¯ 1 + ¯h2 ) (g1 − g˜2 )G2 + (˜ g1 − g˜1 )(F1 − F2 − h 2 2r 1 1 ¯ 2 )g1 + (g1 − g2 − g˜1 + g˜2 )(F2 − ¯h2 ) − 3 (Q21 − Q22 )(F2 − h 2r 2r 1 ¯1 + h ¯ 2 )g1 − 1 Q2 (F2 − h ¯ 2 )(g1 − g2 ) − 3 Q22 (F1 − F2 − h 2r 2r3 2 i ¯1 ¯ 1 − i Q2 (g1 − g2 )h − (Q1 − Q2 )g1 h 2r 2r i ¯1 − h ¯ 2 ) − i(F1 − F2 )A01 − iF2 (A01 − A02 ) − Q2 g2 (h 2r ¯ 1 |) ¯ 1 |) ∂V (|h ¯ 2 |) ∂V (|h ∂V (|h −4π(g1 − g2 )r . ¯ ∗ − 4πg2 r ¯ ∗ − ∂h ¯∗ ∂h ∂h 1 1 2

Now, as previously, we use the characteristic χj = χj (u, r) deﬁned by dr = −˜ g1 (χj (u, r), u) du

;

r(0) = r0 .

As previously we can represent Θ as u1 u1 g1 − g˜1 Q22 Θ(u1 , r1 ) = − 3 g − iA01 exp du [ϕ]χ1 du, 2r 2r 0 u χ1 where ϕ is, after rearrangement of terms, 2 Q2 g1 g1 − g˜1 iQ2 g2 ¯ ¯ − − ϕ = (h1 − h2 ) 2r3 2r 2r 2 ¯ ¯ 1 |) ¯1 ∂V (|h Q2 (h2 − F2 ) iQ2 h − 4πr − +(g1 − g2 ) 2r3 2r ∂ ¯h∗1 ¯ 1 )g1 (Q1 + Q2 )(F1 − h ig1 ¯h1 + −(Q1 − Q2 ) 2r3 2r 1 g1 − g˜2 )G2 −i(A01 − A02 )F2 + (˜ 2 1 ¯ 2) + [g1 − g˜1 − (g2 − g˜2 )](F2 − h 2r ¯ 1 |) ∂V (|h ¯ 2 |) ∂V (|h − −4πg2 r . ¯∗ ¯∗ ∂h ∂h 1 2

(2.72)

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

53

We write |ϕ|

≤

=

2 Q2 g1 g1 − g˜1 iQ2 g2 ¯ ¯ − |h1 − h2 | 3 − 2r 2r 2r 2 ¯ 1 |) ¯ 2 − F2 ) iQ2 ¯h1 Q (h ∂V (|h +|g1 − g2 | 2 − 4πr − 2r3 2r ∂ ¯h∗1 ¯ 1 )g1 ¯1 (Q1 + Q2 )(F1 − h ig1 h +|Q1 − Q2 | + 2r3 2r 1 +|A01 − A02 ||F2 | + |˜ g1 − g˜2 ||G2 | 2 ¯ ¯ 1 ¯ 2 | + 4πg2 r ∂V (|h1 |) − ∂V (|h2 |) + |g1 − g˜1 − (g2 − g˜2 )||F2 − h ∂ ¯h∗ ¯∗ 2r ∂h 1 2 (2.73) B1 + B2 + B3 + B4 + B5 + B6 + B7 .

By the similar computations as before we have the following estimates ¯ 2| ≤ ¯1 − h |h

1 r

r

0

|h1 − h2 |ds ≤

y , (1 + u) (1 + r + u)

(2.74)

and ¯1 − h ¯ 2 )| ≤ |h1 − h2 − (h ≤ ≤

¯1 − h ¯ 2| |h1 − h2 | + |h 1 ds 1 r y + r 0 (1 + s + u)2 (1 + r + u)2 2y . (1 + u) (1 + r + u)

(2.75)

Thus, using (2.13), we estimate ¯ 1 |2 − |h2 − h ¯ 2 |2 |h1 − h

¯1 − h ¯ 2 )|(|h1 − h ¯ 1 | + |h2 − h ¯ 2 |) ≤ |(h1 − h2 ) − (h 2xyr ≤ (2.76) 2 3. (1 + u) (1 + r + u)

Thanks to the mean value theorem we have ∞ 1 ¯ 1 |2 − |h2 − h ¯ 2 |2 |h1 − h |g1 − g2 | ≤ 4π s r ∞ sxy ≤ 8π 2 3 ds (1 + u) (1 + s + u) r 4πyx . ≤ 3 (1 + u)2 (1 + r + u)2

(2.77)

54

Dongho Chae

And |¯ g1 − g¯2 |

≤ =

1 r

0

r

|g1 − g2 |ds ≤

2

3r (1 + u)

4πxy 2

4πxy

3 (1 + u) (1 + r + u)

0

Ann. Henri Poincar´e

r

ds 2

(1 + s + u)

.

(2.78)

By the similar computation to (2.7) r ¯ ∗ h1 − h ¯ 1 h∗ − ¯h∗ h2 + h ¯ 2 h∗ |ds s|h |Q1 − Q2 | ≤ 4π 1 1 2 2 0 r ¯1 − h ¯ 2 ||h1 | + |h1 − h2 ||h ¯ 2 |)ds ≤ 8π s(|h 0

≤

8πr2 xy 2

2.

(2.79)

(1 + u) (1 + r + u)

We use (2.77) and (2.79) to estimate r |Q1 − Q2 |g1 + |g1 − g2 ||Q2 | ds |A01 − A02 | ≤ s2 0 r r |Q1 − Q2 | |g1 − g2 ||Q2 | ds + ds ≤ s2 s2 0 0 r r 8πxy ds ds 16π 2 xy ≤ + 2 2 4 (1 + u) 0 (1 + s + u) (1 + u) 0 (1 + s + u)4 3 r u u 2 2 16π xy + r (1 + ) + r(1 + ) 3 2 2 8πrxy + = (1 + u)3 (1 + r + u) (1 + u)7 (1 + r + u)3 20π 2 xy ≤ (2.80) 3. (1 + u) From (2.79) again 1 r 1 2 |Q − Q22 |ds r 0 s2 1

≤ ≤ =

From (2.18) and (2.77) we have 1 r 1 |Q1 |2 |g1 − g2 |ds r 0 s2

r

1 |Q1 − Q2 |(|Q1 | + |Q2 |)ds 2 0 s r s2 8πxy 4 4 ds (1 + u) r 0 (1 + s + u) 32π 2 r2 xy 6 3. 3 (1 + u) (1 + r + u)

1 r

≤ =

64π 3 x5 y 6

r

s2

(2.81)

4 ds

3 (1 + u) r 0 (1 + s + u) 64π 3 x5 yr2 . 9 (1 + u)6 (1 + r + u)3

(2.82)

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

By the mean value theorem we have ¯ 2 |)| ≤ ¯ 1 |) − V (|h |V (|h ≤ ≤ ≤ ≤ 1 r

r

0

∂V ¯1 − h ¯ ¯ 2| ¯ (|th1 + (1 − t)h2 |) |h ∂¯ h 0 1 ¯ 1 + (1 − t)h ¯ 2 |p |h ¯1 − h ¯ 2| |th K0 1

0

¯ 1 |p + |h ¯ 2 |p )|h ¯1 − h ¯ 2| K0 2p (|h p p K0 2 x y p+1

8π r

0

r

p+1

(1 + u) (1 + r + u) K0 2p xp y 4 4, (1 + u) (1 + r + u)

¯ 1 |) − V (|h ¯ 2 |)|gds s2 |V (|h

K0 2p xp y

≤

¯ 1 |)||g1 − g2 |ds s2 |V (|h

r

s2

r (1 + u) 0 (1 + s + u)4 K0 2p+2 r2 xp y 6 3, 3 (1 + u) (1 + r + u)

256π 3 K0 xp+2 y

≤

(2.83)

4

= and

55

r

p+3

ds

s2

4 r (1 + u) 0 (1 + s + u) 256π 3 K0 r2 xp+2 p+3 3. (1 + u) (1 + r + u)

=

(2.84)

ds (2.85)

From the estimates (2.78)–(2.85), we obtain 1 r 1 2 1 r 1 2 |˜ g1 − g˜2 | ≤ |¯ g1 − g¯2 | + |Q − Q |ds + |Q1 |2 |g1 − g2 |ds 2 r 0 s2 1 r 0 s2 r 8π r 2 ¯ 1 |) − V (|h ¯ 2 |)|gds + 8π ¯ 1 |)||g1 − g2 |ds s |V (|h s2 |V (|h + r 0 r 0 32π 2 r2 xy 64π 3 x5 yr2 4πyx + + ≤ 2 2 6 3 6 3 3 (1 + u) (1 + r + u) 3 (1 + u) (1 + r + u) 9 (1 + u) (1 + r + u) K0 2p+5 πr2 xp y 256π 3 K0 r2 xp+2 + 6 3 + 3 (1 + u) (1 + r + u) (1 + u)p+3 (1 + r + u)3 K0 2p+4 πrxp y 128π 3 K0 rxp+2 + + 6 3 p+3 3 3 (1 + u) (1 + r + u) (1 + u) (1 + r + u) C23 (x + x5 + xp + xp+2 )y ≤ . (2.86) 3 (1 + u) (1 + r + u) Similarly to (2.77) we estimate g1 − g¯2 )| ≤ |g1 − g2 − (¯

1 r

0

r

r r

∂ (g1 − g2 ) dsdr ∂r

56

Dongho Chae

≤ ≤ =

Ann. Henri Poincar´e

4π r r 1 ¯ 1 |2 − |h2 − h ¯ 2 |2 |h1 − h r 0 r s r r 8πxy ds dr 2 r (1 + u) 0 r (1 + s + u)3 4πrxy 3 2. (1 + u) (1 + r + u)

(2.87)

Combining (2.87) with (2.81), we have r 1 2 1 1 1 |g1 − g˜1 − (g2 − g˜2 )| ≤ |g1 − g2 − (¯ g1 − g¯2 )| + 2 |Q − Q22 |ds 2r 2r 2r 0 s2 1 r 1 4π r 2 1 2 ¯ 1 |) − V (|h ¯ 2 |)|gds |Q1 | |g1 − g2 |ds + 2 s |V (|h + 2 2r 0 s2 r 0 C24 (x + x5 + xp + xp+2 )y . (2.88) ≤ 3 2 (1 + u) (1 + r + u) From (2.74) and (2.21) we deduce

B1

|Q2 |2 |Q2 | |g1 − g˜1 | + + 2r3 2r 2r 2 4 p+1 C25 (x + x + x ) 3 3. (1 + u) (1 + r + u)

¯1 − h ¯ 2| ≤ |h ≤

(2.89)

Similarly we have the following estimates: ¯ 1 |) ¯ 2 | + |F2 |) |Q2 ||h ¯1| ∂V (|h |Q2 |2 (|h + 4πr + B2 ≤ |g1 − g2 | 2r3 2r ∂ ¯h∗1 ≤

32π 3 rx6 y

32π 2 x5 ry

7 7 6 6 |F2 | (1 + u) (1 + r + u) (1 + u) (1 + r + u) 16πK0 rxp+1 y 8π 2 rx4 y + , p+2 p+2 + 3 (1 + u) (1 + r + u) (1 + u)5 (1 + r + u)5

+

(2.90)

B3

≤ ≤

¯ 1 |) |h ¯ 1| |Q1 | + |Q2 |)(|F1 | + |h |Q1 − Q2 | + 2r3 2r 2 3 2 32π rx4 y 32π rx y |F | + 1 4 4 5 5 (1 + u) (1 + r + u) (1 + u) (1 + r + u) 8πrx2 y + , (1 + u)3 (1 + r + u)3 B4 ≤

8π 2 xy 3

(1 + u) (1 + r + u)2

|F2 |,

(2.91)

(2.92)

Vol. 4, 2003

Solutions to the Coupled Einstein and Maxwell-Higgs System

C23 (x + x5 + xp + xp+2 )y

B5 ≤

B6 ≤

2 (1 + u)3 (1 + r + u)

C24 (x + x5 + xp + xp+2 )y 3

2

(1 + u) (1 + r + u)

|F2 | +

57

|G2 |,

(2.93)

C24 (x2 + x6 + xp+1 + xp+3 )y (1 + u)4 (1 + r + u)3

,

(2.94)

and ﬁnally 2 ∂ V ¯1 − h ¯ ¯ ¯ 2 |dt 4πg2 (th1 + (1 − t)h2 ) |h ∂¯ ∗ ¯ h∂ h 0 ¯ 1 |p−1 + |h ¯ 2 |p−1 )|h ¯ 1 − ¯h2 | 2p+2 πK0 r(|h

B7

≤ ≤ ≤

1

2p+3 πK0 rxp y p+1

(1 + u)

p+1 .

(2.95)

(1 + r + u)

Summarizing the above estimates, we have 7

Bj

j=1

+

≤

C26 y(x2 + x4 + x6 + xp + xp+1 + xp+3 ) (1 + u)3 (1 + r + u)2

2 C27 y(x + x3 + x5 + xp + xp+2 ) maxj=1,2 supr,u≥0 {(1 + r + u) |Fj (u, r)|} 3

4

(1 + u) (1 + r + u)

3

+

C23 y(x + x3 + x5 + xp + xp+2 ) supr,u≥0 {(1 + r + u) |G2 (u, r)|} 3

≤ +

4

2 (1 + u) (1 + r + u) C26 y(x2 + x4 + x6 + xp + xp+1 + xp+3 ) 3

2

(1 + u) (1 + r + u) C28 y(x + x3 + x5 + xp + xp+2 ) max{F1 X , F2 X } 3

4

(1 + u) (1 + r + u)

.

(2.96)

In the previous part of the proof (see paragraphs below (2.71)) we ﬁnd that for each j = 1, 2 Fj X = F (hj )X < hj X = xj ≤ x for x ∈ (0, x0 ) and dj < λ1 (xj ), which we are assuming to hold. Thus, for x ∈ (0, x0 ) we have the estimate 7

Bj

≤

C26 (x2 + x4 + x6 + xp + xp+1 + xp+3 )y 3

+ ≤

2

(1 + u) (1 + r + u)

j=1

C28 (x2 + x4 + x6 + xp+1 + xp+3 )y 3

4

(1 + u) (1 + r + u) C29 (x2 + x4 + x6 + xp + xp+1 + xp+3 )y (1 + u)3 (1 + r + u)2

.

(2.97)

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Now, using the inequality, 1 + r(u) + u2 ≥ k(1 + r1 + u21 ) ≥ 12 (1 + r1 + u1 ) along the characteristics and the formula for the integrals as in (3.56), we estimate

u1

|[ϕ]χ |du ≤

0

≤

u1 0

7 [ Bj ]χ du j=1

C30 (x2 + x4 + x6 + xp + xp+1 + xp+3 )y k 2 (1 + r1 + u1 )2

.

(2.98)

On the other hand, similarly to (2.39), we have u1 Q22 g1 − g˜1 − 3 g − iA01 du ≤ C6 (x2 + x4 + xp+1 ). 2r 2r u χ1

(2.99)

Thus, |Θ(u1 , r1 )| ≤ ≤

u1

u1

exp 0

u

g1 − g˜1 Q2 − 23 g − iA01 2r 2r

χ1

du

|[ϕ]χ1 |du

C30 (x2 + x4 + x6 + xp + xp+1 + xp+3 )y exp[C6 (x2 + x4 + xp+1 )] 2

k 2 (1 + r1 + u1 )

.

We ﬁnally have ΘY ≤ Λ2 (x)y, where we set Λ2 (x) =

C30 (x2 + x4 + x6 + xp + xp+1 + xp+3 ) exp[C6 (x2 + x4 + xp+1 )] . k 2 (x)

The function Λ2 (x) is a smooth, monotone increasing function on [0, x0 ], and Λ2 (0) = 0, and there exists x2 ∈ (0, x0 ] such that Λ2 (x) < 1 for all x in (0, x0 ]. The mapping h → F (h) contracts in Y for hX ≤ x0 . This proves the global existence and uniqueness of solution. From (2.14) and (1.10) we ﬁnd that g → 1 as u → ∞ for each r ≥ 0. This, combined with (2.21), in turn, implies g˜ → 1 as u → ∞. Thus our metric, represented in the form of (0.1), becomes the Minkowski metric written in terms of the Bondi coordinate system. This completes the proof of Theorem 2.1.

3 Completeness along the time-like lines In the previous section we proved that the space-time is complete along the timelike and null geodesics toward the future for small initial data. The corresponding initial Bondi mass and charge decay out to inﬁnity, and no ﬁnal mass and charge remain. In this section we study completeness properties of the space-time in

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the case when the ﬁnal Bondi mass and charge are positive numbers. Our study in this section also closely follows [6], but extends it substantially to the case of “charged black hole”. Throughout this section we assume the Higgs potential function, V (|φ|) ≥ 0. We introduce the local mass function deﬁned by r g˜ Q2 (3.1) m(u, r) = 1− + 2 , 2 g r which represents the total mass inside a ball of radius r at the retarded time u. By the expression in (1.10) g is a monotone nondecreasing function of r, and thus from (1.11) we have 0 < g˜ ≤ g¯ ≤ g ≤ 1. The fact that g˜ ≤ g, in particular, implies m(u, r) ≥ 0. We set sup m(u, r) := M1 (u),

0
inf

0
|Q(u, r)| := Q1 (u)

(3.2)

for each u ∈ [0, ∞). Then, we assume lim M1 (u) = M2 < ∞,

u→∞

lim Q1 (u) = Q2 < ∞

u→∞

(3.3)

exist. Then, there exists u0 > 0 such that M1 (u) + Q1 (u) < ∞ for all u ≥ u0 . Let us introduce the real-valued function R(u) = R(M1 (u), Q1 (u)) deﬁned on [u0 , ∞) by   M1 + M12 − Q21 if M1 > Q1 (3.4) R(u) = M1 if M1 = Q1 .  0 if M1 < Q1 We set R0 = limu→∞ R(u) = R(M2 , Q2 ). The purpose of this section is the proof of the following proposition. Proposition 3.1 Suppose the above assumption described in (3.3) holds. Then, the time-like line r = r1 is complete toward future if r1 > R0 . Proof. Given u ≥ u0 , we consider r1 > R(u). Then, since limr→∞ g˜(u, r) = 1, we have ∞ 1 ∂˜ g (u, r)dr − log g˜(u, r1 ) = g ˜ ∂r r1 ∞ g 2m 2Q2 = dr − 3 − 8πrV 2 r r g˜ r1 −1 ∞ 2m 2Q2 2m Q2 = + − − 8πrV 1 − dr r2 r3 r r2 r1 −1 ∞ 2M1 (u) 2M1 (u) Q21 (u) + ≤ dr 1 − r2 r r2 r1 = − log{f (r1 , M1 , Q1 )}, (3.5)

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where we set

 √ r1 −M1 + M12 −Q21  M1  √ √ exp − log if M1 > Q1  2 2  M12 −Q21   r1 −M1 − M1 −Q1 1 f (r1 , M1 , Q1 )(u) = exp − r12M if M1 = Q1 . −M1     1 1  arctan √r1 −M if M1 < Q1 exp −πM1 + √ 2M 2 2 2 2 Q1 −M1

Q1 −M1

Now, we ﬁx r1 > R0 . Let u1 > u0 be chosen so that |R0 − R(u)| < r1 − R0 if u > u1 . Then, r1 > R(u) for all u > u1 . Since lim f (r1 , M1 , Q1 )(u) = f (r1 , M2 , Q2 ) > 0,

u→∞

we can choose u2 > u1 so that f (r1 , M1 , Q1 )(u) ≥

1 f (r1 , M2 , Q2 ) 2

if u > u2 .

Let u3 > u2 . From the fact g ≥ g˜, we estimate u3 u3 u3 g˜ g(u, r1 )du ≥ g˜(u, r1 )du ≥ f (r1 , M1 , Q1 )(u)du 0 0 u0 u3 1 1 f (r1 , M2 , Q2 )du = f (r1 , M2 , Q2 )(u3 − u2 ) → ∞ ≥ 2 u2 2

(3.6)

(3.7)

as u3 → ∞, i.e., the proper time along the line r = r1 tends to inﬁnity as the parameter u → ∞. This completes the proof of the proposition. Remark 3.1. If we assume only limu→∞ M1 (u) < ∞ instead of (3.2) and (3.3), then by the obvious modiﬁcation of the estimate in (3.5) (just ignoring the terms with Q21 ) we can infer that the time-like line r = r1 is complete toward future if r1 > 2M2 . Remark 3.2. In the case of real scalar ﬁeld φ = φ∗ we have Q = 0, and our R0 becomes the Schwartzschild radius independent of the form of nonlinearity in V (|φ|).

Acknowledgments This research is supported partially by the grant no. 2000-2-10200-002-5 from the basic research program of the KOSEF and the SNU Research fund.

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References [1] L. Andersson and V. Moncrief, On the global evolution problem in 3 + 1 gravity, J. Geom. Phys., in preparation. [2] D. Chae, Global Existence of Spherically Symmetric Solutions to the Coupled Einstein and Nonlinear Klein-Gordon System, Class. Quantum Grav. 18, no. 21, 4589–4605 (2001). [3] Y. Choquet-Bruhat, Solutions C ∞ d’´equations hyperboliques non lin´eaires, C. R. Acad. Sci. Paris 272, 386–388 (1968). [4] Y. Choquet-Bruhat, Solutions globales des ´equations de Maxwell-Dirac-KleinGordon (masses nulles). (French) C. R. Acad. Sci. Paris S`eries. I Math., 292, no. 2, 153–158 (1981). [5] Y. Choquet-Bruhat and D. Christodoulou, Existence of global solutions of the Yang-Mills, Higgs and spinor ﬁeld equations in 3+1 dimensions, Ann. Scient. ´ Norm. Sup., 4e s´erie, t. 14, 481–505 (1981). Ec. [6] D. Christodoulou, The problem of a self-graviting scalar ﬁeld, Comm. Math. Phys., 105, 337–361 (1986). [7] D. Christodoulou, The formation of black holes and singularities in spherically symmetric gravitational collapse, Comm. Pure Appl. Math. 44, no. 3, 339–373 (1991). [8] D. Christodoulou, Bounded variation solutions of the spherically symmetric Einstein-scalar ﬁeld equations, Comm. Pure Appl. Math. 46, no. 8, 1131–1220 (1998). [9] D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar ﬁeld, Ann. of Math. (2), 140, no. 3, 607–653 (1994). [10] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar ﬁeld, Ann. of Math. (2), 149, no. 1, 183–217 (1999). [11] D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical series 41, (1993). [12] D. Christodoulou, S. Klainerman and F. Nicol` o, On a null hypersurface approach to the proof of stability of the Minkowski space, in preparation. [13] P.T. Chr´ usciel, On the uniqueness in the large of solutions to the Einstein equations (“strong cosmic censorship”), Mathematical aspects of the classical ﬁeld theory (Seattle,WA, 1991), Amer. Math. Soc., Providence, RI, 235–273 (1992).

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[14] D.M. Eardley and V. Moncrief, The Global Existence of Yang-Mills-Higgs Fields in 4-Dimensional Minkowski Space, Comm. Math. Phys. 83, 171–191, 193–212 (1982). [15] H. Friedrich, On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, J. Diﬀ. Geom. 34, no.2, 275– 345 (1991). [16] H. Friedrich and A. Rendall, The Cauchy problem for the Einstein equations, Einstein’s Field Equations and their Physical Interpretation (ed. B.G. Schmidt), Springer, Berlin, (2000). [17] J. Ginbre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar ﬁelds in the temporal gauge, Comm. Math. Phys. 82, 1–28 (1981). [18] M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132, 485–509 (1990). [19] S. Hod and T. Piran, Mass inﬂation in dynamical gravitational collapse of a charged scalar ﬁeld, Phys. Rev. Lett. 81, no.8, 1554–1557 (1998). [20] S. Hod and T. Piran, Critical behaviour and universality in gravitational collapse of a charged scalar ﬁeld, Phys. Rev. D55 3485–3496 (1997). [21] S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with ﬁnite energy, Duke Math. J., 74, no.1, 19–44 (1994). [22] S. Klainerman and F. Nicol` o, On local and global aspects of the Cauchy problem in General Relativity, Class. Q. Grav., 16, R73-R157 (1999). [23] A.D. Rendall, Theorems on existence and global dynamics for the Einstein equations, URL: http://xxx.lanl.gov/gr-qc/0203012 . [24] G. Rein and A.D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys. 150, 561–583 (1992). [25] M. Struwe, Globally regular solutions to the u5 Klein-Gordon equation, Ann. Scuola Norm. Pisa, 15, 495–513 (1988). Dongho Chae Department of Mathematics Seoul National University Seoul 151-742 Korea email: [email protected] Communicated by Sergiu Klainerman submitted 18/04/02, accepted 27/11/02

Ann. Henri Poincar´e 4 (2003) 63 – 83 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/01063-21 DOI 10.1007/s00023-003-0122-z

Annales Henri Poincar´ e

On the Algebra of Fluctuation in Quantum Spin Chains Taku Matsui Abstract. We present a proof of the central limit theorem for a pair of mutually non-commuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem.

1 Summary of results In [6] and in [7] D. Goderis, A. Verbeure, and P. Vets investigated a central limit theorem for mixing quantum spin systems. We refer to the limit theorem of D. Goderis, A. Verbeure, and P. Vets as CLT below. In case of a single observable, their theorem is a kind of classical limit theorem for a family of spectral measures of operators appearing in quantum statistical mechanics. However if we consider more than two observables, the limit theorem suggests the emergence of Boson ﬁelds as the algebra of ﬂuctuation of physical observables. Then, it is natural to consider the time evolution states of such Bose ﬁelds appearing in the central limit. In fact, as the states appearing in CLT are quasifree states of Bose ﬁelds, at a heuristic level, the dynamics of the ﬂuctuation operators is the quasifree dynamics (via Bogoliubov automorphisms), i.e., unitary time evolution in the test function space of the Boson ﬁeld. D. Goderis, A. Verbeure, and P. Vets in [8] investigated the construction of the time evolution of the algebra of ﬂuctuation. They tried to introduce the time evolution for the algebra of ﬂuctuation induced by the microscopic dynamics. As they could not prove their limit theorem to any time invariant dense subalgebra, their introduction of the time evolution in an indirect way. They have shown that the quasifree state of the algebra of ﬂuctuation is a KMS state if the state of the microscopic system is KMS. However they failed to prove validity of their assumption for any non-trivial KMS states. After publication of [6], [7] and [8], no concrete example of quantum spin systems has been analyzed. The diﬃculty lies in the mixing assumption in [7] where estimate of decay of correlation of two observables Q1 and Q2 located in two disjoint regions, Λ1 and Λ2 . In many cases we may prove the following estimate: |ϕ(Q1 Q2 ) − ϕ(Q1 )ϕ(Q2 )| ≤ C(Q1 , Q2 ) Q1 Q2 r−d−

(1.1)

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where r is the distance between the supports Λ1 and Λ2 . To show the central limit theorem, we need information of dependence of the constant C(Q1 , Q2 ) on the size and the number of components of Λ1 and Λ2 . In fact, in the proof of D. Goderis, A. Verbeure, and P. Vets, the local observables are supported in many components and the typical assumption they made is as follows: |φ(Q1 Q2 . . . Qn+m ) − φ(Q1 . . . Qn )φ(Qn+1 . . . . . . Qn+m )| ≤ C Q1 Q2 . . . Qn+m r( d + )

(1.2)

where Qk ∈ AΛk (k = 1, 2, . . . n + m) Λk ∩ Λl = ∅(k = l) and r = min{distance{Λi, Λj } i = j = 1, 2, . . . n + m}. The constant C must be independent on n, m and the size of support Λj . D. Goderis, A. Verbeure, and P. Vets never proved their assumption for any (non-product ) Gibbs state and until recently it has been an open question whether the above results really hold for any Gibbs states. In [12], we obtained a diﬀerent proof of CLT of a single observable in quantum spin systems under a slightly diﬀerent mixing condition. The advantages of our results in [12] are as follows. (i) We can prove our assumptions for several mixing states of one-dimensional systems. The assumption of [12] is valid for Gibbs states for ﬁnite range interactions, ground states of the (massive) XY model, quasifree states of CAR algebras. (ii) Our limit theorem is valid for (not strictly local) certain quasi-local observables, which we named exponentially localized observables. In fact, instead of assuming (1.1), we can show CLT under the following condition: |ϕ(Q1 Q2 ) − ϕ(Q1 )ϕ(Q2 )| ≤ C(Q1 ) Q2 r−d−1− . We are not aware of proof that our mixing condition implies the condition of D. Goderis, A. Verbeure, and P. Vets. Our proof is based on an idea of E. Bolthauzen in [2]. As far as strictly local observables concerned, we do not require any estimate of the constant C(Q1 ) and we did not assume the assumption on the 4th moment (CLT4) of [7] neither. To show CLT for quasi-local observables we assume stronger mixing, which is still valid in many one-dimensional examples. In this article, we continue our study of CLT. The new result of this article is CLT for mutually non-commuting operators in the following sense: 1

2

m

lim ϕ(eiT QN eiT QN . . . eiT QN ) = e− N

T2 2

t(

k

Qk ,

k

2

Qk ) −i T2

e

k
s(Qk ,Ql )

.

This CLT is a stronger statement conﬁrming appearance of Boson ﬁelds as the algebra of normal ﬂuctuation. We prove this limit theorem under the same condition

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On the Algebra of Fluctuation in Quantum Spin Chains

65

of [12]. As we are handling a convergence of functionals rather that of measures, we had to add more argument and estimates which are absent both in our previous paper [12] and in commutative cases. Let us mention that Goderis, Verbeure and Vets have already considered the same limit theorem only for strictly local observables and again they could not prove their assumption for any non-trivial Gibbs states. We will see that our CLT is valid for a dense algebra invariant under the time evolution generated by any ﬁnite range translationally invariant Hamiltonian. As a corollary, we can introduce the time evolution for the algebra of normal ﬂuctuation directly. The CLT of this paper implies that quasifree states obtained in CLT is a KMS state of the algebra of ﬂuctuation provided that the microscopic state satisﬁes the KMS condition as well. We concentrate here on the one-dimensional system. We may derive the same CLT for higher-dimensional systems under a similar mixing assumption of states, though the condition looks rather diﬃcult to verify, so we do not present its proof here. In our proof of CLT for Gibbs states, one-dimensionality enters in use of the Ruelle transfer operator technique and the entire analyticity of the time evolution (in Heisenberg Picture of Quantum Mechanics). Next we present our results more precisely. We start with explaining our notation. Here the language of Operator Algebra is used and the reader unfamiliar with this may consult [3] and [4]. By A we denote the UHF C ∗ -algebra d∞ (the inﬁnite tensor product of d by d matrix algebras): C∗ Md (C) . A= Z

Each component of the tensor product above is speciﬁed with a lattice site j ∈ Z. By Q(j) we denote the element of AZ with Q in the jth component of the tensor product and the unit in any other component. For a subset Λ of Z, AΛ is deﬁned as the C ∗ -subalgebra of AZ generated by elements localized in Λ. When ϕ is a state of AZ the restriction to AΛ will be denoted by ϕΛ : ϕΛ = ϕ|AΛ .

(1.3)

Aloc = ∪|Λ|<∞ AΛ

(1.4)

For simplicity we set

where |Λ| is the cardinality of Λ. Let τj be the lattice translation (j shift to the right) determined by τj (Q(k) ) = Q(j+k) for any j and k in Z.

(1.5)

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Time evolution of local observables In this paper, the time evolution of (microscopic) local observables is determined by a translationally invariant ﬁnite range interaction. Our local Hamiltonian HΛ on a ﬁnite volume Λ has the following standard form: HΛ = ψ(X) (1.6) X⊂Λ

where X is an interval [a, b] of Z and ψ(X) = ψ(X)∗ ∈ AX ψ(X) = 0

,

τj (ψ(X)) = ψ(X + j)

if the diameter d(X) of the set X is greater than r.

The time evolution

αΛ t

(1.7)

of the ﬁnite system in a ﬁnite set Λ of Z is determined by itHΛ αΛ Qe−itHΛ . t (Q) = e

From the local time evolution αΛ t , the global time evolution αt is obtained via the thermodynamic limit which converges in the norm topology of A. αt (Q) = lim αΛ t (Q). Λ→Z

Next, we introduce the notion of exponentially localized observables. Definition 1.1 Let θ be a positive constant 0 < θ < 1. Define Q(n) by the following equation: (n) (1.8) Q = inf Q − Qn | Q ∈ A[−n,n] for n positive, n > 0 and we set Q(0) = Q . (n)

In terms of Q

we introduce |Q|θ : |Q|θ =

(n)

Q

θ−n .

(1.9)

n=0,1,2,...

An element Q of A is exponentially localized with rate θ if |Q|θ is finite. The set of all exponentially localized elements with rate θ is denoted by Fθ . We fix an element Qn of A[−n,n] satisfying Qn = Q − Qn . Note that due to compactness of a closed bounded subset of A[−n,n] the inﬁmum in (1.8) is attained by an element. In [11] we used a slightly diﬀerent deﬁnition of Fθ . But they are essentially equivalent. The following invariance of Fθ under the time evolution is a corollary of results due to H. Araki in [1] or due to the Propagation of Estimates due to E. Lieb and D. Robinson (cf. Theorem 6.2.9. and 6.2.11 of [4]) .

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Proposition 1.2 Suppose that the interaction is of finite range. (i) For any θ and θ with θ < θ , there exists a constant C = C(θ, θ ) such that |αt (Q)|θ ≤ C|Q|θ . (ii) For any Q in Fθ , αt (Q) is entire analytic (analytic in the whole complex plane) as a function of t. Due to the above proposition the sets ∪θ Fθ and ∩θ Fθ are invariant under αt .

Gibbs states As usual, the local Gibbs state at the inverse temperature β is deﬁned by (Λ)

ϕβ (Q) =

tr(e−βHΛ Q) . tr(e−βHΛ )

(1.10)

For the ﬁnite range interaction, the following thermodynamics limit exists: ([−N,N ])

lim ϕβ

N →∞

(Q) = ϕβ (Q).

ϕβ is a unique Gibbs state of the time evolution αt and ϕβ satisﬁes the KMS boundary condition: ϕβ (Q1 αiβ (Q2 )) = ϕβ (Q2 Q1 ) for Q1 and Q2 in Fθ . By applying the Ruelle transfer operator technique for UHF algebras (cf. [1], [9], and [11]) we obtain the following estimates for the inﬁnite volume Gibbs state ϕβ . Theorem 1.3 Suppose that the interaction is of finite range. (i) There exist constants C, m > 0 such that [−k+a,b+k] (Q) − ϕβ (Q) ≤ Ce−mk Q ϕβ

(1.11)

for any Q in A[a,b] . (ii) We have exponential decay of correlation for observables localized in the half infinite intervals (−∞, −1] and, [0, ∞). |ϕ(Q1 τj (Q2 )) − ϕ(Q1 )ϕ(Q2 )| ≤ Ke−Mj Q1 Q2 for Q1 in A(−∞,−1] , Q2 in A[0,∞) and j > 0.

(1.12)

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Central limit theorem Suppose that ϕ is a translationally invariant factor state. Consider the local ﬂuctuation QN of an observable Q:   1  QN = √ (τj (Q) − ϕ(Q)) . 2N + 1 |j|≤N If the observable is diagonal, the limit limN QN makes sense as convergence of spectral measures and this is nothing but a classical central limit theorem of mixing systems. The central limit does not converge in neither strong nor weak topology of operators in a Hilbert space. Nevertheless, the following weak limit exists in the GNS space of ϕ

(1.13) w − lim QN , RN = s(Q, R)1 N →∞

where s(Q, R) =

ϕ([τk (Q), R]).

(1.14)

k∈Z

Set t(Q, R) = lim ϕ(QN RN ).

(1.15)

N →∞

Definition 1.4 Let ϕ be a translationally invariant state of A. Suppose that Q in A is selfadjoint. We say that the central limit theorem holds for Q and ϕ if lim ϕ(eiT QN ) = e−

N →∞

T2 2

t(Q,Q)

.

(1.16)

Theorem 1.5 (i) Let ϕ be a translationally invariant state. Suppose there exist positive constants M (j) such that ∞ jM (j) < ∞ j=1

and |ϕ(Q1 τj (Q2 )) − ϕ(Q1 )ϕ(Q2 )| ≤ K Q1 Q2 M (j) for Q1 in A(−∞,−1] , Q2 in A[0,∞) and j > 0. The central limit theorem holds for any selfadjoint strictly local observable Q in Aloc . We have convergence of the following correlation functions for local Q(k) (k = 1, 2, . . . r). lim ϕ(

N →∞

r

1

eiQ(k)N ) = e− 2 t(

k

Q(k),

k

Q(k)) − 2i

e

k
s(Q(k),Q(l))

(1.17)

k=1

(ii) If the translationally invariant state satisfies the exponential mixing condition (1.12), (1.17) is valid for any exponentially localized selfadjoint Q(k) in Fθ . The convergence of (1.17) was not proved in [12]. Our proof is similar to the case of a single observable. We will give our proof in Section 3.

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Time evolution of the algebra of fluctuation As the Gibbs state ϕβ for any ﬁnite range interaction satisﬁes the assumption of Theorem 1.5, we obtain a direct construction of dynamics for the algebra of ﬂuctuation. The set of all selfadjoint exponentially localized elements with rate θ is denoted by Fθreal . Set F = ∪0<θ<1 Fθreal . Due to our assumption on decay of correlation, t(Q, Q) = ϕβ (Qτk (Q)) − ϕβ (Q)2

(1.18)

k∈Z

is ﬁnite and non-negative. Then, t(Q, Q) of (1.15) is a (degenerate) positive semideﬁnite bilinear form. We denote the kernel of t by N . N = {Q ∈ F | t(Q, Q) = 0} . Next we introduce the equivalence relation on F . A and B are equivalent if A − B is in N . By A˜ we denote the equivalence class for A in F . Then s(A, B) of (1.13) ˜ B) ˜ on F˜ where gives rise to a non-degenerate symplectic form s˜(A, F˜ = F /N

(1.19)

˜ B) ˜ = s(A, B). s˜(A,

(1.20)

and As was already discussed in [7] and [8], the central limit gives rise to the Weyl ˜ ˜ algebra W (F , s˜) on the symplectic space F , s˜ . More precisely, we deﬁne W(F˜ , s˜) by the C ∗ -algebra generated by unitaries ˜ satisfying W (Q) ˜

˜

W (Q˜1 )W (Q˜2 ) = W (Q˜1 + Q˜2 )e−i1/2s(Q1 ,Q2 ) .

(1.21)

W(F˜ , s˜) is called the algebra of normal (macroscopic) fluctuations. When the interaction is of ﬁnite range, the global dynamics of A leaves F invariant and we can introduce the dynamics α ˜ t on the Weyl algebra W(F˜ , s˜) via the following equation: ˜ = W (˜ α ˜ t (W (Q) αt (Q)). (1.22) The time evolution α ˜ t has weak continuity in the sense that α ˜ t (Q) is continuous with respect to the topology induced by the bilinear form t(Q, Q). In fact, if we have exponential clustering (1.12), it is not diﬃcult to show lim t(αu (Q) − Q, αu (Q) − Q) = 0.

u→0

(1.23)

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Note that the left-hand side of (1.23) is equal to lim ϕβ ({αt (Q) − Q, τk (Q)}) = 0. t→0

k∈Z

As was argued in [7], the central limit limN ϕ(eiQN ) deﬁnes a quasifree state ϕ˜ of the algebra of normal ﬂuctuations W(F˜ , s˜). We now consider the GNS representation of the algebra of normal ﬂuctuations W(F˜ , s˜) associated with ϕ˜β . Let {π(W(F˜ , s˜)), Ω, H} be the GNS triple for ϕ˜β . Let M be the von Neumann algebra generated by π(W(F˜ , s˜)). As ϕ˜β is invariant under the time evolution α ˜t of the algebra of normal ﬂuctuations W(F˜ , s˜), we have a one ˜t . parameter group of unitary Ut on H which implements α Ut π(Q)Ut∗ = π(˜ αt (Q)),

Ut Ω = Ω.

As a consequence of the continuity (1.23), we see that the adjoint action of Ut gives rise to a weakly continuous one parameter group of automorphisms of M. We denote this dynamics of M by the same symbol α ˜t: α ˜ t (M ) = Ut M Ut∗ for M in M. We state our main result of this paper. Theorem 1.6 Let ϕβ be the unique β-KMS state for a finite range translationally invariant interaction of a one-dimensional quantum spin chain. The quasifree state ϕ˜β is a β-KMS state for the dynamics α ˜t of the von Neumann algebra M at the same inverse temperature β. It is easy to derive the KMS boundary condition of ϕ˜β from the central limit Theorem (1.17) for mutually non-commuting observables. Note that the argument of [8] is based on a weaker version CLT, i.e., CLT only for strict local observables, which makes arguments more complicated. We explain the contents of the rest of this paper now. Section 2 is a mathematical preliminary. In Section 3, we present our proof of Theorem 1.5 (1.17) and that of Theorem 1.6. In Section 4 we make a few comment on unsolved problems. Here, we present our proof of the uniform exponential mixing condition for ﬁnitely correlated states. Even though the result is as expected, we are not aware of any proof published elsewhere. Our proof is a simple application of dual transfer operator.

2 Localization We present some features of exponentially localized elements. Lemma 2.1 Fθ is complete in the norm topology induced by |Q|θ . Any bounded set of Fθ is a compact subset of A in norm topology.

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Lemma 2.2 Let Q be exponentially localized with rate θ. There exists an element ˜ n of A[−n,n] satisfying Q ˜ ˜ n (2.1) Q − Q ≤ 2|Q|θ θn , Q n ≤ Q . ˜ n such that If Q is positive we can find a positive Q ˜ n ˜n ≤ 0 ≤ inf Q ≤ inf Q ≤ Q Q

(2.2)

where inf Q is the infimum of the spectrum of Q. ˜ n is tr (∞,−n]∪[n,∞) (Q) where tr (∞,−n]∪[n,∞) is the partial trace One choice of Q (conditional expectation to A[−n,n] ). Lemma 2.3 Let Q1 and Q2 be exponentially localized, Q1 , Q2 ∈ Fθ . Then the following sum is finite.

τk (Q1 ), Q2 < ∞. (2.3) k∈Z

Proof. Fix

Qin

in A[−n,n] satisfying i Q − Qin = Q(n) .

for i = 1, 2 and set Rni = Qin − Qin−1 (n ≥ 1) , Then

R0i = Qi0 .

i i i Rn ≤ Qn + Qn−1 ≤ (1 + θ−1 )θn |Qi |θ .

Now we have Qi =

∞

Rni .

n=0

As a consequence, k∈Z

[τk (Q1 ), Q2 ]

∞

2 τk (Rn1 ), Rm

≤

n,m=0 k∈Z ∞

≤2

≤2

2 (2(m + n) + 1) Rn1 Rm

n,m=0 ∞

(1 + θ−1 )2 θn θm |Q1 |θ |Q2 |θ < ∞. (2.4)

n,m=0

The following result is due to H. Araki in [1].

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Proposition 2.4 Suppose that the interaction is of finite range. For any Q in Fθ , αt (Q) is entire analytic (analytic in the whole complex plane) as a function of t. For any β > 0 and any θ, there exists a constant M = M (β, θ, r) independent of the interaction ψ such that the following estimate is valid: αz (Q)θ ≤ M F0 (2 |β| h(ψ))|Q|θe4|β|h(ψ)

(2.5)

where any complex number z satisfying |z| ≤ β and h(ψ) =

ψ(X) , d(X)

X:0∈X

F0 (x) = exp[(−r + 1)x + 2

r

k −1 (ekr − 1)].

k=1

Thus this proposition tells us that ∩0<θ<1 Fθ is a dense αz invariant subalgebra of A. Next we show that ∪0<θ<1 Fθ is a dense αt invariant subalgebra of A. Proof of Proposition 1.2 (i). The proof of Proposition 6.2.9 of [4] tells us the following estimate: αt (A) − eitHΛ Ae−itHΛ ≤ A |Λ0 | (e2|t|ψλ − 1) e−λd(j,Λ0 )

(2.6)

j∈Λc

where A is localized in Λ0 , λ is any positive constant, d(j, Λ0 ) is the distance of j from Λ0 and ψλ is a positive constant depending on the interaction ψ and λ. Now take Q from Fθ and in (2.6), we set e−λ = θ0 where θ < θ0 < θ . A = Rm = Qm − Qm−1 in A[−m,m] , Λ0 = [−n, n]. For n larger than m, we obtain αt (Rm )

(n)

≤

2(e2|t|ψλ − 1)(2m + 1) Rm

∞

θ0k−m

k=n

≤

2(e2|t|ψλ − 1)(2m + 1)θn−m

Rm . 1 − θ0

On the other hand, in general, (n)

Q

≤ 2 Q

due to Lemma 2.2. As a consequence, for n smaller than m, n < m, we have αt (Rm )

(n)

≤ 2 Rm ≤ 4θn |Q|θ .

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Then, |αt (Q)|θ ≤ ≤

∞

|αt (Rm )|θ m=1 m ∞ −k (k) θ αt (Rm ) m=1 k=0 +2(e

2|t|ψλ

∞ (2m + 1) −k − 1) Rm θ θ0k−m 1 − θ0

k=m+1

≤ ≤

C1 C2

∞

4(m + 1)(

m=1 ∞

θ −m −m ) θ0 Rm θ0

θ0 −m Rm = C3 |Q|θ .

(2.7)

m=1

Note that supm ((2m + 1)θ

−m

θ0 m ) is ﬁnite.

3 Proof of Theorem 1.5 and 1.6 We present here our proof for the convergence of two point correlation for (1.17). We concentrate the case of exponentially localized observables as this is what we need to deﬁne the time evolution and KMS states for the algebra of normal ﬂuctuation. The idea of proof is as follows. Set N (T ) = ϕ(eiT QN eiT RN ). F(Q,R)

(3.1)

Instead of showing lim FN (Q, R) = e−1/2T

2

t(Q+R,Q+R) −i/2T 2 s(Q,R)

N →∞

e

),

we prove that the following limit vanishes. d FN (Q, R)(T ) + T (t(Q + R, Q + R) + is(Q, R))FN (Q, R)(T ) = 0. lim N →∞ dT (3.2) Note that the above claim is the same as showing the following: lim

N →∞

{ϕ((T (it(Q + R, Q + R) − s(Q, R)) − QN )eiT QN eiT RN ) −ϕ(eiT QN RN eiT RN )} = 0.

N If the second derivative of F(Q,R) (T ) is bounded in a small neighborhood of the N origin uniformly in N , F(Q,R) (T ) and its derivative are equicontinuous. It turns

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∞ out that there is at least one accumulation point F(Q,R) (T ) in the sequence of the functions FN (Q, R)(T ) (N = 1, 2, 3, . . . ). Due to equicontinuity of the ﬁrst and N ∞ the second derivative of F(Q,R) (T ), F(Q,R) (T ) is diﬀerentiable. Then (3.2) implies ∞ F(Q,R) (T ) satisﬁes

d ∞ ∞ F (T ) = T (t(Q + R, Q + R) − is(Q, R))F(Q,R) (T ). dT (Q,R) The unique solution to (3.3) is e−1/2T

2

t(Q+R,Q+R) −i/2T 2 s(Q,R)

e

(3.3)

.

Now we begin the proof of the above claim. Without loss of generality we assume that ϕ(Q) = ϕ(R) = 0. Suppose that A = A∗ B = B ∗ are selfadjoint bounded operators on a Hilbert space. Then

i(A+B) − eiB eiA ≤ A, eiB e

A, eiB ≤ [A, B]

(3.4)

(cf. [7] or [12]). N Lemma 3.1 The first and the second derivative of F(Q,R) (T ) are bounded uniformly in N . N Proof. The second derivative F(Q,R) (T ) has the following expression:

−

d2 N F (T ) dT 2 (Q,R) =

ϕ(QN 2 eiT QN eiT RN ) + ϕ(eiT QN eiT RN RN 2 ) +2ϕ(QN eiT QN eiT RN RN ).

(3.5)

Each term in (3.5) is uniformly bounded in N . We focus on ϕ(QN 2 eiT QN eiT RN ). Due to exponential decay of correlation, we have lim ϕ(QN 2 ) =

N →∞

ϕ(Qτk (Q)) < ∞

k∈Z

where we used ϕ(Q) = 0. Thus we have a constant K1 such that for any N 0 ≤ ϕ(QN 2 ) ≤ K1 .

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By use of (3.4) we can show uniform boundedness of the commutator of QN and eiT RN :

QN , eiT RN ≤ T 2 QN , RN 1 ≤ T2 [Q, τb−a (R)] 2N + 1 0≤|a|,|b|≤N [Q, τk (R)] ≤ K2 < ∞. (3.6) ≤ T2 k∈Z

Then, ϕ(QN 2 eiT QN eiT RN ) ϕ(QN eiT QN eiT RN QN ) ≤

+ ϕ(QN eiT QN eiT RN , QN ) ≤ ϕ(Q2N )1/2 ϕ((eiT RN QN )∗ (eiT RN QN ))1/2

+ QN , eiT RN ≤

K1 + (K1 )1/2 K2 < ∞.

By the same estimate, we see that other terms of (3.5) and the ﬁrst derivative of N F(Q,R) (T ) are uniformly bounded in N . Lemma 3.2 lim ϕ((iT s(Q, R) + (eiT QN RN e−iT QN − RN ))eiT QN eiT RN ) = 0. (3.7)

n→∞

Proof. First note that iT Q

e N RN e−iT QN − RN − iT QN , RN

≤ CT 2 QN , QN , RN .

(3.8)

The right-hand side of (3.8) is bounded from above by CT 2 ≤

1 (2N + 1)3/2

1 CT 2 (2N + 1)1/2

[τj−k (Q), [τi−k (Q), R]]

|i|,|j|,|k|≤N

[τj (Q), [τi (Q), R]] .

(3.9)

|i|,|j|≤2N

If Q and R are local, the non-vanishing terms in the right-hand side of (3.9) are ﬁnite: [τj (Q), [τi (Q), R]] = 0

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for |i| , |j| ≥ r. When Q and R are exponentially localized, the commutator [τi (Q), R] is also exponentially localized and [τj (Q), [τi (Q), R]] ≤ Cθ|j|+|i| .

Thus

[τj (Q), [τi (Q), R]] < ∞

i,j∈Z

and we obtain the following convergence:

lim eiT QN RN e−iT QN − RN − iT QN , RN = 0. N →∞

(3.10)

Finally we show the following convergence which implies the claim of Lemma 3.2.

lim ϕ((s(Q, R) − QN , RN )eiT QN eiT RN ) = 0 (3.11) N →∞

Then,

QN , RN =

1 τi ([τj−i (Q), R]) 2N + 1 |i|≤N |j|≤N

and lim

N →∞

[τj−i (Q), R] =

|j|≤N

[τj (Q), R] .

(3.12)

j∈Z

Note that the right-hand side of (3.12) converges in the norm topology due to exponential localization. We deﬁne A via the following equation: A= [τj (Q), R] . j∈Z

Note that s(Q, R) = ϕ(A). Instead of showing (3.11), we have only to prove   1 lim ϕ(s(Q, R) − lim { τj (A)} eiT QN eiT RN ) = 0. N →∞ N →∞ 2N + 1

(3.13)

|j|≤N

It is not diﬃcult to see that the following identity implies (3.13) 1 lim ϕ(( τj (A) − ϕ(A))2 ) = 0. N →∞ 2N + 1

(3.14)

|j|≤N

The left-hand side of (3.14) is equal to 1 ϕ(τi−j (A)A) − ϕ(A)2 lim 2 N →∞ (2N + 1) |i|,|j|≤N

=

1 ϕ(τi (A)A) − ϕ(A)2 = 0. (3.15) N →∞ (2N + 1) lim

|i|≤N

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The last identity is due to ergodicity of the state ϕ. (3.15) implies the claim of Lemma. We can ﬁnd an increasing sequence m(N ) of positive integers such that √ lim e−m(N ) 2N + 1 = 0 , lim m(N )−1 (2N + 1)1/4 = ∞. (3.16) N →∞

N →∞

We set

Qj,N =

τk (Q) ,

|k|≤N,|k−j|≤m(N )

QN

  −1/2

= βN



|k|≤N

  τk (Q) 

βN =

1/2ϕ({τk (Q) , Qk,N })

|k|≤N

,

−1/2

Qj,N = βN

Qj,N .

Then βN = (2N + 1)(1 + o(1)) due to summability of two point correlation functions. Thus we have only to consider ϕ(eiT QN eiT RN ) and its derivative. Furthermore, in what follows, we assume that t(Q + R, Q + R) = 1 for simplicity of exposition. (The case t(Q + R, Q + R) = 0 can be handled in the same manner.) We set P = Q + R. Lemma 3.3 lim ϕ((iT − P N )eiT QN eiT RN ) = 0.

n→∞

(3.17)

Proof. We start with the following identity: (iT − P N )eiT QN eiT RN −1 iT (1 − βN τk (P )Pk,N )eiT QN eiT RN

= −

−1/2 βN

−

−1/2 βN

−

−1/2 βN

|k|≤N

|k|≤N

τk (P )(1 − e−iT P k,N − iT P k,N )eiT QN eiT RN τk (P ) e−iT P k,N − e−iT Qk,N e−iT Rk,N eiT QN eiT RN

|k|≤N

τk (P )e−iT Qk,N e−iT Rk,N eiT QN eiT RN

|k|≤N

=

(A1 + A2 + A3 )eiT QN eiT RN + A4 .

(3.18)

Using the estimates of proof of Lemma 2.2 in [12] (cf. (2.8) and (2.10) of [12]) we obtain lim ϕ(Aa eiT QN eiT RN ) = 0 N →∞

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for a = 1, 2. On the other hand,

−1/2 A3 ≤ βN τk (P ) T 2 Qk,N , Rk,N |k|≤N

≤

−3/2 T 2 βN (2N

+ 1) P m(N )

[τj (Q), R] .

j∈Z −3/2

When we take N to ∞ we have βN

(2N + 1)m(N ) → 0. As a result

lim ϕ(A3 eiT QN eiT RN ) = 0.

N →∞

We now look at A4 . We claim that 1/2 lim βN e−iT Qk,N e−iT Rk,N eiT QN eiT RN

N →∞

−eiT (QN −Qk,N ) eiT (RN −Rk,N ) = 0 . (3.19)

To see this, we point out the following inequalities: 1/2 −1/2 [τj (Q), R] βN e−iT Rk,N , eiT QN ≤ 2T 2 βN m(N ) j∈Z

1/2 βN

−iT Ak,N iT AN e − eiT (AN −Ak,N ) e −1/2 ≤ T 2 βN m(N ) [τj (A), A] .

(3.20)

j∈Z

for A = P or Q. Again for exponentially localized A, j∈Z [τj (A), A] is ﬁnite. Thus due to choice of m(N ) satisfying (3.16), the right-hand side of (3.20) vanishes in the limit of taking N to ∞. The convergence (3.19) tell us that −1/2

− lim ϕ(A4 ) = lim βN N →∞

N →∞

ϕ(τk (P )eiT (QN −Qk,N ) eiT (RN −Rk,N ) ).

|k|≤N

Now we concentrate on the following limit: lim ϕ(τk (P )eiT (QN −Qk,N ) eiT (RN −Rk,N ) ).

N →∞

(3.21)

We can estimate decay of (3.21) as in [12]. The observable τk (P ) is localized around the site k with exponential tail while eiT (QN −Qk,N ) eiT (RN −Rk,N ) is localized in (−∞, k − m(N )] ∪ [k + m(N ), ∞). We have to approximate Q and R by strictly local elements as in our proof of CLT for nonlocal observables of [12]. (We omit

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the proof as the estimate is the same as that of [12].) We arrive at the following estimate: ϕ(τk (P )eiT (QN −Qk,N ) eiT (RN −Rk,N ) ) ≤ Cα(m(N )). This inequality shows that the limit (3.21) vanishes, and the claim of Lemma is proved. Theorem 1.5 follows from Lemma 4.3 and Lemma 4.4. Next we consider the KMS condition. Theorem 1.7 follows easily from Theorem 1.5 in the same manner as [8]. We repeat the argument brieﬂy. As the state ˜ However as CLT ϕ˜β is quasifree, it suﬃces to show KMS condition for ϕ˜β (W (Q)). is valid only for the selfadjoint part of Fθ we cannot consider the complex time evolution αz in the central limit. In fact, we can use another equivalent condition to the KMS condition (cf. Proposition 5.3.6. of [4]). Proposition 3.4 (i) If the state ϕ is KMS, the time dependent correlation function F (t) = ϕ(Q1 αt (Q2 )) has an analytic extension to the strip Iβ = {z|0 < Imz < β}, F (t) is bounded continuous on I β = {z|0 ≤ Imz ≤ β}, and F (t + iβ) = ϕ(αt (Q2 )Q1 ).

(3.22)

(ii) Conversely if F (t) admits an analytic extension to Iβ and is bounded continuous on I β satisfying (3.22) for any Q1 and Q2 ϕ is a KMS state. Furthermore we have the following bound for F (z) on I β : |F (z)| ≤ Q1 Q2 .

(3.23)

Let ϕβ be the unique Gibbs state of a ﬁnite range interaction. Set FN (t) = ϕβ (e(i(Q1 )N e(iαt (Q2 ))N ). Due to (3.23), for any N we have |FN (t)| ≤ 1. Then we can choose a subsequence of FN (t) which converges to a bounded continuous function F∞ (t) on I β and analytic on Iβ . F∞ (t) satisﬁes the KMS condition on W (F˜ ) which implies the same condition on the von Neumann algebra generated by π(W (F˜ )).

4 Remarks We include here one example of CLT which was not discussed in [12]. We consider ﬁnitely correlated states of M. Fannes, B. Nachtergaele and R. Werner (cf. [5]). The ﬁnitely correlated state is a non-commutative analogue of the (function of) Markov measure.

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Let ϕ be a state of A. Consider the linear functional ϕQ on A(−∞,k] deﬁned by ϕQ (R) = ϕ(QR) Set

for Q in A[k+1,∞) .

Sk (ϕ) = ϕQ | Q ∈ A[k+1,∞) .

(4.1) (4.2)

Definition 4.1 ϕ is a finitely correlated state if Sk defined in (4.2) is a finitedimensional subspace of the set of all linear functionals on A(−∞,k] for any k. Proposition 4.2 Let ϕ be a translationally invariant finitely correlated state. Suppose that ϕ is mixing in the following sense: lim ϕ(Q1 τk (Q2 )) = ϕ(Q1 )ϕ(Q2 ).

k→∞

Then, the uniform exponential mixing (1.11) holds and the central limit theorem is valid. Note that the exponential decay of two point correlation |ϕ(Q1 τk (Q2 )) − ϕ(Q1 )ϕ(Q2 )| ≤ C(Q1 , Q2 )e−Mk is known. What matters here is the constant C(Q1 , Q2 ). Proof. To prove the above proposition, we consider a translationally invariant state ϕ of A and the GNS triple {πϕ (A), Hϕ , Ωϕ } associated with ϕ. Let U be the unitary on Hϕ implementing the shift τ1 . U πϕ (Q)U ∗ = πϕ (τ1 (Q))

U Ω ϕ = Ωϕ

Let P be the projection with the following range: H(−∞,−1] = πϕ (A(−∞,−1] )Ωϕ . As the unitary U ∗ leaves H(−∞,−1] invariant, we have P U ∗ P = U ∗ P. Set N = P πϕ (A[0,∞) )P. As τ1 is implemented by a unitary U on Hϕ , τ1 is extended to N via the formula τ1 (P QP ) = P U QU ∗ P. We deﬁne the completely positive unital map Ldual from N to N via the following equation: Ldual (R) = P U RU ∗ P where R is an element of N . We call Ldual the dual transfer operator. Consider the vector state ψdual of N associated with Ωϕ . ψdual is faithful as N commutes with A(−∞,−1] and Ωϕ is a cyclic vector for A(−∞,−1] in H(−∞,−1] .

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Lemma 4.3 The dimension of N is finite if and only if the state ϕ is finitely correlated. Proof. Set K = P πϕ (A[0,∞) )Ωϕ . Note that ϕQ (R) = (P πϕ (Q∗ )Ωϕ , πϕ (R)Ωϕ ) . By the Schwartz inequality ϕQ is a bounded linear functional on the Hilbert space H(−∞,−1] and due to the Riesz Lemma the set S0 is isomorphic to the subspace K of H(−∞,−1] . To see the above Lemma, it suﬃces to notice that the dimension of K is ﬁnite if and only if ϕ is ﬁnitely correlated. Due to Lemma 4.3, for a ﬁnitely correlated ϕ, the dual transfer operator Ldual is a linear operator on the ﬁnite-dimensional space N . In fact we can extend the dual transfer operator Ldual to a unital completely positive map on the set of bounded linear operators on K. The norm of Ldual is one, Ldual = 1. Furthermore, due to the mixing property of ϕ lim Lndual (A) = ψdual (A)1.

N →∞

The eigenvalue 1 of the dual transfer operator Ldual is non-degenerate and there exists no other peripheral spectrum due to mixing of the state ϕ. Other eigenvalues of Ldual is strictly less than one (cf. [5]). Thus there exist constants C and M such that Lndual (A) − ψdual (A)1 ≤ Ce−Mn A . Then for Q in A(−∞,−1] , R in A[0,∞) and n positive, |ϕ(Aτn (B)) − ϕ(A)ϕ(B)|

≤ ≤

Lndual (P πϕ (B)P ) − ψdual (P πϕ (B)P )1 A

Ce−Mn A B .

(4.3)

In Section 1 we considered Gibbs states of ﬁnite range interactions for onedimensional chains. The formalism can be extended to long range interactions. However, we are unable to prove necessary mixing condition for CLT. To clarify the diﬃculty in our approach we present the results we can prove by our methods. First we replace Fθ with suitable non-commutative H¨older continuous functions. Instead of the weight θn we take n−η with the condition η > 3 and set ∞ (n) η (η) F = Q∈A | Q n < ∞ . n=0

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Assumption 4.4 The interaction ψ is translationally invariant and we assume two conditions: (i) The time evolution is well defined on A lim eitH[−N,N ] Qe−itH[−N,N ] = αt (Q).

N →∞

(ii) The following limit exists and it gives rise to an element of F (η) lim eβ/2H[0,N ] e−β/2H[1,N ] = aβ ∈ F (η) .

N →∞

(4.4)

Proposition 4.5 If Assumption 4.4 is valid, the Gibbs state for the inverse temperature β is unique, satisfying the following mixing: |ϕ(Q1 τn (Q2 )) − ϕ(Q1 )ϕ(Q2 )| ≤ Kn−(η−1) Q1 Q2

(4.5)

for Q1 in A(−∞,−1] , Q2 in A[0,∞) and n > 0. Thus as a corollary, CLT holds if η > 3. Decay of correlation (4.5) can be derived by analysis of the Ruelle transfer operators (cf. [11]), however we have no idea to prove the convergence of (4.4). So far, we have discussed quantum spin models on one-dimensional lattice only. In higher-dimensional lattice we can prove CLT if we have the following mixing for a state ϕ |ϕ(Q1 Q2 ) − ϕ(Q1 )ϕ(Q2 )| ≤ CΛ0 e−Mk Q1 Q2 where the support of Q1 is Λ0 and the distance of the support of Q1 and that of Q2 is k and the constant CΛ0 is independent of the size of the support of Q2 . We are not certain that this estimate (or the (in) dependence of the constant CΛ0 on the size of the support of Q2 ) is valid for high temperature Gibbs states at the moment. It is a non-trivial question whether the above estimate can be shown by the high temperature expansion. Note that we do not require the assumption (CLT4) of [7].

References [1] H. Araki, Gibbs states of the one-dimensional quantum spin chain, Commun. Math. Phys. 115, 477–528 (1988). [2] E. Bolthauzen, On the central limit theorem for stationary mixing random ﬁelds, Ann. Prob. 4, 1047–1050, (1982). [3] O. Bratteli, D. Robinson, Operator algebras and quantum statistical mechanics I, 2nd edition (Springer, 1987). [4] O. Bratteli, D. Robinson, Operator algebras and quantum statistical mechanics II, 2nd edition (Springer, 1997).

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[5] M. Fannes, B. Nachtergaele, R. Werner, Finitely Correlated States on Quantum Spin Chains, Commun. Math. Phys. 144, 443–490 (1992). [6] D. Goderis, A. Verbeure, P. Vets, Non-commutative central limits, Prob. Th. Related Fields 82, 527–544 (1989). [7] D. Goderis, P. Vets, Central limit theorem for mixing quantum systems and the CCR-algebra of ﬂuctuations, Commun. Math. Phys. 122 , 249–265 (1989). [8] D. Goderis, A. Verbeure, P. Vets, Dynamics of Fluctuations for Quantum Lattice Systems, Commun. Math. Phys. 128, 533–549 (1990). [9] V. Golodets, S.V. Neshveyev, Gibbs states for AF-algebras, J. Math. Phys. 234, 6329–6344 (1998). [10] K. Hepp, E. Lieb, On the superradiant phase transition for molecules in a quantized radiation ﬁeld: The Dicke Maser model, Ann. Physics 76, 360–404 (1973). [11] Taku Matsui, On Non-Commutative Ruelle Transfer Operator, Rev. Math. Phys. 13, 1183–1201 (2001). [12] T. Matsui, Bosonic Central Limit Theorem for the One-Dimensional XY Model, to appear in Rev. Math. Phys. [13] S.V Neshveyev, E. Størmer, The variational principle for a class of asymptotically abelian C ∗ -algebras, preprint. [14] W. Parry, M. Pollicot, Zeta function and the periodic orbit structure of hyperbolic dynamics, Ast´erisque 268, 187–188 (1990). [15] D. Ruelle, Statistical Mechanics of a One-Dimensional Lattice Gas, Commun. Math. Phys. 6, 267–278 (1968). [16] Walter F. Wreszinski, Fluctuations in some mean-ﬁeld models in quantum statistics, Helv. Phys. Acta, 46, 844–868 (1973/74).

Taku Matsui Graduate School of Mathematics Kyushu University 1-10-6 Hakozaki Fukuoka 812-8581 Japan email: [email protected] Communicated by Jean-Bernard Zuber submitted 07/02/02, accepted 01/10/02

Ann. Henri Poincar´e 4 (2003) 85 – 126 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/01085-42 DOI 10.1007/s00023-003-0123-y

Annales Henri Poincar´ e

Complex Angular Momentum Diagonalization of the Bethe-Salpeter Structure in General Quantum Field Theory J. Bros and G.A. Viano

Abstract. The Complex Angular Momentum (CAM) representation of (scalar) fourpoint functions has been previously established starting from the general principles of local relativistic Quantum Field Theory (QFT). Here, we carry out the diagonalization of the general t-channel Bethe-Salpeter (BS) structure of four-point functions in the corresponding CAM variable λt , for all negative values of the squared-energy variable t. This diagonalization is closely related to the existence of BS-equations for the absorptive parts in the crossed channels, interpreted as convolution equations with spectral properties. The production of Regge poles equipped with factorized residues involving Euclidean three-point functions appears as conceptually built-in in the analytic axiomatic framework of QFT. The existence of leading Reggeon terms governing the asymptotic behaviour of the four-point function at ﬁxed t is strictly conditioned by the asymptotic behaviour of a global BetheSalpeter kernel of the theory.

1 Introduction In a previous paper [1], we have proved that the existence of analytic structures in the Complex Angular Momentum (CAM) variables is a general property of Quantum Field Theories (QFT) satisfying the basic principles [2] of local commutativity, Poincar´e invariance, spectral condition and temperate ultraviolet behaviour. Considering four-point functions H([k]) of general relativistic scalar ﬁelds in a given two-ﬁeld channel called the “t-channel”, we have in fact shown that appropriate ˜ (a) of H are holomorphic functions of a CAM ˜ (s) and H Laplace-type transforms H variable λt , dual to the (oﬀ-shell) scattering angle Θt of the t-channel. The ana˜ (a) is a half-plane of the form ˜ (s) and H lyticity domain which was obtained for H eλt > NH , where the number NH (NH > 0) corresponds to a certain “degree of temperateness” of H([k]) at large momenta. This domain was obtained for all negative values of the squared total energy t of the channel considered. It is the purpose of our program to investigate under which general conditions ˜ (a) of H may admit meromorpic continuations in a ˜ (s) and H the transforms H domain of the joint complex variables (t, λt ) which constitutes a “bridge” between a) the “primitive set ” {(t, λt ); t < 0; Re λt > NH } obtained in [1] (and from which poles are excluded), and

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b) a complex neighbourhood of a real set of the form {(t, λt ); 0 < t < t0 ; λt > ˜ (s) and H ˜ (a) in the variables (t, λt ) (namely N (t)} in which possible poles of H “Regge poles”) might interpolate bound states of H, supposed to be present in that real set (at integral values λt = 0, 1, . . .). In the framework of analyticity properties implied by the general principles of QFT [16], the existence of a meromorphic continuation of H exhibiting real poles at t > 0 under the two-particle threshold (resp. complex poles in the twoparticle second sheet of the t-plane) is a consequence of the general Bethe-Salpeter structure of QFT, in which the additional postulate of “Asymptotic Completeness” (or “oﬀ-shell unitarity”) plays an essential role [8, 9]. As we have announced it in [3], we shall show that the general Bethe-Salpeter (BS) structure of QFT is also operational for displaying the Regge pole interpolation of bound states (introduced in potential theory in [4]) as a consequence of the basic principles of QFT. In order to perform this program, it is necessary i) to extend the analysis of the general BS structure of ﬁeld theory so as to include the complex angular momentum variables, which requires in a ﬁrst step that we restrict ourselves to the range t < 0, along the line of [1]; ii) to justify the analytic continuation of this structure at t > 0. Part i) is the object of the present paper, while part ii) will be treated in a further paper. The contents of this paper can be described as follows. We start from a t-channel Bethe-Salpeter type structural equation for the four-point function of scalar ﬁelds; the latter is considered in the space of Euclidean energy-momenta which is fully contained in the primitive axiomatic analyticity domain. Such an integral equation involves a (“regularized two-particle-irreducible”) Bethe-Salpetertype kernel B whose introduction is completely justiﬁed on the basis of the general principles of local QFT and (as proved in [8, 9]) the complete two-ﬁeld structure of the theory in the t-channel is encoded in this equation. These results and the corresponding structural equation for the absorptive parts in the crossed s and uchannels [8, 10] are recalled in Section 2. In Section 3, we work out various aspects of these structures in the mass variables and angular variables and ultimately in the CAM variable λt . After having written the BS-equation in the mass variables and angular variables we perform its partial-wave decomposition in the Euclidean region in terms of “oﬀ-shell Euclidean partial-waves”. Next we are interested in performing the analytic continuation of the BS-equation from Euclidean subspace to the real Minkowskian subspace by travelling in the complex angular variables of the t-channel inside the “enlarged analyticity domain” obtained in [1] until reaching the spectral sets of the crossed s and u-channels. This allows one to give a new approach to the (t-channel) BS-equations for the symmetric and antisymmetric combinations of the s and u-channel absorptive parts. In this study, an important role is played by an appropriate class of holomorphic kernels on the complexiﬁed hyperboloid, called “perikernels”, and by the “(c)-convolution product” of such kernels [11], which is diagonalized by the Laplace-type transformation Ld

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introduced in [1]. It follows that the BS-equations for the s and u-channel absorptive parts admit “t-channel Laplace-type transforms” which are themseves Fredholm-type integral equations depending analytically of the CAM variable λt in a half-plane of the form eλt > N . In view of the “Froissart-Gribov equalities” proved in [1] (giving a geometrical version of [6, 7] in the QFT framework), these integral relations provide Carlsonian interpolations [5] in this half-plane of the corresponding BS-equations for the even and odd Euclidean partial waves. In the subsequent part of this paper (Section 4), the theory of Fredholm resolvent equations and speciﬁcally the N D method, N and D being regarded as analytic functions of both variables t and λt , are shown to provide a general framework for generating Regge poles equipped with factorized residues involving Euclidean three-point functions. The existence of corresponding leading “Reggeon terms” in the asymptotic behaviour of H at large s and ﬁxed values of t and of the mass ˜ (a) in a strip of ˜ (s) and H variables necessitates the meromorphic continuation of H the form NB (t) < Re λt < NH : this property is shown to be strictly conditioned by the existence of a “two-particle irreducible kernel” B whose behaviour in the complex s (or u) plane at ﬁxed values of t and of the mass variables is bounded by sNB (t) , with NB (t) smaller than NH . An Appendix is devoted to a survey of the Fredholm theory in complex space with complex parameters [8, 9, 13, 17], which covers the various versions of BS-equations encountered in Sections 3 and 4; it establishes the corresponding analyticity properties and bounds of their solutions in the relevant variables.

2 Bethe-Salpeter-type structure of four-point functions in the t-channel: general properties We are concerned with relativistic local ﬁeld theory in (d + 1)-dimensional spacetime, with d ≥ 1 throughout this section; in the rest of the paper, the interesting structures are relevant for all d ≥ 2. As our main object of study, we consider a general four-point function in complex momentum space Cd+1 , denoted by H ([k]) , which describes the interaction of two local (and mutually local) ﬁelds φ1 and φ2 ; 2 we specify a certain “t-channel”, with squared-total energy t = (k1 + k2 ) asasso ciated with “two-ﬁeld states” of the form φ˜1 (k1 ) φ˜2 (k2 ) f (k1 , k2 ) dk1 dk2 |Ω , |Ω being the vacuum state of this ﬁeld theory and f any appropriate test-function in momentum space. Following [1], we adapt our notations to this t-channel by putting [k] = (k1 , k2 ; k1 , k2 ) , with k1 + k2 = k1 + k2 = K; ki and ki describe incoming and outgoing energy-momentum vectors carried by the ﬁeld φi (i = 1, 2) . We moreover assume that the function H ([k]) is amputated from the four external propagator factors Πi (ki ) , Πi (ki ) , i = 1, 2, Πi denoting the (complete) two-point function of the ﬁeld φi . We introduce the incoming and outgoing relative energymomentum vectors k1 − k2 k − k2 , Z = 1 Z= 2 2

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and we also write H ([k]) ≡ H (K; Z, Z ) as a function of the three independent complex vectors K, Z, Z . We refer to [1] for a short survey of the axiomatic analyticity properties of H ([k]) . A basic fact is that, ﬁxed Lorentz frame LF with energy for any → → → (0) − (0) − (0) − momentum coordinates K = K , K , Z = Z , Z , Z = Z , Z , the function H ([k]) is holomorphic for all real or complex values of the energy→ − − → (0) components K (0) , Z (0) , Z and real values of the momentum-components K , Z , − → Z , except on “cuts” which are deﬁned by “spectral sets” in the various mass and channel-energy variables 2 2 K K ζ1 = k12 = + Z , ζ2 = k22 = −Z , 2 2 2 2 K K 2 2 + Z , ζ2 = k 2 = − Z , ζ1 = k 1 = 2 2 2

2

2

2

s = (k1 − k1 ) = (Z − Z ) , u = (k1 − k2 ) = (Z + Z ) , t = K 2 . → − 2 In each of these expressions, the notation k 2 means k (0) − k 2 for the cor → − responding vector k = k (0) , k varying in Cd+1 . The so-called “absorptive parts” of H are the discontinuities of H ([k]) , denoted by ∆s H, ∆u H, ∆t H, across the cuts deﬁned respectively by the spectral sets Σs (s s0 ) , Σu (u u0 ) , Σt (t t0 ) ; s0 , t0 , u0 denote the corresponding mass thresholds of these spectral sets (here supposed to be strictly positive). It is worthwhile to note that for all massive ﬁeld theories this subset DLF of the primitive axiomatic analyticity domain [16] contains the whole corresponding “Euclidean subspace” of complex energymomentum space, namely the set of all complex conﬁgurations (K, Z, Z ) whose (0) , Z (0) , Z ) are purely imaginary and whose momentum energy-components (K (0) − → → − → − components K , Z , Z are real in the Lorentz frame LF considered. Another aspect which plays an essential role in the following concerns “temperateness properties” of the four-point function H ([k]) , which we assume to be of the form speciﬁed in formula (3.1) of [1]. These properties imply bounds of the following form |H (K; Z, Z )| Cloc (K, Z, Z )(1 + K)NH (1 + Z)NH (1 + Z )NH ,

(2.1)

where NH is a ﬁxed power (NH 0) and the notation k stands for the norm (i) 2 of the complex vector k = (k (0) , · · · , k (d) ), namely k2 = 0≤i≤d |k | ; the function Cloc includes an inverse power dependence with respect to the distance of the point (K, Z, Z ) from the union of the spectral sets, which takes into account the distribution character of the boundary values of H([k]) on the reals. The bound (2.1) holds uniformly in the Euclidean subspace and moreover in the following parts of the axiomatic analyticity domain, which are of basic use for the BS-structure:

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a) the subset DLF used in [8, 9], b) the union of the complex domains D(w,w ,ρ ) in one vector k (for all w, w , ρ ) introduced in [1] (see Propositions 3 and 4 of the latter).

2.1

A recall of axiomatic results on the Bethe-Salpeter structure of four-point functions

In [8, 9], it has been shown that for massive QFT’s satisfying the postulate of “asymptotic completeness of two-particle states” (or “oﬀ-shell unitarity in the two particle spectral region”) and an additional regularity assumption in the energyvariable t (for t ≥ t0 ), the two-particle t-channel analytic structure of H ([k]) is entirely encoded in any Bethe-Salpeter-type integral equation of the following form H (K; Z, Z ) = B (K; Z, Z ) + B (K; Z, Z ) H (K; Z , Z ) G (K; Z ) dZ Γ(K)

(2.2) where the “Bethe-Salpeter kernel” B ([k]) ≡ B (K; Z, Z ) is a four-point function satisfying the same axiomatic analyticity properties and bounds of the form (2.1)1 as H ([k]) and in addition the property of “two-particle irreducibility in the t-channel”. This means that the corresponding absorptive part (or disin the two-particle region, which is of the form continuity) ∆t B of B vanishes 2 2 t;t0 = (m1 + m2 ) t < M , m1 and m2 being the masses of the “lowest poles”

2 ki = m2i , i = 1, 2 in the respective propagators Π1 (k1 ), Π2 (k2 ) of φ1 and φ2 . As a matter of fact, these poles are present in the function G (K, Z ) under the integral of Equation (2.2), which represents a “regularized double-propagator” K with respect to the internal (d+1)-momenta ki = ±Z , i = 1, 2, of the following 2 form: K K (reg) (reg) G (K, Z ) = iΠ1 + Z .Π2 −Z ; (2.3) 2 2 (reg)

(i = 1, 2) is a regularized form of Πi obtained by multiplying the latter by a Πi suitable Pauli-Villars-type factor, equal to 1 on the mass shell. The only required (reg) property is that Πi be a Lorentz-invariant function, namely a function of the squared-mass complex variable ζi = ki2 having the same pole ζi = m2i (with the same residue) as Πi , analytic in the same cut-plane of the form C\{m2i }\[Mi2 , +∞[ and satisfying uniform bounds of the following form −r (reg) 2 (ki ) c(1 + ki ) , (2.4) Πi with r suﬃciently large and c < 1, for ki Euclidean (i.e., ζi = − ki 2 ). 1 These analyticity properties and bounds of B are implied by those of H, except for possible “CDD poles” in K corresponding to the Fredholm alternative, which can be excluded for t (reg) negative and |t| suﬃciently large under a suitable choice of the regularized propagators Πi (see Appendix, Proposition A1).

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The integration cycle Γ (K) in Equation (2.2) is the Euclidean subspace → (0) − , when K, Z and Z are themselves taken in Ed+1 . Ed+1 = Z = iY ,X If r is suﬃciently large with respect to N, Equation (2.2) is a genuine Fredholm integral equation (depending analytically on the vector parameter K) which allows one to deﬁne B ([k]) in terms of H ([k]) , with B also satisfying bounds of the form (2.1). In [8, 9], Equation (2.2) has been shown to extend by analytic continuation to all conﬁgurations (K, Z, Z ) in the subset DLF of the axiomatic domain, provided Γ (K) is distorted from its initial situation Ed+1 = Ed+1 (LF ) in order to remain in the analyticity domain of the integrand and to have an inﬁnite part parallel to Ed+1 . (Of course, by using the Lorentz invariance of the axiomatic domain, Equation (2.2) is also shown to sweep similarly the subsets DLF associated with all Lorentz frames LF .) In the following, we shall call “Feynman convolution” and denote by (B ◦t H) (K; Z, Z ) the integral at the right-hand side of Equation (2.2), which enjoys the axiomatic analyticity properties of a four-point function. Equation (2.2) can thus be rewritten in short: H = B + B ◦t H,

(2.2’)

The analysis of [8, 9] relies on the exploitation of Equation (2.2) as a Fredholm resolvent integral equation on the space Γ (K) , depending analytically of the (vector) parameter K. In particular, the result of this analysis is that H ([k]) can be written as a ratio of the following form NB (K; Z, Z ) (2.5) DB (K 2 )

where NB (K; Z, Z ) is a four-point function and DB K 2 is a two-point function, deﬁned in terms of B and G through the appropriate Fredholm series. NB and DB are proven to admit analytic continuations in the variable t = K 2 across the two-particle region t0 t < M 2 in a ramiﬁed domain around the two-particle threshold t = t0 . The zeros of DB K 2 correspond to poles of H ([k]) , interpreted as bound states or resonances. H (K; Z, Z ) =

2.2

Bethe-Salpeter equations for the absorptive parts in the crossed channels

In the present paper, we shall not exploit Equation (2.2) for the latter meromorphic structure in t occurring at Re t > 0, but for its remarkable implications in the crossed channels, namely the s and u-channels. In fact, it has been shown in [8, 10] that Equation (2.2) implies corresponding integral relations between the absorptive parts ∆s H, ∆u H of H and ∆s B, ∆u B of B, which need to be recalled with some care. In these relations, the Feynman convolution ◦t is replaced by a composition product ♦t involving integration on a certain compact set speciﬁed below. We now write these relations in the following concise form, before giving

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their detailed meaning and interpretation: ∆s H = ∆s B + ∆s B ♦t ∆s H + ∆u B ♦t ∆u H,

(2.6)

∆u H = ∆u B + ∆s B ♦t ∆u H + ∆u B ♦t ∆s H;

(2.7)

In the latter, the absorptive part ∆s H (and similarly for ∆s B) is speciﬁed as follows: H(K;Z,Z ) − lim H(K;Z,Z )] ∆s H(K;Z,Z ) = i [lim 2 2 Im (Z−Z ) = −ε Im (Z−Z ) =ε ε>0 ε→0 ε>0 ε→0 (2.8) The support of ∆s H (or ∆s B), which is the spectral set Σs = {(K, Z, Z ); k1 − k1 = Z − Z real,

2

(Z − Z ) ≥ s0 }

is contained in the union of the following two disjoint sets + Σ+ , s = (K; Z, Z ) ; Z − Z real, Z − Z ∈ V − Σ− , s = (K; Z, Z ) ; Z − Z real, Z − Z ∈ V where V + (resp. V − ) denotes the open forward (resp. backward) cone: 2

k 2 ≡ k (0) − k 2 > 0, k (0) > 0 (resp. k (0) < 0). − + − We shall call ∆+ s H and ∆s H (resp. ∆s B and ∆s B) the restrictions of ∆s H + (resp. ∆s B) to the corresponding disjoint sets Σs and Σ− s .

The absorptive part ∆u H (and similarly for ∆u B) is speciﬁed as follows: ∆u H(K;Z,Z ) = i [lim H(K;Z,Z ) − lim H(K;Z,Z )] 2 2 Im (Z+Z ) = −ε Im (Z+Z ) =ε ε>0 ε→0 ε>0 ε→0 (2.9) The support of ∆u H (or ∆u B), which is the spectral set 2

Σu = {(K, Z, Z ); k1 − k2 = Z + Z real, (Z + Z ) ≥ u0 }, is contained in the union of the following two disjoint sets − Σ+ , u = (K; Z, Z ) ; Z + Z real, Z + Z ∈ V + Σ− . u = (K; Z, Z ) ; Z + Z real, Z + Z ∈ V − The corresponding restrictions of ∆u H (resp. ∆u B) to Σ+ u and Σu are called − + − and ∆u H (resp. ∆u B and ∆u B) .

∆+ uH

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For all complex conﬁgurations (K; Z, Z ) in Σ+ s one introduces the “double-cone” ♦ (Z, Z ) = Z ; Z − Z real, Z − Z ∈ V + , Z − Z ∈ V + and deﬁnes the composition product ♦t as follows: (∆s B ♦t ∆s H) (K; Z, Z ) + = ∆+ s B (K; Z, Z ) ∆s H (K; Z , Z ) G (K; Z ) dZ

(2.10)

♦(Z,Z )

and similarly (∆u B ♦t ∆u H) (K; Z, Z ) + = ∆− u B (K; Z, Z ) ∆u H (K; Z , Z ) G (K; Z ) dZ

(2.11)

♦(−Z ,−Z)

It is important to note (in view of the distribution character of the absorptive parts ∆s,u H and ∆s,u B in the corresponding variables s and u respectively) that these composition products are always well deﬁned in the sense of distributions in view of the compactness of the integration region. Equations (2.10) and (2.11) specify completely the meaning of formula (2.6) in the case when Z − Z ∈ V + . The complete speciﬁcation of formulae (2.6) and (2.7) for conﬁgurations + − (K, Z, Z ) contained respectively in Σ− s , Σu and Σu is obtained by changing the integration region ♦ (Z, Z ) or ♦ (−Z , −Z) of (2.10), (2.11) into an appropriate region of the form ♦ (εZ, ε Z ) or ♦ (ε Z , εZ) , with ε, ε = ±1 and by picking-up ± the corresponding relevant parts ∆± s,u H and ∆s,u B whose choice is dictated by the consistency of support properties. The derivation of formulae (2.6), (2.7), which is obtained by taking the absorptive parts ∆s and ∆u of both sides of Equation (2.2), relies on the following discontinuity formulae; for any pair of four-point functions (F1 , F2 ) , one has: and

∆s (F1 ◦t F2 ) = ∆s F1 ♦t ∆s F2 + ∆u F1 ♦t ∆u F2

(2.12)

∆u (F1 ◦t F2 ) = ∆s F1 ♦t ∆u F2 + ∆u F1 ♦t ∆s F2

(2.13)

These formulae have been derived in [8, 10] by performing suitable distortions of the integration cycle Γ(K) (starting from Ed+1 ) in the Feynman-convolution prod uct (F1 ◦t F2 )(K; Z, Z ) = Γ(K) F1 (K; Z, Z )F2 (K; Z , Z )G(K; Z )dZ . These

distortions, which are performed in the energy variable Z (0) inside the axiomatic domain of the integrand, lead one to fold the cycle Γ(K) around the supports of the absorptive parts ∆s,u F1 and ∆s,u F2 in limiting situations Γ(K) = Γs± (K) (resp. Γu± (K)) where s = es ± iη (resp. u = eu ± iη), with η positive and tending to zero. Taking the discontinuities ∆s (resp. ∆u ) between the convolution integrals on Γs+ (K) and Γs− (K) (resp. Γu+ (K) and Γu− (K)) then reduces the integration cycle to the compact cycle with support ♦ (εZ, ε Z ) previously deﬁned. A nice property enjoyed by the previous formulae (2.12), (2.13) is the following

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Additivity property of spectral sets If two four-point functions F1 ([k]) , F2 ([k]) have their spectral sets Σs and Σu speciﬁed respectively by the conditions s ≥ s1 , u ≥ u1 and s ≥ s2 , u ≥ u2 , then the fol− + − lowing support properties hold, inside each connected component Σ+ s , Σs , Σu , Σu of the spectral sets: a) The support of (∆s F1 ♦t ∆s F2 ) (K, Z, Z ) is contained in the set √ √ 2 2 (K, Z, Z ) ; (Z − Z ) ≥ ( s1 + s2 ) . This directly follows from the deﬁnition (2.10) of ♦t for ∆s F1 ♦t ∆s F2 by using the fact that Z − Z = (Z − Z ) + (Z − Z ) , with (Z − Z )2 ≥ s1 and 2 (Z − Z ) ≥ s2 (Z − Z can be either in V + or in V − ) . b) Similarly, the supports of ∆u F1 ♦t ∆u F2 , ∆s F1 ♦t ∆u F2 , ∆u F1 ♦t ∆s F2 are respectively contained in the sets deﬁned by the conditions: √ √ 2 2 (Z − Z ) ≥ ( u1 + u2 ) , √ √ 2 2 (Z + Z ) ≥ ( s1 + u2 ) , √ √ 2 2 (Z + Z ) ≥ ( u1 + s2 ) (with all the possibilities Z − Z ∈ V ± and Z + Z ∈ V ± ). Formulae (2.6) and (2.7) appear as a pair of coupled Fredholm-type equations (depending on K) which we call “Bethe-Salpeter equations for the crossed-channel absorptive-parts”. A remarkable feature of these equations, which is due to the “additivity property of spectral sets”, is the following Finiteness property of the Bethe-Salpeter equations for crossed-channel absorptive-parts (Theorem 1 of [8]) In any bounded region of the variables s and u, the Bethe-Salpeter equations (2.6) and (2.7) for ∆s H and ∆u H can be solved explicitly by a ﬁnite number of ♦t -composition-products. In fact, by applying the standard iteration procedure to Equation (2.6), one gets the following relations, written for simplicity in the case when s0 = u0 : For s < 4s0 ,

∆s H = ∆s B.

For s < 9s0 ,

∆s H = ∆s B + ∆s B ♦t ∆s B + ∆u B ♦t ∆u B.

For s < 16s0 , ∆s H = ∆s B + ∆s B ♦t ∆s B + ∆u B ♦t ∆u B + ∆s B ♦t ∆s B ♦t ∆s B +∆s B ♦t ∆u B ♦t ∆u B +∆u B ♦t ∆s B ♦t ∆u B +∆u B ♦t ∆u B ♦t ∆s B. (2.14) etc. . . . Equation (2.7) is solved by similar expressions for ∆u H.

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3 Transferring the Bethe-Salpeter structure from complex momentum space to mass and complex-angular-momentum variables In this section, we shall derive alternative versions of the Bethe-Salpeter equations (2.2) and (2.6), (2.7) in which the complex energy-momentum vectors are replaced 1) by squared-mass and complex-angular variables and 2) by squaredmass and complex-angular-momentum (CAM) variables. Throughout these two steps, the four-point function H([k]) is considered as given with its axiomatic analyticity properties and temperate bounds of the form (2.1), which imply (according to the results of [1]) two corresponding sets of properties in the complex-angular and CAM variables. At each step, analogous properties of the Bethe-Salpeter kernel B will appear as derived from those of H by applying the Fredholm method or “N /D-method” in complex-space to the corresponding version of the BetheSalpeter equation. The common point to these various versions will be the occurrence of an integration-space involving radial and longitudinal variables (ρ, w), equivalent to “squared-mass variables” ζ ≡ (ζ1 , ζ2 ), as described below. A uniﬁed presentation of the relevant results of the N /D-method in complex-space, applicable to these various versions is summarized in the Appendix . We now assume that d ≥ 2 and ﬁx once for all the total energy-momentum vector K of the t-channel in such a way that K is space-like (i.e., t√< 0); we

choose a system of space-time coordinates such that K = 0, 0, . . . , 0, −t . For (0) (1) (d) real or complex vectors k = (k (0) , k (1) , . . . , k (d) ), k = (k , k , . . . , k ), the (0) (1) (d) Minkowskian scalar product is k · k = k (0) k − k (1) k − · · · k (d) k and k 2 ≡ k · k. We adopt the parametrization of the vector variables Z, Z in terms of radial, longitudinal and angular variables ρ, w, z and ρ , w , z as in [1], Equation (2.2), namely (3.1) Z = ρz + wK, Z = ρ z + w K,

(0) (1) (0) (1) (d−1) where the vectors z = z , z , . . . , z (d−1) , 0 , z = z , z , . . . , z ,0 are such that: z · K = z · K = 0

and

z 2 = z 2 = −1.

(3.2)

The radial and longitudinal variables (ρ, w) (resp. (ρ , w )) can be equivalently replaced by the “squared-mass variables” ζ = (ζ1 , ζ2 ) , resp. ζ = (ζ1 , ζ2 ) , where 2 2 K K 1 2 1 2 2 ±Z ±Z = −ρ + (w ± ) t, ζ1,2 = = −ρ + (w ± )2 t ζ1,2 = 2 2 2 2 (3.3) and one has: ρ2 =

Λ (ζ1 , ζ2 , t) , 4t

ρ2 =

Λ (ζ1 , ζ2 , t) , 4t

(3.4)

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Complex Angular Momentum Diagonalization of the Bethe-Salpeter

with

2

Λ (α, β, γ) = (α − β) − 2 (α + β) γ + γ 2 and w=

ζ1 − ζ2 , 2t

w =

ζ1 − ζ2 . 2t

95

(3.5)

(3.6)

The variables ρ, w, ρ , w will always be real, with ρ > 0, ρ > 0, which means that (ζ, ζ ) varies in the real region ∆t × ∆t , where ∆t is the parabolic region of (negative) “Euclidean squared-masses” (see Fig. 1 of [1]) ∆t = {ζ = (ζ1 , ζ2 ) ; Λ (ζ1 , ζ2 , t) < 0} . On the contrary, the vector variables z, z or “hyperbolic angular variables” (c) are allowed to vary on the whole complex hyperboloid Xd−1 with equation z 2 = (d)

= 0) and one introduces the variable z 2 = −1 (in the subspace z (d) = z cos Θt = −z.z , Θt being interpreted as the oﬀ-shell (complex) scattering angle in the t-channel.

3.1

Bethe-Salpeter equation in the mass variables and angular variables-partial-wave decomposition in the Euclidean region

In this subsection, we shall assume that the vectors Z, Z remain in the Euclidean subspace Ed+1 which in the parametrization (3.1), (3.2) corresponds to choosing the complex vectors z, z on a unit sphere of dimension d− 1, called “the Euclidean (c) sphere” Sd−1 of Xd−1 , namely (0) (1) (d−1) ,0 , z = iy (0) , x(1) , . . . , x(d−1) , 0 , z = iy , x , . . . , x with (0) (1) (d−1) ω = y (0) , x(1) , . . . , x(d−1) ∈ Sd−1 , ω = y , x , . . . , x ∈ Sd−1 . The Minkowskian scalar product z · z here reduces to z · z = − ω · ω , where ω · ω denotes the Euclidean scalar product in Rd . Then ω · ω = cos Θt with Θt real. We then intend to write the Bethe-Salpeter equation (2.2) in the Euclidean subspace Ed+1 in terms of the radial, longitudinal and angular variables. We shall use the fact that the four-point function H ([k]) of the scalar ﬁelds (φ1 , φ2 ) , and therefore also B ([k]) are invariant under the connected part of the complex Lorentz (c) group SO0 (1, d) . This implies that these functions only depend on the Lorentzinvariant variables (ρ, w, ρ , w , t) , or equivalently (ζ, ζ , t) and z · z = − ω · ω = − cos Θt , the restrictions of these functions to the Euclidean subspace (Ed+1 )3

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being then invariant under the connected orthogonal group SO0 (d + 1). One can thus put (with notations similar to those of [1], Equation (3.24)): H ([k]) ≡ H [t; ρ, w; ρ , w ; ω · ω ] ≡ H (ζ,ζ ,t) ( ω · ω )

(3.7)

B ([k]) ≡ B [t; ρ, w; ρ , w ; ω · ω ] ≡ B (ζ,ζ ,t) ( ω · ω )

(3.8)

We note that in view of Eq. (3.1), the bounds (2.1) on H in (Ed+1 )3 can be (H) rewritten as follows, with a suitable constant CE : |H [t; ρ, w; ρ , w ; ω · ω ] | (H)

1

1

1

≤ CE (1 + |t| 2 )NH (1 + ρ)NH (1 + |w||t| 2 )NH (1 + ρ )NH (1 + |w ||t| 2 )NH . (3.9) (reg) (reg) 2 (k) ≡ Πi In view of the Lorentz invariance of the propagator Πi k of φi (i = 1, 2) , Equation (2.3) can also be rewritten as follows: (reg)

G (K, Z) = G (ζ) = iΠ1

(reg)

[ζ1 ] Π2

[ζ2 ]

(3.10)

or (reg)

G [t; ρ, w] = iΠ1

(reg) (w + 1/2)2 t − ρ2 Π2 (w − 1/2)2 t − ρ2 ,

(3.11)

and the bounds (2.4) then yield correspondingly: 1 1 |G[t; ρ, w]| ≤ c2 [1 + ρ2 + |t|(w + )2 ]−r [1 + ρ2 + |t|(w − )2 ]−r . 2 2 In view of (3.1)–(3.6), the integration measure dZ on Ed+1 reads: √ dZ = i −tρd−1 dρ dw dω = −idµt (ζ) dω with 1 dµt (ζ) = √ 4 −t

Λ (ζ1 , ζ2 , t) 4t

d − 2 2

dζ1, dζ2

(3.12)

(3.13)

(3.14)

We can now give the following alternative form of the BS-equation (2.2) in Euclidean space in terms of mass variables and angular variables; as shown in the Appendix, the bounds (3.9), (3.12) on H and G ensure that this integral relation is a genuine Fredholm equation depending on the parameter t for all t < 0. One obtains: H [t; ρ, w; ρ , w ; ω · ω ] = B [t; ρ, w; ρ , w ; ω · ω ] + (B◦t H) [t; ρ, w; ρ , w ; ω · ω ] ,

(3.15)

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Complex Angular Momentum Diagonalization of the Bethe-Salpeter

where: (B◦t H) [t; ρ, w; ρ , w ; ω · ω ] ∞ ∞ √ d−1 = i −t ρ dρ dw G [t; ρ , w ] 0

−∞

Sd−1

97

dω B [t; ρ, w; ρ , w ; ω · ω ]

×H [t; ρ , w ; ρ , w ; ω · ω ]

(3.16)

or equivalently, by using the mass variables and introducing the convolution product ∗ of SO (d)-invariant kernels a and b on the sphere Sd−1 , namely dω a ( ω · ω ) b ( ω · ω ) , (3.17) (a ∗ b) ( ω · ω ) = Sd−1

−i

∆t

H (ζ,ζ ,t) ( ω · ω ) = B (ζ,ζ ,t) ( ω · ω ) B (ζ,ζ ,t) ∗ H (ζ ,ζ ,t) ( ω · ω ) G (ζ ) dµt (ζ ) .

(3.18)

Let us now introduce the “partial-wave expansion” of invariant kernels a ( ω · ω ) ≡ a (cos θ) on the sphere Sd−1 by the following formulae: a (cos θ) = a ˜ = ωd−1

1 ωd

+1

0≤<∞

(d)

P

−1

(d)

a ˜ hd () P

(cos θ) ,

(3.19)

(cos θ) a (cos θ) (sin θ)d−3 d cos θ

(3.20)

(d)

where the functions P are the “ultraspherical Legendre polynomials”, given by the following integral representation: ωd−2 π (d) d−3 P (cos θ) = (cos θ + i sin θ cos ϕ) (sin ϕ) dϕ, (3.21) ωd−1 0 hd (λ) =

(2λ + d − 2) Γ (λ + d − 2) (d − 2)! Γ (λ + 1)

(3.22)

d/2

π and ωd = 2 Γ(d/2) is the area of the sphere Sd−1 . Equation (3.20) allows us to introduce the “t-channel partial-waves” ˜ (ζ, ζ , t) of the restriction of the four-point function H ([k]) ˜ [t; ρ, w, ρ , w ] ≡ h h 3 to Ed+1 :

˜ (ζ, ζ , t) = ωd−1 h

+1

(d)

d−3

d cos θ,

(3.23)

and similarly for B ([k]): +1 (d) ˜b (ζ, ζ , t) = ωd−1 P (cos θ) B (ζ,ζ ,t) (cos θ) (sin θ)d−3 d cos θ.

(3.24)

−1

−1

P

(cos θ) H (ζ,ζ ,t) (cos θ) (sin θ)

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By now using the “factorization property”2 according to which the partial waves of (a ∗ b) ( ω · ω ) are: (a ∗ b) = ab ,

(3.25)

we can replace the version (3.18) of the Bethe-Salpeter equation by the following set of “Bethe-Salpeter equations for the partial waves” of H ([k]) : ˜ (ζ, ζ , t) = ˜b (ζ, ζ , t) + ˜ (ζ , ζ ; t) G (ζ ) dµt (ζ ) . ˜b (ζ, ζ ; t) h h (3.26) ∆t

Note that each of these integral equations is comparable to a Bethe-Salpeter equation for ﬁeld theory in two-dimensional space-time. In view of the bounds (3.9) for H and (3.12) for G, the integral equations (3.15) and (3.26) appear as Fredholm resolvent equations, to which the results of the Appendix apply (see Proposition A1 with z, z varying in Γ0 for Equation (3.15) and the remark after Proposition A3 for Equation (3.26)).

3.2

Bethe-Salpeter equation for absorptive parts in the mass variables and angular variables: perikernel structure

In Section 2, we have recalled the fact that the axiomatic analyticity domain of H ([k]) (or B ([k])) contains the Euclidean subspace of complex momentum-space and provides a connection between the latter and the real Minkowskian subspace (0) Z, Z . by travelling in the complex energy variables K (0) , Z (0) , Z at ﬁxed K, This allowed one to reach the spectral sets Σs , Σu , to compute the corresponding discontinuities of B ◦t H and thereby to obtain the Bethe-Salpeter equations (2.6) and (2.7) for the absorptive parts of H ([k]) in the s and u-channels. This derivation real). was valid for arbitrary vectors K = (K (0) complex, K 2 For K = (0, K)(K = t < 0), an analyticity property of similar type, but actually diﬀerent since adapted to the mass variables and angular variables (ζ, z) , (ζ , z ) , was established in [1]. In fact, it was proven there that for all (real) values of (ζ, ζ ) in ∆t × ∆t , the “enlarged ” axiomatic analyticity domain of H ([k]) , obtained by geometrical techniques of analytic completion, contains ˆ (ζ,ζ ,K) parametrized by Equations (3.1)–(3.6), in the whole complex manifold Ω (c)

(c)

which (z, z ) varies on Xd−1 × Xd−1 , deprived from “cuts” generated by the spectral sets Σs ((Z − Z )2 ≥ s0 ) and Σu ((Z + Z )2 ≥ u0 ). In other words, for each (ζ, ζ ) ﬁxed, the function H (ζ,ζ ,t) (−z · z ) ≡ H ([k]) considered in Subsection 3.1 as an invariant kernel on the sphere Sd−1 admits an analytic continuation on (c) (c) Xd−1 × Xd−1 deprived from cuts Σs (ζ, ζ , t) and Σu (ζ, ζ , t) (see Equations (3.29), (3.30) below) which describe the traces of the spectral sets Σs and Σu in the 2 Note that the normalizations chosen for writing the deﬁnitions (3.17) and (3.20) of the convolution-product and of the partial waves yield Equation (3.25) without extra-coeﬃcient; they however diﬀer by a factor ωd from the standard normalization.

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Complex Angular Momentum Diagonalization of the Bethe-Salpeter

ˆ (ζ,ζ ,K) . Moreover, since H manifold Ω

(ζ,ζ ,t)

99

only depends on z, z through the

variable cos Θ = −z · z , it is analytic with respect to this variable in the im(c) (c) age of Xd−1 × Xd−1 \(Σs (ζ, ζ , t) ∪ Σu (ζ, ζ , t)), which is a cut-plane of the form C\ {[cosh vs + ∞ [ ∪ ] − ∞, − cosh vu ]} . The class of functions K(z, z ) holomorphic in the previous cut-domains of (c) (c) Xd−1 × Xd−1 (also denoted by K(−z · z ) when they are Lorentz invariant) has been extensively studied in [11, 12] under the name of (invariant) perikernels, and their discontinuities have been characterized as (invariant) “Volterra kernels” on the one-sheeted hyperboloid Xd−1 ; useful results involving these notions will be recalled below. Such a perikernel structure is satisﬁed not only by H([k]) but also by the BS-kernel B([k]), since the latter enjoys the same analyticity domain as H([k]) for ﬁxed K (with t = K 2 < 0); so we shall put similarly B([k]) ≡ B (ζ,ζ ,t) (−z · z ). This perikernel structure of B (ζ,ζ ,t) (−z · z ) will also be reobtained below in two ways by introducing and solving appropriate extensions of the BS-equations (3.18) and (3.26). In the continuation of this program, a basic role is played by the bounds (2.1) which H([k]) is assumed to satisfy in its axiomatic domain, in particular in the sets D(w,w ,ρ ) of Propositions 3, 4 of [1]. In fact, it has been established in Theorem 1 ˆ (ζ,ζ ,K) : of [1] that bounds of the following form hold in each submanifold Ω (H)

|H (ζ,ζ ,t) (cos Θt )| ≤ C(ζ,ζ ,t) eNH |mΘt | | sin eΘt |−nH .

(3.27)

In the latter, nH describes the maximal local order of singularity of the distribution boundary-values of H (ζ,ζ ,t) on the reals. We shall consider here nH as being independent3 of the exponent NH which governs the behaviour of H at inﬁnity (H) according to the assumed bounds (2.1). Concerning the “constant” C(ζ,ζ ,t) , one can check (by following the proof of Theorem 1 of [1]) that it can be taken equal to the uniform bound of H in the Euclidean subspace, namely (see (3.9)): (H)

(H)

1

1

1

C(ζ,ζ ,t) = CE [(1 + |t| 2 ) (1 + ρ) (1 + |w||t| 2 ) (1 + ρ ) (1 + |w ||t| 2 )]NH . (3.28) Absorptive parts: passage to the mass variables and hyperbolic angular variables; convolution products ♦ The absorptive parts ∆s H, ∆u H (resp. ∆s B, ∆u B) deﬁned in Section 2 also appear at ﬁxed (ζ, ζ , t) (or (ρ, w, ρ , w , t)) as being equal to the discontinuities ∆s H (ζ,ζ ,t) (−z · z ) and ∆u H (ζ,ζ ,t) (−z · z ) of H (ζ,ζ ,t) (−z · z ) (and similarly for B (ζ,ζ ,t) ). The supports of these discontinuities, obtained by writing the conditions s = (Z − Z )2 ≥ s0 , u = (Z + Z )2 ≥ u0 in terms of the parametrization (3.1), 3 The occurrence of max(n , N ) in place of N H H H in the exponential factor obtained in the bound (3.25) of [1] is in fact without physical content.

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(3.2) are respectively: (c)

(c)

Σs (ζ, ζ , t) = {(z, z ) ∈ Xd−1 × Xd−1 ; −z · z = cosh v ≥ cosh vs }, 2

with cosh vs = 1 +

2

s0 + (ρ − ρ ) − (w − w ) t 2ρρ v ≥ vs );

(or Θt = iv, (c)

(3.29)

(c)

Σu (ζ, ζ , t) = {(z, z ) ∈ Xd−1 × Xd−1 ; z · z = cosh v ≥ cosh vu }, 2

with cosh vu = 1 +

2

u0 + (ρ − ρ ) − (w + w ) t 2ρρ

(or Θt = π + iv,

(3.30)

v ≥ vu ).

It is easy to check that these sets are respectively contained in the regions 2 2 (z − z ) > 0 and (z + z ) > 0. For z, z real, i.e., belonging to the real onesheeted hyperboloid Xd−1 , these relations express the fact that z is either in the “future” or in the “past” of z (resp.−z ). (Note that the future and the past of z , namely the regions Γ± (z ) = Xd−1 ∩ z ∈ Rd−1 , z − z ∈ V ± are bounded by the cone of generatrices of Xd−1 passing through z .) We conclude that the trace of Σs (ζ, ζ , t) on Xd−1 × Xd−1 is composed of two disjoint sets Σs+ (ζ, ζ , t) ¯ + and and Σ− s (ζ, ζ , t) , corresponding respectively to the conditions z − z ∈ V − z − z ∈ V¯ . Of course, these sets represent the supports of the respective com− ˆ ponents ∆+ s H and ∆s H of ∆s H in the manifold Ω(ζ,ζ ,K) , and this leads one to + distinguish two kernels ∆+ H (z, z ) ≡ ∆ H ([k]) and ∆− s s s H(ζ,ζ ,t) (z, z ) ≡ (ζ,ζ ,t) ∆− s H ([k]) of disjoint supports on Xd−1 , although (in view of Lorentz invariance) they are both represented by the same function of one variable denoted earlier by ∆s H (ζ,ζ ,t) (−z · z ) . Kernels on Xd−1 such as ∆+ (z, z ) , whose support is s H (ζ,ζ ,t) + ¯ contained in the set (z, z ) ∈ Xd−1 × Xd−1 ; z − z ∈ V have been introduced in [14] under the name of “Volterra kernels” on Xd−1 . One introduces similarly ± the kernels ∆± u H (ζ,ζ ,t) (z, z ) with respective supports Σu (ζ, ζ , t) distinguished + by the conditions ± (z + z ) ∈ V¯ , which are represented by the same function ∆u H (ζ,ζ ,t) (−z · z ) . These considerations will now allow us to reinterpret the Bethe-Salpeter equations for the absorptive parts (2.6) and (2.7), after rewriting the latter in terms of the mass variables (ζ, ζ ) and of the “hyperbolic angular variables” z, z , varying on Xd−1 . As a counterpart of our expression (3.16) of (B ◦t H) in Euclidean space, we can in fact rewrite the corresponding composition product ∆s B ♦t ∆s H (see Equation (2.10)) as follows, in the situation where Z is in the future of Z (a

Vol. 4, 2003

Complex Angular Momentum Diagonalization of the Bethe-Salpeter

101

similar expression would be obtained in the opposite situation): (∆s B ♦t ∆s H) [t; ρ, w; ρ , w ; −z · z ] ∞ +∞ √ d−1 ρ dρ G [t; ρ , w ] dw = −t 0

−∞

∆s B [t; ρ, w; ρ , w ; −z · z ] × · · · × ∆s H [t; ρ , w ; ρ , w ; −z · z ] dz ,

♦(z,z )∩Xd−1

(3.31) dz0 . . . dzd−2 where dz denotes the Lorentz-invariant measure dz = on Xd−1 . zd−1 The fact that the integration region on Xd−1 is restricted to the double-cone ♦ (z, z ) is a consequence of the integration prescription {Z ∈ ♦ (Z, Z )} expressing the support properties of ∆s B(K;Z,Z ) and ∆s H(K;Z,Z ) in Equation (2.10). + This results from the implication relation Z − Z ∈ V =⇒ z − z ∈ V¯ + , obvious from the following identities (entailed by Equations (3.1), (3.2)):

2

2

2

2

ρρ (z − z ) = (Z − Z ) + (ρ − ρ ) + (w − w ) |t| and

(3.32)

2

(z − z ) (3.33) 2 Moreover the same support properties of ∆s B and ∆s H also imply that the integrand at the right-hand side of Equation (3.31) vanishes outside a compact subset of the space of integration variables (ρ , w ); the integral (3.31) is therefore well deﬁned independently of the bounds on ∆s B and ∆s H. It is now appropriate to introduce the notion of convolution product ♦ of Volterra kernels on the one-sheeted hyperboloid (see [14, 11]) by the following formula: F1 (z, z ) F2 (z , z ) dz . (3.34) (F1 ♦F2 ) (z, z ) = (Z − Z ) · (z − z ) = (ρ + ρ ) ×

♦(z,z )∩Xd−1

It is to be noted that, due to the compactness of the integration region in (3.34), this convolution product remains meaningful for distribution-like Volterra kernels. For invariant Volterra kernels, which are of the form Fi (z, z ) = fi (−z · z ), + i = 1, 2 (as it is the case here for ∆+ s H (ζ,ζ ,t) , ∆s B (ζ,ζ ,t) ), formula (3.34) takes the following alternative form: 2ωd−2 f1 (cosh v1 ) f2 (cosh v2 ) (F1 ♦F2 ) (z, z ) ≡ (f1 f2 ) (cosh v) = sinh v d−3 v1 ≥0,v2 ≥0 v1 +v2 ≤v

· · · [(cosh v − cosh (v1 + v2 )) (cosh v − cosh (v1 − v2 ))]

d−4 2

d (cosh v1 ) d (cosh v2 ) . (3.35)

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Using the latter and putting ∆s H [t; ρ, w; ρ , w , −z · z ] ≡ ∆s H (ζ,ζ ,t) (−z · z ) (and similarly for ∆s B), we can then rewrite Equation (3.31) as follows: (∆s B ♦t ∆s H) [t; ρ, w; ρ , w ; −z · z ] =

∆s B (ζ,ζ ,t) ∆s H (ζ ,ζ ,t) (−z · z ) G (ζ ) dµt (ζ ) .

(3.36)

∆t

Similar expressions could be written for the three other composition products ∆u B ♦t ∆u H, ∆s B ♦t ∆u H, ∆u B ♦t ∆s H of Equations (2.6), (2.7), in terms of corresponding convolution products of Volterra kernels. From the Euclidean BS-equation to the BS-equation for s and u-channel absorptive parts through contour-distortion of “perikernel convolution products” We now recall the basic relationship which relates the ∗-convolution product of kernels on the “Euclidean” sphere Sd−1 and the ♦-convolution product of Volterra kernels on the one-sheeted hyperboloid Xd−1 (Theorem 2 of [11]). (c)

Being given two perikernels Ki (z, z ) , i = 1, 2, on Xd−1 whose respective discontinuities on the sets {z − z ∈ V ± } , {z + z ∈ V ± } are the Volterra kernels ± (c) ∆± K2 , s Ki (z, z ) , ∆u Ki (z, z ) , there exists a perikernel K denoted by K = K1 ∗ such that: i) the restrictions of K, K1 , K2 , to the “Euclidean” sphere Sd−1 are such that: K|Sd−1 = K1 |Sd−1 ∗ K2 |Sd−1 ,

(3.37)

+ ii) the discontinuities ∆+ s K (z, z ) and ∆u K (z, z ) are given by the following ♦-convolution products: + + − + ∆+ s K = ∆s K1 ♦∆s K2 + ∆u K1 ♦∆u K2 − + + + ∆+ u K = ∆s K1 ♦∆u K2 + ∆u K1 ♦∆s K2 ,

(3.38)

− (similar formulae being satisfied by ∆− s K and ∆u K),

iii) for every (z, z ) in the analyticity domain of K, there exists a class of cycles Γ(z, z ) such that K(z, z ) = K1 (z, z )K2 (z , z )dz , (3.39) Γ(z,z )

Γ(z, z ) being obtained by continuous distortion inside the analyticity domain of the integrand from the special cycle Γ0 (z, z ) ≡ Sd−1 , relevant for the Euclidean configurations (z, z ) ∈ Sd−1 × Sd−1 , iv) if K1 , K2 are invariant perikernels, K is also an invariant perikernel.

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This statement can then be applied for each set of ﬁxed values of (ζ, ζ , ζ , t) to the (invariant) perikernels B (ζ,ζ ,t) (−z · z ) and H (ζ ,ζ ,t) (−z · z ). It im plies the existence of an (invariant) perikernel B (ζ,ζ ,t) ∗(c) H (ζ ,ζ ,t) (−z · z ), whose restriction to the “Euclidean” sphere Sd−1 is correspondingly the kernel B (ζ,ζ ,t) ∗ H (ζ ,ζ ,t) appearing at the right-hand side of Equation (3.18). In view of property iii), one can see that the Bethe-Salpeter equation (3.18) (c) (c) can be analytically continued to all (z, z ) in Xd−1 × Xd−1 , for all t < 0, (ζ, ζ ) in ∆t × ∆t , under the following form: H (ζ,ζ ,t) (z, z ) B (ζ,ζ ,t) ∗(c) H (ζ ,ζ ,t) (z, z ) G (ζ ) dµt (ζ ). (3.40) = B (ζ,ζ ,t) (z, z ) − i ∆t

The latter can be considered as a Fredholm resolvent equation in complex space whose integration space is the product of ∆t by the “ﬂoating cycle” Γ on which the complex points z, z vary. In view of general results of [13] adapted to the present situation in the Appendix, the function B (ζ,ζ ,t) (z, z ) is directly obtained with its full perikernel structure as the solution of the Fredholm equation (3.40): in fact, B (ζ,ζ ,t) (z, z ) can be identiﬁed with the resolvent RH|α=−1 (t; ρ, w, ρ , w , z, z ) of Propositions A1 and A2 (F being replaced by H). In the case when the boundary values of H (ζ,ζ ,t) (z, z ) and its discontinuities are continuous (namely if the bounds (3.27) hold with nH = 0), the Fredholm solution B (ζ,ζ ,t) (z, z ) of Equation (3.40) satisﬁes the same regularity properties and therefore one can apply the property ii) (Equation (3.38)) of the ∗(c) -composition product of perikernels for computing side-by-side the discontinuities of Equation (3.40). This yields: + ∆+ s H (ζ,ζ ,t) = ∆s B (ζ,ζ ,t)

+ ∆t

+ ∆t

(3.41)

+ − + ∆+ s B (ζ,ζ ,t) ♦∆s H (ζ ,ζ ,t) + ∆u B (ζ,ζ ,t) ♦∆u H (ζ ,ζ ,t) G (ζ ) dµt (ζ ) + ∆+ u H (ζ,ζ ,t) = ∆u B (ζ,ζ ,t)

(3.42)

+ + + ∆− s B (ζ,ζ ,t) ♦∆u H (ζ ,ζ ,t) + ∆u B (ζ,ζ ,t) ♦∆s H (ζ ,ζ ,t) G (ζ ) dµt (ζ )

(with similar expressions for ∆− s,u H (ζ,ζ ,t) (z, z )). Then by comparing the convolution products here obtained with the forms (3.31), (3.36) of the ♦t -compositionproduct (2.10), we see that the latter equations are in fact identical to Equations (2.6), (2.7). So in this new presentation making use of perikernel-convolutionproducts, we have reobtained the BS-equations for the absorptive parts in terms of mass variables and hyperbolic angular variables. − We now make use of the fact that both Volterra kernels ∆+ s H (•) , ∆s H (•)

are represented by the same Lorentz-invariant function ∆s H (•) (−z · z ) , with −z ·

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− z = cosh v ≥ 1, while both Volterra kernels ∆+ u H (•) , ∆u H (•) are represented

by the same function ∆u H (•) (−z · z ) , with −z · z = − cosh v ≤ −1. It is then convenient to put ∆ u H (•) (cosh v) = ∆u H (•) (− cosh v) ; similar considerations are done for B (•) . Then, Equations (3.41), (3.42) can be rewritten in terms of convolution products of invariant Volterra kernels (see Equation (3.35)) as follows: ∆s H (ζ,ζ ,t) = ∆s B (ζ,ζ ,t) + · · ·

(3.43)

∆s B (ζ,ζ ,t) ∆s H (ζ ,ζ ,t) + ∆ u B (ζ,ζ ,t) ∆u H (ζ ,ζ ,t) and ∆ u H (ζ,ζ ,t) = ∆u B (ζ,ζ ,t) + · · ·

(3.44)

∆s B (ζ,ζ ,t) ∆ u H (ζ ,ζ ,t) + ∆u B (ζ,ζ ,t) ∆s H (ζ ,ζ ,t). In these equations, all the terms are functions of the variable cosh v varying on the half-line [1, +∞[. Symmetrized and antisymmetrized Bethe-Salpeter equations for the absorptive parts Let us put (s) H (•) = ∆s H (•) + ∆ ∆(s) B (•) = ∆s B (•) + ∆ u B (•) , ∆ u H (•) ,

(3.45)

(a) ∆(a) B (•) = ∆s B (•) − ∆ H (•) = ∆s H (•) − ∆ u B (•) , ∆ u H (•) ,

(3.46)

and

Then by adding-up and subtracting Equations (3.43) and (3.44) side-by-side one obtains the following decoupled Bethe-Salpeter equations for the symmetric and antisymmetric combinations of the s and u-channel absorptive parts: ∆(s) H (ζ,ζ ,t) = ∆(s) B (ζ,ζ ,t) +

∆(s) B (ζ,ζ ,t) ∆(s) H (ζ ,ζ ,t) G(ζ )dµt (ζ ) (3.47)

∆t

and ∆(a) H (ζ,ζ ,t) = ∆(a) B (ζ,ζ ,t) +

∆(a) B (ζ,ζ ,t) ∆(a) H (ζ ,ζ ,t) G(ζ )dµt (ζ ) .

∆t

(3.48)

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The case of distribution-like boundary values This more general case is characterized by a positive exponent nH in the bound (3.27) on H (ζ,ζ ,t) (cos Θt ). Then the function B (ζ,ζ ,t) (cos Θt ) obtained as the Fredholm resolvent of H (ζ,ζ ,t) from Equation (3.40) cannot be proved directly to enjoy a similar power bound near the reals (in view of the occurrence of the factor | sin eΘt |−nH in the constant MΓ inside the argument of the entire function Φ in the bound (A.20)). The fact that such a power and the corresponding distribution character of the boundary values of B (ζ,ζ ,t) still hold true (near the s and u-cuts) is a consequence of the following properties: a) For any Feynman convolution F1 ◦t F2 and for any composition product of perikernels K1 ∗(c) K2 , the discontinuity formulae (2.12), (2.13) and (3.38) can still be justiﬁed (by the corresponding contour-distortion arguments) when the boundary values are not continuous but governed by power bounds near the reals. These formulae then hold as well-deﬁned convolution-type products of distributions with supports in a salient cone (resulting in the double-coneshaped integration region of the ♦t and ♦-products). b) In view of the additivity property of spectral regions (see Section 2), any fourpoint function of the form H ◦t (n+1) = H ◦t · · ·◦t H (n products) has absorptive parts ∆s H ◦t (n+1) and ∆u H ◦t (n+1) admitting thresholds sn and un “of order n”: to make it simple, in the case when s0 = u0 one has sn = un = (n+ 1)2 s0 . c) For every n let Bn+1 (K, Z, Z ) be the solution of the auxiliary equation H ◦t (n+1) = Bn+1 + (−1)n Bn+1 ◦t H ◦t (n+1) . Bn+1 has the same analyticity domain as H ◦t (n+1) , namely it has the same thresholds s = sn , u = un . This results either from the Fredholm-type analysis of [8, 9] or from the present (c) one in the perikernel framework by using contours Γ in Xd−1 (see [11] and the Appendix). d) For every n the following relation is obtained by iterating the equation B = H − B ◦t H and taking into account the deﬁning equation of Bn+1 : B=

n+1

n+1

p=1

p=1

(−1)p−1 H ◦t p +

(−1)n+p Bn+1 ◦t H ◦t p .

(3.49)

Being interested in the boundary values and absorptive parts ∆s B, ∆u B of (n) (n) B in the real regions Rs , Ru where (respectively) s = (Z − Z )2 < sn and 2 u = (Z + Z ) < un , one can see that only the terms of the ﬁrst sum at the righthand side of Equation (3.49) will contribute to these absorptive parts. In fact in view of c) all the terms of the second sum in Equation (3.49) are analytic in the (n) (n) regions Rs and Ru . Therefore in view of a) the existence of power bounds near (n) the reals for B and the formulae for the corresponding absorptive parts in Rs and n+1 (n) Ru are completely governed by the ﬁnite sum p=1 (−1)p−1 H ◦t p . It then also follows that one can apply a) directly to the Feynman convolution B ◦t H in these

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regions and therefore derive Equations (3.41), (3.42) by applying the contourdistortion argument to Equation (3.40) (as in the case of continuous boundary values).

3.3

Complex angular momentum diagonalization of the Bethe-Salpeter equation

We shall now apply the main result of [1] (Theorem 5) to the four-point function H ([k]). This result relies basically on the Laplace-type Ld -transformation (see [12] for a complete study) which associates with each invariant Volterra kernel with moderate growth F (z, z ) ≡ f (−z · z ) on Xd−1 the following analytic function (see Proposition III-3 of [12]): ∞ (d) f (cosh v)Qλ (cosh v) (sinh v)d−2 dv. (3.50) F˜ (λ) = ωd−1 0

(d)

In this equation, Qλ (cosh v) denotes the second-kind Legendre function in dimension d whose integral representation is given by Equation (4.36) of [1]. If |f (cosh v)| is majorized by cst eN v , then F˜ (λ) is proved to be holomorphic in the half-plane (N ) C+ = {λ ∈ C; eλ > N }. According to Theorem 5 of [1], the absorptive parts ∆s H (ζ,ζ ,t) and ∆u H (ζ,ζ ,t) of a function H (ζ,ζ ,t) (cos Θt ) satisfying the bounds (3.27) admit “t˜ s (ζ, ζ ; t, λt ) and H ˜ u (ζ, ζ ; t, λt ) , which are channel Laplace-type transforms” H holomorphic with respect to the complex angular momentum variable λt in the (N ) half-plane C+ H . These Laplace-type transforms, obtained by applying the Ld transformation (3.50) at ﬁxed values of ζ, ζ , t are: ∞ (d) d−2 ˜ Hs (ζ, ζ ; t, λt ) = ωd−1 ∆s H (ζ,ζ ,t) (cosh v) Qλt (cosh v) (sinh v) dv (3.51) 0

˜ u (ζ, ζ ; t, λt ) = ωd−1 H

0

∞

(d)

d−2

∆u H (ζ,ζ ,t) (− cosh v) Qλt (cosh v) (sinh v)

dv

(3.52) In the general case, these formulae have to be understood in the sense of distribu(d) d−2 playing the role of a test-function. tions, namely with Qλt (cosh v) (sinh v) Moreover, the “symmetric and antisymmetric Laplace-type transforms” ˜ (s) = H ˜s + H ˜u H

,

˜ (a) = H ˜s − H ˜u, H

(3.53)

which are deﬁned in terms of ∆(s) H (ζ,ζ ,t) and ∆(a) H (ζ,ζ ,t) (see Equations (3.45), (3.46)) via the similar formulae: ˜ (s),(a) (ζ, ζ ; t, λt ) H ∞ (d) ∆(s),(a) H (ζ,ζ ,t) (cosh v) Qλt (cosh v) (sinh v)d−2 dv (3.54) = ωd−1 0

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enjoy the following “Froissart-Gribov-type equalities” for 2 > NH for 2 + 1 > NH ,

˜ 2 (ζ, ζ , t) ˜ (s) (ζ, ζ , t, 2) = h H

(3.55)

˜ 2+1 (ζ, ζ , t) ˜ (a) (ζ, ζ , t, 2 + 1) = h H

(3.56)

˜ (s) and H ˜ (a) are Carlsonian [5] (i.e., unique) interpolations of the respective sets H ˜ 2+1 ; 2 + 1 > NH ; they indeed satisfy ˜ 2 ; 2 > NH and h of partial waves h (NH +ε)

bounds of the following form in C+

(for all positive ε, ε ) :

˜ (s),(a) (s),(a) (H) n − d−2 +ε (ζ, ζ ; t, λt ) ≤ c(ε,ε ) C(ζ,ζ ,t) |λt − NH | H 2 e−[ eλt −(NH +ε)]v0 H (3.57) where v0 = min (vs (ζ, ζ , t) , vu (ζ, ζ , t)) , vs , vu being the quantities introduced in (3.29), (3.30). ˜ (s) and H ˜ (a) reexpress exactly (after symmetrization and These bounds on H antisymmetrization) the bounds (4.84) established in Theorem 5 of [1]; in particular, the occurrence of the power of |λt − NH | is correlated to the distribution character of ∆(s),(a) H (ζ,ζ ,t) (cosh v) encoded in the bound (3.27). However, we ε,ε of the latter reference under the form now have rewritten the constants Cs,u (s),(a)

(H)

(H)

c(ε,ε ) C(ζ,ζ ,t) , where the quantity C(ζ,ζ ,t) (see Equation (3.28)) contains the full dependence of the bound (3.27) with respect to the mass-variables ζ, ζ , while (s) (a) c(ε,ε ) and c(ε,ε ) are purely numerical constants: the dependence of the bound (H)

(3.57) with respect to C(ζ,ζ ,t) , results from the linearity of the transformations ˜ (s) (ζ, ζ ; t, λt ) and H ˜ (a) (ζ, ζ ; t, λt ) with H (cos Θt ). which associate H (ζ,ζ ,t) ˜ (s) and H ˜ (a) provide interpolations in the variable λt of the respecSince H ˜ 2 (ζ, ζ , t); 2 > NH } tive sequences of even and odd Euclidean partial waves {h ˜ and {h2+1 (ζ, ζ , t); 2 + 1 > NH } of H([k]), it is now natural to consider the corresponding interpolations in λt (for eλt > NH ) of the BS-equations (3.26) for these partial waves, which can be written as follows: ˜ (s),(a) (ζ, ζ ; t, λt ) = B ˜ (s),(a) (ζ, ζ ; t, λt ) + H

˜ (s),(a) (ζ, ζ ; t, λt ) H ˜ (s),(a) (ζ , ζ ; t, λt ) G (ζ ) dµt (ζ ) . B

(3.58)

∆t

˜ (s) and We have thus obtained two decoupled Bethe-Salpeter-type equations for B (a) (s) (a) ˜ in terms of H ˜ and H ˜ for B respectively, in which the Fredholm integration space reduces to the two-dimensional real region ∆t of the plane of squared-mass (N ) variables ζ = (ζ1 , ζ2 ) , while (t, λt ) are parameters varying in R− × C+ H .

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˜ (s) and In view of the mass-dependence of the uniform bounds (3.57) on H (a) 4 ˜ H , the Fredholm method can again be applied in the equivalent, but simpler form speciﬁed in the Appendix, where the variable integration space {ζ ∈ ∆t } is replaced by the fixed integration space {(ρ , w ) ∈ R+ × R}. This proves the exis˜ (s) (ζ, ζ ; t, λt ) and B ˜ (a) (ζ, ζ ; t, λt ) of the integral equations tence of the solutions B (3.58), depending meromorphically on the parameters t and λt for (t, λt ) varying (N ) in R− × C+ H . ˜ (a) are interpolations in the λt ˜ (s) and B By construction, these functions B plane of the corresponding Euclidean partial waves {˜b2 (ζ, ζ , t); 2 > NH } and {˜b2+1 (ζ, ζ , t); 2 + 1 > NH } of the Bethe-Salpeter kernel B([k]), namely there holds the following Froissart-Gribov-type equalities: for 2 > NH , for 2 + 1 > NH ,

˜ (s) (ζ, ζ ; t, 2) = ˜b2 (ζ, ζ , t), B

(3.59)

˜ (a) (ζ, ζ ; t, 2 + 1) = ˜b2+1 (ζ, ζ , t), B

(3.60) ˜ (s) According to the results of Proposition A3, in which F is replaced by H ˜ (a) ), we can make the following remark on the corresponding resolvent (resp., H ˜ (s) (resp. B ˜ (a) ). RF |α=−1 = B ˜ (s),(a) (ζ, ζ ; t, λt ) lead one to a choice Remark. The general bounds (3.57) on H n − d−2

of the function C(λt ) of Proposition A3 proportional to |λt − NH | H 2 , which ˜ (s),(a) (ζ, ζ ; t, λt ) × D ˜ (s)(a) implies bounds of the type (A.22) on B H |α=−1 (t, λt ). These bounds are comparable to (3.57) concerning their dependence in the mass variables, but contain in general no temperate dependence with respect to λt in (N ) C+ H ; this also corresponds to the impossibility of controlling a priori the poles ˜ (s),(a) in the limit |λt | → ∞. of B In the following, we shall always be led to assume that B satisﬁes an extraassumption of temperateness which leads us to the following statement Proposition 1 If ∆(s) B (ζ,ζ ,t) (cosh v) and ∆(a) B (ζ,ζ ,t) (cosh v) have a behaviour ˜ (s) and at infinity which is governed by eNB v (with NB ≥ NH ), then the solutions B (a) ˜ B of the BS-equation (3.58) are holomorphic functions of λt in the half-plane (NB ) C+ , and coincide there with the following Ld -transforms ∞ (s),(a) (d) ˜ B (ζ, ζ ; t, λt ) = ωd−1 ∆(s)(a) B (ζ,ζ ,t) (cosh v)Qλt (cosh v)(sinh v)d−2 dv 0

(3.61) Moreover, the BS-equations (3.58) then appear as “diagonalized forms” of the original BS-equations (3.47), (3.48). 4 Note

(H)

that in the bound (3.57) the mass dependence has to be majorized by C(ζ,ζ ,t) since the exponential factor (also mass-dependent through v0 ) has no better uniform majorant than (H) 1. In the treatment of the Appendix one then uses a majorant of C(ζ,ζ ,t) of the form (A.1) as explained there.

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For simplicity, we give the proof of this result for the case when ∆(s),(a) B (ζ,ζ ,t) are locally integrable functions. Applying the Ld -transformation (3.50) (N )

side-by-side to Equations (3.47), (3.48), one obtains for λt in C+ B (in view of Equations (3.54), (3.61)): ∞ (d) ˜ (s),(a) (ζ, ζ ; t, λt ) = B ˜ (s),(a) (ζ, ζ ; t, λt ) + ωd−1 H Qλt (cosh v)(sinh v)d−2 × · · · 0

   ∆(s),(a) B (ζ,ζ ,t) ∆(s),(a) H (ζ ,ζ ,t) (cosh v) G (ζ ) dµt (ζ ) dv

(3.62)

∆t

We shall apply the result of [12] (Proposition III-2) according to which the Ld -transform of the convolution product F1 ♦F2 of invariant Volterra kernels Fi (z, z ) = fi (−z · z ) , i = 1, 2, namely f1 f2 (in view of our deﬁnition (3.35)), is equal to the product F˜1 (λ) × F˜2 (λ) . More precisely (see Proposition II-2i) of [12]), if the kernels Fi (i = 1, 2) are regular functions satisfying norm conditions of the form ∞ e−N v |fi (cosh v)|dv < ∞, (3.63) gN (fi ) = 0

then their Volterra convolution product satisﬁes a bound of the form gN (f1 f2 ) ≤ cN gN (f1 ) gN (f2 ) < ∞,

(3.64)

where cN is a numerical constant. Correspondingly, the product F˜1 × F˜2 is holo(N ) morphic in C+ and one has: ∞ (d) ˜ ˜ |F1 (λ)F2 (λ)| ≤ cst |(f1 f2 )(cosh v)| |Qλ (cosh v)|(sinh v)d−2 dv 0

≤ c(ε) gN (f1 f2 ),

(3.65)

(N +ε)

(for all ε > 0 and a suitable choice of the uniformly in any half-plane C+ constant c(ε) ): this majorization follows from the exponential decrease property of (d) the function Qλ (cosh v) (see [12]). Assuming that the absorptive parts ∆(s),(a) H, ∆(s),(a) B satisfy norm conditions of the form (3.63) with the mass dependence given by Equation (3.28), (H) namely gNB (∆(s),(a) H (ζ,ζ ,t) ) = cH C(ζ,ζ ,t) (as it is implied by (3.27) if nH = 0) and gNB (∆(s),(a) B (ζ,ζ ,t) ) = cB C(ζ,ζ ,t) , (cH and cB being numerical constants), then it follows from (3.64), (3.65) that the repeated integral ∞ (d) ∆(s),(a) B (ζ,ζ ,t) ∆(s),(a) H (ζ ,ζ ,t) (cosh v) Qλt (cosh v)(sinh v)d−2 dv

∆t

0

· · · × G (ζ ) dµt (ζ )

(3.66)

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is absolutely convergent and therefore equal to the double integral at the righthand side of Equation (3.62). The integral over v in (3.66) is the Ld -transform of a ˜ (s),(a) (ζ, ζ ; t, λt ) H ˜ (s),(a) (ζ , ζ ; t, λt ) Volterra convolution, equal to the product B (NB )

in C+

. This shows that Equation (3.62) coincides with the corresponding BS˜ (s),(a) coincides with the equation (3.58) for the Laplace transforms, so that B B) ˜ (s),(a) to the half-plane C(N restriction of B . +

The case when ∆(s),(a) B (ζ,ζ ,t) are distributions can be treated similarly, since the convolution product F1 ♦F2 of distribution-like invariant Volterra kernels is again transformed by Ld in the corresponding product F˜1 (λ)× F˜2 (λ), these holomorphic functions admitting now a power bound in |λ| instead of being bounded.

4 On the Bethe-Salpeter generation of Regge poles in general quantum field theory In Section 3 we have exhibited how the BS-type structure resulting from the general (axiomatic) framework of QFT can be expressed in terms of the squared-mass and angular variables, and then in terms of the squared-mass and complex angular momentum variables; at each step, this was done by considering the kernel B (resp. ˜ (s),(a) ) as determined by H via the Fredholm method. In the present section, we B shall adopt a more exploratory viewpoint by assuming that in the ﬁeld theory under consideration, the kernel B([k]) = B[t; ρ, w, ρ , w ; cos Θt ] satisﬁes “better properties” than H, in a sense to be speciﬁed below. We then intend to draw the consequences of these additional assumptions on B for the structure of H.

4.1

Local generation of Regge poles

We consider again the diagonalized form (3.58) of the Bethe-Salpeter structure of a given four-point function H, assuming that the conditions of Proposition 1 are satisﬁed with NB = NH . Equation (3.58) is then valid as an identity between (N ) holomorphic functions of λt in the half-plane C+ H , for all values of (ζ, ζ ) in ˜ (s),(a) satisfy bounds which are similar to those ∆t × ∆t and negative t. Moreover B (s),(a) ˜ on H , as far as their dependence on the mass variables ζ, ζ is concerned, namely these bounds contain (as in (3.57)) a factor C(ζ,ζ ,t) of the form (3.28). ˜ (s),(a) can be analytically Let us now make the additional assumption that B continued in some disk of the λt -plane, centered on the border line (eλt = NH ) (N ) of C+ H and still satisfy similar bounds including the factor C(ζ,ζ ,t) in that disk. Let D be the intersection of the latter with the closed left-hand plane eλt ≤ NH ; we can consider the analytic continuation of (3.58) in D, these equations being ˜ (s) (resp. H ˜ (a) ) in terms now regarded as Fredholm-resolvent equations deﬁning H ˜ (a) ) through expressions of the form: ˜ (s) (resp. B of B ˜ (s),(a) (ζ, ζ ; t, λt ) = NB˜ (s),(a) (ζ, ζ ; t, λt ) . H DB˜ (s),(a) (t, λt )

(4.1)

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In Equation (4.1), the notations of Proposition A3 have been simpliﬁed by putting NB˜ (s),(a) (ζ, ζ ; t, λt ) = NB˜ (s),(a) |α=1 (t, λt ; ρ, w, ρ , w ) and DB˜ (s),(a) (t, λt ) = DB˜ (s),(a) |α=1 (t, λt ). ˜ (a) are holomorphic in C(NH ) , the resolution (4.1) provides a ˜ (s) and H Since H + ˜ (s) and H ˜ (a) in D, whose poles are given respecmeromorphic continuation of H tively by the zeros of DB˜ (s) (t,λt ) and DB˜ (a) (t,λt ) : the locations of the latter in the λt -plane will be denoted respectively by λt = λj (s) (t), λt = λj (a) (t). Remark. The variable t being kept real (and negative) at the present step of our program, there is no point of proving the analytic dependence of the previous zeros with respect to t; however, this analyticity will naturally appear at a further step, once one has extended the results of [1] to a relevant set of complex values of t by techniques of analytic completion. For simplicity, we shall only consider the case when there is a unique zero λ(s) (t) (resp. λ(a) (t)) of DB˜ (s) (t,λt ) (resp. DB˜ (a) (t,λt ) ): in D, which we suppose to be ˜ (a) admit these values λ(s) (t), λ(a) (t) as ˜ (s) and H a simple zero; correspondingly, H simple poles, which can be called Regge poles, and we introduce the corresponding residue functions: ˜ (s),(a) (ζ, ζ ; t, λ(s),(a) (t)) Res H ˜ (s),(a) (ζ, ζ ; t, λt )]|λ =λ(s),(a) (t) . (4.2) = [(λt − λ(s),(a) (t)) × H t

Factorization property of the Regge pole residues ˜ (a) ˜ (s) and Res H We claim that, as a result of Fredholm theory, the functions Res H can be written in the following form: (s) ˜ (s) (ζ, ζ ; t, λ(s) (t)) = p (ζ, ζ , t) , Res H (s) β (t)

(4.3)

(a) ˜ (a) (ζ, ζ ; t, λ(a) (t)) = p (ζ, ζ , t) , Res H (a) β (t)

(4.4)

where p(s) and p(a) are (for each t) kernels of finite rank satisfying the projector relations p(s) ◦t p(s) = p(s) , p(a) ◦t p(a) = p(a) . This can be seen as follows. ˜ (s) , Making use of the expression (A.16) of the Fredholm resolvent RB˜ (s) of B and calling α = α(t, λt ) the zero of DB˜ (s) (t, λt ; α) whose restriction to α = 1 yields ˜ (s) (λt being thus a solution of the implicit equation the pole λt = λ(s) (t) of H α(t, λt ) − 1 = 0), one can show (see, e.g., Section 3.3 of [17] and references therein to standard results of Fredholm theory) that RB˜ (s) (ζ, ζ ; t, λt , α) =

p(s) (ζ, ζ , t) + RB ˜ (s) (ζ, ζ ; t, λt , α), α − α(t, λt )

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where p(s) is of ﬁnite rank and RB ˜ (s) holomorphic in α at α = α(t, λt ). Formula ∂α (t, λ(s) (t)). The analysis would be (4.3) then follows by putting β (s) (t) = − ∂λ t more involved in the case of multiple poles (not considered here).

By assumption, we shall consider as “generic” the case when the Regge pole terms are characterized by projectors of rank one; for the case of Hermitian ﬁelds φ1 , φ2 considered here, these terms can always be written under the following form: p(s) (ζ, ζ , t) = ϕ(s) (ζ, t) × ϕ(s) (ζ , t), (4.5) (4.6) p(a) (ζ, ζ , t) = ϕ(a) (ζ, t) × ϕ(a) (ζ , t), (s) 2 (a) 2 with ∆t |ϕ (ζ, t)| dµt (ζ) = ∆t |ϕ (ζ, t)| dµt (ζ) = 1. The Hermitian character of the right-hand side of Equations (4.5), (4.6) is in fact implied by the symmetry ˜ (s),(a) (ζ, ζ ; t, λt ) together with the assumption ˜ (s),(a) (ζ, ζ ; t, λt ) = H property H (s),(a) (t) are real. To be more precise, if H([k]) is the four-point function of that λ two Hermitian scalar ﬁelds φ1 , φ2 , it satisﬁes in its analyticity domain the sym

metry relation H(K; Z, Z ) = H(K; Z , Z). The latter implies the following ones (for real values of t, ζ, ζ ): H (ζ,ζ ,t) (cos Θt ) = H (ζ,ζ ,t) (cos Θt ) and therefore also ˜ (s),(a) (ζ, ζ ; t, λt ). It follows ˜ (s),(a) (ζ, ζ ; t, λt ) = H (in view of Equation (3.54)): H that the poles λt = λj (s),(a) (t) resulting from Equation (4.1) can only be produced either at real values or at pairs of complex conjugate values. However in the range t < 0 considered here, the case of real poles appears to be more “physical”, as it can be illustrated by simple models of BS-kernels, and this justiﬁes our reality assumption on λ(s),(a) (t). Remark. The functions ϕ(s),(a) (ζ, t) which depend on the three Lorentz invariants ζ1 , ζ2 , t can be interpreted (from the geometrical viewpoint) as three-point functions deﬁned in the Euclidean domain of ﬁeld theory, namely in the set of conﬁgu(0) rations (k1 , k2 , k3 = k1 + k2 ) such that ki = (iqi , pi ), i = 1, 2, 3, whose invariants ζ1 = k12 , ζ2 = k22 and t = (k1 + k2 )2 vary in the set {(ζ, t); ζ ∈ ∆(t), t < 0}.

4.2

Asymptotic assumption on the Bethe-Salpeter kernel and generation of dominant Reggeon terms in the four-point functions

In this subsection, we shall assume that B([k]) = B[t; ρ, w, ρ , w ; cos Θt ] satisﬁes increase properties in the variable cos Θt = −z · z which are “better than” those of H. We then intend to show that, for any value of t for which this assumption is ˜ (a) which will pro˜ (s) and H made, there exists a possible Regge pole structure of H duce corresponding asymptotically dominant “Reggeon terms” in the four-point function H([k]). For the negative values of the channel variable t only considered in the present paper, it appears diﬃcult to give a general ﬁeld-theoretical support to this type of assumption, although it is naturally satisﬁed by standard approximations of the Bethe-Salpeter kernel B such as the relativistic counterpart of

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Yukawa-type potentials. As a matter of fact, it is for t varying in some positive interval that we shall be able to produce more general arguments in favour of such assumption on B, but this can only be relevant in our program after the analytic continuation in the variable t has been performed (in a further work). However, since the mathematical analysis is independent of the value of t (playing the role of a ﬁxed parameter), we think it appropriate to present it below as “a generic procedure for the generation of asymptotically dominant Reggeon terms in the field-theoretical framework”. More speciﬁcally, we shall now assume that B satisﬁes the following bounds (B) involving a given exponent NB and a positive constant CE : i) for all u ((0 ≤ u ≤ π and π ≤ u ≤ 2π, the jumps at u = 0 and u = π being taken into account), ∞ (B) e−NB v |B (ζ,ζ ,t) (cos(u + iv))|dv ≤ C(ζ,ζ ,t) , (4.7) 0

ii)

(B)

sup

−1≤cos Θt ≤1

|B (ζ,ζ ,t) (cos Θt )| ≤ C(ζ,ζ ,t) ,

(4.8)

with (B)

(B)

1

1

1

C(ζ,ζ ,t) = CE [(1 + |t| 2 ) (1 + ρ) (1 + |w||t| 2 ) (1 + ρ ) (1 + |w ||t| 2 )]NB . (4.9) In these assumptions, the real number NB is supposed to satisfy the conditions max(−1, − d−2 2 ) < NB < NH . So, if we compare the previous assumptions on B with the corresponding bounds (3.27), (3.28) on H, we see that they diﬀer under two respects: a) the increase at inﬁnity in the cos Θt -plane is governed by eNB |mΘt | (dominated by eNH |mΘt | ), b) the use of an L1 -bound instead of a uniform bound (as for H in (3.27)) (c) is motivated by its convenience for the (c) -convolution formalism on Xd . Here, the local order of singularity of the boundary values and discontinuities ∆(s),(a) B of B is encoded in their L1 -character with respect to the variable v = mΘt together with their uniform dependence on sin eΘt . We then have: Proposition 2 The previous assumptions i), ii) on B imply the analyticity property (N ) ˜ (s) , with respect to λt in C+ B and the following majorizations for the transforms B ˜ (a) of ∆(s) B, ∆(a) B: B (B)

˜ (s),(a) (ζ, ζ ; t, λt )| ≤ C Ψ(|mλt |), |B (ζ,ζ ,t)

(4.10)

where Ψ denotes a bounded positive function on [0, ∞[, tending to zero at infinity. (N ) These majorizations hold uniformly for all (ζ, ζ ) ∈ ∆t ×∆t , t < 0 and λt ∈ C+ B .

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The proof of the latter relies on the fact that the action of the Ld -transformation on ∆(s),(a) B factorizes as follows (see [12], formulae (III-1), (III-3)): ∞ (s),(a) B(ζ, ζ ; t, w)dw, ˜ B (ζ, ζ ; t, λt ) = e−λt w ∆(s),(a) (4.11) v

with

ωd−2 e−

d−2 2

B(ζ, ζ ; t, w) = · · · ∆(s),(a)

w

0

∆(s),(a) B (ζ,ζ ,t) (cosh v)[2(cosh w − cosh v)]

d−4 2

sinh vdv.

(4.12)

It can be seen (see corollary of Proposition II-6 in [12]) that in view of the condition NB > −1 the L1 -bound on ∆(s),(a) B (ζ,ζ ,t) (cosh v) deduced from (4.7), namely 0

∞

(B)

e−NB v |∆(s),(a) B (ζ,ζ ,t) (cosh v)| dv ≤ C(ζ,ζ ,t)

(4.13)

(s),(a) B(ζ, ζ ; t, w). Then implies the same L1 -bound (up to a constant factor) on ∆ since Equation (4.11) represents a usual Fourier-Laplace transformation, the announced bound (4.10) readily follows from the Lebesgue theorem.

Remark. A similar result holds if the bound (4.13) is satisﬁed only in the sense of distributions: this is the case, e.g., if B contains a simple or multiple pole of the form (cos Θt − cosh v0 )−q , interpreted as a “particle-exchange” (or Yukawa-type) contribution for q = 1, or as a more singular “gluon-type exchange” contribution for q > 1. In view of Proposition 2, the considerations of Subsection 4.1 apply to the ˜ (a) in the full half-plane eλt > ˜ (s) and B analytic continuations of the kernels B ˜ (s) and H ˜ (a) admit meromorphic continuations of the form NB . It follows that H (4.1) in the strip NB < eλt ≤ NH . In this strip, there will occur possible Regge poles equipped with factorized residues of the form described by Equations (4.3)– ˜ (s),(a) , the last (4.6). Moreover we notice that, in view of the bounds (4.10) on B statement of Proposition A3 can be applied. It follows that, under our assumptions ˜ (s) and H ˜ (a) are conﬁned in a on B, the poles λt = λj (s) (t), λt = λj (a) (t) of H bounded region of the form {λt ; NB < eλt ≤ NH , |mλt | < νΨ (t)}. Finally, by (B) applying the majorization (A.22) with C(λt ) = CE Ψ(|mλt |), we obtain bounds of the following form which hold uniformly for all (ζ, ζ ) ∈ ∆t × ∆t , NB < eλt ≤ NH , (and t ≤ −ε, for any given ε, ε > 0): ˜ (s),(a) (ζ, ζ ; t, λt )| ≤ Cˆ (B) Ψ(|mλt |), |DB (s),(a) (t, λt ) × H (ζ,ζ ,t) (B)

(4.14)

where Cˆ(ζ,ζ ,t) is given by Equation (A.22). Besides, for |mλt | > A(t), with A(t) suﬃciently large, and NB < eλt ≤ NH , one can satisfy the inequality |DB (s),(a)

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(t, λt ) − 1| < 12 (in view of (A.14)), and therefore the bound (4.14) can be advantageously replaced by ˜ (s),(a) (ζ, ζ ; t, λt )| ≤ 2Cˆ (B) Ψ(|mλt |). |H (ζ,ζ ,t)

(4.15)

Reggeon structure of the four-point function We shall now show that the previous properties of meromorphic continuation of ˜ (s) and H ˜ (a) imply the existence of a peculiar structure of the four-point function H H([k]) ≡ H (ζ,ζ ,t) (cos Θt ) as a function of t and cos Θt . This will be done by starting from the inversion formula (see Equation (4.89) in Theorem 5 of [1]), which allows one to reexpress H (ζ,ζ ,t) (cos Θt ) for all cos Θt in the cut-plane Π = ˜ s = H˜ (s) +H˜ (a) and H ˜u = C \ {[cosh v0 , +∞[ ∪ ] − ∞, − cosh v0 ]} in terms of H 2

˜ (s) −H ˜ (a) H . 2

In view of the bounds (3.27) on H and correspondingly of the bounds ˜ ˜ (a) in C(NH ) , formula (4.89) of [1] can be applied5 and yields: (3.57) on H (s) ,H + H (ζ,ζ ,t) (cos Θt ) =− −

1 2iωd 1 2iωd

NH +ε+i∞

NH +ε−i∞

(d)

(d)

(d)

(d)

˜ (s) (ζ,ζ ;t,λ) hd (λ)[Pλ (cos(Θt − εΘt π) + Pλ (cos(Θt )] dλ H 2sinπλ

NH +ε+i∞

˜ (a) (ζ,ζ ;t,λ) hd (λ)[Pλ (cos(Θt − εΘt π) − Pλ (cos(Θt )] dλ H 2sinπλ NH +ε−i∞ 1 ˜ (d) + (4.16) h (ζ, ζ , t) hd ()P (cos Θt ). ωd 0≤≤NH

(d)

Γ(λ+d−2) In the latter, hd (λ) = 2λ+d−2 (d−2)! × Γ(λ+1) , ε(Θt ) = sgn(eΘt ) and Pλ (cos Θt ) denotes the ﬁrst-kind Legendre function in dimension d, whose integral representation is given by Equation (4.64) of [1]. The absolute convergence of the integral at the right-hand side of Equation (4.16) is ensured for all cos Θt in Π (uniformly in cos Θt ) by appropriate exponential decrease properties of the integration kernels with respect to the variable ν = mλ. These properties result from the following bounds on the Legendre (d) functions Pλ (obtained, e.g., from formulae (II.85), (II.86) of [12] by a reﬁnement of the argument of Proposition II-12): h (N + iν)P (d) (cos Θ ) t d N +iν ≤ Cd (cos Θt )emax(N,−N −d+2)|mΘt| e−|ν|(π−| eΘt |) , sin π(N + iν) (4.17) 5 The choice of the line eλ = N H + ε (ε arbitrarily small) for the integration cycle in Equation (4.15) is more correct than the prescription (eλ = NH ) given in Equation (4.89) of (N ) [1], which necessitates uniform bounds in the closure of + H .

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where the function Cd (cos Θt ) is uniformly bounded at inﬁnity in the cut-plane C\] − ∞, −1]. ˜ (s) and H ˜ (a) By now taking advantage of the meromorphic continuation of H in the strip NB < eλt ≤ NH together with the uniform bound (4.15) on these functions when |mλt | tends to inﬁnity, one can shift the integration line in formula (4.16) from its initial position to the line eλt = NB , provided one extracts residue terms corresponding to all the poles λt = λj (s) (t), λt = λj (a) (t) contained in the strip. Assuming there exists only one simple pole at λt = λ(s) (t) real for ˜ (s) and one simple pole at λt = λ(a) (t) real for H ˜ (a) , the expression (4.16) of H H (ζ,ζ ,t) (cos Θt ) can be replaced by: H (ζ,ζ ,t) (cos Θt ) π ˜ (s) (ζ, ζ ; t, λ(s) (t)) × · · · = − Res H ωd (d)

(d)

hd (λ(s) (t)) [Pλ(s) (t) (cos(Θt − εΘt π) + Pλ(s) (t) (cos(Θt )] 2 sin πλ(s) (t) −

π ˜ (a) (ζ, ζ ; t, λ(a) (t)) × · · · Res H ωd (d)

(d)

hd (λ(a) (t)) [Pλ(a) (t) (cos(Θt − εΘt π) − Pλ(a) (t) (cos(Θt )] 2 sin πλ(a) (t) − −

1 2iωd 1 2iωd

NB +i∞

NB −i∞ NB +i∞

NB −i∞

(d)

(d)

(d)

(d)

˜ (s) (ζ, ζ ; t, λ) hd (λ)[Pλ (cos(Θt − εΘt π) + Pλ (cos(Θt )] dλ H 2 sin πλ ˜ (a) (ζ, ζ ; t, λ) hd (λ)[Pλ (cos(Θt − εΘt π) − Pλ (cos(Θt )] dλ H 2 sin πλ 1 ˜ (ζ, ζ , t) hd ()P (d) (cos Θt ). + (4.18) h ωd 0≤≤NB

˜ (s),(a) are of the form given by Equations (4.2)– In the latter, the functions Res H (4.6). The important feature of Equation (4.18) concerns the asymptotic behaviour in the cos Θt -plane of the various terms at its right-hand side. In view of the dependence on mΘt of the bound (4.17), it follows that the two integrals as well as the last term at the right-hand side of Equation (4.18) are functions of cos Θt which are bounded at inﬁnity by | cos Θt |NB . The ﬁrst two terms at the right-hand side of Equation (4.18) then appear as the leading terms giving the asymptotic behaviour at inﬁnity of H (ζ,ζ ,t) (cos Θt ). This asymptotic behaviour exhibits rates (d)

(d)

of increase which are those of Pλ(s) (± cos Θt ) and Pλ(a) (± cos Θt ), namely respec(s)

(a)

tively | cos θt |λ (t) and | cos θt |λ (t) , with NB < λ(s),(a) (t) ≤ NH . These leading terms in Equation (4.18) will be called Reggeon terms. To summarize, we can state:

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Theorem. Let H([k]) = H (ζ,ζ ,t) (cos Θt ) be the analytic four-point function of two (mutually) local Hermitian scalar fields whose Bethe-Salpeter structure in the tchannel is encoded in a BS-kernel B([k]) = B (ζ,ζ ,t) (cos Θt ). Let us assume that for some range of negative values of t (or possibly the whole half-line t < 0), B satisfies bounds of the form (4.7), (4.8), (4.9). Then in a generic situation involving only one Reggeon term of each symmetry type, H (ζ,ζ ,t) admits a representation of the following form, valid for all (ζ, ζ ) in ∆t × ∆t and cos Θt in Π: H (ζ,ζ ,t) (cos Θt ) (s)

=−

π ϕ(s) (ζ,t) × ϕ(s) (ζ ,t) hd (λ ωd β (s) (t)

(d)

(d)

(t)) [Pλ(s) (t) (cos(Θt − εΘt π) + Pλ(s) (t) (cos(Θt )] 2sinπλ(s) (t)

(a)

π ϕ(a) (ζ,t) × ϕ(a) (ζ ,t) hd (λ − ωd β (a) (t)

(d)

(d)

(t)) [Pλ(a) (t) (cos(Θt − εΘt π) − Pλ(a) (t) (cos(Θt )] 2sinπλ(a) (t)

+H (ζ,ζ ,t) (cos Θt ),

(4.19)

where the residual term H (ζ,ζ ,t) (cos Θt ) is bounded at infinity by Cst| cos Θt |NB in Π, while the first two terms exhibit leading behaviours governed respectively by (s) (a) | cos Θt |λ (t) and | cos Θt |λ (t) , with λ(s) (t) > NB and λ(a) (t) > NB .

5 Conclusion In this paper, we have established the following results: a) the general Bethe-Salpeter structure of four-point functions of scalar ﬁelds in a given t-channel has been shown to be diagonalized in the corresponding complex angular momentum variable λt for all negative values of t and of the squared-mass variables corresponding to Euclidean conﬁgurations in complex momentum space. More speciﬁcally, Carlsonian interpolations of the Euclidean Bethe-Salpeter equations for even and odd partial waves have been constructed in a half-plane of the form Re λt > N , starting from the corresponding Bethe-Salpeter equations for the s and u-channel absorptive parts. b) any information on B leading to continue analytically these interpolations in some region of the left-hand half-plane Re λt ≤ N results in the “potential generation” of Regge poles, equipped with a factorized residue structure involving Euclidean three-point functions. c) the existence of a meromorphic continuation of these interpolations in a strip of the form Nb < Re λt ≤ N (with the relevant factorization property of the residues of the poles) has been shown to hold if and only if the Bethe-Salpeter kernel satisﬁes bounds of the form | cos Θt |NB in the complex plane of the oﬀ-shell scattering angle Θt , which are better than those on the complete four-point function H, namely iﬀ NB < NH .

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Here our analysis has been done for negative values of t by relying on the analyticity domains resulting from the general principles of quantum ﬁeld theory. It is expected to hold similarly for t in some positive interval ]0, t0 [, and then to produce the desired interpolation of possible bound states in the t-channel, after one has justiﬁed the necessary properties of analytic continuation of H (in the spirit of [15]). In the general arguments of our axiomatic approach, the unspeciﬁed number NH was introduced (see [1]) as a consequence of the temperateness assumption in the ﬁeld-theoretical framework. In a speciﬁc model of ﬁeld theory, the actual asymptotic behaviour of H in the cos Θt -plane would be governed by a precise function NH = NH (t). Then the crucial question of the validity of Reggeon terms with asymptotic dominance in the four-point function H of a general quantum ﬁeld theory appears to rely basically on the knowledge of the exponent NB = NB (t) which governs the asymptotic behaviour of B: such Reggeon terms may appear for all values of t such that NB (t) < NH (t). More questionable is the justiﬁcation of the inequality NB (t) < NH (t), which would require further investigations in the present ﬁeld-theoretical framework. In the absence of global information on general (regularized or renormalized) versions of the two-particle irreducible kernel B, one can think of exploiting various contributions to B deﬁned in terms of generalized Feynman convolutions enjoying the graph property of “two-particle irreducibility”. The simplest ones are of course those associated with “particle-exchange” graphs, which are interpreted as relativistic Yukawa-type potentials; they do satisfy the previous inequality (with NB = −1) and have been widely exploited in the literature, since they directly transfer to particle physics the initial procedure of Regge pole production in potential theory. However, much more complicated contributions to B (of higher-order in the coupling constants) are imposed by the Feynman convolution structure of ﬁeld theory; typical examples of those have been studied by Mandelstam [18] and others (see, e.g., [19] and references therein). The treatment of such contributions which these authors have given by using the methods of (maximally analytic) S-matrix theory, has led them to introduce “Regge cuts” in the λt -plane, whose eﬀect in some situations seems to wipe out the dominant asymptotic behaviour of the potential-type Regge poles introduced at ﬁrst. Although no mechanism of production of Regge cuts appears in our general ﬁeld-theoretical framework, it can be seen in our approach that the contributions considered by the previous authors cannot satisfy the inequality NB (t) < NH (t) for negative t, although they hopefully do it for positive t. Of course, such results on partial contributions to B do not allow one to conclude the non-validity of the Bethe-Salpeter generation of Reggeon leading terms at negative t; more global (non-perturbative) information on the ﬁeld models would be needed for investigating the relevant asymptotic properties of B.

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Appendix

We give a uniﬁed treatment of the various versions of BS-equation presented in Section 3, in which the integration space either contains or is identical with the space of mass variables {ζ = (ζ1 , ζ2 ) ∈ ∆t } equivalent to {(ρ, w); ρ ≥ 0, w ∈ R} (see Equations (3.3)–(3.6)). One is led to use the N /D-resolution of Fredholm resolvent equations in a complex analysis framework, in the following way. F (t, ; ρ, w, ρ , w , •, • ) will denote a t-dependent kernel acting on the integration space I = {(ρ, w, •); ρ ≥ 0, w ∈ R, (• ∈ Γ)} (((ρ, w, •), (ρ , w , • )) ∈ I × I). The notation • (• ) stands for possible additional variables z (z ) which vary on (c) a (d − 1)-cycle with compact support Γ of Xd−1 : this is the case for the kernel H[t; ρ, w, ρ , w ; −z · z ] ≡ H (ζ,ζ ,t) (−z · z ) (resp. B[· · · ] ≡ B (ζ,ζ ,t) (· · · )) of Section 3.1 (see Equations (3.7), (3.8)). Alternatively, the kernel may also depend on the (N ) complex parameter λt varying in a half-plane C+ , here represented by : this is ˜ (s),(a) [t, λt ; ρ, w, ρ , w ] ≡ H ˜ (s),(a) (ζ, ζ ; t, λt ) (or B ˜ (s),(a) ) the case for the kernels H (see Equation (3.54)). Our treatment of the BS-equation is valid for the whole range t < 0 and allows one to study the regularity of the solution for t tending to −∞ rather than for t tending to zero. Indications for the treatment of the latter are given at the end. F is assumed to satisfy a bound of the following form: ˆ

|F (t, ; ρ, w, ρ , w , •, • )| ≤ C( ,•,• ) (1 + |t|)N (1 + ρ)N (1 + |w|)N (1 + ρ )N (1 + |w |)N (A.1) and one considers the following Fredholm resolvent equation: 1

RF (t, ; ρ, w, ρ , w , •, • ; α) = F (t, ; ρ, w, ρ , w , •, • ) + iα|t| 2 × · · · I

(A.2)

F (t, ; ρ, w, ρ , w , •, • )RF (t, ; ρ , w , ρ , w , • , • ; α) G[t; ρ , w ]ρ

d−1

dρ dw (d• );

in the latter, the weight G[t; ρ, w] is assumed to satisfy the following uniform bound: |G[t; ρ, w]| ≤ c2r |t|−r (1 + ρ)−2r (1 + |w|)−2r , ˆ + 1 and c2r < 1. with r > max N + d2 , N 2

(A.3)

One checks that the relevant bounds (3.9), (3.27), (3.28) on H and (4.1)–(4.3) ˆ = 3N for all t < 0 if on B imply bounds of the form (A.1) with respectively N 2 ˆ = N for all t < −1 if N < 0 (case N = NB ). Similarly the bound N ≥ 0 and N 2 (3.12) on G implies (A.3) for all t < 0 (with cr = 4r c).

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According to the standard Fredholm argument, one introduces the following entire series in α: NF (t, ; ρ, w, ρ , w , •, • ; α) =

∞

αp , p!

N (p) (t, ; ρ, w, ρ , w , •, • )

p=0

DF (t, ; α) =

∞

D(p) (t, )

p=0

where N (0) = F,

D(0) = 1,

(A.4)

αp , p!

(A.5)

and for p ≥ 1 : p

N (p) (t, ; ρ, w, ρ , w , •, • ) = (−i)p |t| 2 × · · · F (t, ; ρ, w, ρ , w , •, • ) F (t, ; ρi , wi , ρ , w , •i , • ) Ip

F (t, ; ρ, w, ρj , wj , •, •j ) × ··· F (t, ; ρi , wi , ρj , wj , •i , •j ) 1≤i,j≤p

G[t; ρ1 , w1 ] · · · G[t; ρp , wp ]ρ1 d−1 dρ1 dw1 (d•1 ) · · · ρp d−1 dρp dwp (d•p ), p (p) p 2 |F (t, ; ρi , wi , ρj , wj , •i , •j )|1≤i,j≤p × · · · D (t, ) = (−i) |t|

(A.6)

Ip

G[t; ρ1 , w1 ] · · · G[t; ρp , wp ]ρ1 d−1 dρ1 dw1 (d•1 ) · · · ρp d−1 dρp dwp (d•p ).

(A.7)

In the latter, we have used a shortened notation for determinants of order p + 1 (resp. p) under the integral at the right-hand side of (A.6) (resp. (A.7)). The convergence of the integrals at the right-hand side of Equations (A.6), (A.7) is ensured by the bounds (A.1), (A.3); in fact, by introducing the functions F (t, ; ρ, w, ρ , w , •, • ) =

F (t, ; ρ, w, ρ , w , •, • ) C( ,•,• ) (1 + |t|)Nˆ (1 + ρ)N (1 + |w|)N (1 + ρ )N (1 + |w |)N

,

which are bounded in modulus by 1, one can rewrite the determinants under the integrals of (A.6), (A.7) respectively as follows: ˆ

p+1 N (p+1) C( ,•,• (1 + ρ)N (1 + |w|)N (1 + ρ )N (1 + |w |)N × · · · ) (1 + |t|)

 

 2N

(1 + ρi )

1≤i≤p

 ˆ

p Np  C( ,•,• ) (1 + |t|)

1≤i≤p

F ·· F ·j F i· F ij

2N 

(1 + |wi |)

,

(A.8)

1≤i,j≤p

 (1 + ρi )2N (1 + |wi |)2N  |F ij |1≤i,j≤p .

(A.9)

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In the latter the determinants are similar to those of Equations (A.6), (A.7), with F replaced by F ; therefore, in view of Hadamard’s majorization, they are p+1 p respectively bounded in modulus by (p + 1) 2 and p 2 . It then follows from (A.8), (A.9) together with the bound (A.3) on G that the integrals in (A.6), (A.7) are absolutely convergent and deﬁne functions N (p) and D(p) satisfying the following bounds for all p ≥ 0 : ˆ

|N (p) (t, ; ρ, w, ρ , w , •, • )| ≤ C( ,•,• ) (1 + |t|)N (1 + ρ)N (1 + |w|)N (1 + ρ )N × · · · !

· · · (1 + |w |) (p + 1) N

p+1 2

p M( ,Γ)

p |D(p) (t, )| ≤ p 2 M( ,Γ) p

ˆ

(1 + |t|)N

"p ,

1

|t|r− 2 ! " ˆ p (1 + |t|)N 1

|t|r− 2

.

(A.10)

(A.11)

In the latter, M( ,Γ) denotes a constant (independent of t), which is expressed as follows: ∞ ∞ M( ,Γ) = c2r C( ,Γ) × (AreaΓ) (1 + ρ)2(N −r) ρd−1 dρ (1 + |w|)2(N −r) dw, 0

−∞

(A.12)

where we have put C( ,Γ) = sup(•,• )∈Γ×Γ C( ,•,• ) . It now follows from the bounds (A.10), (A.11), that the entire series (A.4), (A.5) are majorized in the whole complex α-plane by convergent series; NF and DF are thus deﬁned as entire functions of α satisfying the following bounds: ˆ

|NF (t, ; ρ, w, ρ , w , •, • ; α)| ≤ C( ,•,• ) (1 + |t|)N (1 + ρ)N (1 + |w|)N (1 + ρ )N × · · · # · · · (1 + |w |)N Φ

ˆ

|α|M( ,Γ)

(1 + |t|)N

ˆ

|DF (t, ; α) − 1| ≤ Φ |α|M( ,Γ)

Φ(z) =

∞ zp p=1

,

1

|t|r− 2

#

where

$

p!

(1 + |t|)N 1

|t|r− 2

(A.13)

$ ,

p

p2 .

(A.14)

(A.15)

The solution of the Fredholm resolvent equation (A.2) is then obtained as a meromorphic function of α (in C), namely: RF (t, ; ρ, w, ρ , w , •, • ; α) =

NF (t, ; ρ, w, ρ , w , •, • ; α) . DF (t, ; α)

(A.16)

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In the applications of this result to the solutions of BS-type equations (of the form H = B + B ◦ H), the Fredholm parameter α is ﬁxed at either value −1 or +1, according to whether H or B is considered as given, so that one has: B=

NH|α=−1 = RH|α=−1 , DH|α=−1

H=

NB|α=1 = RB|α=1 . DB|α=1

(A.17)

In all cases, it is therefore important to justify the fact that the functions DH|α=−1 (t, ) and DB|α=1 (t, ) are not identically equal to zero, which is done below by considering large values of |t|: in this connection, the crucial assumption for solving BS-type equations is the choice of a “regularized double-propagator” G[t; ρ, w] satisfying the bounds (A.3). a) BS-equation in the mass variables and complex angular variables We apply the previous analysis to the case when F ≡ F (t; ρ, w, ρ , w , z, z ), with (c) t < 0 and z, z varying on a (d − 1)-cycle Γ of Xd−1 . F is assumed to satisfy a bound of the general form (A.1), with a constant C( ,•,• ) ≡ C(z,z ) , appropriately chosen (according to (3.27)) as follows: C(z,z ) = eN |mΘt | × | sin eΘt |−N

(A.18)

(with cos Θt = −z · z ). The support of Γ is equal to the Euclidean sphere Sd−1 in its “initial situa(c) tion” Γ0 (see Section 3.1); it is then distorted in Xd−1 (e.g., in the way described in the study of ∗(c) -convolution of perikernels in [11]), being always submitted (c) (c) to the condition that Γ × Γ belongs to Xd−1 × Xd−1 \ (Σs (ζ, ζ , t) ∪ Σu (ζ, ζ , t)) for all (ζ, ζ , t) (or (t, ρ, w, ρ , w )). This condition takes into account the fact that F is (for each (t, ρ, w, ρ , w )) holomorphic in the domain of a perikernel on (c) Xd−1 and expresses the requirement that in all the integrals (A.6) the integration space in (z1 , . . . , zp ) should belong (for each (t, ρ, w, ρ , w )) to the holomorphy domain of the integrand. The use of such integrals (A.6), with integration cycles “ﬂoating in complex space” implies the fact (see [13], Proposition 1) that NF (t; ρ, w, ρ , w , z, z ; α) is for each ﬁxed values of (t, ρ, w, ρ , w , α) a holomorphic function of (z, z ) in the domain covered by the distortion of Γ × Γ: according to (c) (c) [11], this is the full domain D(per) ≡ {(z, z ) ∈ Xd−1 × Xd−1 ; |z · z | ∈ / [+1, +∞[} of a general perikernel. We now notice that since the integrals D(p) (t) (see Equation (A.7)) do not contain the external variables z, z , it is not worthwhile to distort the integration cycle Γ in the latter. Γ can be kept equal to Γ0 , so that the expression (A.12) yields a ﬁxed numerical constant M(Γ0 ) ≡ M0 , relevant for the bounds (A.11) on the functions D(p) ( (AreaΓ0 ) being equal to the area of the sphere Sd−1 ). It then follows from the bound (A.14) that one has: # $ ˆ (1 + |t|)N ; (A.19) |DF |α=±1 (t) − 1| ≤ Φ M0 1 |t|r− 2

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since Φ is an entire function which vanishes at the origin (see Equation (A.15)), ˆ −r+ 1 < 0, there exists a value t = t1 < 0 such that for all t ∈]−∞, t1 ] and since N 2 the function DF |α=±1 (t) does not vanish. (Note that t1 can be made arbitrarily close to zero if the constant cr in (A.3) can be taken arbitrarily small.) This shows: Proposition A1

The solution

RF |α=±1 (t; ρ, w, ρ , w , z, z ) =

NF |α=±1 (t; ρ, w, ρ , w , z, z ) DF |α=±1 (t)

of the BS-equation with given kernel F (t; ρ, w, ρ , w , z, z ) and weight G[t; ρ, w] satisfying the bounds (A.1), (A.3), is well defined for t ≤ t1 , ρ, ρ ≥ 0, w, w real, as a holomorphic function of z, z in the domain D(per) . Under the assumption (A.18) on F , it satisfies a bound of the following form: ˆ

|DF |α=±1 (t) × RF |α=±1 (t; ρ, w, ρ , w , z, z )| ≤ C(z,z ) (1 + |t|)N (1 + ρ)N (1 + |w|)N # N

× · · · (1 + ρ ) (1 + |w |) Φ N

ˆ

M(Γ)

(1 + |t|)N 1

|t|r− 2

$ .

(A.20)

If one now makes use of the assumption that F (t; ρ, w, ρ , w , z, z ) ≡ F ([k]) satisﬁes the axiomatic analyticity properties of a four-point function [16], one obtains as a by-product of [8, 9] the following results: i) There exists a complex neighborhood V of the half-line {t; t < 0} in which DF |α=±1 (t) admits an analytic continuation, with DF |α=±1 (t) = 0 for et < t1 (for some t1 < 0). ii) There exists a complex neighborhood of the following set: {(t,ρ,w,ρ ,w ,z,z ); t ∈ V, ρ ≥ 0, w ∈ R, ρ ≥ 0, w ∈ R, (z, z ) ∈ Sd−1 × Sd−1 }, in which NF |α=±1 (t; ρ, w, ρ , w , z, z ) admits an analytic continuation. These results are obtained (according to [8, 9]) by performing a small distortion of the Euclidean integration contour Ed+1 (here reprsented by the set {(ρ, w, z); ρ ≥ 0, w ∈ R, z ∈ Sd−1 }) inside the axiomatic analyticity domain of F and by using the corresponding properties of analytic continuation of the functions NF and DF . By now putting together the latter results and those of Proposition A1 and by applying a standard technique of analytic completion (see, e.g., Appendix A of [1]), one obtains: Proposition A2 The function NF |α=±1 (resp. RF |α=±1 ) admits an analytic (resp. meromorphic) continuation in a complex neighborhood of the set {(t, ρ, w, ρ , w , z, z ); t ∈ V, ρ ≥ 0, w ∈ R, ρ ≥ 0, w ∈ R, (z, z ) ∈ D(per) }.

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b) BS-equation with dependence on the complex angular momentum variable As a second application, we consider the case when F ≡ F (t, λt ; ρ, w, ρ , w ), (N ) F being holomorphic with respect to λt in C+ for all t < 0, ρ ≥ 0, ρ ≥ 0, w, w real. By exploiting the bounds (A13) and (A14) similarly as in a) and by taking into account the dependence C( ,•,• ) = C(λt ) of the constant C( ,•,• ) introduced in (A1), we obtain: Proposition A3 Let F satisfy the following uniform bound: ˆ

|F (t, λt ; ρ, w, ρ , w )| ≤ C(λt )(1+|t|)N (1+ρ)N (1+|w|)N (1+ρ )N (1+|w |)N (A.21) for t < tˆ (with tˆ < 0). N

(t,λ ;ρ,w,ρ ,w )

t of the Then the solution RF |α=±1 (t, λt ; ρ, w, ρ , w ) = F |α=±1 DF |α=±1 (t,λt ) BS-equation with given kernel F (t, λt ; ρ, w, ρ , w ) and with weight G[t; ρ, w] satisfying the bound (A3), is well defined for all t, with t < tˆ, as a meromorphic (N ) function of λt in C+ , satisfying a uniform majorization of the following form:

# C(λt )Φ

|DF |α=±1 (t, λt ) × RF |α=±1 (t, λt ; ρ, w, ρ , w )| ≤ · · · $ ˆ (1 + |t|)N ˆ M 0 C(λt ) (1+|t|)N [(1+ρ)(1+|w|)(1+ρ)(1+|w |)]N (A.22) 1 r− 2 |t|

In the general case when C(λt ) is a locally bounded function, there exists a value t = tC < 0 depending on C(λt ) such that for each t < tC , RF |α=±1 is holomorphic in a region of the form {λt ; eλt > N, |λt − N | < λC (t)}, with λC (t) increasing with |t| and tending to infinity for t → −∞. Moreover, the following specification holds: Let C(λt ) = Ψ(|mλt |), where Ψ denotes a bounded positive function on [0, ∞[, tending to zero at infinity. Then there exists a value t = tΨ ≤ tˆ such that for each t < tΨ , RF |α=±1 is holomorphic (N ) in C+ . Moreover, for tΨ ≤ t < tˆ, RF |α=±1 is holomorphic in a region of the form {λt ; eλt > N, |mλt | > νΨ (t)}, with νΨ (t) decreasing with |t| and such that νΨ (tΨ ) = 0. Remark. The treatment of the BS-equation (3.26) for partial waves is contained in the latter statement for λt = > N with f˜ (t; ρ, w, ρ , w ) = F (t, ; ρ, w, ρ , w )). Treatment near t = 0: In a range of the form tˆ ≤ t ≤ 0, it is preferable to introduce 1 1 the variables W = |t| 2 w, W = |t| 2 w , instead of w, w in Equation (A.2), and to substitute to (A.1) and (A.3) the following bounds on the corresponding functions ˆ which are consequences (now for tˆ ≤ t ≤ 0) of the relevant bounds (3.9), Fˆ and G, (3.27), (3.28) on H, (4.1)–(4.3) on B and (3.12) on G: |Fˆ (t, ; ρ, W, ρ , W , •, • )| ≤ C( ,•,• ) (1 + |t|) 2 [(1 + ρ)(1 + |W |)(1 + ρ)(1 + |W |)]N (A.23) ˆ ρ, W ]| ≤ c2r 4r (1 + ρ)−2r (1 + |W |)−r , |G[t; (A.24) N

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By repeating all the previous computations with these new variables, one would obtain that for r > max(N + d2 , 2N + 1), the Fredholm formulae are still applicable in the closed interval tˆ ≤ t ≤ 0, the right-hand side of (A.14) being now replaced N

by Φ |α|M( ,Γ) (1 + |t|) 2 . In particular, the analysis of the possible localization of the poles in the λt -plane given in Proposition A3 can be extended to all values of t, with t ≤ 0.

References [1] J. Bros and G.A. Viano, “Complex angular momentum in general quantum ﬁeld theory”, Annales Henri Poincar´e 1 (2000) p. 101–172. [2] R.F. Streater and A.S. Wightman, “PCT, Spin and Statistics and all that”, W.A. Benjamin, New York, 1964. [3] J. Bros and G.A. Viano, “Complex angular momentum in axiomatic quantum ﬁeld theory”, in Rigorous methods in particle physics, S. Ciulli, F. Scheck, W. Thirring Eds., Springer tracts in Mod. Phys. 119, 53–76 (1990). [4] T. Regge, Nuovo Cimento 14, 951 (1959) and Nuovo Cimento 18, 947 (1960). [5] R.P. Boas, “Entire Functions”, Academic Press, New York, 1954. [6] M. Froissart, “Asymptotic behaviour and subtractions in the Mandelstam representation”, Phys. Rev. 23, 1053–1057 (1961). [7] V.N. Gribov, “Partial waves with complex angular momenta and the asymptotic behaviour of the scattering amplitude”, J. Exp. Theor Phys. 14, 1395 (1962). [8] J. Bros, “Some analyticity properties implied by the two-particle structure of Green’s functions in general quantum ﬁeld theory”, in Analytic methods in mathematical physics, R.P. Gilbert, R.G. Newton eds, Gordon and Breach, New York (1970) p. 85–133. [9] J. Bros and M. Lassalle, Commun. Math. Phys. 54, 33 (1977). [10] J. Bros and M. Lassalle, Ann. Inst. H. Poincar´e 27, 279 (1978). [11] J. Bros and G.A. Viano, “Connection between the algebra of kernels on the sphere and the Volterra algebra on the one-sheeted hyperboloid: holomorphic perikernels”, Bull. Soc. Math. France 120, 169–225 (1992). [12] J. Bros and G.A. Viano, “Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint”, Forum Math. I 8, 621–658 (1996), II 8, 659–722 (1996), III 9, 165–191 (1997).

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[13] J. Bros and D. Pesenti, J. Math. pures et appl. 58, 375 (1980). [14] J. Faraut and G.A. Viano, “Volterra algebra and the Bethe-Salpeter equation” J. Math. Phys. 27, 840–848 (1986). [15] A. Martin, Nuovo Cimento 42, 930 (1965) and 44, 1219 (1966). [16] J. Bros, H. Epstein and V. Glaser, “Some rigorous analyticity properties of the four-point function in momentum space”, Nuovo Cimento 31, 1265–1302 (1964). [17] J. Bros and D. Pesenti, “Fredholm resolvents of meromorphic kernels with complex parameters: a Landau singularity and the associated equations of type U in a non-holonomic case” J. Math. pures et appl. 62, 215–252 (1983). [18] S. Mandelstam, Nuovo Cimento 30, 1127 and 1148 (1963). [19] P.D.B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge University Press, London, (1977). J. Bros Service de Physique Th´eorique CEA-Saclay F-91191 Gif-sur-Yvette Cedex France email: [email protected] G.A. Viano Istituto Nazionale di Fisica Nucleare Sezione di Genova Dipartimento di Fisica dell’Universit` a di Genova I-16146 Genova Italy email: [email protected] Communicated by Klaus Fredenhagen submitted 13/12/01, accepted 10/12/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 127 – 136 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/010127-10 DOI 10.1007/s00023-003-0124-x

Annales Henri Poincar´ e

Interface Models and Boundary Conditions Salvador Miracle-Sole and Jean Ruiz Abstract. We study certain aspects of the thermodynamic formalism of interface models. Under appropriated conditions, we prove a conjecture proposed by Spohn [1] few years ago, and as a consequence the validity of exact results on the equilibrium shape associated to certain Solid-On-Solid models.

Interface models have been introduced as simple models for a microscopic description of the phase separation surface between coexisting phases. In these models the interface is represented as the graph of a function deﬁned on a reference plane. At each site i of a ﬁnite square lattice Λ ⊂ L = Z2 an integer variable φ(i) is assigned which represents the height of the interface at this site. A statistical mechanical model is obtained by deﬁning the energy of each conﬁguration φ = {φ(i)}. The standard examples are of the form U (φ(i) − φ(j)) (1) HΛ (φ) = |i−j|=1

with U (r) ≥ 0. Thus, the case U (r) = r2 corresponds to the discrete Gaussian model, while for U (r) = |r| one obtains the solid-on-solid (SOS) model. Restricted SOS models, in which U (r) = +∞, except for a ﬁnite number of values of r, can also be considered. We assume e−βU(k) = K(β) < ∞ (2) k∈Z

if β ≥ 0. The weight of a given conﬁguration, at the inverse temperature β, is proportional to the Boltzmann factor exp − βHΛ (φ) . These models provide an approximate description of the microscopic interface separating two phases at equilibrium, such as the positively and negatively magnetized phases of the three-dimensional Ising model. Actually the SOS model may be obtained as the limit of the anisotropic Ising model with nearest neighbour interactions, when we let the interaction parameter, in the vertical direction, tend to inﬁnity. Here the Ising model is deﬁned inside a box with boundary conditions which enforce an interface with a given average slope between the positive and negative phases. Moreover, if all interaction parameters tend to inﬁnity or, equivalently, if β tends to inﬁnity, the interface is then described by a restricted SOS model. The statistical mechanics of interface models is, naturally, rather diﬀerent from that of bounded spin systems. Indeed, the thermodynamic free energy of

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such models represents an interfacial free energy per unit projected area. It thus depends on the chosen boundary condition. Moreover, the interface can be rough, as it is the case in many situations, implying that the Gibbs states do not exit. For such models, a detailed analysis related to these two properties is still lacking, as it is pointed out by Spohn in Ref. [1] (see Appendix B). Concerning the properties of Gibbs states, he proposed a study of the equilibrium measure in terms of the heights diﬀerences satisfying a local constraint. He conjectured in particular that for a given ﬁxed boundary condition, the corresponding Gibbs state is unique (see also Ref. [2]). Concerning the free energy, one is naturally led to introduce a conjugate Gibbs ensemble with respect to the interface boundaries. As it is conjectured in ref. [1], the free energy associated to this conjugate ensemble should coincide with the Legendre transform of the interfacial free energy. It is the aim of this article to examine certain aspects of the corresponding thermodynamic formalism. To this end, we shall ﬁrst introduce the appropriated deﬁnitions for the free energies associated to the diﬀerent Gibbs ensembles. For a class of appropriated conditions, we shall then prove the validity of this last mentioned conjecture. We take Λ = Λ(N1 , N2 ) as a rectangular box of sides 2N1 and 2N2 centered at the origin, i.e., as the set of sites i = (i1 , i2 ) ∈ L such that |i1 | ≤ N1 and |i2 | ≤ N2 . Its area is denoted by |Λ| = 4N1 N2 . The boundary ∂Λ is the set of sites i ∈ Λ such that |i1 | = N1 or |i2 | = N2 . It is understood in equation (1) that the bond {i, j} belongs to Λ and can intersect or be included in ∂Λ. In order to deﬁne the free energy of the macroscopic interface corresponding to a given average slope p = (p1 , p2 ), we introduce the Gibbs ensemble E clos (p, Λ) which consists of all conﬁgurations, in the box Λ, satisfying the (“closed”) boundary conditions (3) φ(i) = φ¯p (i) = [p · i], i ∈ ∂Λ where p · i = p1 i1 + p2 i2 and [ · ] represents the integer part. The partition function is exp − βHΛ (φ) (4) Z clos (p, Λ) = φ∈E clos (p,Λ)

where the sum runs over all conﬁgurations in Λ satisfying conditions (3). The associated free energy per unit projected area is deﬁned as f clos (p, Λ) = −

1 ln Z clos (p, Λ). β |Λ|

(5)

Theorem 1 The thermodynamic limit of (5), which deﬁnes the projected surface tension f (p), exists. It is a convex and Lipschitz continuous function of p in the interior of the eﬀective domain of f . The validity of the above statements is known. See, for instance, Ref. [3], for a proof of them in a more general setting. The eﬀective domain of a convex function f is the set dom f = {p : f (p) < ∞}.

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The surface tension, which represents the interfacial free energy per unit area of the mean interface, is τ (p) = (1 + p21 + p22 )−1/2 f (p). The convexity of f is equivalent to the fact that the surface tension τ satisﬁes a stability condition called the pyramidal inequality (see Refs. [3], [4]). The boundary conditions considered above can be interpreted as a “canonical” constraint. We are going to discuss the conjugate Gibbs ensemble, which can be viewed as a “grand canonical” ensemble with respect to the interface boundaries. It is useful to introduce the variables ξ() = ξ(i, j) = φ(i) − φ(j) associated to the oriented bonds of the lattice = {i, j}, |i − j| = 1. Note that − = {j, i} and ξ() = −ξ(−). The admissible conﬁgurations ξ, being the gradient of φ, satisfy ∈λ ξ() = 0 for any closed loop λ in Λ. Equivalently, ξ(i, j) + ξ(j, k) + ξ(k, ) + ξ(, i) = 0

(6)

for every plaquette (elementary square loop) P = {i, j, k, } in Λ. Periodic boundary conditions in the box Λ are deﬁned with respect to the ξ variables. Namely, for all i1 = −N1 , . . . , N1 − 1, i2 = −N2 , . . . , N2 − 1, it is assumed that φ(N1 , i2 + 1) − φ(N1 , i2 ) = φ(i1 + 1, N2 ) − φ(i1 , N2 ) =

φ(−N1 , i2 + 1) − φ(−N1 , i2 ), φ(i1 + 1, −N2 ) − φ(i1 , −N2 ).

We introduce the boundary terms S1 (φ) = ξ() = −N1 ≤i2 ≤N1

∈1 (Λ)

S2 (φ)

=

ξ() =

∈2 (Λ)

(7)

φ(N1 , i2 ) − φ(−N1 , i2 ) φ(i1 , N2 ) − φ(i1 , −N2 )

(8)

−N2 ≤i1 ≤N2

where 1 (Λ) and 2 (Λ) are the sets of all bonds in Λ parallel to the i1 and to the i2 axis, respectively, oriented according to increasing coordinates. The grand canonical prescription, which is convenient to consider, consists in adding to the energy a term of the form (9) x1 S1 (φ) + x2 S2 (φ) where x = (x1 , x2 ) ∈ R2 represent the slope chemical potentials. The associated partition function and free energy are Ξfree (x, Λ) = exp − βHΛ (φ) + x1 S1 (φ) + x2 S2 (φ) (10) φ,φ(0)=0

ϕfree (x, Λ)

= −

1 ln Ξfree (x, Λ) β |Λ|

(11)

in the case of free boundary conditions. To break the translation symmetry, we pinned the height φ(0) at zero.

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The periodic partition function is deﬁned as the sum in (10) with the constraint (7) on the conﬁgurations. This function and the corresponding free energy are denoted (12) Ξper (x, Λ), ϕper (x). From the deﬁnitions it can be proved that the thermodynamic limit of (11) exists. We shall not discuss this point here and analyze instead the problem of the equivalence of the conjugate Gibbs ensembles described by Z and Ξ. Notice that (10) and (11) are not deﬁned as the equivalent ensemble of (12) with closed boundary conditions. We introduce the set E per (p, Λ), of conﬁgurations in Λ which satisfy the boundary conditions φ(−N1 , i2 ) − φ(N1 , i2 ) = [2N1 p1 ],

φ(i1 , −N2 ) − φ(i1 , N2 ) = [2N2 p2 ]

(13)

for all i1 = −N1 + 1, . . . , N1 , i2 = −N2 + 1, . . . , N2 (and also φ(0) = 0), contains the set (3) used in the deﬁnition of the surface tension, and describes also a set of interfaces with average slope p. Under periodic boundary conditions, condition (13) can equivalently be written as S1 (φ) = [|Λ|p1 ],

S2 (φ) = [|Λ|p2 ].

(14)

In this case, (13) is satisﬁed as soon as it is satisﬁed for some i1 and some i2 . This set of conﬁgurations, satisfying conditions (7) and (14), deﬁnes the canonical Gibbs ensemble which corresponds to the grand canonical ensemble described above. We denote by exp − βHΛ (φ) (15) Z per (p, Λ) = φ∈E per (p,Λ),φ(0)=0

f per (p, Λ) = −

1 ln Z per (p, Λ) β|Λ|

(16)

the associated “canonical” partition function and free energy. Our purpose is now to prove that the two partition functions Z per and Z clos , with periodic and closed boundary conditions, deﬁne the same free energy. Theorem 2 The thermodynamic limit of (16) exists and coincides with the projected surface tension lim f per p, Λ(N1 , N2 ) = f (p). (17) N1 ,N2 →∞

This theorem is proved in the Appendix. In the next theorem we study the grand canonical ensemble (with respect to the interface boundaries), deﬁned in equation (10), with periodic boundary conditions. We introduce the Legendre transform −ϕ(x) = sup p · x − f (p) (18) p

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¯ and ∂D, Theorem 3 Let D = {x : ϕ(x) > −∞} and write, respectively, Dint , D, for the interior, the closure, and the boundary of the convex set D. Then, for x ∈ Dint , ¯ for x ∈ R2 \ D, for x ∈ ∂D,

lim

ϕper (x, Λ) = ϕ(x),

lim

ϕper (x, Λ) = −∞,

N1 ,N2 →∞ N1 ,N2 →∞

lim sup ϕper (x, Λ) ≤

N1 ,N2 →∞

lim sup x →x,x ∈Dint

ϕ(x ).

(19)

Proof. Since, from Theorem 2, we know that f = limN1 N2 →∞ f per , the above statements, together with the relation (18) between the free energies, express the thermodynamic equivalence of the Gibbs ensembles with partition functions Z per (p) and Ξ per (x), and can be proved in the same way as Theorem 4 in Ref. [3]. These relations imply that the surface z = ϕ(x1 , x2 ) gives, according to the Wulﬀ construction, or its equivalent, the Andreev construction, the equilibrium shape of the crystal associated to the system (see [5], [3]). We next examine the particular case of the horizontal interfaces (slope p = 0), in which stronger properties can be proved. Theorem 4 The following limits exist and coincide lim

N1 ,N2 →∞

ϕfree (0, Λ(N1 , N2 )) =

lim

N1 ,N2 →∞

ϕper (0, Λ(N1 , N2 )) = ϕ(0).

(20)

Moreover, ϕ(0) = f (0).

(21)

This theorem is proved in the Appendix. These results apply, in particular, also to the restricted solid-on-solid models. Of particular interest are some of these models which are exactly solvable, in which the height diﬀerences ξ(i, j) = φ(j) − φ(i) (22) for nearest neighbour sites, are restricted to have only two values. One of these models is the body-centered solid-on-solid model (bcsos) of van Beijeren [6], deﬁned on a square lattice with the restriction ξ(i, j) = ±1. Another is the triangular Ising solid-on-solid model (tisos) of Bl¨ ote and Hilhorst [7], in which the height variables are associated to the sites of a triangular lattice and ξ(i, j) is allowed to take the values 1 or −2. These two models appear in the description of the ground state interfaces of the Ising model on a body-centered cubic lattice with nearest and next-nearest neighbour interactions [8]. The latter model describes also the ground state interfaces of the usual Ising model on a cubic lattice with nearest neighbour interactions [7], [9]. Taking into account the compatibility condition (6) between the diﬀerence variables ξ(i, j), it can easily be seen that there is a one to one correspondence between the set of conﬁgurations of the bcsos model and the set of conﬁgurations of

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the six-vertex model, the compatibility condition being equivalent to the ice rule. Similarly, the conﬁgurations of the tisos model are in one to one correspondence with the ground state conﬁgurations of the Ising antiferromagnet on a triangular lattice. This explains why the free energy of the six-vertex model or of the triangular Ising model depends on the boundary conditions. Theorem 4 proves then the equivalence between the free and the periodic boundary conditions for these models in the case of symmetric interactions with respect to the axes. The crystal shape associated to these particular models was obtained from the (“grand canonical”) partition function (12). In this way the van Beijeren model is equivalent to a six-vertex model with polarizations, and the Bl¨ote-Hilhorst model to a zero-temperature triangular Ising antiferromagnet with external ﬁelds. The equilibrium shape of the corresponding crystals is directly related to the free energy of these models and may be exactly computed. See the original work by Jayaprakash et al. [10] and Nienhuis et al. [11] for a more detailed discussion, including the study of the shape of the facets of these crystals and their roughening transitions. Theorem 3 above proves then the rigorous validity of these exact results.

Appendix To prove Theorems 2 and 4 we ﬁrst establish the following two lemmas. For concreteness we shall consider the solid-on-solid models. The same proof applies to other interface models, as the discrete gaussian model. Its extension to restricted SOS models is explained in the remark after the proof of Theorem 2. Lemma 1 The partition function Z per satisﬁes the subadditivity property 2 Z per p, Λ(2N1 , N2 ) ≥ Z per p, Λ(N1 , N2 ) K(β)−2N2 .

(23)

Proof. Consider the rectangular box Λ(N1 , N2 ) with the speciﬁed conﬁgurations ¯ of bonds ξ¯ = {ξ(), ∈ ∂2 Λ} on the two sides parallel to the i2 -axis of the ¯ be the partition function Z per with these imposed rectangle, and let Z per (p, Λ | ξ) constraints. Then ¯ Z per (p, Λ) = Z per (p, Λ | ξ). (24) ξ¯

If we paste two such boxes to form a (2N1 , N2 ) rectangle then 2 ¯ Z per p, Λ(2N1 , N2 ) | ξ¯ ≥ Z per p, Λ(N1 , N2 ) | ξ¯ eβHλ (ξ)

(25)

since we can always impose the conﬁguration ξ¯ on each component of the box Λ(2N1 , N2 ) and the energy associated to their common side λ is ¯ = ¯ Hλ (ξ) U ξ() (26) ∈λ

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From the Schwartz inequality, we have

2 Z per p, Λ(N1 , N2 ) | ξ¯

ξ¯

≤

2 ¯ ¯ Z per p, Λ(N1 , N2 ) | ξ¯ eβHλ (ξ) e−βHλ (ξ) ξ¯

(27)

ξ¯

and, on the other hand, ξ¯

¯

e−βHλ (ξ) ≤

¯

e−βU(ξ())

2N2

2N2 = K(β)

(28)

¯ ξ()

¯ by considering the ξ(), ∈ λ as independent variables. The application of inequalities (25), (27) and (28) to equation (24) gives the proof of the Lemma. Now let Λ0 = Λ(N1 , N2 ) be an arbitrary, but henceforth ﬁxed rectangle and form the standard sequence Λk = Λ 2k N1 , 2k N2 with k integral. Using the deﬁnition (16), we have, by arguing as in the proof of (28), f per(p, Λk ) ≥ −

1 ln K(β) β

(29)

and, as a consequence of Lemma 1, f per(p, Λk+1 ) ≤ f per (p, Λk ) + 2−k

1 1 1 ln K(β). + 2N1 2N2 β

(30)

Hence, the sequence {f per(p, Λk )}, k = 0, 1, . . ., is essentially a decreasing sequence and since it is bounded below it has a limit. Next, it will be convenient to restrict the set of conﬁgurations on Λ in such a way that the interface be contained in a parallelepiped of height M . Namely, we impose the condition |φ(i) − p · i| ≤ M. (31) We denote by Z per(p, Λ, M ) and f per (p, Λ, M ) the associated partition functions and free energies. Lemma 2 With the above notations, we have f per(p, Λk , M ) ≤ f per (p, Λ0 , M ) + 2

1 1 1 ln K(β). + 2N1 2N2 β

(32)

Proof. Since Lemma 1 is still valid for the restricted set of conﬁgurations (30), we obtain the lemma from equation (27), by iteration. Proof of Theorem 2. We ﬁrst compute the partition function (3) on a rectangle of sides 2(2k N1 + 1) and 2(2k N2 + 1), which contains the standard rectangle Λk and

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has the same center. We denote by B the set of bonds which join the boundary of this rectangle to the boundary of Λk . The sum below is over the set of conﬁgurations in Λk which satisfy condition (31). This set of conﬁgurations is furthermore restricted by conditions (7), (14) and φ(0) = 0. This gives the ﬁrst inequality in the expression below. The second inequality follows from condition (28) which implies |φ(j) − φ(i)| ≤ 2M . k 1, 2k N2 + 1) Z clos p, Λ(2 N1 + exp − β |φ(j) − φ¯p (i)| − |φ(j) − φ(i)| − HΛk (φ) ≥ ≥

φ {i,j}∈B {i,j}∈∂Λ exp − β2k (2N1 + 2N2 )4M Z per(p, Λk , M ).

Taking the logarithms and dividing by −β|Λk |, we get

1 1 clos per −k αk f (p, Λk ) ≤ f (p, Λk , M ) + 2 4M + 2N1 2N2

(33)

(34)

where αk = (2k N1 + 1)(2k N2 + 1)2−2k (N1 N2 )−1

(35)

which tends to 1 when k → ∞. Then, from Lemma 2,

1 1 αk f clos (p, Λk ) ≤ f per (p, Λ0 , Mk ) + 2−k 4Mk + 2N1 2N2

1 1 1 ln K(β). + + 2 2N1 2N2 β

(36)

This equation holds for any M . Taking M = Mk in such a way that Mk → ∞ and 2−k Mk → 0, when k → ∞, we obtain

1 1 1 ln K(β). (37) f (p) ≤ f per (p, Λ0 ) + 2 + 2N1 2N2 β Since, on the other side f per (p, Λ) ≤ f clos (p, Λ), as a direct consequence of their deﬁnitions, the theorem follows from the last inequality. Remark. It is easy to see that Lemma 1 is still valid for restricted SOS models and hence also Lemma 2, provided that the expression (1 + e−β )/(1 − e−β ), in (25) and (34), is replaced by the number 2. This comes from the fact that equation (28) now ¯ = 0. The proof of Theorem 2 has to be reviewed since an expression reads Hλ (ξ) analogous to the ﬁrst inequality (25) cannot be obtained simply by increasing by 2 the length of the sides of the box. In order that the required conﬁguration could be admissible in the restricted SOS models we have to increase it by Lk = 2Mk . But, since 2−k Lk → 0 when k → ∞, this does not aﬀect the thermodynamic limit and, hence, the theorem can be proved similarly.

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Proof of Theorem 4. From the corresponding deﬁnitions it follows that Ξfree (0, Λ) ≥ Ξper (0, Λ) ≥ Z clos (0, Λ).

(38)

The appropriate converse inequalities can be established by arguing as in the proof of Theorem 2, itself a consequence of Lemmas 1 and 2. The main point is to prove the validity of Lemma 1 for the partition function Ξfree . Consider the partition function with free boundary conditions in the rectangular box Λ(2N1 , N2 ), obtained by pasting two boxes Λ(N1 , N2 ) along one of the sides parallel to the i2 -axis. Let λ be this common line, inside the box Λ(2N1 , N2 ), ¯ ¯ and let {ξ()} on the bonds belonging to λ. We denote by ξ be the conﬁguration free ¯ 0, Λ(N1 , N2 ) | ξ the partition function over all conﬁgurations in Λ(N1 , N2 ) Ξ whose restriction to one of the sides parallel to the i2 -axis coincides with the given ¯ and having free boundary conditions on the other three sides. conﬁguration ξ, Because of the symmetry of the system with respect to the λ-axis, we have free 2 ¯ Ξ 0, Λ(N1 , N2 ) | ξ¯ eβHλ (ξ) (39) Ξfree 0, Λ(2N1, N2 ) ≥ ξ¯

¯ is given by (26). Then, inequality (39) allows us to derive Lemma 1, where Hλ (ξ) the subadditivity property, for the function Ξfree . All the other steps in the proof of Theorem 2 follow in the same way as above and lead to the conclusion that the ﬁrst limit in (20) is equal to f (0). Then inequalities (38), Theorems 1 and 3 imply that this limit coincides with the second limit in (20) and with ϕ(0). This ends the proof of the theorem.

References [1] H. Spohn, Interface motion in models with stochastic dynamics, J. Stat. Phys. 71, 1081 (1993). [2] T. Funaki and H. Spohn, Motion by mean curvature from the GinzburgLandau ∇φ interface model, Commun. Math. Phys. 185, 1–36 (1997). [3] A. Messager, S. Miracle-Sole and J. Ruiz, Convexity properties of the surface tension and equilibrium crystals, J. Stat. Phys. 67, 449 (1992). [4] R.L. Dobrushin and S.B. Shlosman, Thermodynamic inequalities and the geometry of the Wulﬀ construction, in “Ideas and Methods in Mathematical Analysis, Stochastics and Applications”, S. Albeverio, S.E. Fenstad, H. Holden and T. Lindstrom eds., Cambridge University Press, Cambridge 1991. [5] A.F. Andreev, Faceting phase transitions of crystals, Sov. Phys. JETP 53, 1063 (1981). [6] H. van Beijeren, Exactly solvable models for the roughening transition of a crystal surface, Phys. Rev. Lett. 38, 993 (1977).

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[7] H.W. Bl¨ ote and H.J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A 15, L631 (1982). [8] R. Koteck´ y and S. Miracle-Sole, Roughening transition for the Ising model on a bcc lattice. A case in the theory of ground states, J. Stat. Phys. 47, 773 (1987). [9] S. Miracle-Sole, Facet shapes in a Wulﬀ crystal, in: “Mathematical Results in Statistical Mechanics”, S. Miracle-Sole, J. Ruiz, V. Zagrebnov (eds.), pp. 83–101, World Scientiﬁc, Singapore, 1999. [10] C. Jayaprakash, W. F. Saam and S. Teitel, Roughening and facet formation in crystals, Phys. Rev. Lett. 50, 2017 (1983). [11] B. Nienhuis H.J. Hilhorst and H.W. Bl¨ ote, Triangular SOS models and cubic crystal shapes, J. Phys. A 17, 3559 (1984). Salvador Miracle-Sole and Jean Ruiz Centre de Physique Th´eorique CNRS Luminy, Case 907 F-13288 Marseille Cedex 9 France email: [email protected] email: [email protected] Communicated by Vincent Rivasseau submitted 07/02/02, accepted 19/12/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 137 – 193 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/010137-57 DOI 10.1007/s00023-003-0125-9

Annales Henri Poincar´ e

Low Temperature Analysis of Two-Dimensional Fermi Systems with Symmetric Fermi Surface G. Benfatto, A. Giuliani and V. Mastropietro Abstract. We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivity.

1 Introduction and main results 1.1

Motivations

The unexpected properties of recently discovered materials, showing high-Tc superconductivity and signiﬁcative deviations from Fermi liquid behavior in their normal phase (i.e., above Tc ) [VLSAR], provides the main physical motivation for the search of well-established results on models for interacting non-relativistic fermions, describing the conduction electrons in metals. One can consider such models not only in d = 3, but also in d = 1, 2, to describe metals so anisotropic that the conduction electrons move essentially on a chain or on a plane. Renormalization Group (RG) methods provide a powerful technique for studying such models. While in d = 1 RG methods were applied since long time [So] and many rigorous results up to T = 0 were established (see for instance [BGPS], [BoM], [BM] and [GM] for an updated review), in d > 1 the application of RG methods is much more recent and started in [BG], [FT]. At the moment RG methods seem unable to get a rigorous control of such models in d > 1 up to T = 0, for the generic presence of phase transitions (for instance to a superconducting state) at low temperatures (unless such phase transitions are forbidden by a careful choice of the dispersion relation, see [FKT]). On the other hand, RG methods seem well suited to obtain rigorous information on the behavior of d > 1 models at temperatures above Tc , and to clarify the microscopic origin of Fermi or non-Fermi liquid behavior in the normal phase. One can write, in the weakly interacting case, an expansion for the Schwinger functions based on RG ideas; the ﬁnite temperature acts as an infrared cut-oﬀ so that each perturbative order is trivially ﬁnite; the mathematical non-trivial problem is to prove that the expansion is convergent, and it turns out that such problem is more and more diﬃcult as the temperature of the system decreases. Indeed, if λ is the interaction strength, the cancellations due to the anticommutativity properties of fermions allow quite easily to prove convergence of naive perturbation theory for T ≥ |λ|α , for some constant α > 0. On

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the other hand, the critical temperature in the weak coupling case at which phase −1 transitions are expected is O(e−(a|λ|) ) where a is a constant essentially given by the second order contributions [AGD] of the expansion, i.e., it is exponentially small and so quite smaller than |λ|α , if λ is small enough. In [FMRT] and [DR] the perturbative expansion convergence was proved for the eﬀective potential up to exponentially small temperatures, in the d = 2 Jellium model, describing fermions in the continuum, with dispersion relation ε(k) = |k|2 /(2m) and a rotation invariant weak interaction. One of the main diﬃculties of the proof is that non-perturbative bounds are naturally obtained in coordinate space, while one has to exploit the geometric properties of the Fermi surface (i.e., the set of momenta k such that ε(k) = µ), which are naturally investigated in momentum space. In [FMRT] and [DR] an expansion based on RG is considered, such that only the relevant (but not the marginal) terms are renor−1 malized; this has the eﬀect that one has convergence for T ≥ e−(c|λ|) , where c is related to an all order bound, hence it is expected to be much bigger than a. The proof uses in a crucial way the rotation invariance of the Jellium model, an hypothesis which is indeed quite unrealistic (it corresponds to completely neglecting the eﬀect of the lattice). The aim of this paper is to prove convergence of the perturbative expansion for the two point Schwinger function, in the case of an interacting system of fermions in a lattice or in the continuum. Since the interaction modiﬁes the Fermi surface, we write the dispersion relation ε0 (k) of the free model in the form ε0 (k) = ε(k) + δε(k) and try to choose the counterterm δε(k), which becomes part of the interaction, as a suitable function of the original interaction, so that the Fermi surface of the interacting system is the set F = {k : ε(k) = µ}. We can face this problem if ε(k) satisﬁes some conditions, implying mainly that F is a smooth, convex curve, symmetric with respect to the origin. We prove convergence for weak coupling and up to temperatures T ≥ exp{−(c0 |λ|)−1 }, where c0 is a constant whose explicit expression is given in (4.31) and depends only on the size of the second order contributions of the perturbative RG expansion. In order to get this type of result, we consider an expansion in which both the relevant and marginal terms are renormalized. In fact, if one does not renormalize the marginal terms, one obtains the bound T ≥ exp{−(c|λ|)−1 }, the constant c being related to an all order bound, like in [DR]; of course it is expected that c >> c0 .

1.2

The model

There are two main classes of models of interacting fermions, depending whether the Fermi operators space coordinates are continuous or discrete variables. Our analysis deals with both such possibilities, so we give the following deﬁnitions. 2

1) Continuum models. In such a case, given a square [0, L]2 ∈ R , the inverse temperature β and the (large) integer M , we introduce in Λ = [0, L]2 × [0, β] a lattice

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ΛM , whose sites are given by the space-time points x = (x0 , x) = (n0 a0 , n1 a, n2 a), a = L/M , a0 = β/M , n1 , n2 , n0 = 0, 1, . . . , M − 1. We also consider the set 2 n n ∈ Z , n = (n1 , n2 ), D of space-time momenta k = (k0 , k), with k = 2π L , 2π 1 −M ≤ ni ≤ M − 1 and k0 = β (n0 + 2 ), n1 , n2 , n0 = 0, 1, . . . , M − 1. With each ε , ε, σ ∈ {+, −}. The lattice k ∈ D we associate four Grassmanian variables ψˆk,σ ΛM is introduced only for technical reasons so that the number of Grassmanian variables is ﬁnite, and eventually, before sending L to inﬁnity, the (essentially trivial) limit M → ∞ is taken. 2

2) Lattice models. In such a case, given [0, L]2 ∈ Z , the inverse temperature β and the (large) integer M , we introduce in Λ = [0, L]2 × [0, β] a lattice ΛM , whose sites are given by the space-time points x = (x0 , x) = (n0 a0 , n1 , n2 ), a0 = β/M , n1 , n2 = 0, . . . , L − 1 and n0 = 0, 1, . . . , M − 1; this deﬁnition is obtained from the previous one by deﬁning a = 1. In such a case D is a set of space-time momenta 1 2π n k = (k0 , k), with k0 = 2π β (n + 2 ), n ∈ Z, −M ≤ n ≤ M − 1; and k = L , 2

n ∈ Z , n = (n1 , n2 ), −[ L2 ] ≤ ni ≤ [ (L−1) 2 ]. With each k ∈ D we associate four ε ˆ Grassmanian variables ψk,σ , ε, σ ∈ {+, −}. All the models are deﬁned by introducing a linear functional P (dψ) on the ε , such that Grassmanian algebra generated by the variables ψˆk,σ

P (dψ)ψˆk−1 ,σ1 ψˆk+2 ,σ2 = L2 βδσ1 ,σ2 δk1 ,k2 gˆ(k1 ) , gˆ(k) =

(1.1)

C¯0−1 (k) , −ik0 + ε(k) − µ

where ε(k), the dispersion relation of the model, is a function strictly positive for k = 0 and equal to 0 for k = 0, µ is the chemical potential and C¯0−1 (k) is the ultraviolet cut-oﬀ. In the case of lattice models we choose C¯0−1 (k) = 1, while for continuum models the function C¯ −1 (k) is deﬁned as C¯ −1 (k) = H ε(k) − µ 0

0

where H(t) ∈ C ∞ (R) is a smooth function of compact support such that, for example, H(t) = 1 for t < 1 and H(t) = 0 for t > 2. We introduce the propagator in coordinate space: 1 −ik·(x−y) − + e g ˆ (k) = lim ψy,σ , (1.2) P (dψ)ψx,σ M→∞ L2 β M→∞

g L,β (x − y) ≡ lim

k∈D

where the Grassmanian field ψxε is deﬁned by ± ψx,σ =

1 ˆ± ±ik·x ψk,σ e . L2 β k∈D

(1.3)

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The “Gaussian measure” P (dψ) has a simple representation in terms of the “Lebesgue Grassmanian measure” + − dψˆk,σ dψˆk,σ , (1.4) Dψ = k∈D,σ=± −1 C0 ( k)>0

deﬁned as the linear functional on the Grassmanian algebra, such that, given a − + monomial Q(ψˆ− , ψˆ+ ) in the variables ψˆk,σ , ψˆk,σ , its value is 0, except in the case − + Q(ψˆ− , ψˆ+ ) = k,σ ψˆk,σ ψˆk,σ , up to a permutation of the variable, in which case its value is 1. We deﬁne         1 + ˆ− −1 ˆ ¯ P (dψ) = N Dψ · exp − 2 C0 (k)(−ik0 + ε(k) − µ)ψk,σ ψk,σ ,   L β k∈D,σ=±     ¯ −1 C 0

(k)>0

(1.5) ¯ with N = k∈D,σ=± [(L β) (−ik0 + ε(k) − µ)C0 (k)]. The Schwinger functions are deﬁned by the following Grassmanian functional integral P (dψ)e−V(ψ)−N (ψ) ψxε11 ,σ1 . . . ψxεnn,σn , S(x1 , ε1 , σ1 ; . . . , xn , εn , σn ) = lim lim L→∞ M→∞ P (dψ)e−V(ψ)−N (ψ) (1.6) where 1 + − N (ψ) = 2 νˆ(k, λ)ψk,σ ψk,σ , (1.7) L β k∈D,σ=±

and, if we use

2

V(ψ) = λ

−1

−1 C0 ( k)>0

dx and δ(x0 − y0 ) as short-hands for σ,σ

x∈ΛM

a0 a2 and a−1 0 δx0 ,y0 ,

+ − + − ψx,σ ψy,σ dxdyδ(x0 − y0 )vσ,σ (x − y )ψx,σ ψy,σ ,

(1.8)

vσ,σ (x) being smooth functions such that maxσ,σ dx (1 + |x|2 ) |vσ,σ (x)| is bounded. Note that νˆ(k, λ) is related to the counterterm δε(k) introduced in §1.1 by the relation δε(k) = C¯0−1 (k)ˆ ν (k, λ) In order to make more precise the model, we have to specify some properties of the dispersion relation. We will assume that ε(k) veriﬁes the following properties (whose consequences are discussed in §7. From now on c, c1 , c2 , . . ., will denote suitable positive constants. 1. There exists e0 such that, for |e| ≤ e0 , ε(k) − µ = e deﬁnes a regular C ∞ convex curve Σ(e) encircling the origin, which can be represented in polar

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coordinates as p = u(θ, e)er (θ) with er (θ) = (cos θ, sin θ). Note that e0 < µ, since ε(k) > 0 for k = 0 and ε(0) = 0; moreover u(θ, e) ≥ c > 0 and, if r(θ, e) is the curvature radius, (1.9) r(θ, e)−1 ≥ c > 0 . 2. e0 is chosen so that, if k ∈ Σ(e) and |e| ≤ e0 , then C¯0−1 (k) = 1. 3. If |e| ≤ e0 , then

p) · er (θ) ≤ c2 . 0 < c1 ≤ ∇ε(

(1.10)

4. The following symmetry relation is satisﬁed ε( p) = ε(− p) ,

(1.11)

implying that the curves Σ(e) are symmetric by reﬂection with respect to the origin. We will call ΣF ≡ Σ(0) the Fermi surface and we will put u(θ, 0)er (θ) = pF (θ) and u(θ) ≡ u(θ, 0) = | pF (θ)|. Remarks. The Grassmanian functional integrals (1.6) are equal, in the limit M → ∞, to the Schwinger functions of an Hamiltonian model of fermions in two dimensions, expressed in terms of fermionic creation or annihilation operators. Among the dispersion relations which are in the class we are considering is that of the Hubbard model, deﬁned in a lattice with local interaction vσ,σ (x − y ) ≡ δσ,−σ δx,y (without the counterterm) and ε(k) = 2 − cos k1 − cos k2 , and that of the Jellium model, deﬁned in the continuum with ε(k) = |k|2 /2m. The index σ is the spin index; in the following it will not play any role and it will be omitted to shorten the notation. We are mainly interested in the two point Schwinger function S(x − y) ≡ S(x, −; y, +), with S(x, −; y, +) given by (1.6). For λ = 0 and νˆ(k, λ) = 0, S(x−y) is equal to the propagator (1.2), hence its Fourier transform is singular at k0 = 0 (which is not an allowed value at ﬁnite temperature) and ε(k) = µ. As we said in §1.1, we want to ﬁx νˆ(k, λ) so that the location of this singularity does not change for λ = 0; this allows to study the model as a perturbation of the model with λ = 0. Our goal is to prove the following theorem. Theorem 1.1 There exist two positive constants ε and c0 , the last one only depending on ﬁrst and second order terms in the perturbative expansion, and a continuous function νˆ(k, λ) = O(λ), such that, for all |λ| ≤ ε and T ≥ exp{−(c0 |λ|)−1 }, ˆ S(k) = gˆ(k)(1 + λSˆ1 (k)) ,

(1.12)

where gˆ(k) is the free propagator at ﬁnite β (i.e., it is equal to the Fourier transform of limL→∞ g L;β (x − y), see (1.2)) and |Sˆ1 (k)| ≤ c, for some constant c. In

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the continuum case with ε(k) = |k|2 /2m and v(r) = v˜(|r|), there exists another constant c1 such that, if |λ| ≤ ε and T ≥ exp{−(c1 |λ|)−1 }, νˆ(k, λ) = ν(λ) is a constant. However, in order to simplify the discussion, we shall not really study (even if we would be able to do that) the ﬁnite volume case and the convergence as L → ∞, but we shall study directly the formal thermodynamical limit of our expansions and we shall introduce a few other technical simpliﬁcations, which leave unchanged the relevant properties of the models. See the remark at the end of §2.1 for a precise deﬁnition of the model for which we do prove Theorem 1.1. This theorem says that the two point Schwinger function of the interacting system is close to the free one, for weak interactions and up to exponentially small temperatures; the condition on the temperature is not technical, as at temperatures low enough phase transitions are expected and a result like (1.12) cannot hold. The theorem is proved by an expansion similar to the one in [BG], in which the relevant and the marginal interactions are renormalized at any iteration of the ˆ Renormalization Group. One writes S(k) in terms of a set of running coupling functions, which obey recursive equations, the beta function of the model. We prove ˆ that the expansions of S(k) and of the beta function are convergent, if the running coupling functions are small in a suitable norm; the convergence proof is based on the tree expansion and the determinant bounds used for instance in [BM] and on a suitable generalization to the present problem of the sector counting lemma of [FMRT]. Finally we show, by choosing properly the counterterm νˆ(k, λ) and by solving iteratively the beta function, that the running coupling functions are small up to temperatures exponentially small T ≥ exp −(c0 |λ|)−1 ; c0 is expressed in terms of a few terms of ﬁrst and second order, see (4.31) below, so much closer to the expected value for the onset of superconductivity. Our non perturbative deﬁnition of the beta function is interesting by itself, as it could be used to detect the main instabilities of the model at lower temperatures. In order to complete our program, we should prove that νˆ(k, λ) and ε(k) can be chosen in a space of functions with the same diﬀerentiability properties and that the relation ε0 (k) = ε(k) + νˆ(k, λ) can be solved with respect to ε(k), given ε0 (k) and λ. This would imply that the introduction of the counterterm is only a technical trick, but does not restrict the class of allowed dispersion relations; for example one could consider the Hubbard model away from half-ﬁlling. We did not yet get this result, mainly because our bounds can only show that νˆ(k, λ) is a continuous function of compact support, whose Fourier transform is summable, while ε(k) has to be a bit more regular than a twice diﬀerentiable function. A similar problem appears in [FKT] in which a result similar to Theorem 1.1 above is proved in a class of asymmetric Fermi surfaces (the asymmetry makes an equation like (1.12) valid up to T = 0). It is likely that an improvement in the diﬀerentiability properties of the counterterm could be obtained by applying the more detailed analysis on the derivatives of the self-energy introduced in [DR].

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This problem is not present in the Jellium model, where, by using rotational invariance (so that both the free and the interacting Fermi surfaces are circles), one can choose νˆ(k, λ) as a constant with respect to k; this is the last statement in the theorem, already proved in [DR]. In order to get this result in a simple way, we chose to give up the “close to optimal” upper bound on the critical temperature; in fact the constant c1 depends on an all order bound, like in [DR]. However, even in this case, our result has some interest, since we get it without being involved in the delicate one particle irreducibility analysis of [DR].

2 Renormalization Group analysis We start, for clarity reasons, by studying the free energy of the model. In this section we write an expansion for it in terms of a set of running coupling functions and we show that the expansion is convergent if the running coupling functions are small enough in a suitable norm.

2.1

The scale decomposition

The free energy is deﬁned as EL,β

1 = − 2 log L β

P (dψ (≤1) ) ≡ P (dψ),

P (dψ (≤1) )e−V

(1)

,

(2.1)

V (1) ≡ V + N .

Note that our model has an ultraviolet cut-oﬀ in the k momentum, but the k0 variable is unbounded in the limit M → ∞. Hence, it is convenient to decompose the ﬁeld as ψ (≤1) = ψ (+1) + ψ (≤0) , where ψ (+1) and ψ (≤0) are independent ﬁelds whose covariances have Fourier transforms with support, respectively, in the ultraviolet region and the infrared region, deﬁned in the following way. Item 1) in the list of properties of the dispersion relation given in §1.2 implies 1 that, if H0 (t) is a smooth function of t ∈ R , such that 1 if t < e0 /γ , H0 (t) = (2.2) 0 if t > e0 , γ > 1 being a parameter to be ﬁxed below, then, since C¯0−1 (k) = 1 if |ε(k)−µ| ≤ e0 , C¯0−1 (k) = C0−1 (k)

=

f1 (k)

=

C0−1 (k) + f1 (k) , H0 k02 + [ε(k) − µ]2 , . C¯0−1 (k) 1 − H0 k02 + [ε(k) − µ]2

(2.3)

The covariances g (+1) and g (≤0) of the ﬁelds ψ (+1) and ψ (≤0) are deﬁned as in (1.8), with f1 (k) and C0−1 (k) in place of C¯0−1 (k).

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If we perform the integration of the ultraviolet ﬁeld variables ψ (+1) , we get 0 ≤0 −L2 βEL,β −L2 βE0 =e (2.4) e P (dψ (≤0) )e−V (ψ ) , where V (0) (ψ (≤0) ), the eﬀective potential on scale 0, is given by an expression like (2.12) below and E0 is deﬁned by the condition V (0) (0) = 0. The analysis of the ultraviolet integration is far easier than the infrared one. It can be done by the same procedure applied below for the infrared problem, by making a multiscale expansion of the u.v. propagator g (1) (x), based on an obvious smooth partition of the interval {|k0 | > 1}. In this way, one can build a tree expansion for V (0) , with endpoints on scale M > 0, similar to the infrared tree expansion, to be described below, see Fig. 1 and following items 1)–6). It is easy to see that there is no relevant or marginal term on any scale > 0, except those which are obtained by contracting two ﬁelds associated with the same space-time point in a vertex located between an endpoint and the ﬁrst non-trivial vertex following it (i.e., the tadpoles). However the sum over the scales of this type of terms, which is not absolutely convergent for M → +∞, can be controlled by using the explicit expression of the single scale propagator, since there is indeed no divergence, but only a discontinuity at x0 = 0 for x = 0. We shall omit the details, which are of the same type of those used below for the infrared part of the model. Let us now consider the infrared integration; it will be performed, as usual, by an iterative procedure. Note ﬁrst that we can write H0 (t) =

0

f˜h (t) ,

(2.5)

h=−∞

where f˜h (t) = H0 (γ −h t) − H0 (γ −h+1 t) is a smooth function, with support in the interval [γ h−2 e0 , γ h e0 ], and γ > 1 is the scaling parameter. In order to simplify some calculations, we will put in the following γ = 4, but this choice is not essential. Since |k0 | ≥ π/β, ∀k ∈ D, if we deﬁne hβ = max{h ≤ 0 : γ h−1 e0 < π/β} ,

(2.6)

we have the identity C0−1 (k) =

0

fh (k)

,

fh (k) ≡ f˜h

k02 + [ε(k) − µ]2 .

(2.7)

h=hβ

We associate with the decomposition (2.7) a sequence of constants Eh , h = hβ , . . . , 0, and a sequence of eﬀective potentials V (h) (ψ) such that V (h) (0) = 0 and (h) (≤h) −L2 βEL,β −L2 βEh ) =e , (2.8) P (dψ (≤h) )e−V (ψ e

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where P (dψ ≤h ) is the fermionic integration with propagator g (≤h) (x) = with Ch−1 (k) =

Ch−1 (k) 1 e−ikx , 2 L β −ik + ε( k) − µ 0 k∈D h

(2.9)

−1 fj (k) = Ch−1 (k) + fh (k) .

(2.10)

j=hβ

The deﬁnition (2.8) implies that EL,β = Ehβ −

1 log L2 β

P (dψ (≤hβ ) )e−V

(hβ ) (ψ (≤hβ ) )

.

(2.11)

If we neglect the spin indices and we put ε1 = · · · = εn = +, εn+1 = · · · = ε2n = −, we can write the eﬀective potentials in the form 2n ∞ (h) (h) (≤h) (≤h)εi V (ψ )= ψxi (2.12) W2n (x1 , . . . , x2n ) . dx1 . . . dx2n n=1

i=1

Remark. The terms in the right-hand side of (2.12) are well deﬁned at ﬁnite M and L, as elements of a ﬁnite Grassmanian algebra, but have only a formal meaning for M = L = ∞. However, one can prove that the kernels, as well as EL,β , have well-deﬁned limits as M and L go to inﬁnity. Such result is achieved by studying a suitable perturbative expansion of these quantities and by proving that they are uniformly (in M and L) convergent and, in the case of the kernels, that they have fast decaying properties in the x variables; see [BM] for a complete analysis of this type in the one-dimensional case. However, since this procedure is cumbersome and diﬃcult to describe rigorously without making obscure the main ideas, which have nothing to do with the details related with the ﬁnite values of M and L, we shall discuss in the following only the formal limit of our expansions and we shall prove that the kernels as well as the free energy constants Eh are well deﬁned. For similar reasons, we shall also consider k0 as a continuous variable and we shall take into account the essential infrared cut-oﬀ related with the ﬁnite temperature value, by preserving the deﬁnition (2.10) of the cut-oﬀ functions. This means, in particular that, from now on 1 1 → dk . (2.13) L2 β (2π)3 D k∈D

Moreover, we shall suppose that the space coordinates are continuous variables, both in the continuum and lattice models. This means that, from now on, dx will 3 denote the integral over R . Finally, we shall still use the symbol L2 β to denote the formally inﬁnite space-time volume in the extensive quantities like L2 βEh .

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The localization procedure

Let us now describe our expansion, which is produced by using an inductive procedure. First of all, we deﬁne an L operator acting on the kernels in the following way: (h)

1. LW2n = 0 if n ≥ 3. ˜ i = (˜ 2. If n = 2 and we put x = (x1 , . . . , x4 ), xi = (xi,0 , xi ), x xi,0 , xi ), δ(x0 ) = δ(x1,0 − x2,0 )δ(x1,0 − x3,0 )δ(x1,0 − x4,0 ) (h) (h) LW4 (x) = δ(x0 ) d(˜ x0 \˜ x1,0 )W4 (˜ x) . (2.14) Note that, because of translation invariance, this deﬁnition is independent of the choice of the localization point, that is the point whose time coordinate is not integrated (x1 in (2.14)). (h)

3. If n = 1 and we put (by using translation invariance) W2 (x1 , x2 ) = ˜ (h) (x1 − x2 ), W 2 (h) ˜ (h) (t, x1 − x2 ) LW2 (x1 , x2 ) = δ(x1,0 − x2,0 ) dt W 2 ˜ (h) (t, x1 − x2 ) . +∂x2,0 δ(x1,0 − x2,0 ) dt t W (2.15) 2 The deﬁnition of L is extended by linearity to V (h) , so that we can write (h) (≤h) ) = dx1 dx2 δ(x1,0 − x2,0 )γ h νh (x1 − x2 )ψx(≤h)+ ψx(≤h)− LV (ψ 1 2 + dx1 dx2 δ(x1,0 − x2,0 )zh (x1 − x2 )ψx(≤h)+ ∂x2,0 ψx(≤h)− 1 2 + dxλh (x)δ(x0 )ψx(≤h)+ ψx(≤h)+ ψx(≤h)− ψx(≤h)− , (2.16) 1 2 3 4 (h) ˜ (h) (t, x1 − x2 ) and where λh (x) = d(x0 \x1,0 )W4 (x), γ h νh (x1 − x2 ) = dtW 2 ˜ (h) (t, x1 − zh (x1 − x2 ) = − dt t W x ). Note that, in the term containing zh (x1 − 2 2 (≤h)+ (≤h)− (≤h)+ (≤h)− x2 ), we can substitute ψx1 ∂x2,0 ψx2 with −[∂x1,0 ψx1 ]ψx2 . The functions λh , νh and zh will be called the running coupling functions of scale h or simply the coupling functions. It is useful to consider also the representation of LV (h) (ψ (≤h) ) in terms of the Fourier transforms, deﬁned so that, for example, dk −ik(x1 −x2 ) ˆ (h) (h) W2 (k) , e (2.17) W2 (x1 , x2 ) = (2π)3 3 dki −iεi ki (xi −x4 ) ˆ (h) (h) W4 (k1 , k2 , k3 ) . W4 (x1 , x2 , x3 , x4 ) = e (2.18) (2π)3 i=1

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We can write LV (h) (ψ (≤h) ) =

dk (≤h)+ (≤h)− [γ h νˆh (k) − ik0 zˆh (k)]ψk ψk (2π)3

+

4 dki (≤h)+ (≤h)+ (≤h)− (≤h)− ψ ψk2 ψk3 ψk4 (2π)3 k1 i=1

·

ˆh (k1 , k2 , k3 )δ(k1 + k2 − k3 − k4 ) , λ

(2.19)

ˆ h (k1 , k2 , k3 ) = W ˆ (h) ((0, k1 ), (0, k2 ), (0, k3 )), γ h νˆh (k) = W ˆ (h) (0, k), zˆh (k) = where λ 4 2 (h) ˆ (0, k). i∂k0 W 2 We also deﬁne R ≡ 1 − L; by using (2.15), we get: (h)

˜ (h) (x1 − x2 ) − δ(x1,0 − x2,0 )W ¯ (h) (0, x1 − x2 ) W 2 2 (h) ¯ (2.20) i∂x1,0 δ(x1,0 − x2,0 )∂k0 W2 (0, x1 − x2 ) ,

RW2 (x1 , x2 ) = − ¯ (h) (k0 , x) = where W 2

(h)

˜ (t, x). Furthermore dt eik0 t W 2 (h)

(h)

(h)

¯ (0, x) , RW4 (x) = W4 (x) − δ(x0 )W 4 (h)

(2.21)

(h)

¯ (k , x) is the Fourier transform of W (x) with respect to the time where W 0 4 4 coordinates.

2.3

The sector decomposition

We now further decompose the ﬁeld ψ (≤h) , by slicing the support of Ch−1 (k) as in [FMRT]. Let H2 (t) be a smooth function on the interval [−1, +1], such that 1 if |t| < 1/4 H2 (t) = , H2 (t) + H2 (1 − t) = 1 if 1/4 < t < 3/4, (2.22) 0 if |t| > 3/4; and let us deﬁne, if ω is an integer in the set Oh ≡ {0, 1, . . . , γ −(h−1)/2 − 1} (recall that γ = 4) and h ≤ 0, h 1 h γ− 2 ¯ (t − θh,ω ) ζh,ω (t) = H2 , θh,ω = π(ω + )γ 2 . (2.23) π 2 It is easy to see that ζ¯h,ω (t) can be extended to the real axis as a periodic function of period 2π, that we can use to deﬁne a smooth function on the one-dimensional 1 torus T , to be called ζh,ω (θ); moreover ω∈Oh

ζh,ω (θ) = 1

,

1

∀θ ∈ T .

(2.24)

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On the other hand, the properties of ε(k) assumed in §1.2 imply that, if Ch−1 (k) = 0, k = u(θ, e)er (θ) with e = ε(k) − µ. Hence, we can write (≤h)± ψx(≤h)± ≡ e±ipF (θh,ω )x ψx,ω , P (dψ (≤h) ) = P (dψω(≤h) ) , (2.25) ω∈Oh

ω∈Oh

(≤h)

) is the Grassmanian integration with propagator −1 1 −i[k(x−y)− pF (θh,ω )( x− y)] Ch (k)ζh,ω (θ) dke gω(≤h) (x − y) = . (2π)3 −ik0 + ε(k) − µ

where P (dψω

(2.26)

If we insert the left-hand side of (2.25) in (2.12), we get ∞ (h) εi pF (θh,ω ) x (≤h))ε V e ψω = ω∈Oh

·

dx1 . . . dx2n

2n

n=1 ω1 ,...,ω2n ∈Oh

e

εi i pF (θωh,i ) xi

i ψx(≤h)ε i ,ωi

(h)

W2n (x1 , . . . , x2n ). (2.27)

i=1

By using (2.10), we can write εi pF (θh,ω ) x (≤h)ε −(L+R)V (h) ψω ω∈Oh e P (dψω(≤h) )e ω∈Oh

=

P (dψ (≤h−1) )

P (dψω(h) )

ω∈Oh

·e

(≤h−1)ε −(L+R)V (h) ψx +

ω∈Oh

pF (θh,ω ) x (h)ε eεi ψx,ω

(2.28)

,

(h)

where P (dψω ) is the integration with propagator Fh,ω (k) 1 gω(h) (x) ≡ dke−i(kx−pF (θh,ω )x) , (2π)3 −ik0 + ε(k) − µ Fh,ω (k) = fh (k)ζh,ω (θ) .

(2.29) (2.30)

The support of Fh,ω (k) will be called the sector of scale h and sector index ω. (h) In order to compute the asymptotic behavior of gω (x) it is convenient to introduce a coordinate frame adapted to the Fermi surface in the point pF (θh,ω ). By using the deﬁnitions of §1.2 and putting et (θ) = (− sin θ, cos θ), we deﬁne −1 pF (θ) u (θ)er (θ) + u(θ)et (θ) d pF (θ) d = , τ (θ) = dθ dθ u (θ)2 + u(θ)2 u(θ)er (θ) − u (θ)et (θ) n(θ) = . (2.31) u (θ)2 + u(θ)2

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Moreover, given any k belonging to the support of Fh,ω (k), we put k = pF (θh,ω ) + k n(θh,ω ) + k τ (θh,ω ) = pF (θh,ω ) + k ; 1 2

(2.32)

h

it is easy to verify that |k1 | ≤ Cγ h , |k2 | ≤ Cγ 2 , see Lemma 7.3 in §7 for details. By using (2.32), we can rewrite (2.29) as gω(h) (x)

1 ≡ (2π)3

dk0 dk e−i(k0 x0 +k x)

Let us now put

Fh,ω (k0 , pF (θh,ω ) + k ) . −ik0 + ε( pF (θh,ω ) + k ) − µ

x = x1n(θh,ω ) + x2 τ (θh,ω ) ;

(2.33)

(2.34) (h)

the following lemma gives a bound on the asymptotic behavior of gω (x), which is very important in our analysis, as in [FMRT]. It will be proved in §7. Lemma 2.1 Given the integers N, m, n0 , n1 , n2 ≥ 0, with m = n0 + n1 + n2 , there exists a constant CN,m such that 3

|∂xn00 ∂xn1 ∂xn2 gω(h) (x)| 1 2

≤

1

CN,m γ 2 h γ (n0 +n1 + 2 n2 )h 1

1 + (γ h |x0 | + γ h |x1 | + γ 2 h |x2 |)N

.

(2.35)

Remark. Lemma 2.1 holds also for non C ∞ Fermi surfaces: it is suﬃcient the conn n dition that the derivatives of ε(k) diverge “not too fast” (i.e., that ∂ n /∂k 1 1 ∂k 2 2 1 −h(n + n −1) 1 2 2 [ε(k) − µ] = O(γ )).

2.4

The tree expansion

Our expansion of V (h) , 0 ≥ h ≥ hβ is obtained by integrating iteratively the ﬁeld variables of scale j ≥ h + 1 and sector index ω = 1, . . . , γ −h/2 and by applying at each step the localization procedure described above, which has the purpose of summing together the relevant contributions of the same type. It is well known (see for instance the reviews [G], [BG1] or [GM] for a tutorial introduction) that the result of this iteration can be expressed as a sum over trees. We assume the reader familiar with this formalism and we simply give here some deﬁnitions in order to ﬁx the notation. 1) Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree (see Fig. 1), so that r is not a branching point. n will be called the order of the unlabeled tree and the branching points will be called the non-trivial vertices. The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. Two unlabeled trees are identiﬁed if they can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. It is

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v r

v0

h h+1

hv

0

+1 +2

Figure 1: A possible tree of the expansion for the eﬀective potentials. then easy to see that the number of unlabeled trees with n endpoints is bounded by 4n . We shall consider also the labeled trees (to be called simply trees in the following); they are deﬁned by associating some labels with the unlabeled trees, as explained in the following items. 2) We associate a label h ≤ 0 with the root and we denote Th,n the corresponding set of labeled trees with n endpoints. Moreover, we introduce a family of vertical lines, labeled by an integer taking values in [h, 2], and we represent any tree τ ∈ Th,n so that, if v is an endpoint or a non-trivial vertex, it is contained in a vertical line with index hv > h, to be called the scale of v, while the root is on the line with index h. There is the constraint that, if v is an endpoint, hv > h + 1. The tree will intersect in general the vertical lines in set of points diﬀerent from the root, the endpoints and the non-trivial vertices; these points will be called trivial vertices. The set of the vertices of τ will be the union of the endpoints, the trivial vertices and the non-trivial vertices. Note that, if v1 and v2 are two vertices and v1 < v2 , then hv1 < hv2 . Moreover, there is only one vertex immediately following the root, which will be denoted v0 and cannot be an endpoint (see above); its scale is h + 1. Finally, if there is only one endpoint, its scale must be equal to h + 2. 3) With each endpoint v of scale hv = +2 we associate one of the two contributions to V (1) (ψ (≤1) ), and a set xv of space-time points (the two corresponding integration variables); we shall say that the endpoint is of type λ or ν, respectively. With each endpoint v of scale hv ≤ 1 we associate one of the three terms appearing in (2.16) and the set xv of the corresponding integration variables; we shall say that the endpoint is of type ν, z or λ, respectively. Given a vertex v, which is not an endpoint, xv will denote the family of all space-time points associated with one of the endpoints following v.

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Moreover, we impose the constraint that, if v is an endpoint, hv = hv + 1, if v is the non-trivial vertex immediately preceding v. 4) If v is not an endpoint, the cluster Lv with scale hv is the set of endpoints following the vertex v; if v is an endpoint, it is itself a (trivial) cluster. The tree provides an organization of endpoints into a hierarchy of clusters. 5) The trees containing only the root and an endpoint of scale h + 1 will be called the trivial trees; note that they do not belong to Th,1 , if h ≤ 0 (see the end of item 3 above), and can be associated with the three terms in the local part of V (h) . 6) We introduce a ﬁeld label f to distinguish the ﬁeld variables appearing in the terms associated with the endpoints as in item 3); the set of ﬁeld labels associated with the endpoint v will be called Iv . Analogously, if v is not an endpoint, we shall call Iv the set of ﬁeld labels associated with the endpoints following the vertex v; x(f ) and ε(f ) will denote the space-time point and the ε index, respectively, of the ﬁeld variable with label f . If hv ≤ +1, one of the ﬁeld variables belonging to Iv carries also a time derivative ∂0 if the corresponding local term is of type z, see (2.16). Hence we can associate with each ﬁeld label f an integer m(f ) ∈ {0, 1}, denoting the order of the time derivative. Note that m(f ) is not uniquely determined, since we are free to choose on which ﬁeld exiting from a vertex of type z the derivative falls, see comment after (2.16); we shall use this freedom in the following. If h ≤ 0, the eﬀective potential can be written in the following way: ˜h+1 = V (h) (ψ (≤h) ) + Lβ E

∞

V (h) (τ, ψ (≤h) ) ,

(2.36)

n=1 τ ∈Th,n

where, if v0 is the ﬁrst vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ with root v0 , V (h) (τ, ψ (≤h) ) is deﬁned inductively by the relation (−1)s+1 T ¯ (h+1) Eh+1 [V (τ1 , ψ (≤h+1) ); . . . ; V¯ (h+1) (τs , ψ (≤h+1) )] , s! (2.37) and V¯ (h+1) (τi , ψ (≤h+1) ) V (h) (τ, ψ (≤h) ) =

a) is equal to RV (h+1) (τi , ψ (≤h+1) ) if the subtree τi is not trivial; b) if τi is trivial and h ≤ −1, it is equal to one of the three terms contributing to LV (h+1) (ψ (≤h+1) ) or, if h = 0, to one of the two terms contributing to V (1) (ψ ≤1 ). T Eh+1 denotes the truncated expectation with respect to the measure

ω

P (dψω(h+1) ),

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that is ∂p T Eh+1 (X1 ; . . . ; Xp ) ≡ log ∂λ1 . . . ∂λp

ω

Ann. Henri Poincar´e

P (dψω(h+1) )eλ1 X1 +···+λp Xp

. λi =0

(2.38) This means, in particular, that, in (2.37), one has to use for the ﬁeld variables the sector decomposition (2.25). We can write (2.37) in a more explicit way, by a procedure very similar to that described, for example, in [BM]. Note ﬁrst that, if h = 0, the right-hand side of (2.37) can be written more explicitly in the following way. Given τ ∈ T0,n , there are n endpoints of scale 2 and only another one vertex, v0 , of scale 1; let us call v1 , . . . , vn the endpoints. We choose, in any set Ivi , a subset Qvi and we deﬁne Pv0 = ∪i Qvi ; then we associate a sector index ω(f ) ∈ O0 with any f ∈ Pv0 and we put Ωv0 = {ω(f ) : f ∈ Pv0 }. We have V (0) (τ, ψ (≤0) ) = V (0) (τ, Pv0 , Ωv0 ) , (2.39) V (0) (τ, Pv0 , Ωv0 ) = (1)

Kτ,Pv (xv0 ) = 0

Pv0 ,Ωv0 (1)

≤0 dxv0 ψ˜Ω (Pv0 )Kτ,Pv (xv0 ) , v 0

0

n 1 T ¯(1) E1 [ψ (Pv1 \Qv1 ), . . . , ψ¯(1) (Pvn \Qvn )] Kv(2) (xvi ) , i n! i=1

where we use the deﬁnitions (∂0 is from now on the time derivative) (≤h) m(f ) (≤h)ε(f ) ψ˜Ωv (Pv ) = eiε(f )pF (θh,ω(f ) )x(f ) ∂0 ψx(f ),ω(f ) , h ≤ 0 ,

(2.40) (2.41)

(2.42)

f ∈Pv

ψ¯(1) (Pv ) = Kv(2) (xvi ) i

=

f ∈Pv

(1)ε(f )

ψx(f )

,

(2.43)

λv(x − y)δ(x0 − y0 ) if vi is of type λ and xvi = (x, y), ν(x − y )δ(x0 − y0 ) if vi is of type ν,

(2.44)

and we suppose that the order of the (anticommuting) ﬁeld variables in (2.43) is suitable chosen in order to ﬁx the sign as in (2.41). Note that the terms with ˜1 , while the others Pv0 = ∅ in the right-hand side of (2.39) contribute to Lβ E contribute to V (0) (ψ (≤0) ). We now write V (0) as LV (0) + RV (0) , with LV (0) deﬁned as in §2.2 (it represent, in the usual RG language, the relevant and marginal contributions to V (0) (ψ (≤0) )), and we write for RV (0) a decomposition similar to the previous one, with RV (0) (τ, Pv0 , Ωv0 ) in place of V (0) (τ, Pv0 , Ωv0 ); this means that we modify, according to the representation (2.20), (2.21) of the R operation, the kernels (0) (1) Wτ,Pv (xPv0 ) = d(xv0 \xPv0 )Kτ,Pv (xv0 ) , (2.45) 0

0

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where xPv0 = ∪f ∈Pv0 x(f ). In order to remember this choice, we write (≤0) (1) RV (0) (τ, Pv0 , Ωv0 ) = dxv0 ψ˜Ωv (Pv0 )[RKτ,Pv (xv0 )] . 0

0

153

(2.46)

It is not hard to see that, by iterating the previous procedure, one gets for V (h) (τ, ψ (≤h) ), for any τ ∈ Th,n , the representation described below. We associate with any vertex v of the tree a subset Pv of Iv , the external ﬁelds of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the vertices immediately following it, then Pv ⊂ ∪i Pvi ; if v is an endpoint, Pv = Iv . We shall denote Qvi the intersection of Pv and Pvi ; this deﬁnition implies that Pv = ∪i Qvi . The subsets Pvi \Qvi , whose union Iv will be made, by deﬁnition, of the internal ﬁelds of v, have to be non-empty, if sv > 1. Moreover, we associate with any f ∈ Iv a scale label h(f ) = hv and, if h(f ) ≤ 0, an index ω(f ) ∈ Oh(f ) , while, if h(f ) = +1, we put ω(f ) = 0. Note that, if h(f ) ≤ 0, h(f ) and ω(f ) single out a sector of scale h(f ) and sector index ω(f ) associated with the ﬁeld variable of index f . In this way we assign h(f ) and ω(f ) to each ﬁeld label f , except those which correspond to the set Pv0 ; we associate with any f ∈ Pv0 the scale label h(f ) = h and a sector index ω(f ) ∈ Oh . We shall also put, for any v ∈ τ , Ωv = {ω(f ), f ∈ Pv }. Given τ ∈ Th,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with all the constraints; we shall denote Pτ the family of all these choices and P the elements of Pτ . Analogously, we shall call Oτ the family of possible values of Ω = {ω(f ), f ∈ ∪v Iv }. Then we can write V (h) (τ, ψ (≤h) ) = V (h) (τ, P, Ω) . (2.47) P∈Pτ ,Ω∈Oτ

V

(h)

(τ, P, Ω) can be represented as (≤h) (h+1) (h) V (τ, P, Ω) = dxv0 ψ˜Ωv (Pv0 )Kτ,P,Ω (xv0 ) , 0

(2.48)

(h+1)

with Kτ,P,Ω (xv0 ) deﬁned inductively (recall that hv0 = h + 1) by the equation, valid for any v ∈ τ which is not an endpoint, sv 1 (h ) (h ) [K (hv +1) (xvi )] EhTv [ψ˜Ω1v (Pv1 \Qv1 ), . . . , ψ˜Ωsvv (Pvsv \Qvsv )] , sv ! i=1 vi (2.49) (hv ) ˜ where Ωi = {ω(f ), f ∈ Pvi \Qvi } and ψΩi (Pvi \Qvi ) has a deﬁnition similar to (2.42), if hv ≤ 0, while, if hv = +1, is deﬁned as in (2.43). (2) Moreover, if v is an endpoint, Kv (xv ) is deﬁned as in (2.44) if hv = 2, otherwise  if v is of type λ,  λhv −1 (x)δ(x0 ) Kv(hv ) (xv ) = γ hv −1 νhv −1 (x − y )δ(x0 − y0 ) if v is of type ν, (2.50)  zhv −1 (x − y)δ(x0 − y0 ) if v is of type z, (h )

v Kτ,P,Ω (xv ) =

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where xv = (x1 , x2 , x3 , x4 ) if v is of type λ, and xv = (x, y) in the other two cases. If vi is not an endpoint, (h +1)

v Kv(hi v +1) (xvi ) = RKτi ,P (i) ,Ω(i) (xvi ) ,

(2.51)

where τi is the subtree of τ starting from v and passing through vi (hence with root the vertex immediately preceding v), P(i) and Ω(i) are the restrictions to τi of P and Ω. The action of R is deﬁned using the representations (2.20), (2.21) of the regularization operation, as in (2.45), (2.46). Remark. In order to simplify (2.42) and the following discussion, we now decide to use the freedom in the choice of the ﬁeld that carries the ∂0 derivative in the endpoints of type z, so that, given any vertex v, which is not an endpoint of type z, m(f ) = 0 for all f ∈ Pv . (2.47) is not the ﬁnal form of our expansion, since we further decompose V (h) (τ, P, Ω), by using the following representation of the truncated expectation in the right-hand side of (2.49). Let us put s = sv , Pi ≡ Pvi \Qvi ; moreover we order in an arbitrary way the sets Pi± ≡ {f ∈ Pi , ε(f ) = ±}, we call fij± their − ), elements and we deﬁne x(i) = ∪f ∈P − x(f ), y(i) = ∪f ∈P + x(f ), xij = x(fi,j i i s s + − + yij = x(fi,j ). Note that i=1 |Pi | = i=1 |Pi | ≡ n, otherwise the truncated expectation vanishes. A couple l ≡ (fij− , fi+ j ) ≡ (fl− , fl+ ) will be called a line joining the ﬁelds with labels fij− , fi+ j and sector indices ωl− = ω(fl− ), ωl+ = ω(fl+ ) and connecting the points xl ≡ xi,j and yl ≡ yi j , the endpoints of l. Moreover, we shall put ml = m(fl− ) + m(fl+ ) and, if ωl− = ωl+ , ωl ≡ ωl− = ωl+ . Then, it is well known (see [Le], [BGPS], [GM] for example) that, up to a sign, if s > 1, (h) (h) EhT (ψ˜Ω1 (P1 ), . . . , ψ˜Ωs (Ps )) − + ∂0ml g˜ω(h) (x − y )δ dPT (t) det Gh,T (t) , = l l ω ,ω l T

where

l∈T

l

l

g˜ω(h) (x) = e−ipF (θh,ω )x gω(h) (x) ,

(2.52)

(2.53)

T is a set of lines forming an anchored tree graph between the clusters of points x(i) ∪ y(i) , that is T is a set of lines, which becomes a tree graph if one identiﬁes all the points in the same cluster. Moreover t = {ti,i ∈ [0, 1], 1 ≤ i, i ≤ s}, dPT (t) is a probability measure with support on a set of t such that ti,i = ui · ui for some s family of vectors ui ∈ R of unit norm. Finally Gh,T (t) is a (n − s + 1) × (n − − m(fij )+m(f + ) (h)

i j s + 1) matrix, whose elements are given by Gh,T g˜ωl (xij − ij,i j = ti,i ∂0 − + yi j )δω− ,ω+ with (fij , fi j ) not belonging to T . l l In the following we shall use (2.52) even for s = 1, when T is empty, by interpreting the right-hand side as equal to 1, if |P1 | = 0, otherwise as equal to det Gh = EhT (ψ˜(h) (P1 )).

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If we apply the expansion (2.52) in each non-trivial vertex of τ , we get an expression of the form (≤h) (h) (h) V (τ, P, Ω) = dxv0 ψ˜Ωv (Pv0 )Wτ,P,Ω\Ωv ,T (xv0 ) 0

T ∈T

≡

0

V (h) (τ, P, Ω, T ) ,

(2.54)

T ∈T

where T is a special family of graphs on the set of points xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v. Note that any graph T ∈ T becomes a tree graph on xv0 , if one identiﬁes all the points in the sets xv , for any vertex v which is also an endpoint. Remarks. An important role in this paper, as in [FMRT], will have the remark that, thanks to momentum conservation and compact support properties of propagator Fourier transforms, V (h) (τ, P, Ω) vanishes for some choices of Ω. This constraint will be made explicit below in a suitable way, see (2.79). (h) Note also that Wτ,P,Ω\Ωv ,T (xv0 ), as underlined in the notation, is independent 0

of Ωv0 , so that V (h) (τ, P, Ω, T ) depends on Ωv0 only through the external ﬁelds sector indices.

2.5

Detailed analysis of the R operation (h)

The kernels Wτ,P,Ω\Ωv ,T (xv0 ) in (2.54) have a rather complicated expression, 0 because of the presence of the operators R acting on the tree vertices, which are not endpoints. In order to clarify their structure, we have to further expand each term in the right-hand side of (2.54), by a procedure similar to that explained in [BM]. We start this analysis by supposing that |Pv0 | > 0 (otherwise there is no R operation acting on v0 ) and by considering the action of R on a single contribution to the sum in the right-hand side of (2.54). This action is trivial, that is R = I, by deﬁnition, if |Pv0 | > 4 or, since R2 = R, if v0 is a trivial vertex (sv0 = 1) and |Pv0 | is equal to |Pv¯ |, v¯ being the vertex (of scale h + 2) immediately following v0 . Hence there is nothing to discuss in these cases. Let us consider ﬁrst the case |Pv0 | = 4 and note that, by the remark following (2.51), m(f ) = 0 for all f ∈ Pv0 . If Pv0 = (f1 , f2 , f3 , f4 ), with ε(f1 ) = ε(f2 ) = ˜ i = (x1,0 , xi ), ω(fi ) = ωi , + = −ε(f3 ) = −ε(f4 ), and we put x(fi ) = xi , x pF,i = pF (θh,ωi ), we can write, by using (2.21), 4 RV (h) (τ, P, Ω, T ) = dx e i=1 εi pF,i xi W4 (x) · (x2,0 − x1,0 )ψx(≤h)+ [∂ˆ1 (x1,0 )ψx(≤h)+ ]ψx(≤h)− ψx(≤h)− 1 ,ω1 2 ,ω2 3 ,ω3 4 ,ω4 (≤h)+

+(x3,0 − x1,0 )ψx(≤h)+ ψx˜ 2 ,ω2 [∂ˆ1 (x1,0 )ψx(≤h)− ]ψx(≤h)− 1 ,ω1 3 ,ω3 4 ,ω4

(≤h)+ (≤h)− ˆ1 (≤h)− + (x4,0 − x1,0 )ψx(≤h)+ ψ ψ [ ∂ (x )ψ ] , 1,0 x4 ,ω4 ˜ 2 ,ω2 ˜ 3 ,ω3 x x 1 ,ω1

(2.55)

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(h)

where W4 (x) is the integral of Wτ,P,Ω,T (xv0 ) over the variables xv0 \x, up to a sign, and ∂ˆ1 (x0 ) is an operator deﬁned by

∂ˆ1 (x0 )F (y) =

1

0

ds∂0 F (ξ0 (s), y ) ,

ξ0 (s) = x0 + s(y0 − x0 ) .

(2.56)

Similar expressions are obtained, if the localization point (see comment after (2.14)) is changed. Let us now consider the case |Pv0 | = 2. If only one of the external ﬁelds of v0 carries a ∂0 derivative, the action of R would not be trivial. However, we can limit this possibility to the contribution corresponding to the tree with n = 1, whose only endpoint is of type z, which gives no contribution to RV (h) . In fact, if there is more than one endpoint, at most one of the ﬁelds of any endpoint of type z can belong to Pv0 , so that we can use the freedom in the choice of the ﬁeld which carries the derivative so that m(f ) = 0 for both f ∈ Pv0 (see remark after (2.51)). Hence, we have to discuss only the case m(f ) = 0 for both f ∈ Pv0 ; if we put Pv0 = (f1 , f2 ), x(fi ) = xi , ω(fi ) = ωi , pF,i = pF (θh,ωi ), we can write (2.57) RV (h) (τ, P, Ω, T ) (≤h)+ ˆ2 (≤h)− = dxdyei(pF,1 x−pF,2 y) (y0 − x0 )2 W (x − y)ψx,ω [∂ (x0 )ψy,ω ], 1 2 (h)

where W (x1 − x2 ) is the integral of Wτ,P,Ω,T (xv0 ) over the variables xv0 \(x1 , x2 ), up to a sign, and ∂ˆ2 (x0 ) is an operator deﬁned by (≤h)ε ∂ˆ2 (x0 )ψy,ω = 2

0

1

(≤h)ε

ds(1 − s)∂02 ψξ0 (s),y,ω

,

ξ0 (s) = x0 + s(y0 − x0 ) .

(2.58)

(≤h)+ Instead of (2.57), one could also use a similar expression with [∂ˆ2 (y0 )ψx,ω1 ] (≤h)− (≤h)+ (≤h)− ψy,ω2 in place of ψx,ω1 [∂ˆ2 (x0 )ψy,ω2 ]. We shall distinguish these two diﬀerent choices by saying that we have taken x, in the case of (2.57), or y, in the other case, as the localization point. By using (2.49) and (2.52), we can also write

RV (h) (τ, P, Ω, T ) (2.59) 1 (≤h) = dxv0 dPTv0 (t) R[ψ˜Ωv ,α (Pv0 )] (yα,0 − xα,0 )b(|Pv0 |) 0 sv0 ! α∈A

·

l∈Tv0

∂0ml g˜ω(h+1) (xl l

− yl )δω− ,ω+ det G

h+1,Tv0

l

l

(t)

sv0

[Kv(h+2) (xvi )] , i

i=1

where A is a set of indices containing only one element, except in the case |Pv0 | = 4, (≤h) when |A| = 3, and xα , yα are two points of xv0 . Moreover, R[ψ˜Ωv ,α (Pv0 )] = 0

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(≤h) ψ˜Ωv (Pv0 ), except if |Pv0 | = 4 or |Pv0 | = 2 and m(f ) = 0 for both f ∈ Pv0 ; in 0 these cases, its expression can be easily deduced from (2.55) and (2.57). Finally, b(p) is an integer, equal to 1, if p = 4, equal to 2, if p = 2, and equal to 0 otherwise. We would like to apply iteratively equation (2.55) and (2.57), starting from v0 and following the partial order of the tree τ , in all the τ vertices with |Pv | = 4 or |Pv | = 2 and m(f ) = 0 for f ∈ Pv . However, in order to control the combinatorics, it is convenient to decompose the factor (yα,0 − xα,0 )b(|Pv0 |) in the following way. Let us consider the unique subset (l1 , . . . , lm ) of Tv0 , which selects a path joining the cluster containing xα with the cluster containing yα , if one identiﬁes all the points in the same cluster; if this subset is empty (since xα and yα belong to the same cluster), we put m = 0. If m > 0, we call (¯ vi−1 , v¯i ), i = 1, m, the couple of vertices whose clusters of points are joined by li . We shall put x2i−1 , i = 1, m, equal to the endpoint of li belonging to xv¯i−1 , x2i equal to the endpoint of li belonging to xv¯i , x0 = xα and x2m+1 = yα . These deﬁnitions imply that there are two points of the sequence xr , r = 0, . . . , m ¯ = 2m + 1, possibly coinciding, in any set xv¯i , i = 0, . . . , m; these two points are the space-time points of two diﬀerent ﬁelds belonging to Pv¯i . Then, we can write

yα,0 − xα,0 =

m ¯

(xr,0 − xr−1,0 ) .

(2.60)

r=1

If we insert (2.60) in (2.59), the right-hand side can be written as the sum over a set Bv0 of diﬀerent terms, that we shall distinguish with a label αv0 ; note that |Bv0 | ≤ 3(2sv0 − 1)2 . We get an expression of the form 1 (≤h) RV (h) (τ, P, Ω, T ) = dxv0 dPTv0 (t)R[ψ˜Ωv ,α (Pv0 )] 0 sv0 ! αv0 ∈Bv0 (xl,0 − yl,0 )bl (αv0 ) ∂0ml g˜ω(h+1) (xl − yl )δω− ,ω+ · l l

l∈Tv0

· det Gh+1,Tv0 (t)

l

sv0 (i) (i) [(x0 − y0 )bvi (αv0 ) Kv(h+2) (xvi )] , i

(2.61)

i=1

where we called (x(i) , y(i) ) the couple of points which, in the previous argument, belong to xvi and bl (αv0 ), bvi (αv0 ) are integers with values in {0, 1, 2}, such that their sum is equal to b(|Pv0 |). Let us now see what happens, if we iterate the argument leading to (2.61). Let us suppose, for example, that |Pv1 | = 2, that the action of R is not trivial on v1 and that b ≡ bv1 (αv0 ) > 0. In this case, if we exploit the action of R in the (1) (1) form of (2.57), we have an overall factor (x0 − y0 )m , m = 2 + b, which multiplies (h+2) Kv1 (xv1 ). Hence, if we expand this factor, by using an equation similar to (2.60), we get terms with some propagator multiplied by a factor (xl,0 − yl,0 )bl , with bl > 2. If we further iterate this procedure, we can end up with an expansion,

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where some propagator is multiplied by a factor (xl,0 − yl,0 )bl with bl of order |h|, which would produce bad bounds. However, we can avoid very simply this diﬃculty, by noticing that, if we insert (2.20) in an expression like Jb = dxdyF1 (x)F2 (y)(y0 − x0 )b RW (x − y) , (2.62) we get, by a simple integration by part, if b = 2, J2 = dxdyF1 (x)F2 (y)(y0 − x0 )2 W (x − y) ,

(2.63)

that is the R operation can be substituted by the identity, while, if b = 1, we get J1 = dxdyF1 (x)[∂ˆ1 (x0 )F2 (y)](y0 − x0 )2 W (x − y) , (2.64) where ∂ˆ1 (x0 ) is the operator deﬁned by (2.56). This means that, if b = 1, the action of R only increases the power of (y0 − x0 ) by one unit. Note that, in (2.64) one could substitute F1 (x)[∂ˆ1 (x0 )F2 (y)] with −[∂ˆ1 (y0 )F1 (x)]F2 (y); we shall again distinguish these two diﬀerent choices by saying that we have taken x, in the case of (2.64), or y, in the other case, as the localization point. Even simpleris the situation, when |Pv1 | = 4. In fact, if we insert (2.21) in an expression like dxF (x)(y0∗ − x∗0 )RW4 (x), y ∗ and x∗ being two points of x, we get (2.65) dxF (x)(y0∗ − x∗0 )RW4 (x) = dxF (x)(y0∗ − x∗0 )W4 (x) , so that, even in this case, the power of the “zero” cannot increase. There are in principle two other problems. First of all, one could worry that there is an accumulation of the operators ∂ˆq (dimensionally equivalent to a derivative of order q) on a same line, if this line is aﬀected many times by the R operation in diﬀerent vertices. Moreover, since the deﬁnition of the ∂ˆq (x0 ) operators depends on the choice of the localization point x, it could happen that there is an “interference” between the R operations in two diﬀerent vertices, which would make more involved the expansion. However, one can show, by the same arguments given in §3.3 and §3.4 of [BM] in the one-dimensional case, that these problems can be avoided by using the freedom in the choice of the localization point and, mainly, the fact that some regularization operations are not really present. Let us consider, for example, the ﬁrst term in the right-hand side of (2.55) and note that, if we sum it over the sector indices, we get, in terms of Fourier transforms, an expression of the type dk

4

(≤h),εi δ(k1 + k2 − k3 − k4 ) ψˆki

i=1

ˆ 4 (k1 , (0, k2 ), k3 ) . ˆ 4 (k1 , k2 , k3 ) − W · W

(2.66)

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(≤h),+ ˆ 4 (k1 , However, if f¯ is the label of the ﬁeld ψx2 , it is easy to see that W (0, k2 ), k3 ) = 0, if there is a vertex v¯ > v0 with four external legs, such that f ∈ Pv¯ and f is aﬀected by the R operation in v¯. Hence, in this case, we can (≤h),ε substitute the ﬁrst term in the braces of (2.55) with i ψxi ,ωi i . We refer to §3.3 and §3.4 of [BM] for a complete analysis of this problem, whose ﬁnal result is that the action of R on all the vertices of τ will produce terms where the propagators related with the lines of T are multiplied by a factor (xl,0 − yl,0 )bl with bl ≤ 2 and (after that) are possibly subject to one or two operators ∂ˆq , q = 1, 2. Moreover, some of the external lines belonging to Pv0 can be aﬀected from one operator ∂ˆq , as a consequence of the action of R on v0 or some other vertex v > v0 . Finally, the lines involved in the determinants may be aﬀected from one operator ∂ˆq . We introduce an index α to distinguish these diﬀerent terms and, given α, we shall denote by ∂ˆqα (f ) the diﬀerential operators acting on the external lines of Pv0 or the propagators belonging to T , as a consequence of the regularization procedure. All the previous considerations imply that RV (h) (τ, P, Ω, T ) = 0, if |Pv0 | = 4 and n = 1 (that is there is only an endpoint of type λ and no internal line associated with v0 ) or Pv0 = (f1 , f2 ) and m(f1 ) + m(f2 ) = 1 (since this can happen only if n = 1 and the endpoint is of type z, as a consequence of the freedom in the choice of the ﬁeld carrying the derivative in the endpoints of type z) or m(f1 )+m(f2 ) = 0 and n = 1. In all the other cases, we can write RV (h) (τ, P, Ω, T ) in the form (≤h) (h) (2.67) RV (τ, P, Ω, T ) = dxv0 Wτ,P,Ω\Ωv0 ,T,α (xv0 )R[ψ˜Ωv ,α (Pv0 )] , 0

α∈AT

where (≤h) (Pv0 )] 0 ,α

R[ψ˜Ωv

=

f ∈Pv0

(≤h)ε(f )

eiε(f )pF (θh,ω(f ) )x(f ) [∂ˆqα (f ) ψ]xα (f ),ω(f ) ,

(2.68)

and, up to a sign, Wτ,P,Ω\Ωv0 ,T,α (xv0 ) n ! hi = Kv∗ (xvi∗ ) i

i=1

v

1 sv !

dPTv (tv ) det Ghαv ,Tv (tv )

(2.69)

not e.p. " qα (fl− ) qα (fl+ ) bα (l) ml (hv ) ˆ ˆ δω+ ,ω− ∂ (xl,0 )∂ (yl,0 )[(xl,0 − yl,0 ) ∂0 g˜ωl (xl − yl )] , · l∈Tv

l

l

where “e.p.” is an abbreviation of “endpoint” and, together with the deﬁnitions used before, we are using the following ones: 1. AT is a set of indices which allows to distinguish the diﬀerent terms produced by the non-trivial R operations and the iterative decomposition of the zeros;

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2. v1∗ , . . . , vn∗ are the endpoints of τ and hi = hvi∗ ; 3. bα (v), bα (l), qα (fl− ) and qα (fl+ ) are positive integers ≤ 2; 4. if qα (fl− ) > 0, xl,0 denote the time coordinate of the point involved, together with xl , in the corresponding R operation, see (2.58) and (2.56), otherwise ∂ˆ0 (xl,0 ) = I; v ,Tv hv ,Tv (tv ) 5. if v is a non-trivial vertex (so that sv > 1), the elements Ghα,ij,i j of Gα are of the form v ,Tv Ghα,ij,i j = ti,i − qα (fij )

· ∂ˆ0

(2.70)

qα (f + ) m(f − ) m(f − ) (x l,0 )∂ˆ0 i j (y l,0 )∂0 l ∂0 l g˜ω(hl v ) (xij

− yi j )δω− ,ω+ ; l

l

if v is trivial, Tv is empty and dPTv (tv ) det Ghαv ,Tv (tv ) has to be interpreted as 1, if |Iv | = 0 (Iv is the set of internal ﬁelds of v), otherwise it is the determinant of a matrix of the form (2.70) with ti,i = 1.

2.6

Modiﬁcation of the running coupling functions

We want now to introduce a diﬀerent representation of the running coupling functions λh , νh , zh , in order to include in the new deﬁnitions the momentum constraints on the external lines of the corresponding vertices. To remember these constraints in the iterative calculation of the (so modiﬁed) running coupling functions will play an essential role. Note that, if we substitute (2.69) in (2.67) and we express the whole integral in Fourier space, the Fourier transform of Kvh∗i (xvi∗ ) is multiplied by the factor i

f ∈Pv∗ ∩Pv0

≤h 0 ,ε(f ) ψˆk(fv),ω(f )

i

Fh(f ),ω(f ) (k(f )) .

(2.71)

f ∈Pv∗ \Pv0 i

In order to use this property, we deﬁne, for any h ≤ 0 and ω ∈ Oh , the s-sector Sh,ω (see §2.1 and §2.3 for related deﬁnitions) as 2

Sh,ω = {k = ρer (θ) ∈ R : |ε(k) − µ| ≤ γ h e0 , ζh,ω (θ) = 0} .

(2.72)

Note that the deﬁnition of s-sector has the property, to be used extensively in the following, that the s-sector Sh+1,ω of scale h + 1 contains the union of two s-sectors of scale h: Sh+1,ω ⊇ {Sh,2ω ∪ Sh,2ω+1 }, as follows from the deﬁnition of ζh,ω , see (2.23). ≤hv0 ,ε(f ) We now observe that the ﬁeld variables ψˆk(f ),ω(f ) have the same supports −1 as the functions Chv (k(f )) ζhv0 ,ω(f ) (θ(f )) and h(f ) ≤ hi − 1, ∀f ∈ Pvi∗ ; hence in 0 ˆ h∗i (kv∗ ) by F˜h −1,˜ω(f ) (k), the expression (2.69), we can freely multiply K vi

i

f ∈Pv∗ i

i

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where F˜h,ω (k) is a smooth function = 1 on Sh,ω and with a support slightly greater than Sh,ω , while ω ˜ (f ) ∈ Ohi −1 is the unique sector index such that Sh(f ),ω(f ) ⊆ Shi −1,˜ω(f ) . In order to formalize this statement, it is useful to introduce the following deﬁnition. Let G(x) be a function of 2p variables x = (x1 , . . . , x2p ) with Fourier trans ˆ k), where ˆ k), deﬁned so that G(x) = dk(2π)−4p exp(−i 2p εiki xi )G( form G( l=1 ε1 , . . . , εp = −εp+1 = . . . = −ε2p = +1. Then, we deﬁne, given h ≤ 0 and a family σ = {σi ∈ Oh , i = 1, . . . , 2p} of sector indices, 2p dk −i 2p εiki xi ˜ ˆ k) . l=1 Fh,σi (ki ) G( e (2.73) (F2p,h,σ ∗ G)(x) = (2π)4p i=1 In order to extend this deﬁnition to the case h = 1, when the sector index can take only the value 0, we deﬁne F˜1,0 (k) as a smooth function of compact support, equal to 1 on the support of C¯0−1 (k), deﬁned in §1.2. ˜ i = {˜ ω (f ), f ∈ Pvi∗ } and we deﬁne, for any Hence, if we put pi = |Pvi∗ |, Ω family σ = {σ(f ) ∈ Ohi −1 , f ∈ Pvi∗ } of sector indices of scale hi − 1, labeled by ˜ i is a particular example of such a family), the set Pvi∗ (Ω ˜ h∗i (xv∗ ) = Fp ,h −1,σ ∗ K h∗i (xv∗ ) , K (2.74) v ,σ v i i i i i

i

˜ h∗i (xv∗ ). If v ∗ is of type ν, z or we can substitute in (2.69) each Kvh∗i (xvi∗ ) with K i ˜i i vi ,Ω i ˜ h∗i (xv∗ ) can be written as γ hi −1 δ(x0,v∗ )˜ ∗ λ, K ν ( x ), δ(x zhi −1,σ (xvi∗ ) or hi −1,σ vi 0,vi∗ )˜ vi ,σ i i ˜ ˜ h −1,σ (xv∗ ) δ(x0,vi∗ ) λhi −1,σ (xvi∗ ) respectively. ν˜hi −1,σ (xi −yi ), z˜hi −1,σ (xi −yi ) and λ i i will be called the modiﬁed coupling functions. (mod) We shall call Wτ,P,Ω,T,α (xv0 ) the expression we get from Wτ,P,Ω\Ωv0 ,T,α (xv0 ) by the substitution of the running coupling functions with the modiﬁed ones. Note (mod) that Wτ,P,Ω,T,α (xv0 ) is not independent of Ωv0 , unlike Wτ,P,Ω\Ωv0 ,T,α (xv0 ), and (mod)

that Wτ,P,Ω,T,α (xv0 ) is equal to Wτ,P,Ω\Ωv0 ,T,α (xv0 ), only if |Pv0 | = 0; however, the previous considerations imply that, if p0 = |Pv0 | > 0, (mod) (xv0 ) = Fp0 ,h,Ωv0 ∗ Wτ,P,Ω\Ωv0 ,T,α (xv0 ) , Fp0 ,h,Ωv0 ∗ Wτ,P,Ω,T,α (2.75) a trivial remark which will be important in the discussion of the running coupling functions ﬂow in §4.

2.7

Bounds for the eﬀective potentials and the free energy

Given a vertex v of a tree τ and an arbitrary family S¯ = {Sjf ,σf , f ∈ Pv } of s-sectors labeled by Pv , we deﬁne   ¯ = χ ∀f ∈ Pv , ∃k(f ) ∈ Sj ,σ : χv (S) ε(f )k(f ) = 0 , (2.76) f f f ∈Pv

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where χ(condition) is the function = 1 when condition is veriﬁed, and = 0 in the opposite case. Moreover, given a set P of ﬁeld labels, we denote by S(P ) the special family of s-sectors labeled by P , deﬁned as S(P ) = {Sh(f ),ω(f ) , f ∈ P } .

(2.77)

The previous considerations imply that EL,β ≤

0

∞

Jh,n (0, 0) ,

(2.78)

h=hβ −1 n=1

with Jh,n (2l0 , q0 )

=

τ ∈Th,n

·

∗

P∈Pτ :|Pv0 |=2l0 , f ∈Pv0 qα (f )=q0

T ∈T α∈AT Ω∈Oτ

(mod) d(xv0 \x∗ ) Wτ,P,Ω,T,α (xv0 ) ,

χv (S(Pv ))

v

(2.79)

where x∗ is an arbitrary point in xv0 , l0 is a non-negative integer and ∗Ω∈Oτ diﬀers from Ω∈Oτ since one ω index, arbitrarily chosen among the 2l0 ω’s in Ωv0 , is not summed over, if l0 > 0, otherwise it coincides with Ω∈Oτ . Remarks. Note that we could freely insert [ v χv (S(Pv ))] in (2.79), because of the constraints following from momentum conservation and the compact support properties of propagator’s Fourier transform. Note also that, if l0 = 0, given τ ∈ Th,n , the number of internal lines in the lowest vertex v0 (of scale h + 1) has to be diﬀerent from zero. Hence, in order to prove that the free energy and the eﬀective potentials are well deﬁned (in the limit L → ∞ and β not “too large”), we need a “good” bound of Jh,n (2l0 , q0 ). In order to get this bound, we shall extend the procedure used in[BM] for the analysis of the one-dimensional Fermi systems, which we shall refer to for some details (except for the sum over the sector indices, which is a new problem). An important role has the following bound for the determinants appearing in (2.69): sv | det Ghαv ,Tv (tv )| ≤ c i=1 |Pvi |−|Pv |−2(sv −1) sv sv 3 · γ hv 4 ( i=1 |Pvi |−|Pv |−2(sv −1)) γ hv i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )] + − + − (2.80) · γ −hv l∈Tv [qα (fl )+qα (fl )+m(fl )+m(fl )] , where, if P ⊂ Iv0 , we deﬁne qα (P ) = f ∈P qα (f ) and m(P ) = f ∈P m(f ). The proof of (2.80) is based on the well-known Gram-Hadamard inequality, stating that, if M is a square matrix with elements Mij of the form Mij = Ai , Bj ,

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Low Temperature Analysis of Two-Dimensional Fermi Systems

163

where Ai , Bj are vectors in a Hilbert space H with scalar product < ·, · > and induced norm || · ||, then | det M | ≤ ||Ai || · ||Bi || . (2.81) i

Let H = R

|Oh |

3

⊗ R ⊗ L2 (R ); it can be shown that s

(h )

(h )

v v ,Tv , vω+ ⊗ ui ⊗ Bx(fv + Ghα,ij,i j = vω − ⊗ ui ⊗ A x(f − ),ω − l

ij

|O |

l

),ωl+ i j

l

,

(2.82)

where vω ∈ R h , ω ∈ Oh , and ui ∈ R , i = 1, . . . , s, are unit vectors such that (h ) (h ) vω · vω = δω,ω , ui · ui = ti,i ; moreover, Ax(fv − ),ω , Bx(fv + ),ω are deﬁned so that: s

ij

q (fi+ j )

−

qα (f ) α ∂ˆ0 ij (x l,0 )∂ˆ0 (h )

(h )

= Ax(fv − ),ω , Bx(fv + ij

l

i j

l

i j

l

m(fl− ) m(fl+ ) (hv ) ∂0 g˜ωl (xij

(y l,0 )∂0 ≡ ),ω l

− yi j )

(2.83)

dk ∗(hv ) (h ) A − (k)Bx(fv + ),ω (k) , l (2π)3 x(fij ),ωl i j

with ||Ai || · ||Bi || satisfying the same dimensional bound as the left side of (2.83). For example, if qα (fij− ) = qα (fi+ j ) = 0, one can put, Fhv ,ωl ikx m(f − ) m(f + ) v) (k) = e A(h 2 (ik0 ) l (ik0 ) l x,ωl k02 + ε(k) − µ (hv ) ikx ik0 + ε(k) − µ . Bx,ω (k) = e F (2.84) h ,ω v l l Using Lemma 2.1 and (2.81), we easily get (2.80). The next step is to bound by 1 the integrals over the probability measures dPTv appearing in (2.69). After that, we bound the integral n 1 ∗ ˜ h∗i (xv∗ ) K (2.85) d(xv0 \x ) ˜i i v , Ω i sv ! i=1 v not e.p. qα (f − ) + q (f ) α m b (l) (h ) l l l α v ∂ˆ0 · (x l,0 )∂ˆ0 (y l,0 )[(xl,0 − yl,0 ) ∂0 g˜ωl (xl − yl )] . l∈Tv

We can take from §3.15 of [BM] the identity (independent of the dimension): d(xv0 \x∗ ) = drl , (2.86) l∈T ∗

where T ∗ is a tree graph obtained from T = ∪v Tv , by adding in a suitable (obvious) way, for each endpoint vi∗ , i = 1, . . . , n, one or more lines connecting the spacetime points belonging to xvi∗ . Moreover rl = (ξ0 (tl ) − η0 (sl ), xl − yl ) (see (2.56)), if l ∈ ∪v Tv , and rl = xl − yl , if l ∈ T ∗ \ ∪v Tv .

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Hence (2.85) can be written as

Jτ,P,T,α

l∈T ∗ \∪v Tv

with Jτ,P,T,α =

v not

e.p.

1 sv !

n h i ˜ ∗ (xv∗ ) , drl K ˜i i vi ,Ω

(2.87)

i=1

drl

(2.88)

l∈Tv

q (f + ) q (f − ) · ∂ˆ0α l (x l,0 )∂ˆ0α l (y l,0 )[(xl,0 − yl,0 )bα (l) ∂0ml g˜ω(hl v ) (xl − yl )] . By using Lemma 2.1, we can bound each propagator, each derivative and each zero by a dimensional factor, so ﬁnding 1 Jτ,P,T,α ≤ cn c2(sv −1) γ −hv l∈Tv bα (l) sv ! v not e.p. + − + − (2.89) · γ −hv (sv −1) γ hv l∈Tv [qα (fl )+qα (fl )+m(fl )+m(fl )] . Let us now deﬁne, for any set of ﬁeld indices P , Oh (P ) = ⊗f ∈P Oh . The next step is to use the following lemma, to be proved in §3. Lemma 2.2 Suppose that there exist two constants C1 and Cν such that the modiﬁed coupling functions satisfy the following conditions: i) if |Pv | = 4, then ∗ ˜ h −1,σ (xv )| ≤ 2C1 |λ|γ − 12 (hv −1) , (2.90) d(xv \x∗ )|λ v σ∈Ohv −1

∗

means that one of the sector indices is not summed over; where ii) if |Pv | = 2 and xv = (x1 , x2 ), then ∗ νhv −1,σ (x1 − x2 )| ≤ 2C1 Cν |λ| , dx1 |˜

(2.91)

σ∈Ohv −1 ∗

zhv −1,σ (x1 − x2 )| ≤ C1 |λ| . dx1 |˜

(2.92)

σ∈Ohv −1

Consider a tree τ ∈ Th,n , a graph T ∈ T and the corresponding tree graph T ∗ , deﬁned as after (2.86). Then n ∗ hi ˜ δω+ ,ω− drl Kv∗ ,Ω˜ (xvi∗ ) χv (S(Pv )) l l i i ∗ Ω∈Oτ

v∈τ

l∈Tv

l∈T \T

i=1

Vol. 4, 2003

≤

Low Temperature Analysis of Two-Dimensional Fermi Systems 1

cn |λ|n γ − 2 h[m4 (v0 )+χ(Pv0 =∅)]

γ (hi −1)χ(v is of type ν)

i=1

·

n

165

1 1 1 γ [− 2 m4 (v)+ 2 (|Pv |−3)χ(4≤|Pv |≤8)+ 2 (|Pv |−1)χ(|Pv |≥10)] , (2.93)

v not e.p.

where m4 (v) denotes the number of endpoints of type λ following the vertex v. v |Pvi | − |Pv | − 2(sv − 1) ≤ 4n, v (sv − 1) = n − 1 and |AT | ≤ cn , Since si=1 (2.80), (2.88) and Lemma 2.2 imply that |Jh,n (2l0 , q0 )| ≤ (c|λ|)n · τ ∈Th,n

τ :|Pv0 |=2l0 , P∈P qα (f )=q0 f ∈P v0

T ∈T

1

γ − 2 h(m4 (v0 )+χ(l0 =0))

n

γ (hi −1)χ(v is of type ν)

i=1

1 sv 3 γ hv 4 ( i=1 |Pvi |−|Pv |−2(sv −1)) · sv ! v not e.p. sv · γ hv i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )] γ −hv l∈Tv bα (l) γ −hv (sv −1)

1

1

(2.94)

1

· γ [− 2 m4 (v)+ 2 (|Pv |−3)χ(4≤|Pv |≤8)+ 2 (|Pv |−1)χ(|Pv |≥10)] . Note now that the constraints on the values of qα (f ) and bα (l) imply, as shown in detail in §3.11 of [BM], that

hv

v not e. p.

sv

qα (Pvi \Qvi ) + h q0 =

h(f )qα (f ) ,

(2.95)

f ∈Iv0

i=1

γ h(f )qα (f )

f ∈Iv0

γ −hα (l)bα (l) ≤

γ −z(v)

(2.96)

v not e.p.

l∈T

!

where

2 if |P (v)| = 2 , (2.97) 1 if |P (v)| = 4 , 0 otherwise. Moreover, since the freedom in the choice of the ﬁeld carrying the derivative in the endpoints of type z was used (see remark a few lines before (2.57)) so that m(Pv ) = 0, if v is not an endpoint, and the ﬁeld with m(f ) = 1 belonging to the endpoint v is contracted in the vertex immediately preceding v, whose scale is hv − 1, we have the identity z(v) =

v not e.p.

γ hv

sv

i=1

[m(Pvi \Qvi )] =

n i=1

γ (hi −1)χ(v is of type z) .

(2.98)

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Ann. Henri Poincar´e

Putting together the previous bounds and supposing that the hypothesis (2.90), (2.91), (2.92) of Lemma 2.2 are veriﬁed, we ﬁnd that

Jh,n (2l0 , q0 ) ≤ (c|λ|)n

τ ∈Th,n

n

1

1

γ −h[ 2 m4 (v0 )+ 2 χ(l0 =0)+q0 ]

τ :|Pv0 |=2l0 , T ∈T P∈P qα (f )=q0 f ∈P v0

1 hv [ 34 (si=1 v |Pvi |−|Pv |)− 52 (sv −1)] γ s ! v v not e. p. i=1 −z(v)− 12 m4 (v)+ 12 (|Pv |−3)χ(4≤|Pv |≤8)+ 12 (|Pv |−1)χ(|Pv |≥10)] [ ·γ . (2.99)

·

γ

(hi −1)χ(|P (v)|=2)

On the other hand, if m2 (v) denotes the number of endpoints of type ν or z following v, we have, if v˜ is not an endpoint, the identities s v |Pvi | − |Pv | = 4m4 (˜ v ) + 2m2 (˜ v ) − |Pv˜ | , v≥˜ v v not e. p.

i=1

(sv − 1) =

m4 (˜ v ) + m2 (˜ v) − 1 ,

(2.100)

v≥˜ v

which, together with (2.99) imply that Jh,n (2l0 , q0 )

τ ∈Th,n

P |Pv0 |=2l0

T ∈T

v not e.p.

≤ (c|λ|)n γ h[−q0 +δext (2l0 )]

(2.101) 1 δ(|Pv |) γ , sv !

where δ(p) = −χ(2 ≤ p ≤ 4) p p + 1− χ(6 ≤ p ≤ 8) + 2 − χ(p ≥ 10) , 4 4 δext (p) =

1 5 3 − p − χ(p = 0) . 2 4 2

(2.102) (2.103)

Since δ(|Pv |) < 0, for any vertex v, which is not an endpoint, a standard argument, see [BM] or [GM], allows to show that

τ ∈Th,n

P |Pv0 |=2l0

T ∈T

v not e.p.

1 δ(|Pv |) γ ≤ cn . sv !

The bounds (2.101) and (2.104) imply the following theorem.

(2.104)

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167

Theorem 2.1 If conditions (2.90), (2.91), (2.92) are satisﬁed, then Jh,n (2l0 , q0 ) ≤ (c|λ|)n γ h[−q0 +δext (2l0 )] .

(2.105)

Remark. We will prove in §4 that, if |λ| is small enough and c0 log β|λ| ≤ 1, where c0 is a constant depending only on ﬁrst and second order contributions of perturbation theory, it is possible to choose ν˜1 (x) so that the modiﬁed running coupling functions satisfy the hypotheses of Lemma 2.2, (2.90), (2.91) and (2.92). So, in that case, we see from Theorem 2.1 that limL→∞ EL,β does exist and is of order λ.

3 Proof of Lemma 2.2 3.1

The sector counting lemma

In order to present the proof of Lemma 2.2, we need to introduce some new deﬁnitions. 1. Given a tree τ and P ∈ Pτ , we shall call χ-vertices the vertices v of τ , such that Iv (the set of internal lines, that is the lines contracted in v) is not empty. We shall also call Vχ the family of all χ-vertices, whose number is of order n. 2. Given h ≤ 0 and a set of ﬁeld indices P , we deﬁne Oh (P ) = ⊗f ∈P Oh and we shall call σ = {σf ∈ Oh , f ∈ P } the elements of Oh (P ). 3. Given h ≤ 0 and σ ∈ Oh (P ), we deﬁne

Sh (σ) = {Sh,σf , σf ∈ σ}.

4. Given a set of ﬁeld indices P and two families of s-sectors labeled by P , S (i) = {Sj (i) ,σ(i) , f ∈ P }, i = 1, 2, we shall say that S (1) ≺ S (2) , if Sj (1) ,σ(1) ⊂ f

f

f

Sj (2) ,σ(2) , for any f ∈ P . f

f

f

The main point in the proof is the following lemma, which is an extension of that proved in [FMRT] in the Jellium case; see §7 for a proof. Lemma 3.1 Let h , h, L be integers such that h ≤ h ≤ 0. Let v be a vertex of a tree τ , such that |Pv | = L and f1 a ﬁxed element of Pv . Then, given the sector index σf1 ∈ Oh , and a set σ ∈ Oh (Pv \f1 ), the following bound holds: σ ∈O (Pv \f1 ) h (σ )≺ h (σ) h

S

S

' ( χv {Sh ,σf1 } ∪ Sh (σ ) ≤

cL γ c

h−h 2

(L−3)

, if L ≥ 4, , if L = 2 .

(3.1)

168

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Ann. Henri Poincar´e

Proof of Lemma 2.2

First of all, we note that

χv (S(Pv )) =

v

χv (S(Pv )) .

(3.2)

v∈Vχ

Let us consider ﬁrst the case Pv0 = ∅ and let v˜0 be the ﬁrst χ-vertex following the root (possibly equal to v0 ); note that Pv˜0 = Pv0 and that h(f ) = h for any f ∈ Pv˜0 . In the following it will also very important to remember that Ω is the family of all sector indices ω(f ) associated with the ﬁeld labels f and that ω(f ) ∈ ﬁeld Oh(f ) , h(f ) being the scale of the propagator connected to the corresponding ¯ is a subset of Ω, ¯ will variable, see §2.4. In agreement with this deﬁnition, if Ω Ω ¯ denote the sum over ω(f ) ∈ Oh(f ) , for any f ∈ Ω. Let us call f0 the ﬁeld whose sector index ω(f0 ) ∈ Oh is ﬁxed in the sum over Ω. We rewrite the sector sum in the left-hand side of (2.93) as: ∗ Ω

=

∗ Ωv˜0 Ω\Ωv˜0

∗

σ v˜ ∈Ohv˜ (Pv˜0 \f0 )

Ωv ˜0 : (σ v S(Pv ˜0 ) ˜0 \f0 )≺ hv ˜0

Ω\Ωv˜0

= 0

0

S

.

(3.3)

Then, for any ﬁxed σ v˜0 ∈ Ohv˜0 (Pv˜0 \f0 ), we bound the product of χv functions as

χv (S(Pv )) ≤ χv˜0 (S(Pv˜0 ))

χv (S˜v,˜v0 ) ,

(3.4)

v∈{Vχ \˜ v0 }

v∈Vχ

where ) * S˜v,˜v0 = S Pv \(Pv˜0 \f0 ) ∪ Shv˜0 ,σf ∈ Shv˜0 (σ v˜0 ), f ∈ Pv ∩ (Pv˜0 \f0 ) .

(3.5)

In other words, for any v = v˜0 , we relax the sector condition by allowing the external ﬁelds of v, which are also external ﬁelds of v˜0 and are not equal to f0 , to have a momentum varying, instead than in the original sector, of scale h, in that of scale hv˜0 containing it. Let us now observe that the modiﬁed running coupling functions do not depend on Ωv˜0 , if σ v˜0 is ﬁxed, as it follows from deﬁnition (2.74); hence the only remaining dependence on Ωv˜0 is in χv˜0 (S(Pv˜0 )). It follows, by using Lemma 3.1 for |Pv˜0 | ≤ 8 and the trivial bound ∗ Ωv ˜0 S(Pv ˜0 \f0 )≺ hv ˜0 ) ˜0 (σv

S

1

1 ≤ cγ 2 (hv˜0 −h)(|Pv˜0 |−1) ,

(3.6)

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for |Pv˜0 | ≥ 10, that we can bound the sum over Ωv˜0 , for any ∗

169

Shv˜0 (σ v˜0 ), as

χv˜0 (S(Pv˜0 ))

Ωv ˜0 S(Pv (σ v ˜0 \f0 )≺ ˜0 ) hv ˜0

S

1

1

≤ cγ (hv˜0 −h)[ 2 (|Pv˜0 |−3)χ(4≤|Pv˜0 |≤8)+ 2 (|Pv˜0 |−1)χ(|Pv˜0 |≥10)] .

(3.7)

We are thus left with the problem of bounding a sum similar to the initial one, but with all the external sector indices on scale hv˜0 instead of h. We shall do that by iterating the previous procedure, in a way which depends on the structure of the tree τ and of the graph T ; the iteration stops at the endpoints, where we can use the hypotheses (2.90), (2.91) and (2.92). To describe this inductive procedure, we establish, for any vertex v ∈ Vχ , a partial ordering of the sv vertices v1 , . . . , vsv ∈ Vχ immediately following v on τ , by assigning a root to the tree graph T ∗ and to each anchored tree graph Tv . We decide that the root of T ∗ is the space-time point containing f0 ; then we assign a direction to the lines of the tree graph T ∗ , the one which goes from the root towards the leaves. Finally we decide that the root of Tv is the vertex which the line of Tv enters, where v is the χ-vertex immediately preceding v ∈ Vχ , if f0 ∈ Pv ; otherwise, the root of Tv is the vertex containing the root of T ∗ , see Fig. 2.

v3

v0

3

f0

v2 1

v1

Figure 2: A possible cluster structure corresponding to a tree τ of the expansion for the eﬀective potentials such that sv˜0 = 3. The set Tv˜0 is formed by the lines 1 and 3 . The lines diﬀerent from 1 and 3 and not belonging to Pv˜0 have to be contracted into the Lesniewski determinants. The left-hand side of (2.93) is bounded by the product of the right-hand side of (3.7) and the following quantity:   n ˜ hi ˜   χv Sv,˜v0 δω+ ,ω− drl Kv∗ ,Ω˜ (xvi∗ ) , (3.8) ˜ v,˜ v>˜ v0 ,v∈Vχ Ω v0

l∈T

l

l

l∈T ∗ \T

i=1

i

i

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Ann. Henri Poincar´e

where ˜ v,˜v0 = {Ωv \Ωv˜0 } ∪ {σf ∈ Ohv˜ , f ∈ Pv ∩ (Pv˜0 \f0 )} . Ω 0

(3.9)

˜ v,˜v0 and that, if v is a Note that there is no sector index associated with f0 in Ω ˜ v,˜v0 χ-vertex immediately following v˜0 on τ , all the sector indices included in Ω belong to Ohv˜0 , since in this case the ﬁelds associated with Pv \Pv˜0 are contracted on scale hv˜0 . We now consider the sv˜0 χ-vertices immediately following v˜0 and we reorder the expression (3.8) in the following way:    sv˜0    χv (S˜v,˜v0 ) δω+ ,ω−  (3.8) =  χvj (S˜vj ,˜v0 ) l l j=1

·

˜ v,˜ ∪v≥vj Ω v0

vi∗ ≥vj

v>vj v∈Vχ

l∈∪v≥vj Tv



˜ hi ∗ ) (x δω+ ,ω− , drvi∗ K v ˜ i v ∗ ,Ω i

i

l

(3.10)

l

drl , where Tvi∗ denotes the subset of the tree graph T connecting the set xvi∗ ; ii) if sv = 1, l∈Tv δω+ ,ω− has to be thought as l l equal to 1. We now choose a leave of Tv0 (v1 or v3 in Fig. 2), say v ∗ , and we consider the sv˜0 appearing in the right-hand side of (3.10) corresponding factor in the product j=1 to v ∗ , together with the line l∗ ∈ Tv0 entering v ∗ (1 or 3 in Fig. 2). We can associate with v ∗ the following quantity, which is independent of all the other leaves and of the sector indices associated with the lines of Tv0 :   ∗ ∗   ∗ ˜ ∗ ∗ [v ] =  χv (S˜v,˜v0 ) χv (Sv ,˜v0 ) where: i)

∗

drvi∗ is equal to

l∈Tv˜0

l∈Tv∗ i

˜ v∗ ,˜ Ω v0

·



δω+ ,ω− 

l∈∪v≥v∗ Tv

l

l

˜ v,˜ ˜ ∪v>v∗ Ω v0 \Ωv∗ ,˜ v0

i:vi∗ ≥v ∗

v>v∗ v∈Vχ

 h ˜ ∗i (xv∗ ) , drvi∗ K ˜ i v ,Ω i

i

(3.11)

∗ where Ω˜ v∗ ,˜v means that we do not sum over the sector index associated with l∗ . 0 In order to bound the expression in the right-hand side (3.11), we have to distinguish two cases ∗ ∗ (a) v ∗ is an endpoint. In this case Ω˜ v∗ ,˜v = σ∈Ωh ∗ −1 and χv∗ (S˜v∗ ,˜v0 ) = 1, 0 v since the corresponding constraint is already included in the deﬁnition of the modiﬁed coupling functions, so that the expression to bound is simply: ∗ ˜ hv ∗ (3.12) drv∗ K v ∗ ,σ (xv ∗ ) . σ∈Ωhv∗ −1

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171

Hence, conditions (2.90), (2.91), (2.92) imply that 1

[v ∗ ] ≤ c|λ|γ − 2 (hv∗ −1)χ(|Pv∗ |=4) γ (hv∗ −1)χ(v

∗

is of type ν)

.

(3.13)

(b) v ∗ is not an endpoint. In this case, by the remark following (3.9), the expression in the right-hand side of (3.11) has exactly the same structure as the left-hand side of (2.93), which we started the iteration from; one has only to substitute v˜0 with v ∗ , h with hv˜0 and hv˜0 with hv∗ . Hence we can bound the right-hand side of (3.11) by extracting a factor 1

1

cγ (hv∗ −hv˜0 )( 2 (|Pv∗ |−3)χ(4≤|Pv∗ |≤8)+ 2 (|Pv∗ |−1)χ(|Pv∗ |≥10))

(3.14)

and we end up with an expression similar to (3.8), the line l∗ acting now as an external ﬁeld, since there is only one sector sum associated with it, thanks to the factor δω+∗ ,ω−∗ present in the right-hand side of (3.10). l

l

It is now completely obvious that we can iterate the previous procedure, for each leave of Tv˜0 , ending up with a bound of the left-hand side of (2.93) of the form   1 1 γ (hv −hv )( 2 (|Pv |−3)χ(4≤|Pv |≤8)+ 2 (|Pv |−1)χ(|Pv |≥10))  (c|λ|)n  ·

v∈Vχ n

γ

− 12 (hi −1)χ(vi∗ is of type λ) (hi −1)χ(vi∗ is of type ν)

γ

,

(3.15)

i=1

where v is the χ-vertex immediately preceding v on τ , if v > v˜0 , or the root, if v = v˜0 . On the other hand, given v ∈ Vχ , Pv¯ = Pv if v < v¯ ≤ v. Moreover, n

∗

1

γ − 2 (hi −1)χ(vi

is of type λ)

1

= γ − 2 hm4 (v0 )

1

γ − 2 m4 (v) ,

(3.16)

v not e.p.

i=1

where m4 (v) is the number of end points of type λ following vertex v on τ . It follows that (3.15) can be written in the form (c|λ|)n γ ·

− 12 hm4 (v0 )

n

γ

(hi −1)χ(vi∗

is of type ν)

i=1 1 1 1 γ [− 2 m4 (v)+ 2 (|Pv |−3)χ(4≤|Pv |≤8)+ 2 (|Pv |−1)χ(|Pv |≥10)] ,

(3.17)

v not e.p.

which proves Lemma 2.2 in the case |Pv0 | > 0. The case Pv0 = ∅ is treated in a similar way. The only real diﬀerence is that one has to sum over all sector indices. However, since the set of internal ﬁelds Iv0

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Ann. Henri Poincar´e

is necessarily not empty (our deﬁnitions imply that, in this case, v˜0 = v0 ), we can choose in an arbitrary way one ﬁeld f0 ∈ Iv0 and let it play the same role of the selected external ﬁeld of v0 in the previous iterative procedure. Of course, the ﬁrst iteration step, which produced before the “scale jump” factor in the right-hand side of (3.7), is now missing, but this is irrelevant, since that factor is equal to 1 if |Pv0 | = 0. All the other steps are absolutely identical, but, at the end of the iteration, we end up with the sector sum related with f0 ; this produces a factor 1 1 γ − 2 hv0 = γ − 2 (h+1) . This completes the proof.

4 The ﬂow of running coupling functions In this section we prove that, if λ is small enough, under a suitable choice of − 1 the counterterm ν(k) and up to temperatures exponentially small T ≥ e c0 |λ| , the running coupling functions are uniformly bounded, so that the free energy is analytic in λ.

4.1

The expansion for LV (h) (ψ (≤h) )

By using (2.36), (2.47) and (2.54), we get LV ·

(h)

(ψ

(≤h)

)=

∞ n=1 τ ∈Th,n

P∈Pτ : |Pv0 |=2,4

Ω∈Oτ T ∈T

(≤h) (h) dxv0 ψ˜Ωv (Pv0 )LWτ,P,Ω\Ωv 0

0 ,T

(xv0 ) ,

(4.1)

˜ i = (˜ where, if Pv0 = (f1 , . . . , f4 ) and we put x(fi ) = xi = (xi,0 , xi ), x xi,0 , xi ) and x∗ is any point in xv0 , (h) (h) LWτ,P,Ω\Ωv ,T (x) = δ(x0 ) d(˜ x0 \˜ x∗0 )Wτ,P,Ω\Ωv ,T (˜ x) , (4.2) 0

0

while, if Pv0 = (f1 , f2 ) and m(Pv0 ) = m(f1 ) + m(f2 ) = 0, (h) (h) LWτ,P,Ω\Ωv ,T (x1 , x2 ) = δ(x1,0 − x2,0 ) d˜ x1,0 Wτ,P,Ω\Ωv 0

0 ,T

˜2) , (˜ x1 , x

(4.3)

and ﬁnally, if Pv0 = (f1 , f2 ), m(Pv0 ) = 1, (h)

LWτ,P,Ω\Ωv ,T (x1 , x2 ) = δ(x1,0 − x2,0 ) 0 (h) ˜2) . x1,0 − x ˜2,0 )Wτ,P,Ω\Ωv ,T (˜ x1 , x · d˜ x1,0 (˜ 0

(4.4)

Note that there is no other case to consider, since, as a consequence of the freedom in the choice of the ﬁeld carrying the derivative in the endpoints of type z, there

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173

is no contribution to the eﬀective potential with n ≥ 2 and a derivative acting on the external ﬁelds of v0 , before the application of the L operator. Let us consider ﬁrst the contributions to the right-hand side of (4.1) coming from the trees with n = 1. These trees have only two vertices, v0 (of scale h + 1) and the endpoint v ∗ , whose scale has to be equal to h + 2. If we impose the further condition that Pv∗ = Pv0 , the sum of these terms is equal to LV (h+1) (τ, ψ (≤h) ). In order to control the ﬂow of the running coupling functions, we need a “good bound” of the remaining terms. Let us consider a contribution to the right-hand side of (4.1), such that n ≥ 2 or n = 1 and Pv∗ = Pv0 . By proceeding as in §2.5, it is easy to show that (h) (L) LWτ,P,Ω\Ωv ,T (xv0 ) = Wτ,P,Ω\Ωv ,T,α (xv0 ) , (4.5) 0

α∈AT

0

(L)

where AT is a suitable set of indices and Wτ,P,Ω\Ωv ,T,α (xv0 ) can be represented as 0 in (2.69). There is indeed a small diﬀerence, because of the delta function and the integral appearing in (4.2), (4.3) and (4.4), but it can be treated without any new problem. Moreover, by the considerations of §2.6, if we insert (4.5) in the right(L) (L,mod) hand side of (4.1), we can substitute Wτ,P,Ω\Ωv ,T,α (xv0 ) with Wτ,P,Ω,T,α (xv0 ), 0 obtained by using the modiﬁed running coupling functions in place of the original ones. As before, these modiﬁed functions are not constant with respect to Ωv0 . We can prove the following Theorem, analogous to Theorem 2.1. Theorem 4.1 If conditions (2.90), (2.91), (2.92) are satisﬁed, given a couple of integers (p, m) equal to (2, 0), (2, 1) or (4, 0), we have:

∗∗

τ ∈Th,n

P∈Pτ : |Pv0 |=p,m(Pv0 )=m

∗ T ∈T α∈AT Ω∈Oτ

(L,mod) d(xv0 \x∗ ) Wτ,P,Ω,T,α (xv0 )

≤ (c|λ|)n γ h[δext (p)−m] , with δext (p) deﬁned by (2.103) and Pv∗ = Pv0 .

(4.6) ∗∗

means that, if n = 1 and v ∗ is the endpoint,

Proof. We can repeat step by step the proof of Theorem 2.1 and use the remark that, in the identity (2.95), q0 = m.

4.2

The beta function

The discussion of §4.1 and the deﬁnition of modiﬁed running coupling functions (MRCF in the following) of §2.6 (see in particular (2.75)) imply that ˜ h,σ (x) = (F4,h,σ ∗ λh+1 )(x) + (F4,h,σ ∗ β 4,0 )(˜ ˜ 1 ; x) , λ h+1 vh+1 , . . . , v

(4.7)

2,0 ˜ 1 ; x) , ν˜h,σ (x) = γ (F2,h,σ ∗ νh+1 )(x) + (F2,h,σ ∗ βh+1 )(˜ vh+1 , . . . , v

(4.8)

174

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Ann. Henri Poincar´e

2,1 ˜ 1 ; x) , z˜h,σ (x) = (F2,h,σ ∗ zh+1 )(x) + (F2,h,σ ∗ βh+1 )(˜ vh+1 , . . . , v

(4.9)

˜ h is the set of the corresponding MRCF, x = (x1 , . . . , xp ), where vh ≡ (λh , νh , zh ), v σ = (σ1 , . . . , σp ), with σi ∈ Oh−1 and p = 4 in (4.7), p = 2 in (4.8) and (4.9). p,m (vh , . . . , v1 ; x) is deﬁned by the equation Finally, the beta function βh+1 p,m (vh , . . . , v1 ; x) = γ −h·χ(p=2,m=0) βh+1

·

∞

∗∗

n=1 τ ∈Th,n

P:|Pv0 |=p m(Pv0 )=m

T ∈T Ω\Ωv0 α∈AT

(4.10) (L)

d(x0 \x∗0 )Wτ,P,Ω\Ωv

0 ,T,α

(x) .

Note that, given a tree contributing to the right-hand side of (4.10), we can substitute the RCF with the MRCF in all endpoints except those containing one p,m ) is indeed a function of the of the external ﬁelds of v0 . However (Fp,h,σ ∗ βh+1 MRCF, as we made explicit in the right-hand side of (4.7)–(4.9) and p,m ˜ 1 ; x) = γ −h·χ(p=2,m=0) (Fp,h,σ ∗ βh+1 )(˜ vh+1 , . . . , v

·

T ∈T

Ω: Ωv0 =σ

∗∗

∞ n=1 τ ∈Th,n

P |Pv0 |=p, m(Pv0 )=m

(L,mod)

d(x0 \x∗0 ) (Fp,h,σ ∗ Wτ,P,Ω,T,α )(x) .

(4.11)

α∈AT

Iterating (4.7), (4.8) and (4.9) we ﬁnd, for h ≤ 0, ˜ h,σ (x) = λ

˜1 )(x) + (F4,h,σ ∗ λ

1

˜ 1 ; x) , (4.12) (F4,h,σ ∗ βj4,0 )(˜ vj , . . . , v

j=h+1

ν˜h,σ (x) =

γ

−h+1

+

(F2,h,σ ∗ ν˜1 )(x)

1

˜ 1 ; x) , γ −h+j−1 (F2,h,σ ∗ βj2,0 )(˜ vj , . . . , v

(4.13)

j=h+1

z˜h,σ (x) =

(F2,h,σ ∗ z˜1 )(x) +

1

˜ 1 ; x) , (4.14) (F2,h,σ ∗ βj2,1 )(˜ vj , . . . , v

j=h+1

˜ 1 (x) = where, ignoring in the notation the spin dependence of v(x), see (1.8), λ (F4,1,0 ∗ λ1 )(x), with λ1 (x) = −λv(x1 − x2 )δ(x3 − x1 )δ(x4 − x2 ), and ν˜1 (x) = (F2,1,(0,0) ∗ ν1 )(x). Furthermore z˜1 (x) = z1 (x) = 0 and ν1 (x) must be suitably chosen. We note that it is possible to choose the functions F˜h,σ (k) appearing in the deﬁnition of the operators (Fp,h,σ ∗ ·), see (2.73), in such a way that, if h ≤ 0, 1 ˜ |ε(k) − µ| ≤ e0 γ h ⇒ Fh,σ (k) = 1 . 2 σ∈Oh

(4.15)

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175

In order to simplify the following discussion, we shall suppose that the property (4.15) is satisﬁed. Moreover we deﬁne Oh,p = ⊗pi=1 Oh . ¯ < 0, the MRCF are well deﬁned for h ¯ ≤ Theorem 4.1 implies that, given h h ≤ 1, if λ and ν˜1 (x) are small enough. We want to show that, given λ small enough and log β ≤ c0 |λ|−1 , it is possible to choose ν˜1 (x) so that the MRCF are well deﬁned for hβ ≤ h, with hβ deﬁned by (2.6). We shall try to ﬁx ν˜1 (x) in such a way that γ −hβ +1 ν˜1 (x)+

1

γ −hβ +j−1

j=hβ +1

1 4

σj ∈Oj,2

˜ 1 ; x) = 0 , (4.16) (F2,j,σj ∗βj2,0 )(˜ vj , . . . , v

so that (4.13) becomes: h

ν˜h,σ (x) = −

γ −h+j−1

j=hβ +1

·

1 4

∗ σj ∈Oj,2

F2,h,σ ∗ F2,j,σj ∗ βj2,0 (˜vj , . . . , v˜ 1 ; x) ,

(4.17)

∗ where, given σ = (σ1 , σ2 ) ∈ Oh,2 , σj ∈Oj,2 is the sum restricted to the σ j = (σ 1 , σ 2 ) ∈ Oj,2 such that Sh,σi ∩ Sj,σ i = ∅, i = 1, 2. In order to present our results, we have to introduce a few other deﬁnitions. 2 Given h ≤ 1 and ω ∈ Oh , we denote by Dh,σ ∈ R the support of F˜h,σ (k). 2p Moreover, if p = 2, 4, we call Mh,p the space of functions Gσ (x) : Oh,p × R → R, such that 1) for any σ ∈ Oh,p , Gσ (x) is translation invariant; ˆ σ (k) of Gσ (x), deﬁned so that, 2) for any σ ∈ Oh,p , the Fourier transform G Gσ (x) =

p dk −i k· x ˆ G e ( k) δ( εiki ) , σ (2π)2p i=1

(4.18)

with ε1 = −ε2 = +, if p = 2, and ε1 = ε2 = −ε3 = −ε4 = +, if p = 4, is a continuous function with support in the set ⊗pi=1 Dh,σi . Given G ∈ Mh,p , we shall say that Gσ (x) is the σ-component of G. These deﬁnitions are such that ν˜h,σ (x) and z˜h,σ (x) are the σ-components of two functions ˜ h,σ (x) is the σ-component of a function ν˜h and z˜h belonging to Mh,2 , while λ ˜ h ∈ Mh,4 . λ We shall deﬁne a norm on the set Mh,p by putting ||G||h,p = sup (4.19) d(x\xj )|Gσ (x)| . i,σi ∈Oh 2

R σ\σi ∈Oh,p−1

j, xj ∈

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Finally we shall deﬁne h ≤ 1}, such that the norm

Ann. Henri Poincar´e

Mp as the set of sequences G = {Gh ∈ Mh,p, hβ ≤ ||G||p = max ||Gh ||h,p hβ ≤h≤1

(4.20)

˜ ≡ {λ ˜ h , hβ ≤ h ≤ 1} is well is ﬁnite. We want to prove that the sequence λ deﬁned as an element of M4 , while the sequences ν˜ ≡ {˜ νh , hβ ≤ h ≤ 1} and z˜ ≡ {˜ zh , hβ ≤ h ≤ 1} are two elements of M2 . We begin our analysis by “decoupling” equations (4.12) and (4.14) from (4.13), that is we imagine that, in the right-hand side of (4.12) and (4.14), ν˜ is an arbitrary element of M2 , acting as a parameter. We want to look for a solution ˜ ν ) ∈ M4 , z˜(˜ ν ) ∈ M2 ). We shall prove the following lemma. (λ(˜ Lemma 4.1 There exist positive constants C1 and C2 , depending only on ﬁrst and second order terms in our expansion, such that, given two positive constants C3 ≥ C1 and C4 , there exists λ0 so that, if |λ| ≤ λ0 , 2C2 C3 max{1, C4−1 }|λ||hβ | ≤ 1

(4.21)

and ||˜ ν ||2 , ||˜ ν ||2 ≤ C3 |λ|, then, for hβ ≤ h ≤ 1, 1

˜ ν )h ||h,4 ≤ 2C1 |λ|γ − 2 h ||λ(˜

,

||˜ z (˜ ν )h ||h,2 ≤ C1 |λ| ,

(4.22)

1

˜ ν )h − λ(˜ ˜ ν )h ||h,4 ||λ(˜

≤

C4 γ − 2 h max ||˜ νh − ν˜h ||h,2 ,

||˜ z (˜ ν )h − z˜(˜ ν )h ||h,2

≤

C4 max ||˜ νh − ν˜h ||h,2 .

j>h

j>h

(4.23)

Proof. Note that, if F¯h,ω (x) is the Fourier transform of F˜h,ω (k), then F¯h,ω (x) ≤

3

CN γ 2 h

N , h h 2 1 + γ |x1 | + γ |x2 |

(4.24)

so that dx F¯h,ω (x) ≤ cF for some constant cF independent of h and ω. It follows that there exists a constant C1 , such ˜ 1 ||1,4 ≤ 2C1 |λ|γ −1/2 ||λ

,

˜ 1 ||h,4 ≤ C1 |λ|γ −h/2 , ||F4,h,σ ∗ λ

(4.25)

having used also Lemma 3.1 for the second inequality. We shall prove inductively that, if ||˜ ν ||2 ≤ C3 |λ|, with C3 ≥ C1 , then − 12 h ˜ and ||˜ z (˜ ν )h ||h,2 ≤ C1 |λ|. This bound is satisﬁed for ||λ(˜ ν )h ||h,4 ≤ 2C1 |λ|γ h = 1, by the ﬁrst inequality of (4.25) and the fact that z˜1 = 0; let us suppose that it is true for any j > h. Then, by using (4.12), (4.14), the second inequality

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Low Temperature Analysis of Two-Dimensional Fermi Systems (4,0)

of (4.25), Theorem 4.1 and the fact that βj contributions, we ﬁnd ˜ ν )h ||h,4 ||λ(˜

≤

C1 |λ|γ

− 12 h

+γ

− 12 h

(2,1)

and βj

||˜ z (˜ ν )h ||h,2

≤

do not have ﬁrst order

1 ∞ 2 C2,λ C1 C3 |λ| + (c|λ|)n , n=3

j=h+1 1

177

C2,z C1 C3 |λ|2 +

∞

(c|λ|)n .

(4.26)

n=3

j=h+1

˜ ν )h ||h,4 Hence, if λ small enough and 2|λ||hβ |C3 max{C2,λ , C2,z } ≤ 1, then ||λ(˜ − 12 h ≤ 2C1 |λ|γ and ||˜ z (˜ ν )h ||h,2 ≤ C1 |λ|, up to h = hβ . ν ||2 ≤ C3 |λ|, then the bounds (4.23) We still have to prove that, if ||˜ ν ||2 , ||˜ ˜ ν )1 − λ(˜ ˜ ν )1 = 0, are veriﬁed. We shall again proceed by induction, by using that λ(˜ ˜ 1 is independent of ν˜, and that z˜(˜ since λ ν )1 = 0. Then, if we suppose that the bound is true for any j > h, we ﬁnd ˜ ν )h − λ(˜ ˜ ν )h ||h,4 ||λ(˜

1

≤ γ − 2 h max ||˜ νj − ν˜j ||j,2 j>h 1 ∞ n n−1 C˜2,λ C3 max{1, C4 }|λ| + · c |λ| n=3

j=h+1

||˜ z (˜ ν )h − z˜(˜ ν )h ||h,2

ν˜j ||j,2

≤ max ||˜ νj − (4.27) j>h 1 ∞ C˜2,z C3 max{1, C4 }|λ| + · cn |λ|n−1 . n=3

j=h+1

Hence, if λ small enough and 2|λ||hβ |C3 max{1, C4−1 } max{C˜2,λ , C˜2,z } ≤ 1, the bound is veriﬁed up to h = hβ . The constant C2 appearing in the condition (4.21) can be chosen as C2 = max{C2,λ , C2,z , C˜2,λ , C˜2,z }. We want now to show that there is indeed a solution of the full set of equations (4.12)–(4.14), satisfying condition (4.16). Theorem 4.2 If |λ| is small enough there exists a constant c0 such that, for c0 |λ| log β ≤ 1, it is possible to choose ν˜1 (x) so that the MRCF satisfy the hypothesis of Lemma 2.2, (2.90), (2.91) and (2.92). Proof. In order to prove the theorem, it is suﬃcient to look for a ﬁxed point of the operator T : M2 → M2 , deﬁned in the following way, if ν˜ ≡ T(˜ ν ): ν˜h = −

h j=hβ +1

γ −h+j−1

1 4

∗ ' σj ∈Oj,2

F2,h,σ ∗ F2,j,σj ∗ βj2,0 (˜vj (˜ν ), . . . , v˜ 1 (˜ν ); x) , (4.28)

˜ ν ), ν˜, z˜(˜ ˜ j (˜ where v ν ) = (λ(˜ ν )).

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We want to prove that it is possible to choose the constant Cν ≥ 1, so that, if C1 is the constant deﬁned in Lemma 4.1 and |λ| is small enough, the set F = {˜ ν ∈ M2 : ||˜ ν ||2 ≤ 2C1 Cν |λ|} is invariant under T and that T is a contraction on it. This is suﬃcient to prove the theorem, since M2 is a Banach space, as one can easily show. By using Theorem 4.1 and Lemma 4.1 (with C3 = 2C1 Cν ), we see that, if |λ| is small enough and 4C2 C1 Cν max{1, C4−1 }|λ||hβ | ≤ 1 (C4 will be chosen later), ||˜ νh ||h,2

≤

h

γ

−h+j−1

γ

h−j 2

∞

C1,ν C1 |λ| +

j=hβ +1

c |λ| n

n

,

(4.29)

n=2

h−j

where γ 2 is, up to a constant, a bound for the number of sectors σ ∈ Oj with non empty intersection with a given σ ∈ Oh , h ≥ j and C1,ν is a constant C √ and depending on the ﬁrst order contribution (i.e., the tadpole). So, if Cν ≥ γ−1,ν γ ∞ n n c |λ| ≤ C C |λ|, then ||˜ ν || ≤ 2C C |λ|. 1,ν 1 1 ν h n=2 We then show that T is a contraction on F . In fact, given ν˜1 , ν˜2 ∈ F, by using again Theorem 4.1 and Lemma 4.1, we see that, under the same conditions supposed above, − ν˜2,h || ≤ ||˜ ν1,h

h

γ −h+j−1 γ

h−j 2

j=hβ +1

· max ||˜ ν1,i − ν˜2,i || C1,ν C4 + i≥j

∞

c |λ| n

n−1

,

(4.30)

n=2

∞ n n−1 so that, if ≤ C1,ν C4 /2 and C4 = (2Cν )−1 , then ||˜ ν1 − ν˜2 || ≤ n=2 c |λ| 3 2 ν1 − ν˜2 ||, if 8C2 C1 Cν |λ||hβ | ≤ 1. So the stated result follows and the constant 4 ||˜ c0 can be chosen, by using (2.6) and by supposing that e0 ≤ π, as 2 c0 = 2(log γ)−1 C1 C1,ν max{C2,λ , C2,z , C˜2,λ , C˜2,z } .

(4.31)

Remark. We have proved that the ﬂow of the MRCF remains bounded and small −1 up to temperatures T ≥ e−(c0 |λ|) with c0 given by (4.31). Note that C1,ν is a bound for ﬁrst order contribution to ν˜h in the norm || · ||h,2 , see (4.19); more exactly is a bound for the contribution from the tree with one endpoint to the second addendum of the right-hand side of (4.13). In the same way γ −h/2 C2,λ is a bound in the norm || · ||h,4 of the ﬁrst non trivial contribution (a second order one) ˜ h , and so on. Finally C1 is equal to c4 dx(1 + |x|2 )|v(x)|, if c is the constant to λ appearing in the sector counting Lemma 3.1.

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5 The two point Schwinger function In this section we deﬁne a convergent expansion for the two point Schwinger function which allows us to ﬁnd its large distance asymptotic behavior and to complete the proof of Theorem 1.1. The Schwinger functions can be derived by the generating function deﬁned as + − + − W(φ) = log P (dψ)e−V(ψ)−N (ψ)+ dx[φx ψx +ψx φx ] , (5.1) where the variables φσx are deﬁned to be Grassmanian variables, anticommuting with themselves and ψxσ . In particular the two point Schwinger function is given by ∂2 S(x − y) = . (5.2) + − W(φ) ∂φx ∂φy φ=0 We can get a multiscale expansion for W(φ), by a procedure very similar to that used for the free energy, by taking into account that the interaction contains a new term, linear in ψ and φ. This novelty has the consequence that new terms appear in the expansion, containing one or more φ ﬁelds linked to the corresponding graphs through a single scale propagator. In order to study S(x − y), it is suﬃcient to analyze the structure of the terms with one or two φ ﬁelds. Let us consider ﬁrst the terms produced after integrating the scales greater or equal to h + 1 and linear in φ. These terms can be obtained by taking one of the contributions V (h) (τ, P) ≡ Ω V (h) (τ, P, Ω) to the eﬀective potential on scale h and by linking one of its external lines, say f¯, with the φ ﬁeld through a propagator of scale j ≥ h + 1, to be called the external propagator. However, one has to be careful in the choice of the localization point in the vertices v such that f¯ ∈ Pv and |Pv | ≤ 4 (so that the action of R in v is not trivial); we choose it as that one which connects f¯ with the φ ﬁeld (hence no derivative can act on the external propagator, when one exploits the eﬀect of the R operations as in §2.5). This choice has the aim of preserving the regularizing eﬀect of the R operation, based on the fact that, if a ﬁeld acquires a derivative as a consequence of the R operation on scale i, then it has to be contracted on a scale j < i, so producing an improvement of order γ −(i−j) in the bounds. Note also that, because of the localization operation, the scale j of the external propagator can be higher of the scale of the endpoint v¯, such that f¯ ∈ Pv¯ . The situation is diﬀerent in the terms with two φ ﬁelds, connected through two external propagators of scale jx and jy greater than h and involving two ψ ﬁelds, of labels fx and fy . There are two diﬀerent types of contributions. The ﬁrst type is associated with trees τ satisfying the following conditions: 1. the root has scale hr ≥ h, 2. Iv0 (the set of internal lines in the vertex immediately following the root) is not empty,

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3. there is no external line in v0 , except fx and fy , the lines contracted in the external propagators. These terms are produced, in the iterative integration procedure, at scale hr + 1 and, after that scale are constant with respect to the integration process. The other type of terms is associated with trees such that 1. the root has scale h, 2. |Pv0 | > 2. These terms depend on the integration ﬁeld ψ (≤h) , so that they are involved in the subsequent integration steps. Given a tree τ (of any type) with two φ ﬁelds, the corresponding contributions to W(φ) are obtained in a way slightly diﬀerent from that described in the case of the eﬀective potential. Given jx and jy , larger or equal to h + 1, select two ﬁeld labels fx and fy and call v¯ the higher vertex, of scale ¯h, such that ¯ ≤ min{jx , jy }, 1. h 2. fx and fy belong to Pv¯ . Let C be the path on τ connecting v¯ with v0 . Given v ∈ C, we avoid to apply there the localization procedure, because the R operation, no matter we choose the localization point, would give rise to terms with a derivative acting on the external propagators (which is not convenient, see above). In all other vertices of τ the localization procedure is deﬁned as in the case of the free energy expansion, by suitably choosing the localization point in the vertices following v¯ and containing fx or fy , as explained above. Then we substitute fx and fy with two external propagators of scale jx and jy , respectively. Note that these propagators can acquire a derivative, as a consequence of the R operation acting on a vertex v, only if hv is greater or equal to their scale (jx or jy ). The previous considerations imply that S(x − y) is given by the following sum: S(x − y) = g(x − y) +

1

¯ h−1

∞

¯ r ¯ h,h h=h β hr =hβ −1 n=1 τ ∈Tn

¯

Sτ,P (x − y) ,

(5.3)

P

where the family of labeled trees Tnh,hr and the families of external lines Pv can be described as in §2, with the following modiﬁcations (see Fig. 3). ¯ and jy ≥ h, ¯ and a 1) There are two ﬁeld labels, fx and fy , two scale labels jx ≥ h ¯ vertex v¯ such that hv¯ = h, fx , fy ∈ Pv¯ and there is no other vertex v > v¯ such that hv ≤ min{jx , jy } and fx , fy ∈ Pv ; we shall call vx and vy the endpoints (possibly coinciding) that fx and fy belong to. Note that we are not introducing the sector decomposition for the external propagators and that the vertex v¯ can be lower than the higher vertex preceding both vx and vy (opposite to what happens in Fig. 3).

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181

vx v¯ r

vy

v0

¯ h

hr

+1 +2

Figure 3: An example of tree contributing to S(x − y). 2) Given fx and fy , let C be the path on the tree (see dashed line in Fig. 3), connecting v¯ with the lowest vertex v0 , of scale hr + 1. If v ∈ C and v = v0 , |Pv | ≥ 4, while |Pv0 | = 2. ¯ Given τ ∈ Tnh,hr and P, we have Sτ,P (x − y) = g (jx ) ∗ Wτ,P,jx ,jy ∗ g (jy ) (x − y) , (5.4) where ∗ means the convolution in x space and Wτ,P,jx ,jy diﬀers from the kernel (h +1) (hr +1) (hr +1) (hr ) = (τ, P, Ω) (see (2.48) and note that Kτ,P,Ω Kτ,Pr Ω\Ωv0 Kτ,P,Ω of V does not depend on Ωv0 ) only because no R operation acts on the vertices of C. ˆ We now consider the Fourier transform S(k) of S(x−y), which can be written in the form: ˆ (5.5) S(k) = gˆ(k) 1 + λSˆ1 (k) , where gˆ(k) is the free propagator. In order to prove Theorem 1.1, we have to show that Sˆ1 (k) is a bounded function. Let us deﬁne hk = max{h : gˆ(h) (k) = 0}. By using (5.4), it is easy to see that λˆ g (k)Sˆ1 (k) =

hk

x ,jy } ∞ min{j

¯ h=h β

jx ,jy =hk −1 n=1

¯ h−1

¯ r hr =hβ −1 τ ∈T h,h n

Sˆτ,P,jx ,jy (k) ,

(5.6)

||Sτ,P,jx ,jy ||1 ,

(5.7)

P

implying that |Sˆ1 (k)| ≤ c|λ|−1 γ hk ·

sup

x ,jy } ∞ min{j

jx ,jy =hk −1,hk n=1

¯ h=h β

where ||.||1 denotes the L1 norm.

¯ h−1

¯ r hr =hβ −1 τ ∈T h,h n

P

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¯ We can bound P ||Sτ,P,jx ,jy ||1 by proceeding as in §2.7. Since τ ∈Tnh,hr the combinatorial problems are of the same nature, we can describe in a simple way the result by dimensional arguments. We can take as a reference the bound of Jhr ,n (2, 0), see (2.79) and (2.101), that is the bound of the L1 norm of the eﬀective potential terms with two external lines on scale hr and no external derivative, and multiply it by a factor γ −jx −jy , which comes from the external propagators (the derivatives possibly acting on them are absorbed in the “gain factors” γ −(h−h ) , produced by the localization procedure, so that they do not give any contribution to the ﬁnal bound). There are two relevant diﬀerences. 1) There is no regularization on the vertices with four external lines belonging to C. This implies that one “looses” a factor γ −1 , with respect to the bound (2.101), for each vertex v ∈ C such that |Pv | = 4. 2) The external propagators sectors are not on the scale hr , but they are exactly ﬁxed. Hence, we have to modify the momentum conservation constraint (2.76) in the tree vertices v such that fx or fy belong to Pv , in order to remember this condition when we bound the sector sums. Then, we have to prove a lemma similar to Lemma 3.1, by substituting one sector sum with the constraint that one momentum is exactly ﬁxed. It is not hard to see, by using Lemma 7.5 and by proceeding as in §7.4, that we get a bound of the same type of that of Lemma 3.1. The previous considerations, together with the bound (2.101), allow to prove that

¯

P

τ ∈Tnh,hr

·

||Sτ,P,jx ,jy ||1 ≤ (c|λ|)n γ hr −2hk

¯

τ ∈Tnh,hr

P |Pv0 |=2

T ∈T

v not e. p.

1 δv∗ γ , sv !

(5.8)

/ C, otherwise δv∗ = δ(|Pv |)+χ(|Pv | = 4). By using (2.102), where δv∗ = δ(|Pv |), if v ∈ it is easy to see that, if we deﬁne δ˜v = δv∗ − 1/2, if v ∈ C and δ˜v = δv∗ otherwise, δ˜v < 0 for all v ∈ τ . Hence, the bound (2.104) is still valid, if we put δ˜v in place of δ(|Pv |), and we get |Sˆ1 (k)|

≤ cγ

−hk

hk

¯ h−1

¯

γ hr γ (h−hr )/2

¯ h=h β hr =hβ −1

≤ c

hk ¯ h=h β

γ

¯ −(hk −h)

¯ h−1

¯

γ −(h−hr )/2 ≤ c .

(5.9)

hr =hβ −1

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6 The rotation-invariant case We consider now the Jellium model, which is deﬁned in the continuum with ε(k) = y ) = v˜(|x − y |), implying rotation invariance symmetry. In |k|2 /(2m) and v(x − pF (θ)| does not depend on θ and the two point contribution to particular, pF = | ˆ (h) (k0 , k) = dxW ˜ (h) (x) exp(ikx), see (2.17) and the line the eﬀective potential W 2 2 (h) (h) before (2.15), is of the form W2 (k0 , |k|), where W2 (k0 , ρ) is a function of two variables. We show that in such a case we can choose the counterterm νˆ(k) as a − 1 constant ν, if the temperature T is big enough, i.e., T ≥ e c1 |λ| , where c1 is a constant depending on a bound to all orders of multiscale perturbation theory. In order to get this result, we must change the localization deﬁnition, so that (h)

1. LW2n = 0 if n ≥ 2; ˆ (h) (k0 , k) = W (h) (0, pF ) ≡ γ h νh . 2. if n = 1, LW 2

2

We want now to analyze the properties of the R operator. If we put, as in (2.32), for any k ∈ Sh,ω , ω ∈ Oh , k = k + pF (θh,ω ), we can write ˆ (h) (k0 , k) = RW 2 =

d (h) tk0 , |tk + pF (θh,ω )| W2 dt 0 1 (h) dt k0 ∂k0 W2 (tk0 , ρ(t) ) 1

dt

(6.1)

0

+

(tk1 + pF )k1 + t(k2 )2 (h) ∂ρ W2 (tk0 , ρ(t) ) , ρ(t)

≡ v ·τ (θh,ω ) (see (2.31), where, for any vector v , we are deﬁning v1 ≡ v ·n(θh,ω ), v2 recalling that now er (θ) = n(θ), et (θ) = τ (θ)) and ρ(t) ≡ (tk1 + | pF |)2 + (tk2 )2 . (h) It is easy to see that the term ∂ρ W2 (tk0 , ρ(t) ) in (6.1) can be rewritten in the following form: (h) ˆ (h) (tk0 , tk + pF (θh,ω )) ∂ρ W2 (tk0 , ρ(t) ) = cos θ(t)∂k1 W 2 ˆ (h) (tk0 , tk + pF (θh,ω )) + sin θ(t)∂k2 W (6.2) 2 tk + pF tk ˜ (h) (y)eitk0 y0 +i(tk +pF (θh,ω ))y , = dy iy1 1 + iy2 2 W 2 ρ(t) ρ(t)

where θ(t) is the angle between n(θh,ω ) and tk + pF (θh,ω ). Substituting (6.2) in F (θh,ω )), (6.1) we get, if pω = (0, p dk (≤h)+ (≤h)− (h) (≤h) ˆ (h) (k) )= RW ψˆ ψˆ RV (ψ 2 (2π)3 k−pσ ,σ k−pω ,ω σ,ω∈Oh 1 dk (≤h)+ ˆ ˆ(≤h)− ˜ (h) (y)ei(tk +pω )y = dt ψ ψ dyW 2 3 k +pω −pσ ,σ k ,ω (2π) 0 σ,ω∈Oh

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(tk1 + pF )k1 + t(k2 )2 tk1 + pF tk iy1 + 2 iy2 · ik0 y0 + . ρ(t) ρ(t) ρ(t)

(6.3)

Let us deﬁne the operators Di (t), i = 1, 2, so that dk iεk x tk1 + pF (tk1 + pF )k1 + t(k2 )2 ˆ(≤h)ε (≤h)ε ψk ,ω , D1 (t)ψx,ω = iε e (2π)3 ρ(t) ρ(t) dk iεk x tk2 (tk1 + pF )k1 + t(k2 )2 ˆ(≤h)ε (≤h)ε = iε e (6.4) D2 (t)ψx,ω ψk ,ω . (2π)3 ρ(t) ρ(t) Hence, (6.3) can be written as RV (h) (ψ (≤h) ) = −

σ,ω∈Oh

1

dt

0

dx

˜ (h) (y − x)ψ (≤h)+ dy eipσ y−ipω x W y,σ 2 (≤h)−

· [(y0 − x0 )∂0 + (y1 − x1 )D1 (t) + (y2 − x2 )D2 (t)] ψξ(t),ω 1 ˜ (h) (y − x) = dt dx dy eipσ y−ipω x W 2 σ,ω∈Oh

(6.5)

0

(≤h)+

(≤h)− , · [(y0 − x0 )∂0 + (y1 − x1 )D1 (t) + (y2 − x2 )D2 (t)] ψη(t),σ ψx,ω

where η(t) ≡ ξ(t) ≡

y + t(x − y) x + t(y − x) .

(6.6)

It is easy to prove the following dimensional bound. Lemma 6.1 Given non negative integers N, n0 , n1 , n2 , m = n0 + n1 + n2 , there exists a constant CN,m , such that 3

|∂0n0 D1n1 D2n2 gω(h) (x)| ≤ CN,m

3

γ h( 2 +n0 +n1 + 2 n2 )

h 1 + (γ h x0 )2 + (γ h x1 )2 + (γ 2 x2 )2

N

,

(6.7)

where Din denotes the product of n factors Di (tj ), j = 1, . . . , n. Remark. Each operator Di (t) improves the bound of the covariance by a factor at least γ h ; this is what we need to obtain the right dimensional gain from renor malization operations, which also produce a factor γ −h on a scale h > h. This is a consequence of rotational invariance; in fact a naive Taylor expansion would apparently produce a term of the form (y2 − x2 )∂x2 , which would give rise to a “bad factor” γ −h +h/2 in the bounds. We can now repeat the analysis of the previous sections, in a much more simple context. In fact it is easy to see that it is possible to ﬁx ν1 in such a way that

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νh stay bounded for hβ ≤ h ≤ 1. Furthermore we can easily perform the bounds for the nth -order contributions to the kernel of the eﬀective potentials or to the two point Schwinger functions. In both cases we ﬁnd that, unless for the external dimensional factors, the nth -order contributions are bounded by (c|λ|)n (log β)n−1 , where the diverging factor (log β)n−1 is due to the choice of not localizing the four-legs clusters and of localizing the two-legs clusters only at the ﬁrst order. So the result of Theorem 1.1 in the rotational invariant case easily follows.

7 Some technical lemmata In this section, we ﬁrst prove a few geometrical properties of the dispersion relation, see §7.1, and the consequent bound on single scale propagators given in Lemma 2.1, see §7.2. In §7.3 we prove a parallelogram lemma, i.e., an implicit function type theorem stating that, given a vector b varying in a small neighborhood of pF (θ¯1 ) + pF (θ¯2 ), with |θ¯1 − θ¯2 | > 0, (see (7.23) below), b can be uniquely written as pF (θ1 ) + pF (θ2 ), with θi varying in a small interval around θ¯i . This is the key result we need in order to prove the sector counting Lemma 3.1.

7.1

Geometrical properties of the dispersion relation 2

Let B = { p ∈ R : |ε( p) − µ| ≤ e0 }; the hypotheses on ε( p) described in §1.2 imply that there is a C ∞ diﬀeomorphism between B and the compact set A = T1 × [−e0 , e0], deﬁned by p = q (θ, e) = u(θ, e)er (θ)

,

(θ, e) ∈ A .

(7.1)

Moreover, the symmetry property (1.11) implies that q (θ + π, e) = −q(θ, e) ,

(7.2)

a property that will have an important role in the following. Let us now introduce some more geometrical deﬁnitions, which we shall need in the following. For any ﬁxed e, we can locally deﬁne the arc length s(θ, e) on Σ(e); we shall denote ∂/∂s the partial derivative with respect to s, at ﬁxed e, and we shall sometime use the prime to denote the partial derivative with respect to θ. If τ (θ, e) = ∂ p(θ, e)/∂s is the unit tangent vector at Σ(e) in q(θ, e), we have s (θ, e)τ (θ, e) = s (θ, e) =

∂ p (θ, e) = u (θ, e)er (θ) + u(θ, e)et (θ) , ∂θ u (θ, e)2 + u(θ, e)2 ,

(7.3)

where et (θ) = (− sin θ, cos θ). Analogously, if n(θ, e) is the outgoing unit normal vector at Σ(e) in q(θ, e) and 1/r(θ, e) is the curvature (which satisﬁes the convexity condition (1.9)), we

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have s (θ, e)n(θ, e) p ∂2 (θ, e) ∂θ2

= u(θ, e)er (θ) − u (θ, e)et (θ) , s (θ, e)2 n(θ, e) . = s (θ, e)τ (θ, e) − r(θ, e)

(7.4)

Lemma 7.1 The angle α(θ, e) between n(θ, e) and er (0) is a monotone increasing 1 function of θ, such that, if ||θ1 − θ2 || denotes the distance on T . c1 ||θ2 − θ1 || ≤ ||α(θ2 , e) − α(θ1 , e)|| ≤ c2 ||θ2 − θ1 || ;

(7.5)

moreover, α(θ + π, e) − α(θ, e) = π. Proof. By using (7.3) and (7.4) and Taylor expansion, one can easily prove that, if αi = α(θi , e), s (θ1 , e) + O(θ2 − θ1 )2 , (7.6) r(θ1 , e) (θ2 − θ1 )2 s2 (θ1 , e) cos(α2 − α1 ) = n(θ2 , e) · n(θ1 , e) = 1 − + O(θ2 − θ1 )3 , 2 r2 (θ1 , e) sin(α2 − α1 ) = n(θ2 , e) · τ (θ1 , e) = (θ2 − θ1 )

which implies (7.5) for |θ2 − θ1 | small, hence even for any value of θ2 − θ1 , together with the monotonicity property. The fact that α(θ + π, e) − α(θ, e) = π is a trivial consequence of (7.2). q (θ, 0) the generic point of the Fermi surface ΣF ≡ We denote by pF (θ) = Σ(0). Moreover, to simplify the notation, from now on we shall in general suppress the variable e when it is equal to 0; for example, we shall put pF (θ) = u(θ)er (θ). Let us consider an s-sector Sh,ω , see (2.72). Lemma 7.2 If p = ρer (θ) ∈ Sh,ω , h ≤ 0, ω ∈ Oh , then |ρ − u(θ)| ≤ cγ h

,

|θ − θh,ω | ≤ πγ h/2 .

(7.7)

Proof. The bound on θ follows directly from the deﬁnition of Sh,ω . On the other hand, the identity ε( p) − µ = ε(ρer (θ)) − ε(u(θ)er (θ)) =

u(θ)

(ρer (θ)) , dρ er (θ)∇ε

(7.8)

ρ

and the property (1.10) of ε( p), easily imply the bound on ρ − u(θ, 0).

The following lemma shows that, if p ∈ Sh,ω , the diﬀerence between p and pF (θh,ω ) is of order γ h in the direction normal to ΣF in the point pF (θh,ω ), while it is of order γ h/2 in the tangent direction.

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Lemma 7.3 If p ∈ Sh,ω , h ≤ 0, ω ∈ Oh , then p = pF (θh,ω ) + k1 n(θh,ω ) + k2 τ (θh,ω ) Moreover,

,

|k1 | ≤ cγ h

,

|k2 | ≤ cγ h/2 .

∂ h/2 ε( p (θ ) + k n (θ ) + k τ (θ )) . F h,ω 1 h,ω 2 h,ω ≤ cγ ∂k2

(7.9)

(7.10)

p − pF (θ)| ≤ cγ h . Hence, to prove (7.9), Proof. If p = ρer (θ), by Lemma 7.2 | pF (θh,ω )]n(θh,ω )| ≤ cγ h and |[ pF (θ) − it is suﬃcient to prove that |[ pF (θ) − h/2 pF (θh,ω )]τ (θh,ω )| ≤ cγ . These bounds immediately follow from the following ones, which can be easily proved, by using (7.3), (7.4) and some Taylor expansions: pF (θ2 )] · n(θ2 ) = O(θ1 − θ2 )2 , [ pF (θ1 ) −

(7.11)

[ pF (θ1 ) − pF (θ2 )] · τ (θ2 ) = O(θ1 − θ2 ) .

(7.12)

It is suﬃcient to put here θ1 = θ, θ2 = θh,ω and to recall that θ − θh,ω = O(γ h/2 ). Let us now observe that, if we derive with respect to θ the identity ε(u(θ, e)· er (θ)) = e, we get, for any p ∈ B, p)τ (θ, e)]s (θ, e) = 0 [∇ε(

⇒

p) = a(θ, e)n(θ, e) , ∇ε(

(7.13)

a(θ, e) being a smooth function, strictly positive by (1.10). Hence, if p ∈ Sh,ω , by using the ﬁrst line of (7.6), (7.13) and the fact that |ε( p) − µ| ≤ cγ h , |θ − θh,ω | ≤ cγ h/2 , ∂ε( p) ∂k2

which proves (7.10).

=

p) · τ (θh,ω ) = a(θ, e)n(θ, e) · τ (θh,ω ) ∇ε(

=

a(θ, e)n(θ) · τ (θh,ω ) + O(γ h ) = O(γ h/2 ) ,

(7.14)

Given p ∈ Sh,ω , we shall also consider the projection on the Fermi surface, deﬁned as p⊥ = pF (θ⊥ ) = p − x n(θ⊥ ) . (7.15) Note that (7.15) has to be thought as an equation for θ⊥ and x, given p ; it is easy to prove that, as a consequence of the condition (1.9), this equation has a smooth unique solution, if e0 is small enough, what we shall suppose from now on. Lemma 7.4 If p = ρer (θ) ∈ Sh,ω , h ≤ 0, and x and θ⊥ are deﬁned as in (7.15), then |x| ≤ cγ h and |θ⊥ − θh,ω | ≤ cγ h/2 .

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Proof. (7.9) and (7.15) imply that k1 k2

= [ pF (θ⊥ ) − pF (θh,ω )] · n(θh,ω ) + xn(θ⊥ ) · n(θh,ω ) , = [ pF (θ⊥ ) − pF (θh,ω )] · τ (θh,ω ) + xn(θ⊥ ) · τ (θh,ω ) .

(7.16) (7.17)

By using the (7.11), (7.12) and (7.6), one can easily complete the proof of the lemma.

7.2

Proof of Lemma 2.1

The bounds on k1 and k2 in (7.9) imply that dpFh,ω (p) ≤ cγ 5h/2 . On the other hand, if Fh,ω (p) = 0, | − ip0 + ε( p) − µ| ≥ cγ h , so that (2.33) implies the (h) (h) bound |gω (x)| ≤ cγ 3h/2 . It is also very easy to prove that |∂ n gˆω (p)/∂pn0 | and (h) |∂ n gˆω (p)/∂k1n | are bounded by cγ −h(n+1) . Hence, using simple integration by (h) n (h) parts arguments, one can show that |xn0 gω (x)| ≤ cγ h(3/2−n) and |x1 gω (x)| ≤ cγ h(3/2−n) . Moreover, it is easy to prove that ∂ε( p) n | , (7.18) |∂ n gˆω(h) (p)/∂k2n | ≤ cγ −h γ −h sup | p ∈Sh,ω ∂k2 n (h)

which implies the bound |x2 gω (x)| ≤ cγ h(3/2−n/2) . Finally, by using Lemma 7.3, it is easy to prove that the previous bounds have to be multiplied by γ mh , if one (h) (h) m (h) m substitutes gω (x) with ∂ m gω (x)/∂xm 0 or ∂ gω (x)/∂x1 , while they have to (h) (h) be multiplied by γ mh/2 if gω (x) is changed in ∂ m gω (x)/∂x2 m . The bound (2.35) is a simple consequence of the previous considerations.

7.3

The parallelogram lemma

Let us consider the map F, deﬁned on the two-dimensional torus 2 2 in R , such that, if (θ1 , θ2 ) ∈ T and b = F (θ1 , θ2 ), then

T2 , with values

b = pF (θ1 ) + pF (θ2 ) .

(7.19)

The diﬀerential J(θ1 , θ2 ) of F is a matrix, whose columns coincide with s (θ1 )τ (θ1 ) and s (θ2 )τ (θ2 ). Then Lemma 7.1 implies that det J = 0, hence F is invertible, around any point (θ1 , θ2 ) ∈ T , where 2

T = {(θ1 , θ2 ) ∈ T : sin(θ1 − θ2 ) = 0} .

(7.20)

Moreover, if ||θ1 − θ2 || = π, b = 0, while, if θ1 = θ2 = θ, b = 2u(θ)er (θ). Finally, T is the union of two disjoint subsets, which are obtained one from the other by exchanging θ1 with θ2 , and each one of them is in a one-to-one correspondence through F with the open set 1

D = { p = ρer (θ) : 0 < ρ < 2u(θ), θ ∈ T } . The following lemma will have an important role in the following.

(7.21)

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Lemma 7.5 Let (θ¯1 , θ¯2 ) ∈ T , b = pF (θ¯1 ) + pF (θ¯2 ), φ = min{||θ¯1 − θ¯2 ||, π − ||θ¯1 − θ¯2 |||} > 0 , r = r1 n(θ¯1 ) + r2 τ (θ¯1 )

,

|r1 | ≤ c1 ηφ

,

|r2 | ≤ η ≤ c2 φ .

(7.22) (7.23)

Then there exist c0 , c¯2 and η0 , such that, if c2 ≤ c¯2 and η ≤ η0 , then b + r ∈ D and b + r = pF (θ1 ) + pF (θ2 ) , |θi − θ¯i | ≤ c0 η . (7.24) Proof. We shall consider only the case φ = ||θ¯1 − θ¯2 ||; the case φ = π − ||θ¯1 − θ¯2 || can be easily reduced to this one, by using the symmetry property (7.2). We shall ¯ ¯ also choose the sign of θ¯1 − θ¯2 , so that φ = θ2 − θ1 . 2 2 Let us deﬁne δi = θi − θ¯i , δ = δ1 + δ2 ; then we can write, by using (7.19), (7.3) and (7.4), if b + r ∈ D (which is certainly true, if r is small enough), r

= =

d pF (θ¯1 ) d pF (θ¯2 ) δ1 + δ2 + O(δ 2 ) dθ dθ δ1 s (θ¯1 )τ (θ¯1 ) + δ2 s (θ¯2 )τ (θ¯2 ) + O(δ 2 ) .

(7.25)

Let us now put δi = ηxi , r1 = ηφ˜ r1 , r2 = η˜ r2 ; condition (7.23) takes the form |˜ r1 | ≤ c1 and |˜ r2 | ≤ 1. Since, by hypothesis, η ≤ c2 φ, the condition b + r ∈ D is satisﬁed, together with (7.24), if and only if the following system of two equations in the unknowns x1 , x2 has a unique solution: x2

=

x1

=

r˜1 φ ¯ s (θ2 ) sin[α(θ¯1 )

+ O(c2 ) , − α(θ¯2 )] r˜2 r˜1 φ cos[α(θ¯1 ) − α(θ¯2 )] − ¯ + O(η) + O(c2 ) , ¯ s (θ1 ) s (θ1 ) sin[α(θ¯1 ) − α(θ¯2 )]

(7.26)

where α(θ) is deﬁned as in Lemma 7.1 and O(c2 ), O(η) are of second order as functions of the xi ’s. By using Lemma 7.1, we see that the right sides of (7.26) are bounded for φ → 0. Hence, by the Dini Theorem, (7.26) allow to uniquely determine x1 and x2 for any φ > 0, given r, if η and c2 are small enough, and |δi | ≤ c0 η, with c0 independent of c2 .

7.4

Proof of Lemma 3.1 (sectors counting lemma)

Let h , h, L be integers such that h ≤ h ≤ 0. Given ω1 ∈ Oh and ω ˜ i ∈ Oh , ˜2, . . . , ω ˜ L ) be the set of the sequences (ω2 , . . . , ωL ), such i = 2, . . . L, let Ah,h (ω1 ; ω that i) Sh ,ωi ⊂ Sh,˜ωi for i = 2, . . . , L; ii) there exists, for i = 1, . . . , L, a vector k (i) ∈ Sh ,ω , so that L k (i) = 0. i i=1

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If L = 2, the momentum conservation ii) and the symmetry property (1.11) immediately imply that |Ah,h (ω1 ; ω ˜ 2, . . . , ω ˜ L )| = 1. Hence, in order to prove Lemma 3.1 it is suﬃcient to consider the case L ≥ 4; we have to prove that |Ah,h (ω1 ; ω ˜2, . . . , ω ˜ L )| ≤ cL γ

h−h 2

(L−3)

.

(7.27)

Let θi ≡ θh ,ωi , so that θi is the center of the θ-interval, which the polar angle of p has to belong to, if p ∈ Sh ,ωi . For any pair (i, j), we deﬁne φi,j = min{||θi − θj ||, π − ||θi − θj ||} .

(7.28)

By a reordering of the sectors, which is unimportant since we are looking for a bound proportional to cL , we can get the condition (recall that L ≥ 4): φ ≡ φL−1,L ≥ φi,j

∀i, j ∈ [2, L] .

,

(7.29)

Note that, given ω ˜ ∈ Oh , |ω ∈ Oh : Sh ,ω ⊂ Sh,˜ω | = γ

h−h 2

.

(7.30)

Hence, given any positive constant c0 , if we deﬁne

h /2 ˜ 2, . . . , ω ˜ L ) : φ ≤ Lc−1 }, A< = {(ω2 , . . . , ωL ) ∈ Ah,h (ω1 , ω 0 γ

we have: |A< | ≤ γ

h−h 2

(L−3)

2 (cLc−1 0 ) ,

(7.31)

(7.32)

2 where (cLc−1 0 ) is a bound on the number of possible choices of ωL−1 and ωL , given ω1 , . . . , ωL−2 . Hence, in order to prove (7.27), it is suﬃcient to prove that, if c0 is small enough, a similar bound is valid for the set

h /2 ˜ 2, . . . , ω ˜ L ) : φ ≥ Lc−1 }. A> = {(ω2 , . . . , ωL ) ∈ Ah,h (ω1 , ω 0 γ

We have |A> | ≤ mL γ

h−h 2

(L−3)

,

(7.33)

(7.34)

where mL is a bound on the number of choices of ωL−1 and ωL , given ω1 , . . . , ωL−2 . In order to get mL , we consider a particular choice of ω2 , . . . , ωL−2 ∈ Oh and we suppose that the set E = {(ωL−1 , ωL ) : (ω2 , . . . , ωL ) ∈ A> } is not empty. Moreover, we deﬁne max φL−1,L , (7.35) φ0 = (ωL−1 ,ωL )∈E

By deﬁnition, for any choice of (ωL−1 , ωL ) ∈ E, we can ﬁnd L vectors k (1) , . . . , k (L) , such that k (i) ∈ Sh ,ω and i L i=1

k (i) = 0 .

(7.36)

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Moreover, by Lemma 7.3, for i = 1, . . . , L, we can write k (i) = pF (θi ) + xi n(θi ) + yi τ (θi ) ,

|xi | ≤ cγ h

,

|yi | ≤ cγ h /2 .

(7.37)

h /2 Hence, since φ0 ≥ Lc−1 , we get 0 γ

h pF (θi )| | pF (θ2 )| sin φi,2 + O(γ 2 ) ≤ cφ0 , |k (i) ∧ pF (θ2 )| = |

for i = 2, . . . , L, and, by using (7.36), L (1) (i) pF (θ2 )| = k ∧ pF (θ2 ) ≤ cLφ0 , |k ∧

(7.38)

(7.39)

i=2

so that φ1,2 ≤ cLφ0 . Lemma 7.1, (7.37) and (7.39) easily imply that k (i) = pF (θi ) + x ¯i n(θ2 ) + y¯i τ (θ2 ), h /2 h /2 |¯ xi | ≤ cφ0 γ h /2 if i > 1 , |¯ yi | ≤ cγ h /2 if i = 1 cLφ0 γ cLγ

if i > 1 . (7.40) if i = 1

Let us now deﬁne a = −

L−2

pF (θi )

,

b = k (L−1) + k (L) ⊥ ⊥

,

r = b − a ,

(7.41)

i=1

where k⊥ denotes the projection on the Fermi surface, see (7.15). By using Lemma 7.4, the momentum conservation (7.36) and (7.40), we get r = r1 n(θ2 ) + r2 τ (θ2 ) ,

|r1 | ≤ cLφ0 γ h /2

,

|r2 | ≤ cLγ h /2 .

(7.42)

Note that the vector a deﬁned in (7.41) is ﬁxed, if the indices ω1 , . . . , ωL−2 (i) are ﬁxed. Hence, if we put pF (θ¯i ) = k⊥ , i = L − 1, L, mL can be calculated by studying the possible solutions of the equation pF (θ¯L ) = a + r , pF (θ¯L−1 ) +

(7.43)

(0) (0) as r varies satisfying (7.42). Let (θ¯L−1 , θ¯L ) be a particular solution of (7.43), (0) such that k (i) ∈ Sh ,ωi , i = L − 1, L, with φL−1,L = φ0 , and put b0 = pF (θ¯L−1 ) + (0) pF (θ¯L ) = a +r0 , so that (7.43) can be written as pF (θ¯L−1 )+pF (θ¯L ) = b0 +(r −r0 ). (0) The deﬁnition of φ0 implies that r − r0 can be represented as r − r0 = r1 n(θ¯L ) + (0) r2 τ (θ¯L ), with |r1 | ≤ cLφ0 γ h /2 and |r2 | ≤ cLγ h /2 . Hence a simple application of Lemma 7.5 shows that the solutions of (7.43) belong, up to a exchange between θ¯L−1 and θ¯L , to a connected set and that mL ≤ cL2 , if c0 ≤ c¯2 /c1 , where c1 is the constant c of (7.42) and c¯2 is deﬁned in Lemma 7.5.

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Acknowledgments We are indebted to M. Disertori and V. Rivasseau for their enlightening explanation of their work [DR].

References [AGD]

A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of quantum ﬁeld theory in statistical physics, Dover, New York (1963).

[BG]

G. Benfatto, G. Gallavotti, Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems, J. Stat. Phys. 59, 541–664 (1990).

[BG1]

G. Benfatto, G. Gallavotti, Renormalization group, Physics Notes 1, Princeton University Press, Princeton (1995).

[BGPS]

G. Benfatto, G. Gallavotti, A. Procacci, B. Scoppola, Beta functions and Schwinger functions for a many fermions system in one dimension, Comm. Math. Phys. 160, 93–171 (1994).

[BM]

G. Benfatto, V. Mastropietro, Renormalization group, hidden symmetries and approximate Ward identities in the XYZ model,Rev. Math. Phys. 13, 1323–1435 (2001).

[BoM]

F. Bonetto, V. Mastropietro, Beta function and anomaly of the Fermi surface for a d = 1 system of interacting fermions in a periodic potential, Comm. Math. Phys. 174, 191–214 (1995).

[DR]

M. Disertori, V. Rivasseau, Interacting Fermi liquid in two dimensions at ﬁnite temperature, I and II, Comm. Math. Phys. 215, 251–290 and 291–341 (2000).

[FMRT]

J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, An inﬁnite volume expansion for many fermions Green functions, Helv. Phys. Acta 65, 679– 721 (1992).

[FT]

J. Feldman, E. Trubowitz, Perturbation theory for many fermion systems, Helv. Phys. Acta 63, 156–260 (1990).

[FKT]

J. Feldman, H. Kn¨ orrer, E. Trubowitz, A two-dimensional Fermi liquid, series of Preprints (2002).

[G]

G. Gallavotti, Renormalization theory and ultraviolet stability for scalar ﬁelds via renormalization groups methods, Reviews of Modern Physics 57, 471–562 (1985).

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193

[GM]

G. Gentile, V. Mastropietro, Renormalization group for fermions: A review on mathematical results, Phys. Reports 352, 273–437 (2001).

[Le]

A. Lesniewski, Eﬀective action for the Yukawa2 quantum ﬁeld theory, Comm. Math. Phys. 108, 437–467 (1987).

[So]

J. Solyom, The Fermi gas model of one-dimensional conductors, Adv. Phys. 28, 201–303 (1958).

[VLSAR] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, A.E. Ruckenstein, Phenomenology of the normal state of Cu-O high temperature superconductors, PRL 63, 1996–1999 (1989).

G. Benfatto and V. Mastropietro Supported by M.I.U.R. Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientiﬁca I-00133, Roma Italy email: [email protected] email: [email protected] A. Giuliani Dipartimento di Fisica Universit` a di Roma “La Sapienza” Piazzale Aldo Moro 5 I-00185, Roma Italy email: [email protected] Communicated by Joel Feldman submitted 05/07/02, accepted 30/09/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 195 – 197 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/010195-3 DOI 10.1007/s00023-003-0126-8

Annales Henri Poincar´ e

Matrices de Diﬀusion pour l’Op´erateur de Schr¨ odinger en Pr´esence d’un Champ Magn´etique. Ph´enom`ene de Aharonov-Bohm Erratum: Annales I.H.P. Vol. 61, n◦ 3, 1994 Fran¸cois Nicoleau

Dans un preprint r´ecent [4], Roux-Yafaev ont obtenu des r´esultats sur la matrice de diﬀusion pour l’op´erateur de Schr¨ odinger avec champ magn´etique qui sont contradictoires avec ceux obtenus dans notre papier [2]. En eﬀet, Roux-Yafaev montrent que, contrairement `a ce qui est ´enonc´e `a tort dans le Th´eor`eme 2 de [2], la matrice de diﬀusion n’est pas forc´ement une perturbation compacte de l’identit´e et que son spectre peut recouvrir le cercle unit´e complexe. Notre erreur initiale impose en fait la correction de la plupart des ´enonc´es de [2] (en particulier le th´eor`eme 6 et son corollaire) s’appuyant sur une approche perturbative. Dans cet erratum, nous montrons comment corriger le papier [2]; nous reprenons toutes les notations de [2], ainsi que la num´erotation des ´equations qui s’y trouvent. L’erreur initiale de [2] provenait d’un mauvais choix de microlocalisation dans la construction des op´erateurs Fourier int´egraux (O.F.I) Jj,A , j = 1, 2 (p. 336, lignes 13–17). Ces O.F.I ´etaient introduits aﬁn d’obtenir une formule de repr´esentation des matrices de diﬀusion (Equation (2.23)–(2.25)). Avec la microlocalisation donn´ee dans [2], la remarque (p. 337, lignes 4–5) est erron´ee. Pour que cette derni`ere soit exacte, il faut consid´erer la microlocalisation suivante ([1]). D´eﬁnition de la microlocalisation Pour j = 1, 2, soient σ0j , σ1j ∈]0, 1[, γ > 0 assez petit, tels que: −1 < σ01 − γ < σ01 < σ11 < σ11 + γ < 0 < σ02 − γ < σ02 < σ12 < σ12 + γ < 1 . j + + ∞ Ψ+ j ∈ C ([−1, 1]), Ψj (τ ) = 1 si τ ∈ [σ1 , 1] et Ψj (τ ) = 0 si τ ∈ [−1, j − − ∞ Ψ− j ∈ C ([−1, 1]), Ψj (τ ) = 1 si τ ∈ [−1, σ0 ] et Ψj (τ ) = 0 si τ ∈ [

σ0j + σ1j ]. 2

σ0j + σ1j , 1]. 2

χ1,j ∈ C ∞ ([−1, 1]), χ1,j (τ ) = 1 si τ ∈ [−1, σ0j − γ] et χ1,j (τ ) = 0 si τ ≥ σ0j . χ2,j ∈ C ∞ ([−1, 1]), χ2,j (τ ) = 1 si τ ∈ [σ1j + γ, 1] et χ2,j (τ ) = 0 si τ ≤ σ1j .

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La phase Φj (x, ξ) de l’ O.F.I Jj,A est donn´ee par (Equation (2.13), [2]) o` u Ψ+ , − (resp. Ψ− ), est remplac´e par Ψ+ , (resp. Ψ ). L’amplitude d (x, ξ) de l’ O.F.I j,A j j Jj,A est donn´ee par (Equation (2.21), [2]) o` u χ1 , (resp. χ2 ), est remplac´e par χ1,j , (resp. χ2,j ). On a alors la formule de repr´esentation des matrices de diﬀusion ([2], Equations (2.23)–(2.25)): ∗ SA (λ) − 1 = −2iπΓ0 (λ)J1,A T2,A Γ∗0 (λ) ∗ R(λ + i0)T2,AΓ∗0 (λ) + 2iπΓ0 (λ)T1,A

: = BA (λ) + CA (λ).

(1.1)

Avec cette nouvelle microlocalisation, CA (λ) est un op´erateur a` noyau C ∞ sur S n−1 × S n−1 . Le terme pr´edominant de la matrice de transition est donc donn´e par BA (λ). Pour l’´etudier, contrairement a` [2], nous ne consid´erons plus BA (λ) comme une perturbation de B0 (λ), mais comme un op´erateur pseudo-diﬀ´erentiel sur la sph`ere; en suivant les techniques de [5], un calcul facile nous montre que BA (λ) est un 0 op´erateur pseudodiﬀ´erentiel sur la sph`ere dans la classe Sρ,1−ρ de H¨ormander, de symbole principal pour | y | 1, | ω |= 1 et < y, ω >= 0: b(ω, y; λ) = e

i

+∞ −∞

1

A(λ− 2 y+sω).ω ds

− 1.

(1.2)

Par cons´equent, modulo un op´erateur compact, SA (λ) est un op´erateur pseudodiﬀ´erentiel sur la sph`ere de symbole principal : s(ω, y; λ) = e

i

+∞ −∞

1

A(λ− 2 y+sω).ω ds

.

(1.3)

En utilisant (1.3), Roux et Yafaev obtiennent en particulier le r´esultat suivant: Proposition 1 ([4], Prop. 6.13) On suppose que pour un point (y, ω) tel que | ω |= 1, < y, ω >= 0,

+∞

lim sup τ →+∞

ou

−∞

+∞

lim inf

τ →+∞

−∞

A(τ y + sω).ω ds = ∞ A(τ y + sω).ω ds = −∞ .

Alors: le spectre de S(λ) recouvre le cercle unit´e.

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References [1] H. Isozaki et H. Kitada, Scattering matrices for two-body Schr¨ odinger operators, Sci. Papers College Arts and Sci., Univ. Tokyo 35, 81–107 (1985). [2] F. Nicoleau, Matrices de diﬀusion pour l’op´erateur de Schr¨ odinger en pr´esence d’un champ magn´etique. Ph´enom`ene de Aharonov-Bohm, Annales I.H.P. Vol 61, 3, 329–346 (1994). [3] P. Roux et D. Yafaev, On the mathematical theory of the Aharonov-Bohm eﬀect, mp-arc 02-165, (2002). [4] P. Roux et D. Yafaev, The scattering matrix for the Schr¨ odinger operator with a long-range electromagnetic potential, mp-arc 02-364, (2002). [5] D. Yafaev, The scattering amplitude for the Schr¨ odinger equation with a long-range potential, Comm. Math. Phys. 191, 183–218 (1998).

Fran¸cois Nicoleau Laboratoire Jean Leray UMR CNRS-UN 6629 D´epartement de Math´ematiques 2, rue de la Houssini`ere BP 92208 F-44322 Nantes cedex 03 France email: [email protected]

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 199 – 215 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020199-17 DOI 10.1007/s00023-003-0127-7

Annales Henri Poincar´ e

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory Abdelmalek Abdesselam

Abstract. The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many diﬀerent angles. We add here another point of view pertaining to the so-called formal inverse approach, that of perturbative quantum ﬁeld theory.

I Introduction The purpose of this modest note, for which we claim no originality except that of connecting apparently unrelated ﬁelds, is to draw the attention of theoretical physicists to one of the major unsolved problems of mathematics [29], viz. the Jacobian conjecture. The question is so simple that it was coined in [3] a problem in “high school algebra”. One can formulate it as follows. Let F : Cn → Cn be a map written in coordinates as F (x1 , . . . , xn ) = (F1 (x1 , . . . , xn ), . . . , Fn (x1 , . . . , xn )) .

(1)

One says that F is a polynomial map if the functions Fi : Cn → C are polynomial. Suppose that the Jacobian determinant def

JF (x1 , . . . , xn ) = det

∂Fi (x1 , . . . , xn ) ∂xj

(2)

is identically equal to a nonzero constant. Show then that F is globally invertible def (for the composition of maps) and that its inverse G = F −1 is also a polynomial map. Since it was ﬁrst proposed in [21] (for n = 2 and polynomials with integral coeﬃcients), this problem has resisted all attempts for a solution. In fact, this seemingly simple problem is quite an embarrassment. Indeed, some faulty proofs have even been published (see the indispensable [6] and [11] for a review). We will show here that the Jacobian conjecture can be formulated in a very nice way as a question in perturbative quantum ﬁeld theory (QFT). We also expect any future progress on this question to be beneﬁcial not only for mathematics, but also for theoretical physics as it would enhance our understanding of perturbation theory.

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II The formal inverse as a one-point correlation function The most tempting, yet unfortunately least developed, line of attack on the Jacobian conjecture is the so-called formal inverse approach. One tries to solve explicitly for x = (x1 , . . . , xn ) in the equation y = F (x), one then ﬁnds a power series expression for x in terms of y = (y1 , . . . , yn ). By the uniqueness of the power series inverse, all one has to do then is to show that it is in fact a polynomial, that is the terms of high degree in the y variables vanish. One of the many reasons this approach is in its infancy is that it took more than two centuries (say from [22] to [6]) to have a workable formula for the formal inverse in the multivariable case. Early contributions can be found in [23, 19, 9, 30, 24, 15]. An important contribution concerning formal inversion is due to Gurjar and Abhyankar [4]. Modern literature on reversion and Lagrange-Good type formulas is huge and we invite the reader to consult [6, 14, 18, 34] for more complete references. The ﬁrst formula for the coeﬃcients of the formal inverse power series G in terms of those of F is due to J. Towber and was ﬁrst published in [35] (see also [17]). In physicists’ terms ours is the following. Claim: (A. A., V. Rivasseau) The formal solution of y = F (x), without any assumption on F except that its linear part is invertible, is the perturbation expansion of the normalized one-point correlation function 1 dφdφ φi e−φF (φ)+φy (3) xi = Z Cn where φ1 , . . . , φn ,φ1 , . . . , φn are the components of a complex Bosonic ﬁeld. The integration is over Cn with the measure n d(Re φi )d(Im φi ) def dφdφ = , (4) π i=1 def

we used the notation φF (φ) =

def

n

def

i=1

Z =

φi Fi (φ1 , . . . , φn ), φy =

dφdφ e−φF (φ)+φy .

n i=1

φi yi , and (5)

Cn

We obtained this expression by solving iteratively the equation y = F (x) thereby generating a tree expansion in the same way one expresses the eﬀective action Γ(φ) in terms of the logarithm W (J) of the partition function in QFT (see [36] for instance). We then determined the Feynman rules of this tree expansion and ﬁnally the “path integral” formulation (3), only to realize that in fact our formula is closely related to the one introduced by G. Gallavotti, following a suggestion of G. Parisi, to express the Lindstedt perturbation series in the context of KAM theory [13]. The right-hand side of (3) is deﬁned, using formal Gaussian integration, as a ratio of formal power series in the y variables with coeﬃcients in C, thanks to the

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formula, also known as “Wick’s theorem”, for the moments of a complex Gaussian measure. In the following, we will state and prove a precise theorem, using some analysis, for the case where Fi (x) = xi − Hi (x), with the Hi (x), 1 ≤ i ≤ n, being homogeneous of the same degree d. Indeed, it is enough to treat the cubic case d = 3, for all dimensions n, in order to prove the Jacobian conjecture in full generality [6]. However, formula (3) is completely combinatorial in nature and its proper setting is in the ring of formal power series with variables corresponding to the ﬁelds φ and φ together with the y’s, over any commutative ring containing the ﬁeld of rationals. One simply has to deﬁne formal Gaussian integration, somewhat in the spirit of [5]. We refer to [1] for a formulation and proof of our claim as a decent mathematical theorem. The latter article will also provide more details on how Feynman diagrams can be useful in algebraic combinatorics and how well they ﬁt in the Joyal theory of combinatorial species [20]. We also refer to [2] for a very simple heuristic proof of the Lagrange-Good multivariable inversion formula, which becomes a fully rigorous and purely combinatorial proof when interpreted using the formalism of [1]. Now let Fi (x) = xi − Hi (x) with Hi (x) written in tensorial notation as Hi (x) =

n 1 wi,j1 ...jd xj1 . . . xjd d! j ,...,j =1 1

(6)

d

so that the 1-contravariant and d-covariant tensor wi,j1 ...jd is completely symmetric in the j indices. Let us write def

φwφd =

n

n

φi wi,j1 ...jd φj1 . . . φjd

(7)

i=1 j1 ,...,jd =1

so that (3) becomes

1 dφdφ φi exp −φφ + d! φwφd + φy . Gi (y) = 1 dφdφ exp −φφ + d! φwφd + φy

(8)

The free propagator is represented as an oriented line i

j

= δij

(9)

for the contraction of a pair φi φj . There are two types of vertices: the w-vertices represented by

d half lines = φwφd

(10)

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and the y-vertices represented by = φy .

(11)

As is well known in QFT, the numerator and denominator of (8) can be calculated by expanding 1 exp φwφd + φy d! and integrating term by term with respect to the normalized complex Gaussian measure dφdφ e−φφ . The result is a sum over all possible Feynman diagrams that can be built from the vertices of (10) and (11) (and the source φi for the numerator) by joining the half-lines of compatible directions. A quick look at the vertices shows that the only possible diagrams are trees connected to the source φi , or vacuum graphs made by an oriented loop of say k ≥ 1 w-vertices to which k(d − 1) trees, whose leaves are y-vertices, are attached. When one factors out the denominator, the only diagrams that remain are made of a single tree with the source φi as its root. Therefore, at least formally, we have Gi (y) =

V ≥0 N ≥0

1 Ai (T ) V !N !

(12)

T

where T is a Cayley tree (viewed as a set of unordered pairs) on a ﬁnite set E = E(V, N ). The latter is chosen, noncanonically, once for each pair (V, N ), and must be the disjoint union of Eroot of cardinality 1, Einternal of cardinality V and Eleaf of cardinality N . T is constrained by the condition that elements of Eroot ∪ Eleaf have valence 1 while those of Einternal have valence d + 1. This automatically enforces the relation (d − 1)V = N − 1 which can be checked by counting the half-lines. Even though we write, in the sequel, seemingly independent sums over V and N , the previous relation is always assumed. We now deﬁne the amplitude Ai (T ). One directs the edges of T towards the root in Eroot . For each such edge l ∈ T , one introduces an index il in the set {1, . . . , n}. One then considers the expression Ai (T , (il )l∈T ) which is the product of the following factors. – For each a ∈ Eleaf , if l(a) is the unique line going from a, we take the factor yil(a) . – For each b ∈ Einternal , if {l1 (b), . . . , ld (b)} is the set of lines coming into b and l0 (b) is the unique line leaving b, we take the factor wil0 (b) ,il1 (b) ...ild (b) . The resulting monomial in the y’s and w’s is Ai (T , (il )l∈T ) by deﬁnition. Finally Ai (T ) is the sum of Ai (T , (il )l∈T ) over all the indices (il )l∈T except the index of the line arriving at the root which is ﬁxed at the value i, the source index.

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For example, with d = 3, the amplitude of the following tree with V = 4 and N =9

i

is n

Ai (T ) =

wi,α1 α2 α3 wα1 ,α4 α5 α6 wα3 ,α7 α8 α9

α1 ,...,α12 =1

wα9 ,α10 α11 α12 yα2 yα4 yα5 yα6 yα7 yα8 yα10 yα11 yα12 .

(13)

Note that, by the Cayley formula for the number of trees with preassigned valences, the sum over T has ((1 + V + N ) − 2)! (V + N − 1)! = terms. (1 − 1)!((d + 1) − 1)!V (1 − 1)!N d!V

(14)

Let us introduce the norms n

def

||w||∞,1 = max

1≤i≤n

|wi,j1 ...jd |

(15)

j1 ,...,jd =1

and def

||y||∞ = max |yi | . 1≤i≤n

(16)

We now have Theorem 1 The series Gi (y) =

V ≥0 N ≥0

1 Ai (T ) V !N ! T

(17)

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is absolutely convergent, provided def

||y||∞ < R =

d! d 2 ||w||∞,1

d−1 (18)

and satisfies, on this domain of convergence, ||G(y)||∞ ≤

||y||∞ 1−

||y||∞ R

(19)

and F (G(y)) = y .

(20)

Proof. One easily proves by bounding the w and y factors in Ai (T , (il )l∈T ) by their moduli and summing the indices, starting with the leaves and progressing towards the root, that V N (21) |Ai (T )| ≤ ||w||∞,1 ||y||∞ for any ﬁxed tree T . Therefore V,N ≥0

1 |Ai (T )| V !N ! T

(dV )!||w||∞,1 V ||y||∞ (d−1)V +1 ≤ V !((d − 1)V + 1)!d!V V ≥0

d−1 V ||w||∞,1 ||y||∞ (dV )! ≤ ||y||∞ V !((d − 1)V )! d!

(22)

(23)

V ≥0

and one simply uses (dV )! ≤ 2dV V !((d − 1)V )!

(24)

to conclude the convergence proof and obtain the bound (19). Now observe that, on the convergence domain Gi (y) = yi +

V,N,T V ≥1

1 Ai (T ) V !N !

(25)

where the last sum is over trees with at least one w-vertex linked directly to the root. This sum can be performed in the following way. One chooses, among the V internal w-vertices, the vertex w0 ∈ Einternal which hooks to the root. This costs a factor V . Then one divides the remaining vertices into an unordered collection of sets E1 , . . . , Ed such that Ei has Vi w-vertices and Ni y-vertices. This costs a factor 1 (V − 1)! N! . d! V1 ! . . . Vd ! N1 ! . . . Nd !

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Finally one sums over all possible trees T1 , . . . Td on E1 ∪ {w0 }, . . . , Ed ∪ {w0 } as before. The corresponding amplitudes do not depend on the location of the sets Ei in E, but only on the cardinalities Vi and Ni . Therefore Gi (y) = yi +

n

V1 ,...,Vd ≥0 N1 ,...,Nd ≥1

i1 ,...,id =1 T1 ,...,Td

1 V !N !

V.(V − 1)!.N ! wi,i1 ...id Ai1 (T1 ) . . . Aid (Td ) d!V1 ! . . . Vd !N1 ! . . . Nd ! n 1 wi,i1 ...id Gi1 (y) . . . Gid (y) = yi + d! i ,...,i =1 1

(26) (27)

d

= yi + Hi (G(y))

(28)

from which (20) follows. As a result the Taylor series of G at the origin is the right compositional inverse of F . Now algebraic combinatorialists might not be too impressed by this since one can readily rewrite formula (12) under the form given by Towber [34] or Singer [28]. So the series expansion of the formal inverse itself is not new. To obtain a real improvement on previous approaches one has to return to the more fundamental equation (3) and really consider the “integrals” appearing in it as, well, integrals on which one can try all the tools of ordinary calculus: integration by parts, change of variables. . . For an example of the mathematical utility of this way of proceeding, see [5]. Remark. Note that the generalized forest formula of Towber (see [34]) can be easily derived from the perturbation expansion of the higher correlation functions αn 1 φα 1 . . . φn , where we used the standard statistical mechanics notation d 1 def 1 Ω(φ, φ) = (29) dφdφ Ω(φ, φ)e−φφ+ d! φwφ +φy . Z

III Comments on the Jacobian conjecture III.1 What does the constant Jacobian condition mean? Suppose that Fi (x) = xi −

n j1 ,...,jd

1 wi,j1 ...jd xj1 . . . xjd d! =1

(30)

is such that JF (x) = 1 for all x. Several conclusions can be drawn from this constraint. One that is due to V. Rivasseau is that our QFT model is self-normalized. In other words Z=1. (31)

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Indeed, by writing the Feynman diagram expansion of Z=

1

dφdφ e−φφ+ d! φwφ

d

+φy

(32)

one can easily show that log Z =

1

tr M (G(y))k k

(33)

k≥1

where M (x) is the matrix with entries Mij (x)

=

∂Hi (x) ∂xj

=

1 (d − 1)! j

def

(34) n

wi,jj1 ...jd−1 xj1 . . . xjd−1

(35)

1 ,...,jd−1 =1

that is Z

= = =

exp (−tr log (I − M (G(y)))) 1 ∂F

det ∂x (G(y)) 1 . JF (G(y))

(36) (37) (38)

In the case where Fi (x) = xi − Hi (x) with the Hi (x) homogeneous of the same degree d, it is easy to show that the Jacobian condition is equivalent to M (x) being nilpotent for all x (see [6]). There are essentially two ways to express this

or

M (x)n = 0 ∀x ∈ Cn

(39)

tr M (x)k = 0 ∀k ≥ 1, ∀x ∈ Cn .

(40)

Equation (39) means that when one considers a chain (or caterpillar) diagram like i

j

(41)

with n w-vertices, its contribution, for ﬁxed i and j, is zero after symmetrization of the indices of the n(d − 1) incoming lower legs.

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Equation (40) means that loop diagrams like

(42)

with k ≥ 1 w-vertices, vanish after symmetrization of the indices of the k(d − 1) incoming legs. The formal inverse approach to the Jacobian conjecture can now be rephrased as the following Problem: Show explicitly that in the polynomial algebra C[w] with indeterminates given by the tensor elements wi,j1 ...jd , the y = 0 connected correlation functions φi φj1 . . . φjN cy=0 belong to the radical of the ideal generated by the symmetrized chains and/or loops, provided the degree N is large enough. This statement is by the Hilbert nullstellensatz equivalent to the Jacobian conjecture. It is even a theorem due to S. Wang [32] in the (d = 2) quadratic case. The proof is non-constructive however, and an explicit combinatorial argument is an urgent desideratum.

III.2 Chains and/or loops? Let c be the ideal of C[w] generated by the symmetrized chains of length n, and let l be the ideal generated by the symmetrized loops of length k ≥ 1. While it is very tempting to work with c, it seems more fundamental to use l. This conclusion is implicit in [34]. Indeed the author uses the diagonal minor sums, i.e., the elementary symmetric functions of the eigenvalues, to express the nilpotence of M (x), instead of the matrix elements of M (x)n . We use the loops, that is the Newton power sums of the eigenvalues, which makes no diﬀerence since our ground ring is C. Note that c ⊂ l: this is the Cayley-Hamilton theorem, i.e., “the Jacobian √ problem for d = 1”! But we also have l ⊂ c, trivially because a nilpotent matrix must have zero eigenvalues and therefore the Newton sums of these eigenvalues are zero. It is very instructive to understand these two elementary statements in a purely combinatorial way. Regarding the ﬁrst inclusion, we were surprised to ﬁnd in the recent literature a combinatorial proof, with a ﬂavor of loop-erased random walk, of the eminently classical Cayley-Hamilton theorem [31]. As for the second inclusion, there is a very nice explicit Fermionic proof by C. de Calan and J. Magnen that, for a generic n × n matrix N , (tr N )k(n−1)+1 is in the ideal generated by the matrix elements of N k . We brieﬂy include this proof here, with due permission from the authors, as an example of application of the Pauli exclusion principle on which we will comment later in Section III.3 .

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First notice that for any integer k ≥ 2, there are numbers a2 , a3 , . . . , ak−1 such that log(1 + X + a2 X 2 + · · · + ak−1 X k−1 ) = X + O(X k ). Indeed, one only def 1 p!

need take ap =

since

log 1 + X + a2 X 2 + · · · + ak−1 X k−1    Xp  = log eX 1 − e−X p! p≥k     Xp (−X)q   = X + log 1 −  p! q!

(44)

q≥0

p≥k

= X + O(X k ) .

(45)

One then lets def

(43)

A = λN +

k−1

ap (λN )p

(46)

p=2

where λ is an auxiliary complex variable. One then considers the complex Fermionic integral n def Z = dµC (ψ, ψ) e i,j=1 ψi Aij ψj (47)

where dµC (ψ, ψ) is the normalized complex Fermionic Gaussian measure with def

complex conjugate ﬁelds ψ and ψ, and with covariance C = In the n × n identity matrix (see the appendix B of [27] or [12] for a deﬁnition). One has Z = det(In + A) = etr

log(In +A)

k

= etr(λN +λ

N k S)

(48)

where S is a power series in λN . One then computes the rth derivative, with respect def to λ, of Z at λ = 0. Here r = n(k−1)+1. From (47), using the pigeonhole principle and the fact that the square of a Berezin variable is zero, one sees that r d Z =0 (49) dλ since one would generate in the integrand a sum of products containing at least n + 1 pairs of the form ψ i ψj . This vanishing phenomenon is what we mean by the “Pauli exclusion principle”. On the other hand one observes that r k k d etr(λN +λ N S) = 0 (50) dλ λ=0

contains a term exactly equal to (tr N )k , while the remaining terms all contain an N k S factor which puts them into the ideal generated by the matrix elements of N k , therefore proving the assertion.

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To see why the ideal c is tempting to work with, we need to recall a theorem, ﬁrst conjectured by Wang in the quadratic case [32], and proved in full generality by O. Gabber (see [6]). Theorem 2 If F : Cn → Cn is globally invertible with polynomial inverse G then deg G ≤ (deg F )n−1

(51)

def

where deg F = max1≤i≤n (deg Fi ) and likewise for G. In our context, this means that the vanishing of the connected correlation functions < φi φj1 . . . φjN >cy=0 should happen as soon as N > dn−1 . Note that this bound is saturated by the well-known triangular example given by Fi (x) = xi − xdi+1 , for 1 ≤ i ≤ n − 1, and Fn (x) = xn . But dn−1 is the maximal number of leaves of those of our trees which have a depth less than or equal to n − 1. If the chains in (41) needed not be symmetrized, the Jacobian conjecture would be trivial! Indeed, a tree with more than dn−1 leaves must have a chain of length at least n, going from the root φi to one of the leaves φjα . This observation, which goes back to [6], was likely the main impetus behind the formal inverse approach. Remark that if we condition the sum over Feynman diagrams for the correlations φi φj1 . . . φjN cy=0 , by requiring the path between the root φi and a specified leaf φjα to be of a certain length ≥ n; the branches will be automatically symmetrized and the result would be zero. The problem is that we cannot know in advance which leaf will be linked to the root by a long chain. In relation to previously used formal inversion formulas, let us mention that it is against QFT wisdom to mix the index space {1, . . . , n} and the abstract space E that labels the vertices, as far as the combinatorics are concerned. From a QFT point of view, which admittedly is only one among many on the Jacobian conjecture, it is unnatural to use sums over colored or planar objects, as this reduces symmetry in the resulting expansion instead of enhancing it. We nevertheless concede the point that planarity can serve to “locate” the long chain, and order the trees accordingly, which is the main ingredient of the combinatorial “tour de force” of [28]. One of the cases treated in the latter article is that of Fi (x) = xi − Hi (x), with the Hi homogeneous of the same degree d and the matrix M (x) nilpotent of order 2. This has already been treated in [6] and [7] for instance, but let us sketch how to prove this result with our QFT model. The argument is adapted from an idea by V. Rivasseau. First perform the translation change of variables φ → φ + y, φ → φ in (3) to get, using Z = 1, (52) Gi (y) = yi + dµ(φ, φ) φi eφH(φ+y) def

where dµ(φ, φ) = dφdφ e−φφ . This unorthodox change of variables used in (52), which treats φ and φ as independent variables and not as complex conjugates of one

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another, can be justiﬁed a posteriori by comparing the diagrammatic expansions on both sides of the equation. If one insists on viewing φ and φ as complex conjugate variables, one needs analyticity of the integrand in both φ and φ as well as a contour deformation to justify this type of change of variables. Note however that φ and φ can also be considered as Fourier dual variables. Indeed, we learnt from A. Grigis an inversion formula similar to (3) based on the latter interpretation (see exercise 3.2 in [16] and also [1]). One can integrate the source φi by parts to get ∂ φH(φ+y) Gi (y) = yi + dµ(φ, φ) e (53) ∂φi = yi + dµ(φ, φ) Hi (φ + y)eφH(φ+y) (54) = yi + dµ(φ, φ) Hi (φ + y)esφH(φ+y) . (55) s=1

Then, interpolate between s = 1 and s = 0 to get Gi (y) = yi + dµ(φ, φ) Hi (φ + y) +

1

ds Ωi (s, y)

(56)

0

where def

Ωi (s, y) =

dµ(φ, φ) Hi (φ + y) φH(φ + y) esφH(φ+y) .

(57)

Notice that the second term of (56) reduces to Hi (y), whereas for the third we have, by integrating the φ by parts n ∂ Hi (φ + y)Hj (φ + y)esφH(φ+y) (58) Ωi (s, y) = dµ(φ, φ) ∂φj j=1 = Ω1i (s, y) + Ω2i (s, y) + Ω3i (s, y) where def

Ω1i (s, y) =

 dµ(φ, φ) 

n

(59) 

Mij (φ + y)Hj (φ + y) esφH(φ+y)

(60)

  n dµ(φ, φ) Hi (φ + y)  Mjj (φ + y) esφH(φ+y)

(61)

j=1

def

Ω2i (s, y) =

j=1

and def

Ω3i (s, y) =

 dµ(φ, φ) Hi (φ + y) 

n

 Hj (φ + y)sφk Mkj (φ + y) esφH(φ+y)

j,k=1

(62)

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Note that Ω2i (s, y) = 0 since it contains nj=1 Mjj (x) = tr M (x) at x = φ + y, and M (x) is nilpotent. Likewise, Ω1i (s, y) and Ω3i (s, y) vanish since the integrand contains a factor of the form n

Mkj (x)Hj (x)

=

j=1

= =

n 1 Mkj (x)Mjl (x)xl d

(63)

M (x)2 kl xl

(64)

1 d

j,l=1 n l=1

0

(65)

where we used Euler’s identity for the homogeneous Hi ’s, and the fact that M (x) is nilpotent of order 2. As a result Gi (y) = yi + Hi (y).

III.3 The Pauli exclusion principle In order to be able to prove the Jacobian conjecture by purely combinatorial means, one needs to exhibit a volume eﬀect similar to the Pauli exclusion principle, as otherwise one would not see the ﬁniteness of the index set {1, . . . , n} within the strictly tensorial Feynman diagrammatic notation where indices are contracted, i.e., summed over. One would love to have Fermions, instead of Bosons, entering the picture. Let us mention three, typically ﬁeld theoretic, ideas that have not been pursued in previous attempts with the formal inverse approach, and which deserve further investigation. One way to introduce Fermions in a purely Bosonic model is to exhibit a supersymmetry. If this could be done, it would probably be the “voie royale” towards understanding the conjecture. Unfortunately we have not been able to make much headway in this direction so far. Let us simply mention a strange feature of our model that hints towards a hidden supersymmetry. As a result of our choice of vertices and the fact that the propagators are directed, the perturbation expansion of the one-point function is reduced to a tree graph expansion. This means that the semi-classical expansion of our model around the “false vacuum” φ = 0, φ = 0 is exact. Besides, the “integrals” in (3) which are supposed to be over CN reduce to the contribution of a single critical point: the “true vacuum” ∂ ∂ (φF (φ)−φy) = 0 and ∂φ (φF (φ)−φy) = 0 that is φ = G(y) obtained by solving ∂φ i i

and φ = 0. This is reminiscent of the Duistermaat-Heckman theorem [10] which is known to involve supersymmetry (see [33]). The Gabber inverse degree bound, together with the previously given example that saturates it, suggest that the sought exclusion principle has to act along the chains from the root to the leaves of the trees but not across, i.e., within generations. This is quite odd in view of the eventual introduction of Fermionic variables in our model. This however hints to the possibility that the problem may

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come from “divergent” two-point subgraphs, i.e., parts of the diagrams that look like

i

j (66)

where the two indices i and j coincide. This leads to the following Question: Is there a way to eliminate these “divergent” pieces by adding, to the “action” φF (φ) − φy, counter-terms that are made of symmetrized loops? This is possible for d = 1, that is for the Cayley-Hamilton theorem, but does not seem to be the consequence of a natural renormalization condition on the twopoint correlation function. It would be interesting to explore this idea using the new point of view on renormalization pioneered by A. Connes and D. Kreimer [8] since one of their motivations was the study of formal diﬀeomorphisms which is clearly related to our subject material. Finally a third approach, which reverses the usual rationale of a Mayer expansion in statistical mechanics [26] and constructive ﬁeld theory [25], is to try to build up as many hardcore constraint factors (1 − δij ) as possible, where i and j are propagator line indices, in the diagrammatic expansion of the formal inverse. This has to be achieved modulo error terms made of symmetrized loops, which will vanish under the Jacobian condition. Again, for the moment, we could only make this method work for the case d = 1.

IV Conclusion We hope to have provided enough evidence that the Jacobian conjecture is a very beautiful combinatorial challenge, where mathematicians, either conceptually or computationally inclined, and theoretical physicists could fruitfully share their knowledge. While future progress on the conjecture itself is still uncertain, there is bound to be beneﬁts from such an interdisciplinary collaboration on this problem.

Acknowledgments The author is grateful to V. Rivasseau for early encouragements and collaboration on this project. Some of the ideas presented here are due to him. We thank D. Brydges, C. de Calan, A. Grigis and J. Magnen for enlighting discussions. We

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also thank J. Feldman for his invitation to the Mathematics Department of the University of British Columbia where part of this work was done. The pictures in this article have been drawn using a software package kindly provided by J. Feldman. Finally the support of the Centre National de la Recherche Scientiﬁque is gratefully acknowledged.

References [1] A. Abdesselam, Feynman diagrams in algebraic combinatorics, math. CO/0212121, preprint, (2002). [2] A. Abdesselam, A physicist’s proof of the Lagrange-Good multivariable inversion formula, math.CO/0208174, preprint, (2002). [3] S.S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 N◦ 6, 409–448 (1976). [4] S.S. Abhyankar, Lectures in algebraic geometry, Notes by Chris Christensen, Purdue Univ., 1974. [5] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. Thurston, The Aarhus integral of rational homology 3-spheres I: A highly non-trivial ﬂat connection on S 3 , Sel. math., New ser. 8, 315–339, (2002). [6] H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 N◦ 2, 287–330, (1982). [7] C. Cheng, T. Sakkalis and S. Wang, A case of the Jacobian conjecture, J. Pure Appl. Algebra 96 N◦ 1, 15–18, (1994). [8] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 N◦ 1, 203–242, (1998). [9] G. Darboux, Sur la s´erie de Laplace, C.R. Acad. Sci. Paris 68, 324–327, (1869). [10] J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 N◦ 2, 259–268, (1982). [11] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Basel, Birkh¨ auser Verlag, 2000. [12] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Fermionic functional integrals and the renormalization group, CRM Monograph Series, 16, Providence RI, AMS, 2002.

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[13] G. Gallavotti, Perturbation Theory, In: Mathematical physics towards the XXI century, 275-294, R. Sen and A. Gersten, eds., Ber Sheva, Ben Gurion University Press, 1994. [14] I.M. Gessel, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A 45 N◦ 2, 178–195, (1987). [15] I.J. Good, Generalization to several variables of Lagrange’s expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56, 367– 380, (1960). [16] A. Grigis and J. Sj¨ ostrand, Microlocal analysis for diﬀerential operators, An introduction, London Mathematical Society Lecture Note Series, 196, Cambridge, Cambridge University Press, 1994. [17] M. Haiman and W. Schmitt, Incidence algebra antipodes and Lagrange inversion in one and several variables, J. Combin. Theory Ser. A 50 N◦ 2, 172–185, (1989). [18] P. Henrici, Die Lagrange-B¨ urmannsche Formel bei formalen Potenzreihen, Jahresber. Deutsch. Math. Verein. 86, 115–134, (1984). [19] C.G.J. Jacobi, De resolutione aequationum per series inﬁnitas, J. Reine Angew. Math. 6, 257–286, (1830). [20] A. Joyal, Une th´eorie combinatoire des s´eries formelles, Adv. Math 42, 1–82, (1981). [21] O.H. Keller, Ganze Cremona-Transformationen, Monats. Math. Physik 47, 299–306 (1939). [22] J.L. de Lagrange, Nouvelle m´ethode pour r´esoudre des ´equations litt´erales par le moyen des s´eries, M´em. Acad. Roy. Sci. Belles-Lettres Berlin, 24 (1770). [23] P.S. Laplace, M´emoire sur l’usage du calcul aux diﬀ´erences partielles dans la th´eorie des suites, M´em. Acad. Roy. Sci. Paris, 99–122 (1777). [24] H. Poincar´e, Sur les r´esidus des int´egrales doubles, Acta. Math. 9, 321–380, (1886). [25] V. Rivasseau, From perturbative to constructive renormalization, Princeton NJ, Princeton University Press, 1991. [26] D. Ruelle, Statistical mechanics: Rigorous results, New York-Amsterdam, W.A. Benjamin, 1969. [27] M. Salmhofer, Renormalization, An introduction, Texts and Monographs in Physics, Berlin, Springer-Verlag, 1999.

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[28] D. Singer, On Catalan trees and the Jacobian conjecture, Electron. J. Combin. 8 N◦ 1, Research paper 2, 35 pp., (2001). [29] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 N◦ 2, 7 (1998). ´ [30] T.J. Stieltjes, Sur une g´en´eralisation de la s´erie de Lagrange, Ann. Sci. Ecole Normale. Paris S´er. 3 2, 93–98 (1885). [31] H. Straubing, A combinatorial proof of the Cayley-Hamilton theorem, Discrete Math. 43 N◦ 2–3, 273–279 (1983). [32] S. Wang, A Jacobian criterion for separability, J. Algebra 65, 453–494, (1980). [33] E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 N◦ 4, 303–368 (1992). [34] D. Wright, Reversion, trees, and the Jacobian conjecture, In: Combinatorial and computational algebra, 249–267, Contemp. Math., 264, Providence RI, Amer. Math. Soc., 2000. [35] D. Wright, The tree formulas for reversion of power series, J. Pure Appl. Algebra 57 N◦ 2, 191–211 (1989). [36] J. Zinn-Justin, Quantum ﬁeld theory and critical phenomena, New York, The Clarendon Press, Oxford University Press, 1989.

Abdelmalek Abdesselam D´epartement de Math´ematiques Universit´e Paris XIII, Villetaneuse Avenue J.B. Cl´ement F-93430 Villetaneuse France email: [email protected] Communicated by Joel Feldman submitted 16/09/02, accepted 3/01/03

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 217 – 237 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020217-21 DOI 10.1007/s00023-003-0128-6

Annales Henri Poincar´ e

One Non-Relativistic Particle Coupled to a Photon Field Christian Hainzl1 Abstract. We investigate the ground state energy of an electron coupled to a photon ﬁeld. First, we regard the self-energy of a “free” electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant α, the leading order term is represented by 2απ −1 (Λ − ln[1 + Λ]). Next we put the electron in the ﬁeld of an arbitrary external potential V , such that the corresponding Schr¨ odinger operator p2 +V has at least one eigenvalue, and show that by coupling to the radiation ﬁeld the binding energy increases, at least for small enough values of the coupling constant α. Moreover, we provide concrete numbers for α, the ultraviolet cut-oﬀ Λ, and the radiative correction for which our procedure works.

1 Introduction and main results In recent times great eﬀort was made to start a program with the goal to put Quantum Electrodynamics (QED) on a ﬁrm mathematical footing. Bach, Fr¨ ohlich, and Sigal were the ﬁrst who proved (in [BFS1, BFS2, BFS3]), at least for small values of various parameters, that atoms and molecules have a ground state in the presence of a quantized radiation ﬁeld. In the one particle case Griesemer, Lieb, and Loss in [GLL] succeeded in removing the restrictions on the parameters. We refer to [GLL] for an extensive list of references concerning this subject. In the present paper we are especially interested in the behavior of one electron coupled to a photon ﬁeld. We consider the “free” case as well as the presence of an arbitrary external potential V with at least one negative eigenvalue in the electron case. We are interested in the ground state energy of such an electron when the coupling constant α is small and the ultraviolet cut-oﬀ parameter Λ is ﬁxed. Without radiation ﬁeld the self-energy of an electron is simply given by the bottom of the spectrum of −∆, which is equal to 0. In the case of coupling to a photon ﬁeld the situation changes dramatically and the self-energy is a complicated function depending on α and Λ. The fact that evaluating this quantity is a highly non-trivial problem was pointed out in [LL] by Lieb and Loss, who, in contrary to us, considered the case of ﬁxed α and large values of the cut-oﬀ parameter Λ. The paper is organized as follows. In the present section we state our main results on the self-energy and on the binding energy of an electron, which will be proved in Sections 3 and 4. In Section 2 we comment on our notation and in 1 Marie

Curie Fellow

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Section 5 we evaluate the radiative correction in the case of a nucleus. We provide concrete numbers for α and Λ for which our procedure works.

1.1

Self-energy of an electron

The self-energy of an electron is described by the Pauli-Fierz operator √ √ T = (p + αA(x))2 + ασB(x) + Hf .

(1.1)

We ﬁx units such that√ = c = 1 and the electron mass m = 12 . The electron charge is then given by e = α, with α ≈ 1/137 the ﬁne structure constant. In the present paper α plays the role of a small, dimensionless number which measures the coupling to the radiation ﬁeld. Our results hold for suﬃciently small values of α. σ is the vector of Pauli matrices (σ1 , σ2 , σ3 ). Recall, the σi are hermitian 2 × 2 matrices and fulﬁll the anti-commutation relations σi σj + σj σi = 2IC2 δi,j . The operator p = −i∇ is the electron momentum while A is the magnetic vector potential. The magnetic ﬁeld is B = curl A. The underlying Hilbert space is H = L2 (R3 ; C2 ) ⊗ Fb (L2 (R3 ; C2 )) where Fb is the bosonic Fock space for the photon ﬁeld. The vector potential is 1 χ(|k|) A(x) = ελ (k) aλ (k)eikx + a∗λ (k)e−ikx dk, 1/2 2π R3 |k|

(1.2)

(1.3)

λ=1,2

and the corresponding magnetic ﬁeld reads 1 χ(|k|) B(x) = (k ∧ iελ (k)) aλ (k)eikx − a∗λ (k)e−ikx dk, 1/2 2π R3 |k|

(1.4)

λ=1,2

where the operators aλ and a∗λ satisfy the usual commutation relations [aν (k), a∗λ (q)] = δ(k − q)δλ,ν ,

[aλ (k), aν (q)] = 0, etc.

(1.5)

The vectors ελ (k) ∈ R3 are any two possible orthonormal polarization vectors perpendicular to k which we assume to be either odd or even, i.e., ελ (−k) = ±ελ (k).

(1.6)

The function χ(|k|) describes the ultraviolet cut-oﬀ on the wave-numbers k. We choose χ to be the Heaviside function Θ(Λ − |k|/lC ), where lC = /(mc) is the Compton wave-length. Therefore Λ = 1/4 corresponds to the photon energy mc2 , which sometimes is considered as a natural upper bound for the cut-oﬀ parameter, since it is the maximum value that guarantees that no pair-production takes place.

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The photon ﬁeld energy Hf is given by Hf = |k|a∗λ (k)aλ (k)dk. λ=1,2

R3

We estimate the leading order in α of the ground state Σα = inf spec T. Recall that α0|A2 |0 = απ −1 Λ2 , where |0 is the vacuum in Fb , describes the vacuum ﬂuctuation of the ﬁeld A. Thus, this term is somehow ab initio included in the self-energy. Therefore it may turn out as a slight surprise that απ −1 Λ2 is not the leading order term of Σα , which relies on the fact that the magnetic ﬁeld √ ασB reproduces the same number but with a negative sign. We show that the leading order term is given by 2απ −1 [Λ − ln(1 + Λ)], which is the content of our ﬁrst theorem. Theorem 1. Let Λ be ﬁxed but arbitrary. Then Σα − 2απ −1 [Λ − ln(1 + Λ)] ≤ C1 α2 (Λ2 + Λ3 ) + C2 α3 (Λ3 + Λ4 ) ln(1 + Λ) (1.7) for some constants C1,2 > 0 independent of α and Λ. It turns out in the proof that one photon is enough to recover the leading order term in α.

1.2

Increase of the binding energy

We consider an electron in an arbitrary external potential, described by a real valued function V , independent of α, which satisﬁes (i) that the negative part of V , V− , is dominated by the kinetic energy p2 = −∆, V− ≤ −ε∆ + Cε ,

(1.8)

for any ε > 0 and for some positive constant Cε , such that the self-adjointness and boundedness from below of p2 + V is guaranteed. Additionally, since we want the essential spectrum to start at 0, we require (ii) that V− tends to 0 at inﬁnity. Furthermore, we assume (iii) that the operator p2 + V has at least one negative energy bound state. The corresponding operator with a quantized radiation ﬁeld reads Hα = T + V,

(1.9)

for which [Hi] guarantees self-adjointness. Recently, Griesemer, Lieb, and Loss [GLL] proved for all values of α, Λ that Hα has an eigenstate, as well as that the binding energy Σα − Eα , with Eα = inf spec Hα ,

(1.10)

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cannot decrease, i.e., Eα − Σα ≤ −e0 ,

(1.11)

2

where −e0 = inf spec [p + V ]. We show that at least for small values of the coupling constant α the binding energy increases, namely (1.11) holds with strict inequality. Theorem 2. Let V satisfy the conditions (i)–(iii), and φ be the ground state of p2 + V with corresponding energy −e0 . Then Eα − Σα ≤ −e0 − αE(V, Λ) + O(α2 )

(1.12)

for some positive number E(V, Λ) depending on V and the cut-oﬀ Λ. Moreover, there exists a number ρ > 0, such that for all α ∈ (0, ρ] Eα − Σα < −e0 .

(1.13)

Remark 1. The equation (1.12) holds for all real V , such that p2 + V is self-adjoint and has a ground state. But only for V satisfying (i)–(iii) we know that Hα has a ground state. We actually check that E(V, Λ) is bounded from above by E(V, Λ) ≤ pφ

2 32π

3

ln[1 + Λ].

(1.14)

In the spinless case (B = 0) it was previously proven ([HVV]) that coupling to a photon ﬁeld can produce a ground state even if the corresponding Schr¨ odinger operator has only continuous spectrum. This phenomenon of enhanced binding was earlier shown in [HS] in the case of dipole approximation (A = A(0)) in the limit of large values of the coupling constant α.

2 Notation Throughout the paper we use the notation A(x) = D(x) + D∗ (x), B(x) = E(x) + E ∗ (x)

(2.1)

for the vector potential, respectively the magnetic ﬁeld. The operators D∗ and E ∗ create a photon with wavefunctions G(k)e−ik·x and H(k)e−ik·x , respectively, where G(k) = (G1 (k), G2 (k)) and H(k) = (H 1 (k), H 2 (k)) are vectors of one-photon states, given by χ(|k|) ελ (k), 2π|k|1/2

(2.2)

−iχ(|k|) k ∧ ελ (k). 2π|k|1/2

(2.3)

Gλ (k) = and H λ (k) =

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Applying Fourier transform, with respect to k, on G(k)e−ik·x , i.e., −1 −ik·x F [G(k)e ](ξ) = eik·(ξ−x) G(k)dk ≡ D∗ |0(ξ − x),

221

(2.4)

R3

we see that D∗ creates a photon depending on ξ − x, where ξ denotes the photon variable in conﬁguration space. If the reader is not used in thinking of a “position” of a photon, he may consider it as a simple unitary transform. Because, now we introduce the variable η = ξ − x and the object we are going to work with is the Fourier transform with respect to η, namely F [D∗ |0(η)](k) = G(k). (2.5) The same result holds for E ∗ if G(k) is replaced by H(k). Notice that in the so chosen notation the electron momentum p = −i∇x of a one photon state D∗ f (x, η) = f (x) ⊗ D∗ |0(η), with η = ξ − x, can be written as pD∗ f (x, η) =

3 i px f (x) ⊗ Di∗ |0(η) − f (x) ⊗ pη D∗ |0(η) i=1

= [px − pη ]D∗ f (x, η).

(2.6)

It turns out to be convenient to denote a general vector Ψ ∈ H as a sequence Ψ = {ψ0 , ψ1 , ψ2 , . . . },

(2.7)

ψn = ψn (x, ξ1 − x, . . . , ξn − x),

(2.8)

where with relative variables ξi − x = ηi . For simplicity, we suppress the polarization of the photons and the spin of the electron.

3 Proof of Theorem 1 3.1

Upper bound

We choose the sequence of trial wave functions √ Ψn = {fn (x) ↑, − α[p2η + Hf ]−1 (σ ↑)E ∗ fn (x, η), 0, 0, . . .},

(3.1)

with η = ξ − x, fn ∈ L2 (R3 ), fn = 1 and ↑ denoting the spin-up vector in C2 . We assume for the electron part fn (x) that pfn → 0 as n → ∞, which means that the electron spreads out to inﬁnity. This is reasonable, for in the case without coupling this sequence recovers the bottom of the spectrum of −∆. Recall, E ∗ fn (x, η) ≡ fn (x) ⊗ [E ∗ |0](η)

(3.2)

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and, due to η = ξ − x, 2 [p + Hf ]E ∗ fn (x, η) = ([px − pη ]2 + |pη |)E ∗ fn (x, η).

(3.3)

Notice, in the one photon case Hf = |pη |. Observe, [p, D] =

λ=1,2

R3

Gλ (k) · [p, eik·x ]aλ (k)dk =

λ=1,2

R3

(Gλ (k) · k)eik·x aλ (k)dk = 0 (3.4)

by our choice of ελ (k). Analogously, [p, D∗ ] = [p, E] = [p, E ∗ ] = 0.

(3.5)

By means of Schwarz’ inequality it is easy to see that by our assumption pfn → 0 lim (Ψn , pAΨn ) = 2 lim (Ψn , pDΨn ) = 0.

n→∞

n→∞

Similarly, if we denote the 1-photon part of Ψ as √ ψ1n = − α[p2η + Hf ]−1 (σ ↑)E ∗ fn , then

(ψ1n , p2 ψ1n ) = (ψ1n , [px + pη ]2 ψ1n ) = (ψ1n , p2η ψ1n ) + Error(px fn ),

(3.6)

(3.7) (3.8)

with limn→∞ Error(px fn ) = 0. Furthermore, we compute n 2 √ ψ1 , [pη + Hf ]ψ1n + 2 α (σ ↑)E ∗ fn , ψ1n = −α (σ ↑)E ∗ fn , [p2η + Hf ]−1 (σ ↑)E ∗ fn = −α0|(σ ↑)E[p2 + Hf ]−1 (σ ↑)E ∗ |0 (3.9) In momentum representation, recall F [E ∗ |0](k) = H(k), the last term in (3.9) equals |H(k) · (σ ↑)|2C2 dk 0|(σ ↑)E[p2 + Hf ]−1 (σ ↑)E ∗ |0 = |k|2 + |k| |H(k)|2 dk = 0|E[p2 + Hf ]−1 E ∗ |0. (3.10) = |k|2 + |k| The equality from ﬁrst to second line holds, because of the anti-commutation relations of σ and the fact that H is purely imaginary. For simplicity, we will use throughout the paper |H(k)|2 = |H λ (k)|2 , |G(k)|2 = |Gλ (k)|2 . (3.11) λ=1,2

λ=1,2

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We evaluate |H λ (k)|2 dk |k|2 + |k| λ=1,2 1 |k|χ(|k|) dk = π −1 [Λ2 − 2[Λ − ln(1 + Λ)]]. = 2 2π |k|2 + |k|

0|E[p2 + Hf ]−1 E ∗ |0 =

Since we know

(3.12)

[D, D∗ ] ≡ DD∗ − D∗ D = π −1 Λ2 ,

(3.13)

A2 = (D + D∗ )2 = π −1 Λ2 + 2D∗ D + D∗ D∗ + DD.

(3.14)

With (Ψn , DDΨn ) = 0 and

limn→∞ (ψ1n , ψ1n )

≤ 2απ

−1

ln(1 + Λ) we arrive at

lim (Ψn , T Ψn ) ≤ απ −1 Λ2 − α0|E[p2 + Hf ]−1 E ∗ |0 + 2π −2 α2 Λ2 ln(1 + Λ)

n→∞

2

+ 2α2 D[p2 + Hf ]−1 (σ ↑)E ∗ |0 . (3.15) The last term in the right-hand side vanishes which can be seen by explicit calculations using the relation λ=1,2

εiλ εjλ = δi,j −

ki kj . |k|2

(3.16)

Namely, if j l n denotes the totally antisymmetric epsilon-tensor, we obtain 3 Gλi (k)Hjλ (k) χ(|k|)εiλ (k) j l n εlλ (k)kn dk = i dk |k|2 + |k| |k|3 + |k|2 λ=1,2 λ=1,2 l,n=1 χ(|k|)δ − ki kl j l n k 3 i,l n |k|2 = i dk = 0. (3.17) 3 2 |k| + |k| l,n=1

Therefore, since (Ψn , Ψn ) = 1 + const.α, we obtain the upper bound Σα ≤ lim (Ψn , T Ψn )/(Ψn , Ψn ) ≤ lim (Ψn , T Ψn ) ≤ n→∞

n→∞

≤ 2απ −1 [Λ − ln(1 + Λ)] + 2π −2 α2 Λ2 ln(1 + Λ). (3.18)

3.2

Lower bound

We have learned in the previous section that an approximate ground state Ψ,

Ψ = 1, satisﬁes (Ψ, T Ψ) ≤ 2απ −1 Λ + 2π −2 α2 Λ2 ln(1 + Λ).

(3.19)

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Since T = [σ · (p +

√

Ann. Henri Poincar´e

αA(x))]2 + Hf ≥ Hf

(3.20)

we immediately get (Ψ, Hf Ψ) ≤ 2απ −1 Λ + 2π −2 α2 Λ2 ln(1 + Λ).

(3.21)

Therefore, by means of Schwarz’ inequality we obtain √ √ 2 2 (3.22) 2 α(Ψ, pAΨ) = 4 α(Ψ, pDΨ) ≤ a pΨ + 4a−1 α DΨ

√ √ 2 2 2 −1 2 α(Ψ, σBΨ) = 2 α(Ψ, σEΨ) ≤ cΛ α Ψ + c (1/Λ ) EΨ , (3.23) for any a, c > 0. Since, by A2 ≥ 0, √ √

pΨ 2 ≤ (Ψ, T Ψ) − (Ψ, Hf Ψ) − 4 α(Ψ, pDΨ) − 2 α(Ψ, σEΨ),

(3.24)

we obtain by (3.22), (3.23), the operator inequalities (e.g., [GLL, Lemma A. 4]) D∗ D ≤ and (3.21)

2 ΛHf , π

E∗E ≤

2 3 Λ Hf , 3π

pΨ 2 ≤ C1 α(Λ + Λ2 ) + C2 α2 (Λ2 + Λ3 ) ln(1 + Λ)

(3.25)

(3.26)

for some suitable constants C1,2 > 0. Equations (3.21) and (3.26) will be decisive to control our error estimates. Recall, A2 = π −1 Λ2 + 2D∗ D + 2DD.

(3.27)

Thus, since D∗ D ≥ 0, 2

(Ψ, T Ψ) ≥ απ −1 Λ2 Ψ + E0 [ψ0 , ψ1 ] +

∞

E[ψn , ψn+1 , ψn+2 ],

(3.28)

n=0

where

√ E0 [ψ0 , ψ1 ] = (ψ1 , Aψ1 ) + 2 α [σE ∗ + 2pD∗ ]ψ0 , ψ1

(3.29)

and E[ψn , ψn+1 , ψn+2 ] = (ψn+2 , Aψn+2 ) √ √ + 2 [ ασE ∗ + 2 αpD∗ ]ψn+1 + αD∗ D∗ ψn , ψn+2 , (3.30) with

A = p 2 + Hf .

(3.31)

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Recall, we can write our approximate ground state Ψ as Ψ = {ψ0 (x), ψ1 (x, η1 ), . . . , ψn (x, η1 , . . . , ηn ), . . . },

(3.32)

with ηi = ξi − x. For convenience we will work in momentum representation F [ψn (x, η1 , . . . , ηn )](l, k1 , . . . , kn ) = ψn (l, k1 , . . . , kn ),

(3.33)

with k = (k1 , . . . , kn ) and l denotes the momentum representation of the electron variable. We consider the term E[ψn , ψn+1 , ψn+2 ]. As a straightforward consequence of Schwarz’ inequality we derive E[ψn , ψn+1 , ψn+2 ]

√

2

≥ − α A−1/2 σE ∗ + A−1/2 pD∗ ψn+1 + αA−1/2 D∗ D∗ ψn . (3.34) Similarly,

2 E0 [ψ0 , ψ1 ] ≥ −α (A−1/2 σE ∗ + 2A−1/2 pD∗ )ψ0

(3.35)

The fact that H(k) is purely imaginary and A commutes with the reﬂection l → −l imply (Ei∗ ψ0 , A−1 Ej∗ ψ0 ) = 0. (3.36) Together with the anti-commutation relations for σ we infer (σE ∗ ψ0 , A−1 σE ∗ ψ0 ) = (ψ0 , EA−1 E ∗ ψ0 ).

(3.37)

Now we are going to evaluate the right-hand side of (3.34). These evaluations then are also applied to (3.35). First, we consider the diagonal terms. The most important one, that reproduces the leading order in α, is −α(σE ∗ ψn+1 , A−1 σE ∗ ψn+1 ).

(3.38)

Explicitly, in Fourier representation, n+2 1 F [E ∗ ψn+1 ](l, k1 , . . . , kn+2 ) = √ H(ki )ψn+1 (l, k1 , . . . , k i , . . . , kn+2 ), n + 2 i=1 (3.39) where k i indicates that the ith variable is omitted in ψn+1 . Due to permutational symmetry the expression (3.38) consists of two diﬀerent terms (3.40) −α(σE ∗ ψn+1 , A−1 σE ∗ ψn+1 ) = −α In+1 + IIn+1 ).

The diagonal term In+1 where in the right- as well as left-hand side of (3.38) a photon is created with index i and the mixed term IIn+1 . By similar arguments

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as for (3.35), namely if we take advantage of the fact that H is purely imaginary and use the anti-commutation relations of σ we obtain |H(kn+2 )|2 |ψn+1 (l, k1 , . . . , kn+1 )|2 (3.41) In+1 ≤ dldk1 . . . dkn+2 . n+2 2 [l − n+2 i=1 ki ] + i=1 |ki | n+1 n+2 n+1 We set Q = [l − i=1 ki ]2 + |kn+2 |2 + i=1 |ki | and b = 2[l − i=1 ki ] · kn+2 and expand 1 1 1 1 1 1 1 = + b + b b . (3.42) Q−b Q Q Q Q Q−b Q Plugging this expansion into (3.41), the second term, when integrating over kn+2 , obviously vanishes. Since Q ≥ |kn+2 |2 + |kn+2 | and Q − b ≥ |kn+2 | we estimate In+1 ≤ ψn+1 2 +4

R3

|H(k)|2 dk |k|2 + |k|

n+1 |H(kn+2 )|2 |kn+2 |2 |l − i=1 ki |2 |ψn+1 (l, k1 , . . . , kn+1 )|2 [|kn+2 |2 + |kn+2 |]2 |kn+2 | ≤ ψn+1 2 0|EA−1 E ∗ |0 + const.Λ pψn+1 2 . (3.43)

For the mixed term we have |H λ (k1 )||H λ (kn+2 )| IIn+1 ≤ 3(n + 1) n+2 n+2 [l − i=1 ki ]2 + i=1 |ki | λ=1,2 × |ψn+1 (l, k2 , . . . , kn+2 )||ψn+1 (l, k1 , . . . , kn+1 )|dldk1 . . . dkn+2 . (3.44) By means of the one-photon density ρψn+1 (k) = (n + 1) |ψn+1 (l, k, k2 , . . . , kn+1 )|2 dldk2 . . . dkn+1

(3.45)

we infer after applying Schwarz’ inequality to (3.44) |H λ (k1 )||H λ (kn+2 )| |k1 | + |kn+2 | λ=1,2 × ρψn+1 (k1 ) ρψn+1 (kn+2 )dk1 dkn+2 6(n + 1) χ(|k1 |)χ(|kn+2 |) ≤ |k1 |ρψn+1 (k1 ) (2π)2 |k1 | + |kn+2 | × |kn+2 |ρψn+1 (kn+2 )dk1 dkn+2

1/2 3 χ(|k1 |)χ(|kn+2 |) ≤ (k)dk dk dk |k|ρ ψn+1 1 n+2 2π 2 (|k1 | + |kn+2 |)2

IIn+1 ≤ 3(n + 1)

≤ const.Λ2 (ψn+1 , Hf ψn+1 ).

(3.46)

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Therefore, we summarize − α(σE ∗ ψn+1 , A−1 σE ∗ ψn+1 ) ≥ −α0|EA−1 E ∗ |0 ψn+1 2 − const.αΛ pψn+1 + Λ(ψn+1 , Hf ψn+1 ) . (3.47) All other terms in (3.34) are of order O(α2 ) or even of higher order. For the second diagonal term we obtain −α2 (ψn , DDA−1 D∗ D∗ ψn ) ≥ −const.α2 Λ Λ ψn 2 + pψn 2 + (ψn , Hf ψn ) . (3.48) The proof of (3.48), whose strategy is similar to the one for (3.47), will be postponed to Lemma 1 in the appendix. The third diagonal term of (3.34) reads − α(pD∗ ψn+1 , A−1 pD∗ ψn+1 ) =

−α

λ=1,2

n+1 2 Gλ (kn+2 ) · l − i=1 ki |ψn+1 (l, k1 , . . . , kn+1 )|2 × dldk1 . . . dkn+2 n+2 2 n+2 l − i=1 ki + i=1 |ki | Gλ (k1 ) · l − n+2 ki Gλ (kn+2 ) · l − n+2 i=1 i=1 ki + (n + 1) n+2 2 n+2 l − i=1 ki + i=1 |ki | × ψn+1 (l, k1 , . . . , kn+1 )ψn+1 (l, k2 , . . . , kn+2 )dldk1 . . . dkn+2 ≥ −const.αΛ pψn+1 2 + (ψn+1 , Hf ψn+1 ) . (3.49)

For the second term in the right-hand side we used ﬁrst n+2 2 l − i=1 ki ≤ 1, n+2 2 n+2 l − i=1 ki + i=1 |ki |

(3.50)

and afterwards applied Schwarz’ inequality to bound it from below by Gλ (k1 ) Gλ (kn+2 ) ρψn+1 (k1 )ρψn+1 (kn+2 )dk1 dkn+2 ≥ −(n + 1) λ=1,2

≥ −const.

|k|ρψn+1 (k)dk

χ(|k1 |)χ(|kn+2 |) dk1 dkn+2 |k1 |2 |kn+2 |2

which yields the last term in the last line of (3.49).

1/2 , (3.51)

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Next we consider the oﬀ-diagonal terms. Notice, α(pD∗ ψn+1 , A−1 σE ∗ ψn+1 ) n+1 Gλ (kn+2 ) · l − i=1 ki H λ (kn+2 )· = α n+2 2 n+2 λ=1,2 l − i=1 ki + i=1 |ki | × ·ψn+1 , σψn+1 C2 (l, k1 , . . . , kn+1 )dldk1 . . . dkn+2 Gλ (k1 ) · l − n+2 ki H λ (kn+2 )· i=2 + (n + 1) n+2 2 n+2 l − i=1 ki + i=1 |ki |

× ·ψn+1 (l, k2 , . . . , kn+2 ), σψn+1 (l, k1 , . . . , kn+1 )C2 dldk1 . . . dkn+2 . (3.52) The ﬁrst term in the right-hand side vanishes, because the integral is purely imagi1 −1/2 nary. For the second term we use a2|a| and Schwarz’ inequality to bound +b ≤ 2 b it from above by (3.52) ≤ α

|Gλ (k1 )||H λ (kn+2 )| (|k1 | + |kn+2 |)1/2 |k1 |1/2 |kn+2 |1/2 λ=1,2 × ρψn+1 (k1 )|k1 | ρψn+1 (kn+2 )|kn+2 |dk1 dkn+2 ≤ const.αΛ3/2 (ψn+1 , Hf ψn+1 ).

(3.53)

The second oﬀ-diagonal term, 2α3/2 (pD∗ ψn+1 , A−1 D∗ D∗ ψn ), can simply be bounded from above by α(ψn+1 , pDA−1 pD∗ ψn+1 ) + α2 (ψn , DDA−1 D∗ D∗ ψn ),

(3.54)

on which we now apply (3.48) and (3.49). Finally, by means of Lemma 2 in the appendix we estimate the last oﬀdiagonal term by α3/2 (σE ∗ ψn+1 , A−1 D∗ D∗ ψn ) ≤ const.αΛ α ψn+1 2 + (ψn , Hf ψn ) . (3.55) Collecting above estimates and summing over all n we arrive at (Ψ, T Ψ) ≥ 2απ −1 1 − const.α(1 + Λ) Λ2 Ψ 2 − α Ψ 2 0|EA−1 E ∗ |0 2

− const.αΛ(1 + Λ)(Ψ, Hf Ψ) − const.αΛ pΨ . (3.56) By our a priori estimates (3.21) and (3.26) we prove the theorem.

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4 Proof of Theorem 2 It suﬃces to use a cleverly chosen trial wave function. We assume that φ(x) ∈ L2 (R3 ) is the ground state of p2 + V , i.e., (p2 + V )φ = −e0 φ.

(4.1)

Recall, E ∗ φ(x, η) = φ(x) ⊗ [E ∗ (x)|0](η),

D∗ pφ =

3

pix φ(x) ⊗ [Di∗ |0](η)

i=1

are 1-photon functions depending on x and the relative coordinates ξ − x = η. Therefore, in conﬁguration space, where ψ(x, η) denotes one of these functions, we have (cf. the previous section) [(p2 + Hf )ψ](x, η) = ([px − pη ]2 + |pη |)ψ(x, η).

(4.2)

For sake of convenience we deﬁne the operator AV = p2x + p2η + Hf + V + e0 = (p2x + V + e0 ) ⊗ I + I ⊗ (p2η + Hf ),

(4.3)

which acts on L2 (R3 ; C2 ) ⊗ (R3 ; C2 ) and is obviously positive and invertible. Now, we choose our trial wave function Ψ ∈ H as √ √ −1 ∗ ∗ ¯ ¯ ¯ η), 0, 0, . . . }, Ψ = {φ(x), −2 αA−1 αAV σE φ(x, (4.4) V D pφ(x, η) − where φ¯ = φ ↑ and η = ξ − x. We assume φ = 1 and for simplicity we denote the 1-photon part of Ψ as ψ1 . Notice, ¯ η) = φ(x) ⊗ (σ ↑)E ∗ |0(η). σE ∗ φ(x, First, observe that AV φ¯ = 0 yields ¯ AV σE ∗ φ¯ = (p2η + Hf )σE ∗ φ.

(4.5)

Therefore, since AV and p2η + Hf commute by deﬁnition, we infer

and

∗¯ 2 −1 ¯ σE ∗ φ, A−1 V σE φ = (pη + Hf )

(4.6)

¯ σEA−1 σE ∗ φ¯ = φ 2 0|E[p2 + Hf ]−1 E ∗ |0. φ, η V

(4.7)

Thus, we evaluate 2

2

(Ψ, Hα Ψ) = απ −1 Λ2 Ψ − e0 Ψ + (ψ1 , AV ψ1 ) √ ¯ ψ1 + 2α(ψ1 , D∗ Dψ1 ) + 2(ψ1 , px · pη ψ1 ). (4.8) + 2 α (σE ∗ + 2D∗ p)φ,

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Notice, the cross term ¯ = (φ, ¯ σE[p2 + Hf ]−1 D∗ pφ) ¯ ¯ σEA−1 D∗ pφ) (φ, η V

(4.9)

vanishes as in the previous section, because it is purely imaginary. Therefore, by our choice of ψ1 we get √ ¯ ψ1 (ψ1 , AV ψ1 ) + 2 α (σE ∗ + 2D∗ p)φ, 2

∗ 2 −1 ∗ = −4α(D∗ px φ, A−1 E |0. (4.10) V D px φ) − α φ 0|E[pη + Hf ]

Moreover, we have −1 ∗ (ψ1 , px · pη ψ1 ) = α(D∗ px φ, A−1 V px · pη AV D px φ) ¯ [p2 + Hf ]−1 px · pη [p2 + Hf ]−1 σE ∗ φ) ¯ − α(σE ∗ φ, η

η

¯ A−1 px · pη A−1 σE ∗ φ). ¯ + 2α(D∗ px φ, V V

(4.11)

The ﬁrst term in the right-hand side vanishes by integrating over the η-variable (the best way to see it is using the representation in momentum space), due to (1.6) and the fact that the operator AV commutes with reﬂection η → −η, respectively k → −k. The second term vanishes when integrating over the x-variable (notice, (φ, px φ) = 0), and the third term vanishes, because it is again purely imaginary. ∗¯ Furthermore, since DA−1 V σE φ = 0 (see (3.17)), we realize, after straightforward calculation, that α(ψ1 , D∗ Dψ1 ) ≤ const.α2 ln(1 + Λ)2 .

(4.12)

Using above estimates and the fact that απ −1 Λ2 − α0|E[p2η + Hf ]−1 E ∗ |0 = 2απ −1 [Λ − ln(1 + Λ)]

(4.13)

we infer (Ψ, Hα Ψ)/(Ψ, Ψ) ≤ −e0 + 2απ −1 [Λ − ln(1 + Λ)] ∗ 2 2 − 4α(D∗ px φ, A−1 V D px φ) + O(α ) ln(1 + Λ)(Λ + ln(1 + Λ))

(4.14)

which proves the ﬁrst statement of the theorem with ∗ E(V, Λ) = 4α(D∗ px φ, A−1 V D px φ).

(4.15)

∗ The second statement follows by the observation that (D∗ px φ, A−1 V D px φ) ∗ is strictly larger than 0, which is a consequence of the fact that D px φ is a not identically vanishing function ∈ L2 (R3 ⊗ (R3 , C2 )) and AV an invertible operator.

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5 Computation of concrete numbers 5.1

Error for the self-energy

We are going to calculate the error Err of the self-energy, |Σα − 2απ −1 [Λ − ln(1 + Λ)]| ≤ Err(α2 ).

(5.1) 2

First, it is important to estimate the kinetic energy term pΨ for an approximate ground state Ψ with quite good constants. Since we will calculate anyway with values Λ ≤ 1 we can assume (Ψ, T Ψ) ≤ 2απ −1 Λ. By means of (3.22), (3.23) and then applying (3.25) we obtain 2

2απ −1 Λ ≥ (Ψ, T Ψ) ≥ (1 − a) pΨ + (Ψ, Hf Ψ) 2

2

− 4a−1 α DΨ − cαΛ3 − 1/(cΛ3 ) EΨ

2 3 −1 −1 −1 2 ≥ (1 − a) pΨ − cαΛ + 1 − a 8π αΛ − c (Ψ, Hf Ψ), (5.2) 3π where 1 > a > 0. We require the last term [. . . ] to be ≥ 0. For simplicity, we choose c = π2 , then our ﬁrst condition on α reads α≤ and additionally 2

pΨ ≤

aπ 4Λ

2αΛ(1 + Λ2 ) . π(1 − a)

(5.3)

(5.4)

The main contribution to Err stems from the terms (3.47) and (3.49). In fact the third diagonal term (3.48) is negligible compared to (3.47) and (3.49). Evaluating the corresponding integrals yields 2

(3.49) ≤ 2απ −1 Λ pΨ + 2απ −1 Λ(Ψ, Hf Ψ), and

(5.5)

8 3 αΛ pΨ 2 + αΛ2 (Ψ, Hf Ψ). (5.6) 3π 2π In order to incorporate the oﬀ-diagonal terms we, for simplicity, double the sogained value for Err and derive

14 7 2 2 Err ≤ 2 αΛ pΨ + αΛ (Ψ, Hf Ψ) (5.7) 3π 2π

56Λ2 (1 + Λ2 ) + 14π 2 Λ2 ≤ α2 (5.8) 3π 2 (1 − a) (3.43) + (3.46) ≤

where we used (3.21) and (3.26).

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Radiative correction

We consider an electron in the ﬁeld of a nucleus with charge Z, i.e., V =−

Zβ , |x|

where β is the “real” ﬁne structure constant β = 1/137. The ground state energy of the corresponding Schr¨ odinger operator p2 + V is known to be 1 inf spec [p2 + V ] = −e0 = − (βZ)2 . 4 The corresponding radiative correction, obtained in (4.15), is given by ∗ E(V, Λ) = −α4(φ, pDA−1 V pD φ),

(5.9)

where φ = φ(|x|) denotes the ground state of p2 + V . We know 1/2

∇φ(|x|) = ∂r φ(r)er (θ, ϕ) = e0 φ(r)er (θ, ϕ)

(5.10)

when using polar coordinates. Denote with φi the eigenstate of p2 + V with corresponding eigenvalue −ei . Then, by means of (5.10), we obtain, by straightforward computations, ∗ (φ, pDA−1 V pD φ) ≥ 4πe0

where ci =

|ci |2

i≥1

Λ

0

p dp ≡ e0 F (Λ), e0 − ei + p2 + p

φi (r, θ, ϕ)φ(r) cos(θ)r2 drdΩ,

(5.11)

(5.12)

with dΩ = sin θdθdϕ, and the sum runs over all hydrogen eigenstates. Notice, due to textbooks |ci |2 ∼ 2/15 = 23 51 , which in physicists’ words is expressed by “80 percent of the average excitation energy e0 − en AV is achieved by the continuous spectrum”. Moreover, ln[1 + Λ]/ 0

Λ

p dp → 1 e0 − ei + p2 + p

as Λ → ∞. That is why, for simplicity, we take obtain an approximative radiative correction RC = αe0

8π 15

(5.13)

ln[1 + Λ] to evaluate F (Λ) and

32π ln[1 + Λ] 15

(5.14)

for the binding energy. (Indeed, for Λ ∼ 1 these two functions perfectly coincide.)

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Calculating concrete values of α and Λ

We search for values of α and Λ, that guarantee the error of the self-energy Err being smaller than the radiative correction RC . This leads to the condition: α ≤ Min

  

16π 15 e0 ln[1 + Λ] , 28Λ2 [1+Λ2 ] 7 2 + Λ 2 2 3π (1−a) π

 aπ  4Λ 

.

(5.15)

Set Λ = 1/4 which corresponds to a photon energy mc2 /2, that is an enormously high value compared to the binding energy e0 . Therefore, the condition on the coupling parameter, such that the radiative correction RC dominates the error Err, is given by α ≤ 0.85(βZ)2 .

(5.16)

Asking for the nuclear charge numbers Z that guarantees enhanced binding in the physical case α = 1/137, leads to Z ≥ 13. However, in the case of smaller values of Λ ∼ e0 , which seems physically reasonable, we are in a perfect shape and the error of the self-energy is by far negligible compared to the energy shift.

A

Auxiliary lemmas

Lemma 1. Let Ψ ∈ H. Then

Ψ, DD[p2 + Hf ]−1 D∗ D∗ Ψ ≤ 0|DD[p2 + Hf ]−1 D∗ D∗ |0 Ψ 2 + const.Λ Ψ, Hf Ψ + pΨ 2 . (A.1)

Proof. We ﬁx an arbitrary photon number n. Recall, n+2 n+1 1 F [D∗ D∗ ψn ]n+2 = G(kj ) · G(ki )× (n + 2)(n + 1) j=1 i=1

× ψn (l, k1 , . . . , k j , . . . , k i , . . . , kn+2 ), (A.2) where kj indicates that the jth variable is omitted. By permutational symmetry we can distinguish between three diﬀerent terms, ψn , DD[p2 + Hf ]−1 D∗ D∗ ψn = In + IIn + IIIn .

(A.3)

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First, the diagonal part, Gλ (k1 ) · Gν (k2 ) 2 |ψn (l, k3 , . . . , kn+2 )|2 In = dldk1 . . . dkn+2 . l − n+2 ki 2 + n+2 |ki | λ,ν=1,2

i=1

(A.4)

i=1

n+2 2 2 n+2 n+2 If we set Q = l− i=3 ki + k1 +k2 + i=1 |ki | and b = 2 l− i=3 ki · k1 +k2 and use the expansion 1 1 1 1 1 1 1 = + b + b b Q−b Q Q Q Q Q−b Q

(A.5)

then we again see that term vanishes when integrating over k1 , k2 . the second 2 Therefore, with Q ≥ k1 + k2 + |k1 | + |k2 | we arrive at In ≤

λ,ν=1,2

2 Gλ (k1 ) · Gν (k2 )

Ψ

dk1 dk2 |k1 + k2 |2 + |k1 | + |k2 | λ 2 2 G (k1 ) · Gν (k2 ) |k1 | + |k2 | +4 2 |k1 + k2 |2 + |k1 | + |k2 | (|k1 | + |k2 |) 2

n+2 2 × l − ki |ψn (l, k3 , . . . , kn+2 )|2 dldk1 . . . dkn+2 i=3

≤ 0|DD[p2 + Hf ]−1 D∗ D∗ |0 ψn 2 + const.Λ pψn 2 . (A.6) For convenience we deﬁne the operator |D|(x), which can be regarded as the norm of D(x), |D|(x) = (A.7) |Gλ (k)|eik·x aλ (k)dk λ=1,2 ∗

|D| denotes the operator adjoint. Similarly, we can deﬁne |E|(x). Obviously, [GLL, Lemma A. 4] still holds for the “norm” of D and E, namely |D|∗ |D| ≤

2 Hf , π

|E|∗ |E| ≤

2 Hf , 3π

(A.8)

that can be proved analogously to (3.51). For the second term, with p2 + Hf ≥ Hf , we evaluate Gλ (k1 ) · Gν (k2 )Gλ (k1 ) · Gν (kn+2 ) IIn ≤ n n+2 i=1 |ki | λ,ν=1,2 × |ψn (l, k3 , . . . , kn+2 )||ψn (l, k2 , . . . , kn+1 )|dldk1 . . . dkn+2 |G(k1 )|2 dk1 |ψn |, |D|∗ |D||ψn | ≤ const.Λ2 ψn , Hf ψn . (A.9) ≤ |k1 |

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Finally, the term where on one side the indices of the created photons diﬀer completely from the indices on the other side, Gλ (k1 ) · Gν (k2 )Gλ (kn+1 ) · Gν (kn+2 ) IIIn ≤ n n+2 i=1 |ki | λ,ν=1,2 2

× |ψn (l, k3 , . . . , kn+2 )||ψ(l, k1 , . . . , kn )|dldk1 . . . dkn+2 −1/2 −1/2 ≤ |ψn |, |D|∗ Hf |D|∗ |D|Hf |D||ψn | ≤ const.Λ ψn , Hf ψn , (A.10) where we used n+2 i=1

1/2 1/2 n+1 n+2 |ki | ≥ |ki | |ki | , i=1

(A.11)

i=2

the fact that we can write

Hf

−1/2

ψn (l, k1 , . . . , kn ) =

n

−1/2 |ki |

ψn (l, k1 , . . . , kn ),

(A.12)

i=1

and the ﬁrst equation of (A.8). Lemma 2. Let Ψ ∈ H and ﬁx an arbitrary photon number n. Then α3/2 σE ∗ ψn+1 , [p2 + Hf ]−1 D∗ D∗ ψn ≤ const. α2 (Λ + Λ3 ) ψn 2 + α(Λ + Λ2 )(ψn+1 , Hf ψn+1 ) . (A.13) Proof. Obviously, ∗ σE ψn+1 , [p2 + Hf ]−1 D∗ D∗ ψn = In + IIn ,

(A.14)

where In ≤ (n + 1)

H λ (k1 )Gλ (k1 )Gν (k2 ) λ,ν=1,2

|k1 |

× |ψn (l, k3 , . . . , kn+2 )||ψn+1 (l, k2 , . . . , kn+2 )|dldk1 . . . dkn+2 ≤ const.Λ2 ψn 1/2 |G(k2 )| |k2 |ρψn+1 (k2 )dk2 ≤ const. α1/2 Λ3 ψn 2 + α−1/2 Λ2 (ψn+1 , Hf ψn+1 )

(A.15)

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For the second term of (A.14) we obtain 3/2

IIn ≤ (n + 1)

H λ (k1 )Gλ (kn+1 )Gν (kn+2 ) n+2 i=1 |ki | λ,ν=1,2

× |ψn (l, k1 , . . . , kn )||ψn+1 (l, k2 , . . . , kn+2 )|dldk1 . . . dkn+2 H λ (k1 )|ψn (l, k1 , . . . , kn )| 3/2 ≤ (n + 1) n 1/2 | i=1 |ki || λ,ν=1,2 ν λ G (kn+1 )G (kn+2 )|ψn+1 (l, k2 , . . . , kn+2 )| × dldk1 . . . dkn+2 1/2 n+1 |k | i=2 i −1/2

≤ |E|Hf

−1/2

|ψn |

|D|Hf

|D||ψn+1 | , (A.16)

which implies the statement of the lemma by use of (A.8).

Acknowledgment The author has been supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMFCT-2000-00660. Furthermore, he thanks his friend Robert Seiringer for many valuable discussions and Semjon A. Vugalter for initiating the study of this problem.

References [BFS1] V. Bach, J. Fr¨ ohlich, I.-M. Sigal, Mathematical theory of non-relativistic matter and radiation, Lett. Math. Phys. 34, 183–201 (1995). [BFS2] V. Bach, J. Fr¨ ohlich, I.-M. Sigal, Quantum electrodynamics of conﬁned non-relativistic particles, Adv. Math. 137, 299–395 (1998). [BFS3] V. Bach, J. Fr¨ ohlich, I.-M. Sigal, Spectral Analysis for Systems of Atoms and Molecules Coupled to the Quantized Radiation Field, Commun. Math. Phys. 207, 249–290 (1999). [GLL]

M. Griesemer, E.H. Lieb, M. Loss, Ground states in non-relativistic quantum electrodynamics, Inventiones Math. 145, 557–595 (2001).

[Hi]

F. Hiroshima, Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants, Ann. Henri Poincar´e 3, 171 (2002).

[HVV]

Ch. Hainzl, V. Vougalter, S.-A. Vougalter, Enhanced binding in nonrelativistic QED, Commun. Math. Phys. 233, 16–23 (2003).

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[HS]

F. Hiroshima, H. Spohn, Enhanced Binding through coupling to a Quantum Field, Ann. Henri Poincar´e 2 1159 (2001).

[LL]

E.H. Lieb, M. Loss, Self-Energy of Electrons in non-perturbative QED, Diﬀerential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, R. Weikard and G. Weinstein, eds. 279–293, Amer. Math. Soc./Internat. Press (2000). arXiv math-ph/9908020, mp arc 99– 305.

Christian Hainzl Mathematisches Institut LMU M¨ unchen Theresienstrasse 39 D-80333 Munich Germany email: [email protected] Communicated by Gian Michele Graf submitted 19/06/02, accepted 26/12/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 239 – 273 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020239-35 DOI 10.1007/s00023-003-0129-5

Annales Henri Poincar´ e

The Nelson Model with Less Than Two Photons A. Galtbayar, A. Jensen, K. Yajima Abstract. We study the spectral and scattering theory of the Nelson model for an atom interacting with a photon field in the subspace with less than two photons. For the free electron-photon system, the spectral property of the reduced Hamiltonian in the center of mass coordinates and the large time dynamics are determined. If the electron is under the influence of the nucleus via spatially decaying potentials, we locate the essential spectrum, prove the absence of singular continuous spectrum and the existence of the ground state, and construct wave operators giving the asymptotic dynamics.

1 Introduction In this paper we study the spectral and scattering theory for the Hamiltonian |k|a† (k)a(k)dk + Φ(x) HNelson = h ⊗ I + I ⊗ R3

describing the electron coupled to a (scalar) radiation ﬁeld in the Nelson model ([15]), a simpliﬁed model of nonrelativistic quantum electrodynamics. The Hamiltonian acts on the state space deﬁned by HNelson = L2 (R3x ) ⊗ F, where F=

∞

⊗ns L2 (R3k )

n=0

is the boson Fock space, ⊗ns being the n-fold symmetric tensor product; h = − 12 ∆x + V (x), in L2 (R3x ), is the electron Hamiltonian, where V is the decaying real potential describing the interaction between the electron and the nucleus; a(k) and a† (k) are, respectively, the annihilation and the creation operator; |k|a† (k)a(k)dk R3

is the photon energy operator; and the interaction between the ﬁeld and the electron is given by χ(k) −ikx † a (k) + eikx a(k) dk, e Φ(x) = µ |k| R3

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where µ > 0 is the coupling constant, and χ(k) is the ultraviolet cut-oﬀ function, on which we impose the following assumption, using the standard notation k = (1 + k 2 )1/2 . Assumption 1.1. Assume that the function χ(k) is O(3)-invariant, strictly positive, smooth, and monotonically decreasing as |k| → ∞. Moreover, |χ(k)| ≤ Ck−N for a suﬃciently large N . In this paper we study the restriction of HNelson to the subspace with less than two photons. Let P denote the projection onto the subspace H of HNelson given by H = H0 ⊕ H1 ,

H0 = L2 (R3x ),

H1 = L2 (R3x ) ⊗ L2 (R3k ),

which consists of states with less than two photons. Then we consider the Hamiltonian H = P HNelson P on this space. With respect to the direct sum decomposition H = H0 ⊕ H1 , H has the following matrix representation 1 µg| −2∆ + V . H= µ|g − 21 ∆ + V + |k| Here we have deﬁned the operators |g : H0 → H1 and g| : H1 → H0 by (|gu0 )(x, k) = g(x, k)u0 (x), (g|u1 )(x) = g(x, k)u1 (x, k)dk, R3

where the function g(x, k) is given by g(x, k) = We write

χ(k)e−ixk . |k|

χ(k) g0 (k) = |g(x, k)| = . |k|

(1.1)

(1.2)

It is obvious that |g is bounded from H0 to H1 , and that g| is its adjoint. We assume that V is − 21 ∆-bounded with relative bound less than one, so that H is a selfadjoint operator with the domain D(H) = H 2 (R3 ) ⊕ H 2 (R3 ) ⊗ L2 (R3 ) ∩ L2 (R3 ) ⊗ L21 (R3 ) . Here L21 (R3 ) denotes the usual weighted L2 -space, given by L21 (R3 ) = L2 (R3 , k2 dk), and H 2 (R3 ) is the Sobolev space of order 2. Our goal is to describe the dynamics of this model. In what follows u ˆ is the Fourier transform of u with respect to the

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x variables, Dx = −i∂/∂x and Dy are the gradients with respect to x and y, respectively. Here y is the variable dual to k. We denote by H0 the operator H with V ≡ 0, the Hamiltonian for the free electron-photon system. H0 is translation invariant, and it commutes with the total momentum Dx ⊕(Dx +k). Thus, if we introduce the Hilbert space K = C⊕L2 (R3 ) and deﬁne the unitary operator u0 uˆ0 (p) u ˜0 (p) U: H

(1.3) = ∈ L2 (R3p ; K), → u1 u˜1 (p, k) uˆ1 (p − k, k) then, with respect to the decomposition L2 (R3p ; K) = U H0 U ∗ =

⊕

R3

H0 (p)dp,

H0 (p) =

1 2 2p µ|g0

⊕

R3

Kdp, we have

µg0 | , 1 2 2 (p − k) + |k|

(1.4)

where |g0 : C c → cg0 (k) ∈ L2 (R3 ) and g0 | is its adjoint. Our ﬁrst result is the following theorem on the spectrum of H0 (p). We deﬁne for p ∈ R3 : 1 |p|2 for 0 ≤ |p| ≤ 1, 2 1 λc (p) = min3 { 2 (p − k) + |k|} = 2 1 k∈R |p| − 2 for 1 < |p|, and, for (p, λ) in the domain Γ− = {(p, λ) : λ < λc (p)}, deﬁne F (p, λ) = 12 p2 − λ −

1 2 (p

µ2 g0 (k)2 dk . − k)2 + |k| − λ

(1.5)

It will be shown in Section 2 that • The function F (p, λ) is real analytic, and the derivative with respect to λ is negative: Fλ (p, λ) < 0. • There exists ρc > 1 such that the equation F (p, λ) = 0 for λ has a unique solution λ◦ (p), when |p| ≤ ρc , and no solution, when |p| > ρc . • The function λ◦ (p) is O(3)-invariant, real analytic for |p| < ρc , λ◦ (0) < 0, and it is strictly increasing with respect to ρ = |p|. Theorem 1.2. The reduced operator H0 (p) has the following properties: (1) When |p| < ρc , the spectrum σ(H0 (p)) of H0 (p) consists of a simple eigenvalue λ◦ (p) and the absolutely continuous part [λc (p), ∞). The normalized eigenfunction associated with the eigenvalue λ◦ (p) can be given by   1 1  . −µg0 (k) ep (k) = (1.6) −Fλ (p, λ◦ (p)) 1 (p − k)2 + |k| − λ (p) 2

◦

(2) When |p| ≥ ρc , σ(H0 (p)) = [λc (p), ∞) and is absolutely continuous.

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It follows from Theorem 1.2 that the spectrum σ(H0 ) of H0 is given by σ(H0 ) = [Σ, ∞),

Σ = λ◦ (0),

(1.7)

and that it is absolutely continuous. We write B(r) for the open ball {p : |p| < r} and deﬁne Hone =

1 (2π)3/2

eixp h(p)ep (k)dp : h ∈ L2 (Bρc ) ⊂ H

with the obvious Hilbert space structure. The space Hone corresponds to the space of so-called one-particle states ([6]). The operator H0 (p) is a rank two perturbation of 1 2 0 2p , 1 2 0 2 (p − k) + |k| and the Kato-Birman theorem and Theorem 1.2 yield the following theorem on the asymptotic behavior of e−itH0 . f1,0 Theorem 1.3. For any f ∈ H there uniquely exist f1 = ∈ Hone and f2,1,± ∈ f1,1 H1 such that, as t → ±∞, −itH 0 e−itλ◦ (Dx ) f1,0 0 e → 0, f − − it∆/2−it|k| −ikx −itλ◦ (Dx ) f2,1,± e e e f1,1 and the map f → (f1 , f2,1,± ) is unitary from H onto Hone ⊕ L2 (R6 ). This result shows, in particular, that an electron with large momentum |p| > ρc in the vacuum state does not survive. One might associate this phenomenon to Cherenkov radiation, in the sense that the electron of high speed always carries one photon. However, it is not clear how relevant this description is. Usually Cherenkov radiation is described diﬀerently, in a classical electrodynamic context, see for example [12]. When V = 0, we prove the following results. The following assumption on V is too strong for some of our results, however, we always assume it in what follows without trying to optimize the conditions on V . Assumption 1.4. The potential V is real valued, C 2 outside the origin, and V (x), x · ∇V (x) and (x · ∇)2 V (x) are −∆-compact and converge to 0 as |x| → ∞. Under this assumption the spectrum of h = − 12 ∆ + V has an absolutely continuous part [0, ∞). If h has no negative eigenvalues, we let E0 = 0. Otherwise, the eigenvalues are denoted E0 < E1 ≤ · · · < 0. They are discrete in (−∞, 0). Zero may be an eigenvalue, but there are no positive eigenvalues under Assumption 1.4.

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Definition 1.5. (1) The set Θ(H) = {E0 , E1 , . . .} ∪ {Σ} ∪ {0} ∪ {λ◦ (ρc )} is called the threshold set for H. (2) We say that V is short range, if, in addition to Assumption 1.4, it satisﬁes the following condition: V ∈ L2loc (R3 ) and for any 0 < c1 < c2 < ∞

∞ 1

1/2

2

c1 ≤|x|≤c2

|V (tx)| dx

dt < ∞.

Theorem 1.6. Let V satisfy Assumption 1.4 and Σess = min{E0 , Σ}. Then: (1) The spectrum σ(H) of H consists of the absolute continuous part [Σess , ∞) and the eigenvalues, which may possibly accumulate at Θ(H). (2) Assume h has at least one strictly negative eigenvalue, i.e., E0 < 0. Then the bottom of the spectrum inf σ(H) is an isolated eigenvalue of H and inf σ(H) ≤ Σ + E0 (< Σess ). For the possible asymptotic proﬁles of the wave packet e−itH f as t → ±∞, we prove the existence of the following two wave operators. Theorem 1.7. (1) Assume that V is short range. Then, for f ∈ H, the following limits exist: W0± f = lim eitH e−itH0 f . t→±∞

(1.8)

(2) Let φ ∈ L2 (R3 ) be an eigenfunction of h with eigenvalue E: hφ = Eφ. Suppose |φ(x)| ≤ Cx−β for some C > 0 and β > 2. Then, for f ∈ L2 (R3k ) the following limits exist: 0 E,φ itH W± f = lim e . (1.9) t→±∞ e−itE−it|k| φ(x)f (k) We remark that eigenfunctions actually decay exponentially in many cases, but eigenfunctions at a threshold may only decay polynomially. There is a renewed interest in the Nelson model HNelson recently, which was ﬁrst studied in detail in [15], and a large number of papers have appeared (we refer to [5] and [6] and references therein for earlier works and a physical account of the model). In [5] and [6], the spectral and scattering theory of HNelson (V = 0 in [6]) has been studied in detail, when the infrared cut-oﬀ is imposed on the interaction, in addition to the ultraviolet cut-oﬀ. In particular, the essential spectrum is located, the existence of the ground state is proved, and the asymptotic completeness (AC) of scattering is established in the range of energy below the ionization energy ([5]), and in the range of energy, where the propagation speed of the dressed electron is smaller than 1 ([6]), under the additional condition that the coupling constant µ is suﬃciently small. In many papers the atom is modeled by either

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− 12 ∆ + V with compact resolvent (conﬁning potential), or the atom is replaced by a ﬁnite level system (spin-boson Hamiltonian). For the spin-boson Hamiltonian, the spectral and scattering theory has been studied in detail in subspaces with less than three photons in [13], less than four in [19], and with an arbitrary ﬁnite number of photons in [8]. For models with conﬁning potentials, after the works for the exactly solvable model with harmonic potentials ([1]) and its perturbation ([18]), [3] extensively studied the model with general conﬁning potentials and proved in particular (AC), when photons are assumed to be massive (see the recent paper by G´erard [9] for the massless case). In this paper we deal with the model with massless photons, without infrared cut-oﬀ, with arbitrary large coupling constant, and with a decaying potential, which allows ionization of the electron. However, the number of photons is restricted to less than two, and the problem with an inﬁnite number of soft photons is avoided. Nonetheless, the model retains the diﬃculties arising from the singularity of the photon dispersion relation |k|, the diﬀerent dispersion relations of the electron and the photon, and the photon-electron interaction, which is foreign to the mind accustomed to the classical two body interaction. We think, therefore, that the complete understanding of this very simple model is important (and unavoidable) for understanding models of quantum electrodynamics, which are intrinsically more diﬃcult. Finally, let us outline the contents of this paper. In §2 we study in detail the function F (p, z) and show, in particular, the properties stated before in Theorem 1.2. In §3 we prove Theorem 1.2. The existence of the isolated eigenvalue of H0 (p) will be shown by examining its resolvent and identifying the eigenvalues with zeros of F (p, λ); the absolute continuity will be shown by establishing the Mourre estimate for H0 (p) and proving the absence of eigenvalues by elementary calculus. In §4 we then prove Theorem 1.3 via the Birman-Kato theorem and study the propagator e−itH0 in the conﬁguration space. In §5 we prove Theorem 1.6. We show σess (H) = [Σess , ∞) by adapting the “geometric” proof of the HVZ theorem, the corresponding result for N -body Schr¨ odinger operators; we prove that the singular continuous spectrum is absent from H, and that the eigenvalues of H are discrete in R \ Θ(H), by applying the Mourre estimate for H with the conjugate operator A = Ax + Ay , Ax , Ay being the generator of the dilation. This Mourre estimate, however, is not suitable for proving the so-called minimal velocity estimate, an indispensable ingredient for proving (AC) of the wave operators by the now standard methods (cf., e.g., [6]). This is because our Ay = i[|Dy |, y 2 ], and A cannot be directly related to the dynamical variables associated with H. For proving the existence of the ground state it suﬃces to show inf σ(H) ≤ Σ + E0 , since Σess = min{Σ, E0 }, Σ < 0, and furthermore E0 < 0 by assumption. We prove this by borrowing the argument of [10]. Finally in §6, we prove the existence of the wave operators (1.8) and (1.9). The proof of completeness of the wave operators is still missing, mainly because of the aforementioned lack of the minimal velocity estimate.

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2 Properties of the function F (p, λ) In this section we study the function F (p, z), deﬁned for (p, z) ∈ R3 × (C \ [0, ∞)) by (1.5): µ2 g0 (k)2 dk 1 2 F (p, z) = 2 p − z − . (2.1) 1 2 2 (p − k) + |k| − z The following Lemma is obvious. Lemma 2.1. (1) For each z ∈ C \ [0, ∞), F (p, z) is O(3)-invariant with respect to p ∈ R3 . (2) ∓ Im F (p, z) > 0, when ± Im z > 0. (3) Let K ⊂ C \ [0, ∞) be a compact set. Then |F (p, z)| → ∞ as |p| → ∞, uniformly with respect to z ∈ K. We will write F (ρ, z) = F (p, z), ρ = |p|, and Fρ (ρ, z) will denote the derivative of F (ρ, z) with respect to ρ. We will use the notation F (p, z) and F (ρ, z) interchangeably. Let G(p, k) = 12 (p − k)2 + |k|. Then elementary computations show that for each ﬁxed p the function k → G(p, k) has a global minimum, which we denote by λc (p). Due to the invariance, it is only a function of ρ. Thus we will also denote it by λc (ρ). We have λc (ρ) =

1 2 2ρ

ρ−

1 2

for 0 ≤ ρ ≤ 1, for 1 < ρ.

(2.2)

Note that this function is only once continuously diﬀerentiable. We use the notation Γ− for the domain {(p, λ) ∈ R3 × R : λ < λc (|p|)} of R4 also for denoting the corresponding two-dimensional domain Γ− = {(ρ, λ) ∈ R2 : ρ ≥ 0, λ < λc (ρ)}. It is obvious that F (ρ, λ) is real analytic on Γ− with respect to (ρ, λ). Later we will also need the domain Γ+ = {(ρ, λ) ∈ R2 : ρ ≥ 0, λ > λc (ρ)}. Lemma 2.2. The derivatives satisfy Fλ (ρ, λ) < 0 and Fρ (ρ, λ) > 0 in Γ− , and F (ρ, λ) is strictly decreasing with respect to λ and is strictly increasing with respect to ρ. Proof. We set µ = 1 in the proof. Direct computation shows ∂F = −1 − ∂λ

( 12 (p

g0 (k)2 dk < 0. − k)2 + |k| − λ)2

(2.3)

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To prove Fρ > 0, it suﬃces to show that Fp1 (p, λ) > 0, when p1 ≥ 0, p2 = p3 = 0, as F is O(3)-invariant. We compute ∂F = p1 + ∂p1

g0 (k)2 (p1 − k1 )dk = p1 − 1 ( 2 (p − k)2 + |k| − λ)2

g0 (p + k)2 k1 dk . 1 2 ( 2 k + |p + k| − λ)2

(2.4)

The last integral (including the sign in front) can be written in the form

R2

∞ 0

( 12 k 2

g0 (p1 − k1 , k )2 + |(p1 − k1 , k )| − λ)2 −

g0 (p1 + k1 , k )2 ( 12 k 2 + |(p1 + k1 , k )| − λ)2

k1 dk1 dk ,

where k = (k2 , k3 ) ∈ R2 and, for p1 , k1 > 0, g0 (p1 − k1 , k )2 > g0 (p1 + k1 , k )2 , (p1 − k1 )2 + (k )2 ≤ (p1 + k1 )2 + (k )2 . The ﬁrst inequality follows from Assumption 1.1. Thus the integral is positive, and the lemma follows. Remark 2.3. A computation via spherical coordinates yields for (ρ, λ) ∈ Γ− F (ρ, λ) = 12 ρ2 − λ −

2πµ2 ρ

0

∞

g0 (r)2 r log 1 +

2ρr 1 2 2 (r − ρ) + r − λ

dr.

(2.5)

Lemma 2.4. There exist a constant ρc > 1 and a function λ◦ : [0, ρc ] → R with the following properties: (i) λ◦ (0) < 0, (ρc , λ◦ (ρc )) ∈ γ ≡ {(ρ, λc (ρ)) : ρ ≥ 0}, and Ξ = {(ρ, λ◦ (ρ)) : 0 ≤ ρ < ρc } ⊂ Γ− . (ii) (iii) iv) (v)

(2.6)

F (ρ, λ◦ (ρ)) = 0, ρ ∈ [0, ρc ]. λ◦ is real analytic for 0 < ρ < ρc . λ◦ρ (ρ) > 0 for 0 < ρ < ρc . There are no other zeros of F (ρ, λ) in Γ− , than those given by Ξ in (2.6).

Proof. We examine the behavior of F (ρ, λ) on the curve γ. Taking the limit λ ↑ λc (ρ) in (2.5), we have (recall (2.2), and also |p| = ρ) 2πµ2 ∞ 4ρ F (ρ, λc (ρ)) = − |χ(r)|2 log 1 + dr < 0 ρ r + 2(1 − ρ) 0

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for ρ ≤ 1, and it is increasing for ρ > 1 and diverges to ∞ as ρ → ∞. Indeed, we have, using (2.2) and (2.5), 2πµ2 ∞ 4ρr 2 2 1 |χ(r)| log 1 + F (ρ, λc (ρ)) = 2 (ρ − 1) − dr ρ (r − ρ + 1)2 0 for ρ > 1, and it is evident that limρ→∞ F (ρ, λc (ρ)) = ∞. By a change of variable, F (ρ + 1,λc (ρ + 1))

4r(ρ + 1) = |χ(r)| log 1 + dr (r − ρ)2 0 1 2 ∞ ) 4r(1 + ρ 2πµ ρ |χ(ρr)|2 log 1 + dr = 12 ρ2 − ρ+1 0 (r − 1)2 4r(1 + 1ρ ) 1 2πµ2 ∞ 2 = ρ 2ρ − |χ(ρr)| log 1 + dr . 2 ρ+1 0 (r − 1) 1 2 2ρ

2πµ2 − ρ+1

∞

2

This is manifestly increasing for ρ > 0. Thus, there exists a unique ρc > 1 such that F (ρ, λc (ρ)) changes sign from − to + at ρ = ρc . It follows, since F (ρ, λ) in Γ− is decreasing with respect to λ and F (ρ, λ) → ∞ as λ → −∞ that the function λ → F (ρ, λ) has a unique zero λ◦ (ρ) for 0 ≤ ρ ≤ ρc . It satisﬁes (ρ, λ◦ (ρ)) ∈ Γ− for 0 ≤ ρ < ρc and also λ◦ (0) < 0. By the implicit function theorem, λ◦ (ρ) is real analytic, and λ◦ρ (ρ) > 0 for 0 < ρ < ρc . The last statement follows from the explicit formulae above. As above, we will also consider λ◦ as a function of p, through ρ = |p|. The Hessian is given by ∇2p λ◦ (p) = λ◦ρρ (ρ)ˆ p ⊗ pˆ + λ◦ρ (ρ)

1 − (ˆ p ⊗ pˆ) . ρ

Here we write pˆ = p/|p|, and pˆ⊗ pˆ denotes the matrix with entries pˆj pˆk . A straightforward computation yields det ∇2p λ◦ (p) =

1 λ◦ρρ (ρ)(λ◦ρ (ρ))2 . ρ2

(2.7)

Remark 2.5. The second derivative λ◦ρρ (ρ) at ρ = 0 is called the eﬀective mass of the dressed electron. Due to (2.4), we have Fρ (0, λ) = 0, and hence λ◦ρ (0) = −

Fρ (0, λ◦ (0)) = 0. Fλ (0, λ◦ (0))

Using (2.5), one can compute Fρρ (ρ, λ), and then take the limit ρ ↓ 0 to get the result 2πµ2 ∞ r2 − 6r + 6λ◦ (0) Fρρ (0, λ◦ (0)) = 1 − χ(r)2 r 1 2 . (2.8) 3 ( 2 r + r − λ◦ (0))3 0

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The integral in this expression can be evaluated explicitly in the case χ(r) ≡ 1 for r ≥ 0. The value is negative for all λ◦ (0) < 0. Thus we conjecture (perhaps with an additional assumption on χ) that we always have Fρρ (0, λ◦ (0)) > 0 (taking the sign in front of the integral into account). For |µ| small we have this result, without additional conditions on χ. Now using implicit diﬀerentiation and the result Fρ (0, λ◦ (0)) = 0, we ﬁnd λ◦ρρ (0) = −

Fρρ (0, λ◦ (0)) . Fλ (0, λ◦ (0))

(2.9)

We recall from (2.3) that Fλ (0, λ◦ (0)) ≤ −1. Thus we conjecture that we always have a positive eﬀective mass (perhaps with an additional condition on χ). Let us note that λ◦ρ (0) = 0 and the monotonicity of λ◦ (ρ) imply λ◦ρρ (0) ≥ 0.

3 Spectrum of H0 (p) and H0 In this section we ﬁrst carry out the separation of mass in detail, and then we prove Theorem 1.2.

3.1

Separation of the center of mass

It is easy to see that the Hamiltonian of the free electron-photon system 1 −2∆ µg| . H0 = µ|g − 12 ∆ + |k| commutes with the spatial translations u0 (x) u (x + sej ) → iskj 0 , τj (s) : u1 (x, k) e u1 (x + sej , k)

s ∈ R,

Hence H0 and the generators

−i∂/∂xj Pj = 0

0 −i∂/∂xj + kj

j = 1, 2, 3.

(3.1)

of τj (s) can simultaneously be diagonalized. Thus, if we introduce the Hilbert

⊕ space K = C ⊕ L2 (R3 ) and the unitary operator U : H → L2 (R3p ; K) = R3 Kdp

⊕ by (1.3), then we have the direct integral decomposition U H0 U ∗ = R3 H0 (p)dp as in (1.4), where 1 2 p 0 0 µg0 | (3.2) + ≡ H00 (p) + T, H0 (p) = 2 1 2 0 µ|g0 0 2 (p − k) + |k| and T is a rank two perturbation of H00 (p). H0 (p) is essentially the operator known as the Friedrichs model. Thus, it is standard to compute its resolvent and, if we write f˜ , u ˜0 (p, z), , ˜f = ˜ 0 , (3.3) (H0 (p) − z)−1˜f = u˜1 (p, k, z) f1 (k)

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we have u ˜0 (p, z) = u˜1 (p, k, z) =

3.2

1 ˜ f0 − µ F (p, z) 1 2 (p

g0 (k)f˜1 (k)dk , 1 2 2 (p − k) + |k| − z

u0 (p, z) f˜1 (k) µg0 (k)˜ − 1 . 2 + |k| − z − k)2 + |k| − z (p − k) 2

249

(3.4) (3.5)

Proof of Theorem 1.2

We prove here Theorem 1.2 on the spectrum of a reduced operator H0 (p). In what follows ρc is the threshold momentum deﬁned in Lemma 2.4. As was shown in Lemmas 2.2 and 2.4, F (p, z) is an analytic function of z ∈ C \ [λc (p), ∞), it has a simple zero at λ◦ (p), when |p| < ρc , and has no zero, when |p| ≥ ρc . It follows from (3.3)–(3.5) that C \ [λc (p), ∞) z → (H0 (p) − z)−1 is meromorphic with a simple pole at λ◦ (p), if |p| < ρc , and that it is holomorphic, if |p| ≥ ρc . Hence: 1. If |p| < ρc , H0 (p) has an eigenvalue λ◦ (p), and (−∞, λc (p)) \ {λ◦ (p)} ⊂ ρ(H0 (p)), the resolvent set of H0 (p). 2. If |p| ≥ ρc , (−∞, λc (p)) ⊂ ρ(H0 (p)). By virtue of (3.3)–(3.5), we can compute the eigenprojection Ep for H0 (p) associated with the eigenvalue λ◦ (p) as follows: Ep = − s-lim (z − λ◦ (p))(H0 (p) − z)−1 = ep ⊗ ep . z→λ◦ (p)

Thus λ◦ (p) is simple, and ep is a normalized eigenvector, see (1.6) and (2.3). Due to the decomposition (3.2) it is clear that σess (H0 (p)) = [λc (p), ∞). Using Mourre theory [14] we show in Lemma 3.1 that [λc (p), ∞) is an absolutely continuous component of the spectrum, the singular continuous spectrum is empty, and eigenvalues are discrete in this set. In the following Lemma 3.3 we then show that there are no eigenvalues embedded in [λc (p), ∞). This concludes the proof of the theorem. We take η ∈ C ∞ (R3 ) such that η(k) = 1 for |k| > 1 and η(k) = 0 for |k| < 1/2 and deﬁne a vector ﬁeld Xr (k) on R3 by Xr (k) = η(k/r)∇k G(p, k) for a small parameter r > 0. Recall that G(p, k) = 12 (p − k)2 + |k|. We then deﬁne a one-parameter family of auxiliary operators Ar by i 0 0 Ar = , Ar = (Xr (k) · ∇k + ∇k · Xr (k)) , 0 Ar 2 and let D = C ⊕ C0∞ (R3 ). The vector ﬁeld Xr (k) is smooth, Xr (k) = 0 when |k| < r/2, and its derivatives are bounded. Hence it generates a ﬂow k → Φr (t, k) of global diﬀeomorphisms on R3 , such that Φr (t, k) = k for |k| < r/2, and for some c > 0 e−c|t||k| ≤ |Φr (t, k)| ≤ ec|t| |k|,

e−c|t| ≤ ∇k Φr (t, k) ≤ ec|t|

(3.6)

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for all k ∈ R3 and t ∈ R. It follows that if we deﬁne c c Jr (t) , = u(k) det(∇k Φr (t, k))u(Φr (t, k)) then Jr (t) is a strongly continuous unitary group on K, such that Jr (t)D = D, t ∈ R, and it satisﬁes d c 0 −i Jr (t) = . u(k) t=0 Ar u(k) dt Thus Ar is essentially selfadjoint on D, and we denote its closure again by Ar , such that Jr (t) = eitAr . Lemma 3.1. Let I be a bounded open interval such that I ⊂ (λc (p), ∞) \ {p2 /2}. Then there exist r > 0, such that Ar is a conjugate operator of H0 (p) at E ∈ I in the sense of Mourre, viz. (1) D is a core of both Ar and H0 (p). (2) We have that eitAr D(H0 (p)) ⊂ D(H0 (p)), and sup|t|<1 H0 (p)eitAr u < ∞ for u ∈ D(H0 (p)). (3) The form i[H0 (p), Ar ] on D is bounded from below and closable, and the associated selfadjoint operator i[H0 (p), Ar ]0 satisﬁes D(i[H0 (p), Ar ]0 ) ⊇ D(H0 (p)). (4) The form, deﬁned on D(Ar ) ∩ D(H0 (p)) by [H0 (p), Ar ]0 , Ar , is bounded from D(H0 (p)) to D(H0 (p))∗ . (5) There exist α > 0, δ > 0, and a compact operator K, such that P (E, δ)i[H0 (p), Ar ]0 P (E, δ) ≥ αP (E, δ) + P (E, δ)KP (E, δ), where P (E, δ) is the spectral projection of H0 (p) for the interval (E −δ, E +δ). Thus (λc (p), ∞) ⊆ σac (H0 (p)), σsc (H0 (p))∩(λc (p), ∞) = ∅, and the point spectrum of H0 (p) is discrete in (λc (p), ∞) \ {p2 /2}. Proof. We ﬁrst show statements (1) ∼ (4) hold for Ar for any r > 0. (1) is obvious. (2) is also evident because D(H0 (p)) = C ⊕ L22 (R3 ) and the diﬀeomorphisms k → Φr (t, k) satisfy the bound (3.6). On D we compute the commutator 0 0 ≡ L(p). i[H00 (p), Ar ] = 0 i[G(p, k), Ar ]

(3.7)

i[G(p, k), Ar ] = η(k/r)|∇k G(p, k)|2 = η(k/r)|k + kˆ − p|2 ≥ 0,

(3.8)

Here we have

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and it behaves like (|k| + 1)2 for large k. Since g0 (k) = χ(k)/ |k|, χ is smooth and decays rapidly at inﬁnity and Xr (k) = 0 for |k| < r/2, it follows that Ar g0 is C ∞ and rapidly decaying at inﬁnity, such that 0 µAr g0 | i[T, Ar ] = µ|Ar g0 0 has an extension to a bounded rank two operator. Thus i[H0 (p), Ar ] is bounded from below, closable, and the associated selfadjoint operator has the same domain as H0 (p). This proves (3). (4) holds due to (3.8) and the arguments used in establishing (3). To prove (5), ﬁx p ∈ R3 and E ∈ (λc (p), ∞) \ {p2 /2}. (Recall λc (p) is the unique minimum of k → G(p, k) and λc = p2 /2 is attained by k = 0 if |p| ≤ 1, p and λc (p) = |p| − 1/2 is attained by the unique solution k = kc (p) = (|p| − 1)ˆ of ∇k G(p, k) = k + kˆ − p = 0, if |p| ≥ 1, see (2.2).) Then E = G(p, 0) and E = G(p, kc (p)) and, by continuity, there exist r > 0 and δ > 0, such that 0<δ<

1 2

min{|E − 12 p2 |, |E − λc (p)|},

(3.9)

and such that |G(p, k) − E| > 2δ

if

|k| < 2r or |k − kc (p)| < 2r.

(3.10)

Then |G(p, k) − E| ≤ 2δ implies |k| > 2r and |k − kc (p)| > 2r, and hence η(k/r)|∇k G(p, k)|2 = |∇k G(p, k)|2 ≥ r2 .

(3.11)

Indeed, for |p| ≤ 1 + r, |k| > 2r implies ˆ − |p| ≥ 1 + 2r − |p| ≥ r |∇k G(p, k)| ≥ |k + k| and, if |p| ≥ 1 + r, we have p = kc (p) + kˆc (p) and |∇k G(p, k)| = |k + kˆ − kc (p) − kˆc (p)| ≥ |k − kc (p)| > 2r. Thus, if φ0 ∈ C0∞ (R) is such that φ0 (λ) = 0 for |λ| > 2δ, we have 0 0 φ0 (H00 (p) − E) = 0 φ0 (G(p, k) − E) and, by virtue of (3.11), φ0 (H00 (p) − E)L(p)φ0 (H00 (p) − E) ≥ r2 φ0 (H00 (p) − E)2 . Thus, if we choose r and δ as above, statement (5) holds with α = r2 and this δ by virtue of the following lemma.

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Lemma 3.2. Let φ ∈ C0∞ (R), and let L(p) be given by (3.7). Then the operator L(p){φ(H0 (p)) − φ(H00 (p))} is a compact operator. Proof. Let φ˜ be a compactly supported almost analytic extension of φ. Then writing R0 (p, z) = (H00 (p) − z)−1 and R(p, z) = (H0 (p) − z)−1 , we have 1 ˜ ∂z φ(z)R (3.12) φ(H0 (p)) − φ(H00 (p)) = 0 (p, z)T R(p, z)dz ∧ dz, 2πi C see [4, Theorem 8.1]. Here R0 (p, z) commutes with L(p) and L(p)T is a compact operator, as T is of rank two, and Ran T ⊂ D(L(p)). Since (3.12) is the norm limit of the Riemann sums, the lemma follows. Lemma 3.3. We have σpp (H0 (p)) ∩ [λc (p), ∞) = ∅. Proof. We recall from Assumption 1.1 that g0 (k) > 0 for all k ∈ R3 \ {0}, g0 is smooth away from k = 0 and rapidly decaying at inﬁnity. If λ is an eigenvalue, then there exists a non-zero vector (c, f ) ∈ C ⊕ L2 (R3 ), such that

µg0 (k)c +

{ 12 (p

µg0 , f + 12 p2 c = λc, 2

− k) + |k|}f (k) = λf (k).

(3.13) (3.14)

We show that these equations lead to a contradiction, if (|p| , λ) ∈ Γ+ . We write G(p, k) = 12 (p − k)2 + |k| as previously. It is easy to see that Sλ = {k : G(p, k) − λ = 0} has Lebesgue measure zero. (This will be checked in what follows.) This result will imply c = 0, because { 12 (p − k)2 + |k| − λ}f (k) = 0 otherwise, which implies f (k) = 0 almost everywhere, a contradiction. We divide the proof into a number cases. Case 1. Assume |p| > 1 and λ > λc (p) = |p| − 12 . (i) Assume ﬁrst λ = 12 p2 . Then 0 ∈ Sλ , and the gradient ∇k G(p, k) = k − p + kˆ

(3.15)

does not vanish on Sλ (it vanishes only when λ = λc (p)). Therefore Sλ is a smooth hypersurface (of Lebesgue measure zero), and we have f (k) =

cµg0 (k) ∈ L2 (R3 ). G(p, k) − λ

This is a contradiction. (ii) Consider now the subcase λ = 12 p2 . Then besides k = 0 the equation G(p, k) − λ = 12 k 2 − p · k + |k| = 0

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has a root k0 = 0, and around any 0 = k0 ∈ Sλ , Sλ is a smooth hypersurface, because ∇k G(p, k0 ) = 0. Thus f cannot be in L2 (R3 ) by the argument given above. Case 2. Assume |p| ≤ 1 and λ > λc (p) = 12 p2 . Then Sλ does not contain k = 0, ˆ > 1 ≥ |p|. Hence Sλ is again and the gradient (3.15) does not vanish, since |k + k| 2 a nonempty hypersurface, and f ∈ L . Case 3. Consider now the threshold case λ = λc (p). (i) If |p| > 1, then λc (p) is the minimum of R3 k → G(p, k) at the critical point k = kc (p), where the Hessian satisﬁes ∇2k G(p, k) = I +

I − kˆ ⊗ kˆ . |k|

Hence kc (p) is a Morse type critical point, and 0 ≤ G(p, k) − λ ≤ 12 |k − kc (p)|2 near k = kc (p). Hence Sλ = {kc (p)} (obviously of Lebesgue measure zero), |f (k)| =

g0 (k) |cµg0 (k)| ≥C , |G(p, k) − λ| |k − kc (p)|2

and f cannot be square integrable. (ii) If |p| = 1, then λ = λc (p) = 12 p2 = and p,

1 2

and, if we let θ be the angle between k

G(p, k) − λ = −p · k + 12 k 2 + |k| = 12 r2 + r(1 − cosθ) ≥ 0,

r = |k|.

Hence Sλ = {0} is a single point and, by using polar coordinates, we have |cµ|2 g0 (k)2 dk |f (k)|2 dk = |G(p, k) − λ|2 ∞ π 4g0 (r)2 r2 sinθdθdr =C (C = 2π|cµ|2 ) 2 2 0 0 r (r + 2(1 − cosθ)) 2 ∞ 1 = 4C g0 (r)2 dt dr 2 0 0 (r + 2t) ∞ χ(r)2 dr = ∞, = 8C r2 (r + 4) 0 as χ(0) > 0. Thus again λ cannot be an eigenvalue. (iii) Finally we consider the case |p| < 1 and λ = λc (p) = 12 p2 . In this case G(p, k) − λ = 12 k 2 + |k| − p · k ≥ |k|(1 − |p|),

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hence Sλ = {0}, and |G(p, k) − λ| ≤ C|k| for small |k|. Then, f (k) =

cµg0 (k) ∈ L2 (R3 ), G(p, k) − λ

since f (k) has a singularity |k|−3/2 at k = 0.

Remark 3.4. Since the Hamiltonian H0 (p) is essentially a Friedrich Hamiltonian, it is possible to give a diﬀerent proof of Theorem 1.2, by proving the limiting absorption principle for H0 (p) on (λc (p), ∞) \ { 12 p2 }. In order to do this one studies the boundary values limε↓0 F (ρ, λ ± iε) for (ρ, λ) ∈ Γ+ , and then uses the explicit representation for the resolvent given in (3.4) and (3.5). The argument in Lemma 3.3 is then needed only in the cases λ = λc (p) and λ = 12 p2 .

3.3

Resolvent and spectrum of H0

From the equations (3.3), (3.4), and (3.5), we derive the formula for the resolvent: G0 (x, z) −1 f0 (H0 − z) . (3.16) = G1 (x, k, z) f1 Lemma 3.5. Let z ∈ R. Then we have 1 ˆ fˆ0 (p) − µ G0 (p, z) = F (p, z) ˆ 1 (p, k, z) = G

g0 (k)fˆ1 (p − k, k) dk , 1 2 2 (p − k) + |k| − z

ˆ 0 (p + k, z) fˆ1 (p, k) µg0 (k)G − , 1 2 + |k| − z 2 p + |k| − z

1 2 2p

(3.17)

(3.18)

where F (p, z) is given by (2.1). Since λ◦ρ (ρ) > 0 for 0 < ρ < ρc , Theorem 1.2 implies the following theorem, by well-known results on the spectrum of an operator with a direct integral representation, see for example [17, Theorem XIII.85]. Theorem 3.6. The spectrum of H0 is absolutely continuous and is given by σ(H0 ) = [Σ, ∞), where Σ = λ◦ (0) < 0. Remark 3.7. In quantum ﬁeld theory one is often interested in the joint spectrum of (H0 , P ), where the components of the momentum are given in (3.1). Such results follow immediately from the results in this section. In particular, we see that in our case the eigenvalue λ◦ (p) generates an isolated shell in the joint spectrum.

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4 The behavior of e−itH0 In this section we prove Theorem 1.3 and study the asymptotic behavior as t → ±∞ of e−itλ◦ (Dx ) in the conﬁguration space.

4.1

Proof of Theorem 1.3

By virtue of Theorem 1.2, e−itH0 (p) can be decomposed as e−itH0 (p) = e−itλ◦ (p) Ep + e−itH0 (p) Pac (H0 (p)), where we set Ep = 0, when |p| ≥ ρc . Here H0 (p) is a rank two perturbation of H00 (p), H00 (p) has a simple isolated eigenvalue 12 p2 and the absolutely continuous spectrum [λc (p), ∞). The absolutely continuous subspace is Kac (H00 (p)) = {0} ⊕ L2 (R3 ). It follows by the celebrated Kato-Birman theorem (see for example [16]) that the limits s-lim eitH00 (p) e−itH0 (p) Pac (H0 (p)) = Ω± 0 (p) t→±∞

exist, and furthermore that the wave operators Ω± 0 (p) are partial isometries with initial set Kac (H0 (p)) = Pac (H0 (p))K and ﬁnal set {0}⊕L2(R3 ). Thus, as t → ±∞, we have for any ˜f ∈ K −itH0 (p)˜ ˜f (p) (4.1) f − e−itλ◦ (p) Ep ˜f − e−itH00 (p) Ω± e →0 0 K

˜ 2 and ˜f 2 = Ep˜f 2 + Ω± 0 (p)f . If we write, with u ˆ denoting the Fourier transform of u with respect to the x variables as previously, f=

f0 ∈ H, f1

fˆ1,0 (p) ∈K fˆ1,1 (p, k)

and Ep U f (p) =

for

|p| < ρc

then, with the understanding that fˆ1,0 (p) = fˆ1,1 (p, k) = 0, when |p| ≥ ρc , we have U

∗

⊕

e R3

−itλ◦ (p)

Ep dp U f =

e−itλ◦ (Dx ) f1,0 (x) . e−ikx e−itλ◦ (Dx ) f1,1 (x, k)

(4.2)

Since Ep is the one-dimensional projection onto the space spanned by ep (k), we have −1 g0 (k)fˆ1 (p − k, k)dk ˆ ˆ f0 (p) − µ f1,0 (p) = , 1 2 Fλ (p, λ◦ (p)) 2 (p − k) + |k| − λ◦ (p) and fˆ1,1 (p, k) =

1 2 (p

−µg0 (k)fˆ1,0 (p) . − k)2 + |k| − λ◦ (p)

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The operator

⊕

Ann. Henri Poincar´e

Ep dp is the orthogonal projection onto

R3

{h(p)ep (k) : h ∈ L2 (Bρc )} and

∗ U

R

If we write Z± ≡ U ∗

2 2 2 Ep dp U f = f1,0 H0 + f1,1 H1 . 3

⊕

U

⊕

e

−itH00

R3

⊕

R3

then Z ± are unitary from U ∗ ∗

Ω± (p)dp U, 0

⊕ R3

Z ±f =

0

,

f2,1,±

Kac (H0 (p))dp onto {0} ⊕ L2 (R6 ), and

Ω± 0 (p)dp

0 1 Uf = . e−it(− 2 ∆+|k|) f2,1,±

We insert the relation (4.1) into the identity e−itH0 = U ∗ use the identities (4.2) and (4.4). Theorem 1.3 follows.

4.2

(4.3)

H

⊕

R3

(4.4)

e−itH0 (p) dp U , and

Behavior in configuration space 1

As the operator e−it(− 2 ∆+|k|) has been well studied, we concentrate on the operator e−itλ◦ (Dx ) v(x) for the case t > 0. When vˆ ∈ C0∞ (B(ρc )), we may apply the method of stationary phase to 1 e−itλ◦ (p)+ixp vˆ(p)dp. v(t, x) = (2π)3/2 The points of stationary phase are determined by the equation t∇λ◦ (p) = x.

(4.5)

It follows from (2.7) that det ∇2p (p) can vanish only for p with |p| = ρ satisfying λ◦ρρ (ρ) = 0. By the real analyticity of λ◦ (ρ) it follows that these zeros are isolated in (0, ρc ), with 0 and ρc as possible accumulation points. Thus {ρ ∈ (0, ρc ) : λ◦ρρ (ρ) = 0} = {ρj }j=M,...,N , a strictly increasing sequence, or the set is empty. Here we use M and N to distinguish between the following cases. M = 1, if zeros do not accumulate at 0. In that case we also use ρ0 = 0 below. 1 ≤ N < ∞, if the zeros do not accumulate at ρc . In that case we introduce ρN +1 = r < ρc . We take M = −∞ or N = ∞, if the zeros accumulate at 0 or ρc , respectively.

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We consider only the case M = −∞ and N = ∞. The other cases require simple modiﬁcations in the arguments below. Deﬁne Gj = {p ∈ R3 : ρj < |p| < ρj+1 },

j ∈ Z.

(4.6)

We restrict our considerations to a set Gj , j ∈ Z. The equation (4.5) has a unique solution of the form p(x/t) = x ˆρ(|x|/t), where ρ(|x|/t) is the solution of λ◦ρ (ρ) = |x|/t, when xt ∈ ∇λ◦ (Gj ). Assume now vˆ ∈ C0∞ (Gj ). Then v(t, x) can be written in the form 3π

π

t−3/2 eiφ(t,x)−i 4 +i 2 s v(t, x) = vˆ(p(x/t)) + t−1 v1 (x/t) + · · · , 2 1/2 (det ∇p λ◦ (p(x/t)))

(4.7)

where s = 0, if det ∇2p λ◦ (p) > 0 for p ∈ Gj , and s = 1, if det ∇2p λ◦ (p) < 0 for p ∈ Gj . The phase function is deﬁned by φ(t, x) = x · p(x/t) − tλ◦ (p(x/t)),

(4.8)

v1 , v2 , . . . are determined by standard formulae (see [11, Section 7.7]), and for −N −N x x for any N and t large. t ∈ ∇λ◦ (Gj ), |v(t, x)| ≤ CN |t| Lemma 4.1. Let Gj , s and φ(t, x) be deﬁned as above and let fˆ1,0 ∈ C0∞ (Gj ) for some j. Then the functions e−itλ◦ (Dx ) f1,0 (x) and e−itλ◦ (Dx ) f1,1 (x, k) have the following asymptotic expansions as t → ∞ for x ∈ t∇λ◦ (Gj ): 3π

e−itλ◦ (Dx ) f1,0 (x) =

t−3/2 eiφ(t,x)−i 4 +i 2 s ˆ f1,0 (p(x/t)) (det ∇2p λ◦ (p(x/t)))1/2 π

+ t−1 g1 (x/t) + · · · , (4.9)

3π

e

−itλ◦ (Dx )

t−3/2 eiφ(t,x)−i 4 +i 2 s ˆ f1,1 (x, k) = f1,1 (p(x/t), k) (det ∇2p λ◦ (p(x/t)))1/2 π

+ t−1 M1 (x/t, k) + · · · , (4.10)

where g1 (x/t),g2 (x/t), . . ., M1 (x/t, k), M2 (x/t, k), . . ., are deﬁned by standard formulae involving the derivatives of f1,0 and f1,1 . In particular, supports of gj (·) and Mj (·, k) are contained in those of fˆ1,0 (p(·)) and fˆ1,1 (p(·), k) respectively. For x ∈ t∇λ◦ (Gj ), we have for any N , |e−itλ◦ (Dx ) f1,0 (x)| ≤ CN |t|−N x−N ,

(4.11)

|e−itλ◦ (Dx ) f1,1 (x, k)| ≤ CN |t|−N x−N k−N .

(4.12)

The results (4.10) and (4.12) hold for |k| > ε, ε > 0 arbitrary.

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Proof. The formula for e−itλ◦ (Dx ) f1,0 (x) is an immediate corollary of (4.7). The result (4.10) can be proved similarly, since k → fˆ1,1 (·, k) ∈ C0∞ (B(ρc )) is smooth and rapidly decaying, when |k| > ε > 0. Thus we may consider e−ikx e−itλ◦ (Dx ) f1,1 (x, k) as the part of the wave function, which represents the motion of the electron under the dispersion relation λ◦ (p), which is dragging the cloud of photons (however only one photon).

5 Proof of Theorem 1.6 In this section we prove Theorem 1.6 on the spectral properties of the Hamiltonian H. In what follows we mostly use the conﬁguration space representations for the photons. The variable dual to k is denoted by y. We then have (g|u1 )(x) = gˇ0 (x − y)u1 (x, y)dy, (|gu0 )(x, y) = gˇ0 (y − x)u0 (x), where gˇ0 is the inverse Fourier transform of the function given by (1.2). Recall that g0 is O(3)-invariant, such that gˇ0 (x) is also O(3) invariant, gˇ0 is C k for a k depending on N in Assumption 1.1, and it has an asymptotic expansion at inﬁnity: gˇ0 (y) = C0 |y|−5/2 + C1 |y|−7/2 + · · · .

(5.1)

The photon energy is given by the diﬀerential operator |Dy |. We write K0 for the operator H, where the electron-photon interaction is switched oﬀ, viz. h 0 ≡ h ⊕ (h + |Dy |). K0 = 0 h + |Dy | The following two sets of partitions of unity, the one of the electron conﬁguration space R3x , and the other of the electron-photon conﬁguration space R6(x,y) , will play an important role in what follows: χ00 (x)2 + χ01 (x)2 = 1,

x ∈ R3 ,

χ10 (x, y)2 + χ11 (x, y)2 + χ12 (x, y)2 = 1,

(x, y) ∈ R6 ,

where χij satisfy the following properties: (1) χ00 ∈ C0∞ ({|x| < 1}), χ01 ∈ C ∞ (R3 ) and χ10 ∈ C0∞ ({|x|2 + |y|2 < 1}). χ00 (x) = 1 for |x| ≤ 1/4 and χ10 (x, y) = 1 for |x|2 + |y|2 ≤ (1/4)2 . (2) For |x|2 +|y|2 ≥ 1, χ11 , χ12 ∈ C ∞ (R6 ) are homogeneous of degree 0, χ11 (x, y) and χ12 (x, y) vanish in small open cones containing x = 0 and x = y such that, outside a ball of radius 1, they are equal to 1 in open cones containing x = y and x = 0, respectively.

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Such a partition of unity exists, since the linear subspaces {(x, y) : x = 0} and {(x, y) : x = y} do not intersect on the unit sphere of R6 . We deﬁne χ00 χ01 0 0 0 0 χ0 = , χ1 = , χ2 = , (5.2) 0 χ12 0 χ10 0 χ11 so that as operators in H

χ20 + χ21 + χ23 = I.

(5.3)

We denote the commutator AB − BA of operators A and B by [A, B]. Lemma 5.1. Let j ∈ C0∞ (R3 ) and let j (y) = j(y/) for > 0. Then: (1) The operator [|Dy | , j (y)], deﬁned on S(R3 ), extends to a bounded operator on L2 (R3 ). We have [|Dy | , j (y)] B(L2 ) ≤ C−1 , ≥ 1. (2) For any δ > 0, [|Dy | , j (y)] (1 + |Dy |)−δ is compact in L2 (R3 ). (3) Let K be the multiplication by a function K(x, y) such that lim sup K(x, ·)L2 (R3 ) = 0.

R→∞ |x|≥R

Then K(−∆x + 1)−1 : L2 (R3 ) → L2 (R6 ) is compact. Proof. (1) In the momentum representation j is convolution with 3 ˆj(ξ). Thus [|Dy | , j (y)] is an integral operator with the kernel 3 (|ξ| − |η|)ˆj((ξ − η)), which is dominated in modulus by the convolution kernel 3 |ξ − η| |ˆj((ξ − η))|. Statement (1) follows from Young’s inequality. (2) When ε > 0, it is well known that (ε2 − ∆y )1/2 , j (y) (1+|Dy |)−δ is compact. We have |(ε2 + |ξ|2 )1/2 − |ξ|| ≤ ε and (ε2 − ∆y )1/2 , j (y) − [|Dy |, j (y)] B(L2 ) ≤ 2εjL∞ . The compactness of [|Dy |, j (y)] (1 + |Dy |)−δ follows. (3) Let KR (x, y) = χ00 (x/R)K(x, y) and KR be the multiplication operator with KR (x, y). Then, KR → K in the operator norm from L2 (R3 ) to L2 (R6 ) as R → ∞ and KR (−∆x +1)−1 is an operator of Hilbert-Schmidt class because it is an integral operator with the square integrable integral kernel

KR (x, y)e−|x−x | . 4π|x − x | Hence, K(−∆x + 1)−1 is compact.

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Lemma 5.2. (1) The following operators are compact in H for z ∈ R: [(H − z)−1 , χj ], j = 1, 2, ((H − z)−1 − (K0 − z)−1 )χ2 ,

((H − z)−1 − (H0 − z)−1 )χ1 [(K0 − z)−1 , χj ],

j = 1, 2.

(2) Let f ∈ C0∞ (R). Then the following operators are compact in H: [f (H), χj ], (f (H) − f (H0 ))χ1 , (f (H) − f (K0 ))χ2 , [f (K0 ), χj ], j = 1, 2. Proof. By virtue of the Helﬀer-Sj¨ ostrand formula (see (3.12) and [4, Theorem 8.1]), it suﬃces to show the compactness of the operators in (1). (i) We ﬁrst prove that [(H − z)−1 , χj ], j = 1, 2 are compact. Since the proof for the case j = 2 is similar and simpler, we prove it only for j = 1. We have [(H − z)−1 , χ1 ] = (H − z)−1 [H, χ1 ](H − z)−1 and [H, χ1 ] =

1 − 2 ∆χ01 + 12 χ01 ∆ g|χ11 − χ01 g| . |gχ01 − χ11 |g (− 12 ∆ + |Dy |)χ11 − χ11 (− 12 ∆ + |Dy |)

We show that all entries of the matrix on the right are compact operators between appropriate spaces. (a) By the Rellich theorem − 21 ∆χ01 + 12 χ01 ∆ : H 2 (R3 ) → L2 (R3 ) is compact. (b) In the conﬁguration space, |g : u0 (x) → gˇ0 (y − x)u0 (x). We write |gχ01 − χ11 |g = |g(χ01 − 1) − (χ11 − 1)|g. Then, both |g(χ01 − 1)(−∆ + 1)−1 and (χ11 − 1)|g(−∆ + 1)−1 are compact from L2 (R3 ) to L2 (R6 ), because |g(χ01 − 1) and (χ11 − 1)|g are multiplications by gˇ0 (y − x)(χ01 (x) − 1) and gˇ0 (y − x)(χ11 (x, y) − 1), respectively, and they satisfy the condition of Lemma 5.1(3). Indeed, we have ˇ g0 (y − x)(χ01 (x) − 1)L2 (R3y ) = 0 for |x| ≥ 1, because gˇ0 ∈ L2 (R3 ) by virtue of (5.1) and (χ01 (x) − 1) = 0 for |x| ≥ 1, and |ˇ g0 (y − x)(χ11 (x, y) − 1)| ≤ C(1 + |x| + |y|)−5/2 , because (1 − χ11 (x, y)) vanishes in an open cone containing x = y outside the unit ball of R6 . Thus, |gχ01 − χ11 |g is compact from H 2 (R3 ) to L2 (R6 ). (c) (− 12 ∆ + 1)−1 (g|χ11 − χ01 g|) is the adjoint of the operator discussed in step (b) and is compact from L2 (R6 ) to L2 (R3 ). (d) χ11 is homogeneous of degree 0 outside the unit ball and [− 12 ∆x , χ11 ] is a ﬁrst order diﬀerential operator whose coeﬃcients are derivatives of χ11 . It follows that [− 12 ∆x , χ11 ](− 12 ∆x + |Dy | + 1)−1 is compact in L2 (R6 ) by the Rellich compactness theorem. We approximate |Dy | by (ε2 + |Dy |2 )1/2 as

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in the proof of Lemma 5.1. The operator [(ε2 + |Dy |2 )1/2 , χ11 ] is a pseudodiﬀerential operator of order zero in R6 whose symbol decays at inﬁnity with respect to (x, y). Hence, [(ε2 + |Dy |2 )1/2 , χ11 ](− 12 ∆x + |Dy | + 1)−1 is compact in L2 (R6 ), and so is [|Dy |, χ11 ](− 12 ∆x + |Dy | + 1)−1 by the argument of the proof of Lemma 5.1. Thus, [− 21 ∆+|Dy |, χ01 ](− 12 ∆x +|Dy |+1)−1 is compact in L2 (R6 ). Combination of the results (a) ∼ (d) implies that the operator [(H − z)−1 , χ1 ] is compact in H. (ii) Next we show ((H − z)−1 − (H0 − z)−1 )χ1 is compact. We write V˜ = V ⊕ V . Then ((H − z)−1 − (H0 − z)−1 )χ1 = −(H − z)−1 V˜ (H0 − z)−1 χ1 = −(H − z)−1 V˜ χ1 (H0 − z)−1 − (H − z)−1 V˜ [(H0 − z)−1 , χ1 ] By assumption (H −z)−1 V˜ is bounded and [(H0 −z)−1 , χ1 ] is compact in H, as was shown in (i) above. It follows that the second summand on the right is compact. To see that the same holds for the ﬁrst summand, we write V˜ χ1 = (V χ01 ⊕ 0) + (0 ⊕ V χ11 ). Then, V χ01 ⊕ 0 is H0 -compact because V is −∆-compact. Since χ11 vanishes in an open cone about x = 0, V (x)χ11 (x, y) decays as |x| + |y| → ∞. Hence, V (x)χ11 (x, y) is −∆x + |Dy |-compact, viz. 0 ⊕ V χ11 is H0 -compact. It follows that V˜ χ1 (H0 − z)−1 is compact and hence so is ((H − z)−1 − (H0 − z)−1 )χ1 . (iii) To show ((H − z)−1 − (K0 − z)−1 )χ2 is compact, we write it in the form 1 −1 χ12 −1 0 g|(− 2 ∆ + |Dy | − z) −(H − z) . 0 0 It suﬃces to show that (−∆ + 1)−1 g|(− 21 ∆ + |Dy | − z)−1 χ12 is compact from L2 (R6 ) to L2 (R3 ). We write g|(− 12 ∆ + |Dy | − z)−1 χ12 = g|χ12 (− 21 ∆ + |Dy | − z)−1

− g|(− 21 ∆ + |Dy | − z)−1 [− 12 ∆ + |Dy |, χ12 ](− 12 ∆ + |Dy | − z)−1

The argument of (i) (d) above implies [− 12 ∆ + |Dy |, χ12 ](− 12 ∆ + |Dy | − z)−1 is compact in L2 (R6 ). Since χ12 (x, y) = 0 in an open cone containing x = y, (5.1) implies |ˇ g (x − y)χ12 (x, y)| ≤ C(1 + |x| + |y|)−5/2 . It follows that (−∆ + 1)−1 g|χ12 is compact from L2 (R6 ) to L2 (R3 ) because it is the adjoint of χ12 |g(−∆ + 1)−1 which is compact fromL2 (R3 ) to L2 (R6 ) by the argument of (i) (b). (iv) The compactness of [(K0 −z)−1, χj ] is well known, and we omit the details.

5.1

Essential spectrum

In this subsection we show that σess (H) = [min{Σ, E0 }, ∞) by proving the following two lemmas. We write Σess = min{Σ, E0 }. We recall the deﬁnitions, namely Σ = inf σ(H0 ) and E0 = inf σ(h). We impose Assumption 1.4 on V .

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Lemma 5.3. We have [Σess , ∞) ⊆ σess (H). Proof. We prove the lemma by the standard Weyl sequence method. Let us ﬁrst assume λ > Σ. We can choose an orthonormalized sequence un = un0 ⊕ un1 ∈ D(H0 ), such that (H0 − λ)un → 0, because λ ∈ σ(H0 ), and σ(H0 ) is absolutely continuous. Then, for any choice of Rn ∈ R3 , un0 (x − Rn ) ˜ n = −ikRn u , n = 1, 2, . . . e un1 (x − Rn , k) un → 0 as n → ∞, since H0 is translation is still orthonormalized and (H0 − λ)˜ invariant. Due to Assumption 1.4 we can choose Rn such that V un0 (x − Rn )L2 (R3x ) → 0 and V un1 (x − Rn , k)L2 (R6(x,k) ) → 0 as n → ∞. Then we have (H − λ)˜ un → 0, and we conclude λ ∈ σess (H), and then [Σ, ∞) ⊂ σess (H). Next suppose that h has an eigenvalue E < Σ with a normalized eigenfunction φ(x), and let λ > E. We show that λ ∈ σess (H) and hence [E0 , ∞) ⊂ σess (H). Take k0 ∈ R3 such that |k0 | = λ−E, and take a function ψ ∈ C0∞ (R3k ), such that ψ = 1. Set 0 un = , un1 (x, k) = n3/2 φ(x)ψ(n(k − k0 )). un1 (x, k) Then un = 1, and un → 0 weakly as n → ∞. Moreover, we have g|un1 H0 ≤ n−3/2 g0 (k0 + n−1 k)|ψ(k)|dk → 0, (h + |k| − λ)un1 = n3/2 (|k| − |k0 |)ψ(n(k − k0 )) = (|k0 + n−1 k| − |k0 |)ψ(k) → 0 as n → ∞. Hence (H − λ)un → 0, and λ ∈ σess (H).

Lemma 5.4. We have σess (H) ⊆ [Σess , ∞). Proof. We prove that f (H) is a compact operator for any f ∈ C0 (−∞, Σess ) by adapting the geometric proof of HVZ-theorem ([2]), the corresponding result for the N -body Schr¨ odinger operators. We have f (H0 ) = f (K0 ) = 0 by the deﬁnition of Σess . We decompose 2 χ00 0 f (H) = f (H) 0 χ210 2 χ01 0 0 0 + (f (H) − f (H0 )) )) + (f (H) − f (K 0 0 χ212 0 χ211 The ﬁrst summand on the right is compact in H by the Rellich theorem, and so are the others by virtue of Lemma 5.2. Thus f (H) is compact and the lemma follows.

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The Mourre estimate

In this subsection we complete the proof of statement (1) of Theorem 1.6 via the Mourre theory. For this purpose, we ﬁrst prove the following Mourre estimate for the operator H with the conjugate operator 0 Ax A= , 0 Ax + Ay where Ax = 12 (x · Dx + Dx · x) and Ay = 12 (y · Dy + Dy · y). In the momentum representation Ay can be represented by −Ak = − 21 (k · Dk + Dk · k). We write Iδ (λ) = (λ − δ, λ + δ). PH (I) is the spectral projection of H for the interval I. Lemma 5.5. Let λ0 ∈ Θ(H), the threshold set. Then there exist ε > 0, δ > 0 and a compact operator C such that iPH (Iδ (λ0 ))[H, A]PH (Iδ (λ0 )) ≥ εPH (Iδ (λ0 )) + C. Proof. In this proof we take µ = 1 without loss of generality. We compute as a quadratic form on S(R3 ) ⊕ S(R6 ) i[h, Ax ] i(g|(Ax + Ay ) − Ax g|) . i[H, A] = i[h + |Dy |, Ax + Ay ] i(|gAx − (Ax + Ay )|g) A simple computation using eikx (Ax − Ak )e−ikx = Ax − Ak yields i[− 21 ∆ + V, Ax ] = −∆ − x · ∇x V, i[|Dy |, Ay ] = i[|k|, −Ak ] = [k∇k , |k|] = |k|,

(5.4) (5.5)

i(|gAx − (Ax + Ay )|g) = |e−ikx (iAk g0 )(k), i(g|(Ax + Ay ) − Ax g|) = e−ikx (iAk g0 )(k)|.

(5.6) (5.7)

Thus, writing g1 (k, x) = e−ikx (iAk g0 )(k) and W (x) = −x · ∇x V , we obtain −∆x + W g1 | . (5.8) i[H, A] = −∆x + |k| + W |g1 We deﬁne g10 (k) = (iAk g0 )(k) so that g1 (x, k) = e−ixk g10 (k). Note that 1 1 3 1 = k∇k + = , iAk 2 |k| |k| |k| such that g10 (k) has the same property as that of g0 (k), viz. it is a smooth function of |k| = 0 which decays rapidly at inﬁnity and it has a |k|−1/2 singularity at k = 0. In the rest of the proof we ﬁx a function ψ ∈ C0∞ (R), such that ψ(λ) = 1 for |λ| ≤ 1/2 and ψ(λ) = 0 for |λ| ≥ 1 and deﬁne fλ0 ,δ (λ) = ψ((λ − λ0 )/δ)

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for λ0 ∈ R and δ > 0. It follows that i[H, A] can be extended to a selfadjoint operator in H with the domain D(H) and f (H)i[H, A] has a bounded extension for any f ∈ C0∞ (R). We decompose f (H)i[H, A]f (H) as 2

f (H)i[H, A]f (H) =

f (H)i[H, A]f (H)χ2j

(5.9)

j=0

by using the partitions of unity introduced in (5.2) and (5.3). Here f ∈ C0∞ (R). (i) By the Rellich theorem f (H)χ0 is compact. Hence f (H)[H, A]f (H)χ20 is a compact operator in H. (ii) Replacing f (H) by f (H0 ), we write f (H)[H, A]f (H)χ21 = f (H)[H, A]f (H0 )χ21 + f (H)[H, A](f (H) − f (H0 ))χ21 . Here f (H)i[H, A](f (H) − f (H0 ))χ21 is compact by virtue of Lemma 5.2. We then rewrite the ﬁrst term on the right as follows: f (H)[H, A]f (H0 )χ21 = f (H)[H, A][f (H0 ), χ1 ]χ1 + f (H)[[H, A], χ1 ]f (H0 )χ1 + (f (H) − f (H0 ))χ1 [H, A]f (H0 )χ1 + f (H0 )χ1 (W ⊕ W )f (H0 )χ1 + [f (H0 ), χ1 ][H0 , A]f (H0 )χ1 + χ1 f (H0 )[H0 , A]f (H0 )χ1 . (5.10) Here on the right all terms but the last one are compact in H. Indeed, χ1 (W ⊕ W )f (H0 ) is compact by the assumption on V , since χ11 = 0 in an open cone about x = 0 in R6 ; [f (H0 ), χ1 ] and (f (H) − f (H0 ))χ1 are compact by virtue of Lemma 5.2; since W and g1 (k, x) satisfy properties similar to those of V and g(x, k), the proof of Lemma 5.2 implies [i[H, A], χ1 ]f (H0 ) is compact. Thus, f (H)i[H, A]f (H)χ21 = χ1 f (H0 )i[H0 , A]f (H0 )χ1 + C0 ,

(5.11)

where C0 is a compact operator in H. We show that, for any λ0 ∈ Θ(H), there exist ε > 0 and δ0 > 0 such that, for f = fλ0 ,δ with 0 < δ < δ0 , χ1 f (H0 )i[H0 , A]f (H0 )χ1 ≥ εχ1 f (H0 )2 χ1 .

(5.12)

In the direct integral decomposition introduced in (1.4), we have (recall the deﬁnition (1.3) of U ) U f (H0 )i[H0 , A]f (H0 )U ∗ 2 ⊕ p g10 | = f (H0 (p)) f (H0 (p))dp. |g10 (p − k)2 + |k| R3 We divide the proof of (5.12) into three cases.

(5.13)

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(a) Assume Σ < λ0 < 0, λ0 ∈ Θ(H), ﬁrst. Then choose δ0 > 0 such that λ0 +2δ0 < 0 and Σ < λ0 − 2δ0 , and let f = fλ0 ,δ , δ < δ0 . Then Theorem 1.2 implies that f (H0 (p)) is supported in a compact subset of {p : 0 < |p| < ρc }, and f (H0 (p)) = f (λ◦ (p))ep ⊗ ep . We compute the inner product 2 p g10 | ep , (5.14) e |g10 (p − k)2 + |k| p K by using the expression (1.6) for ep . The result is (−Fλ (p, λ◦ (p)))−1 times

g0 (k)2 ((p − k)2 + |k|)dk . ( 12 (p − k)2 + |k| − λ◦ (p))2 d 3θ/2 We recall that g10 (k) = iAk g0 (k) = dθ e g0 (eθ k) and compute p2 − 2

g0 (k)g10 (k)dk + 1 (p − k)2 + |k| − λ◦ (p) 2

θ=0

g0 (k)e3θ/2 g0 (eθ k)dk g0 (k)g10 (k)dk d = 1 1 2 2 dθ θ=0 2 (p − k) + |k| − λ◦ (p) 2 (p − k) + |k| − λ◦ (p) −3θ/2 −θ e g0 (e k)g0 (k)dk d = 1 −θ k)2 + |e−θ k| − λ (p) θ=0 dθ (p − e ◦ 2 g10 (k)g0 (k)dk =− 1 2 2 (p − k) + |k| − λ◦ (p) g0 (k)2 {(p − k)k − |k|}dk . − ( 12 (p − k)2 + |k| − λ◦ (p))2

It follows that 2 1

g0 (k)g10 (k)dk =− 2 2 (p − k) + |k| − λ◦ (p)

g0 (k)2 {(p − k)k − |k|}dk , ( 12 (p − k)2 + |k| − λ◦ (p))2

and (5.14) is equal to 1 −Fλ (p, λ◦ (p))

p2 +

g0 (k)2 (p − k)pdk 1 ( 2 (p − k)2 + |k| − λ◦ (p))2

.

It is easy to check that the quantity inside the parentheses equals p · (∇p F )(p, λ◦ (p)), and (5.14) is exactly equal to λ◦ρ (ρ), which is strictly positive in a compact subset of {p : 0 < |p| < ρc } by virtue of Lemma 2.4. Thus we conclude that for some positive ε > 0 2 ⊕ p g10 | f (H0 (p)) f (H0 (p))dp, ≥ εU f (H0 )2 U ∗ |g10 (p − k)2 + |k| R3 and (5.12) holds in this case.

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(b) Next we let λ0 > Ec ≡ λ◦ (ρc ) > 0. Take δ1 > 0 such that λ0 − 3δ1 > Ec and let f = fλ0 ,δ for 0 < δ < δ1 . Then there exists a compact set Ξ of R3 such that f (H0 (p)) = 0 for all p ∈ Ξ and 0 < δ < δ1 . By virtue of (5.13), we have f (H0 )i[H0 , A]f (H0 ) ≥ (λ0 − δ)f (H0 )2 ⊕ 0 (g10 − g0 )| ∗ +U f (H0 (p)) f (H0 (p))dpU. |(g10 − g0 ) 0 Ξ

(5.15)

Denote by Z the operator represented by the direct integral on the right of (5.15). Since H0 (p) is an analytic family of type A, fλ0 ,δ (H0 (p)) is norm continuous with respect to p. We have fλ0 ,δ (H0 (p)) → 0 strongly as δ → 0, and also fλ0 ,δ (λ)fλ0 ,δ/4 (λ) = fλ0 ,δ/4 (λ). Using the compactness of Ξ, we conclude that the operator norm of Z converges to zero as δ → 0. Thus we conclude that there exists δ0 > 0 such that for 0 < δ < δ0 fλ0 ,δ (H0 )i[H0 , A]fλ0 ,δ (H0 ) ≥ (λ0 − 2δ)fλ0 ,δ (H0 )2 and (5.12) holds also in this case. (c) When 0 < λ0 < Ec , we take δ0 > 0 such that 0 < λ0 − 2δ0 < λ0 + 2δ0 < Ec , and let f = fλ0 ,δ for 0 < δ < δ0 . If δ0 is taken suﬃciently small, then, by virtue of Theorem 1.2, {p ∈ R3 : f (H0 (p)) = 0} consists of two disjoint components Ω1 = {ρ0 < |p| < ρ1 } and Ω2 = {ρ2 < |p| < ρ3 }, 0 < ρ0 < ρ1 < ρ2 < ρ3 < ρc such that H0 (p) is purely absolutely continuous on Iλ,2δ , when p ∈ Ω1 , and f (H0 (p)) = f (λ◦ (p))ep ⊗ ep for p ∈ Ω2 . Then, splitting the direct integral (5.8) into two parts, the one over Ω1 and the other over Ω2 and applying the arguments of (b) and (a), respectively, we obtain fλ0 ,δ (H0 )i[H0 , A]fλ0 ,δ (H0 ) ≥ εfλ0 ,δ (H0 )2 , where ε = min{minp∈Ω2 {λ◦ρ (|p|)}, λ0 − 2δ}. This completes the proof of (5.12). (iii) We now study f (H)i[H, A]f (H)χ22 . As above we write f (H)i[H, A]f (H)χ22 = f (H)i[H, A](f (H) − f (K0 ))χ22 + f (H)i[H, A][f (K0 ), χ2 ]χ2 + f (H)[i[H, A], χ2 ]f (K0 )χ2 + (f (H) − f (K0 ))χ2 i[H, A]f (K0 )χ2 + [f (K0 ), χ2 ]i[H, A]f (K0 )χ2 + χ2 f (K0 )i[H, A]f (K0 )χ2 . We have shown in Lemma 5.2 that (f (H) − f (K0 ))χ2 and [f (K0 ), χ2 ] are compact operators in H. Since i[H, A] has the same form as H, the argument for proving the compactness of f (H)[H, χ2 ]f (H) in (i) of the proof of Lemma 5.2 shows that f (H)[i[H, A], χ2 ]f (K0 ) is also compact. We further use that 0 g1 | 0 0 χ2 f (K0 ) , f (K0 )χ2 = 0 0 0 |g1

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and conclude that f (H)i[H, A]f (H)χ22 = χ2 f (K0 )

i[h, Ax ] 0 f (K0 )χ2 + C2 0 i[h, Ax ] + |Dy |

= 0 ⊕ χ12 f (h + |Dy |)(i[h, Ax ] + |Dy |)f (h + |Dy |)χ12 + C2 , (5.16) where C2 is compact in H. We study f (h + |Dy |)(i[h, Ax ] + |Dy |)f (h + |Dy |). The Fourier transform with respect to the variables k, and the direct integral representation, imply that this operator is unitarily equivalent to the operator f (h + |k|)(i[h, Ax ] + |k|)f (h + |k|) ⊕ = f (h + |k|)(i[h, Ax ] + |k|)f (h + |k|)dk. R3

We use the following lemma due to [7]. Lemma 5.6. Let c = inf σ(h) − 1 and s(λ) be deﬁned by sup(Λ ∩ (−∞, λ]), λ ≥ inf σ(h), s(λ) = c, λ < inf σ(h), where Λ = {E0 , E1 , . . .} ∪ {0}. Then, for any λ0 ∈ R and ε > 0, there exists δ0 > 0 such that f (h + |Dy |)(i[h, Ax ] + |Dy |)f (h + |Dy |) ≥ (λ0 − s(λ0 + ε) − 2ε)f (h + |Dy |)2 for any f = fλ0 ,δ with 0 < δ < δ0 . Suppose now λ0 ∈ Θ and λ0 > inf σ(h) − 1/2. Then, for suﬃciently small ε > 0, we have s(λ0 + ε) = s(λ0 ) < λ0 and, hence, λ0 − s(λ0 + ε) − 2ε ≥ ε. It follows from Lemma 5.6 that there exists δ0 > 0 such that f (h + |Dy |)(i[h, Ax ] + |Dy |)f (h + |Dy |) ≥ εf (h + |Dy |)2

(5.17)

for all 0 < δ < δ0 , f = fλ0 ,δ . Note that (5.17) holds for λ0 ≤ inf σ(h) − 1/2 whenever δ < 1/4, since f (h + |Dy |) = 0 then. From (5.16) and (5.17) we have f (H)i[H, A]f (H)χ22 ≥ εχ2 f (K0 )2 χ2 + C2 .

(5.18)

Combining the results in (i), (ii), and (iii), we see that for any λ0 ∈ Θ, there exist ε > 0 and δ > 0 and a compact operator C3 such that for f = fλ0 ,δ f (H)i[H, A]f (H) ≥ ε(χ1 f (H0 )2 χ1 + χ2 f (K0 )2 χ2 ) + C3 .

(5.19)

Then, using the compactness of (f (H) − f (H0 ))χ1 , [χ1 , f (H0 )], [f (K0 ), χ2 ] and (f (H) − f (K0 ))χ2 again, we derive from (5.19) f (H)i[H, A]f (H) ≥ εf (H) + C4 .

(5.20)

with another compact operator C4 . Lemma 5.5 follows from (5.20) immediately.

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Once the Mourre estimate is established, it is easy to check the conditions of Mourre theory as stated in Lemma 3.1 to conclude the proof of statement (1) of Theorem 1.6. We omit the details here.

5.3

Existence of the ground state

We prove here statement (2) of Theorem 1.6, the existence of the ground state. Since (−∞, Σess ) ∩ σ(H) is discrete, Σess = min{Σ, E0 }, Σ < 0 and E0 < 0 by assumption, it suﬃces to show E ≡ inf σ(H) ≤ Σ + E0 ,

(5.21)

since Σ + E0 < Σess . We prove this by borrowing ideas in [10]. To simplify the notation we assume that µ = 1 below. We denote by f , g the inner product in H. For f = (f0 , f1 ) ∈ D(H), we have 1 2 2 f , Hf = dx 2 |∇x f0 (x)| + V (x)|f0 (x)| g (y − x)f1 (x, y)dxdy + 2 Re f0 (x)ˇ 2 2 1/2 1 + f1 (x, y)|2 dxdy. 2 |∇x f1 (x, y)| + V (x)|f1 (x, y)| + ||Dy | For any ε > 0 there exists f ∈ C0∞ (R3 ) ⊕ C0∞ (R6 ) such that f , H0 f < Σ + ε,

f = 1.

We let φ(x) be the real-valued normalized ground state of − 21 ∆ + V (it exists due to our assumption E0 < 0), viz. (− 12 ∆ + V )φ = E0 φ, and compute φf , Hφf , using the fact that 2

(φf0 , (− 12 ∆)(φf0 ))x = (φf0 , (− 21 ∆φ)f0 )x + (φ, 12 |∇f0 | φ)x , where (·, ·)x denotes the inner product in the x ∈ R3 variable. We ﬁnd that φf , Hφf is equal to 1 2 2 1 2 |∇f0 (x)| + 2 Re(|gf0 , f1 )y + 2 ∇x f1 (x, ·)y +E0 f (x, ·)2y + |Dy |1/2 f1 (x, ·)2y |φ(x)|2 dx = (H0 f , f )y + E0 f (x, ·)2y |φ(x)|2 dx, where (·, ·)y and · y denote the inner product and the norm with respect to the y-variable. We now replace f (x, y) by fz (x, y) = f (x − z, y − z) and change the variables (x, y) → (x + z, y + z), in order to get (H0 f , f )y + E0 f (x, ·)2y |φ(x + z)|2 dx. φfz , Hφfz =

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Here we used the translation invariance of H0 . Now integrate both sides of this equation with respect to z. We then ﬁnd φfz , (H − E0 − Σ − ε)φfz dz = f , (H0 − Σ − ε)f < 0. It follows that there exists z ∈ R3 such that φfz , (H − E0 − Σ − ε)φfz < 0. Since ε is arbitrary, we obtain (5.21).

6 Proof of Theorem 1.7 In this section we prove the existence of the wave operators. In what follows we take µ = 1 to simplify the notation. We begin with Proof of Existence of the Limits (1.9). Since eitH and L2 (R3k ) f → e−itE−it|k| φ ⊗ f ∈ H are isometric operators, it suﬃces to show that the limits exist for every f ∈ C0∞ (R3 \ {0}). For such f the map t → Ft = eitH

0

e−itE−it|k| φ(x)f (k)

is strongly diﬀerentiable, and we can easily compute to obtain d Ft = dt

ft , 0

ft = ie−itE φ(x)

R3

eixk−it|k| g0 (k)f (k)dk.

It suﬃces to show that ft is integrable with respect to t. We estimate the integral ˆ ≥ ||x| − |t||, it follows by with respect to k. Since |∇k (xk − t|k|)| = |x − tk| integration by parts that for any positive N

R3

eixk−it|k| g0 (k)f (k)dk ≤ CN (1 + ||x| − |t||)−N .

It follows, by choosing ε such that 0 < ε < β − 2, that 2 −N ε |ft (x)| dx ≤ CN |t| |φ(x)|2 dx {x : ||x|−|t||>|t|ε} |φ(x)|2 dx + CN {x : ||x|−|t||≤|t|ε }

≤ CN (|t|−N ε + t2+ε−2β ) ≤ CN (|t|−2−2ε + t−β ), for N suﬃciently large. Thus ft is integrable.

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Proof of Existence of the Limits (1.8). By virtue of Theorem 1.2, it suﬃces to prove that the following two limits exist in the strong topology of H: 0 itH , f ∈ L2 (R3x × R3k ). lim e (6.1) 1 t→±∞ eit 2 ∆−it|k| f (x, k) lim e

itH

t→±∞

e−itλ◦ (Dx ) f1,0 , e−ikx e−itλ◦ (Dx ) f1,1

f1 ∈ Hone .

(6.2)

Proof of Existence of the Limit (6.1). Functions of the form N

uj (x)vj (k), j=1

with u ˆj ∈ C0∞ (R3ξ \ {0}) and vj ∈ C0∞ (R3k ), are dense in L2 (R3x × R3k ). Thus it suﬃces to consider f (x, k) = u(x)v(k) with u and v as above. We write again 0 . Ft = eitH it 1 ∆−it|k| e 2 f (x, k) We compute the strong derivative with respect to t. 1 d g|eit 2 ∆−it|k| f g0t (x) . = Ft = i 1 g1t (x, k) dt V eit 2 ∆−it|k| f We estimate g1t (x, k) ﬁrst. We have 1

g1t (x, k) = iV (x)(eit 2 ∆ u)(x)e−it|k| v(k), 1

such that g1t H1 = V eit 2 ∆ u2 v2 . It follows by the well-known estimate for the existence of the wave operator for the two body short potentials (see for example [16]) that g1t H1 is integrable with respect to t. The function g0t (x) can be written in the form 1 g0t (x) = i g0 (k)eikx−it|k| v(k)dk · eit 2 ∆ u(x). By Assumption 1.1 and v ∈ C0∞ , it follows that wt (k) = g0 (k)v(k)e−it|k| belongs to L2 with wt 2 = c0 independent of t. Thus we can estimate g0t as follows, using the fact that the integral term is the inverse Fourier transform of wt (up to a constant), 1

g0t 2 ≤ (2π)3/2 w ˇt 2 eit 2 ∆ u∞ ≤ Cc0 |t|−3/2 u1 . 1

Here we have used the estimate eit 2 ∆ L1 (R3 )→L∞ (R3 ) ≤ c|t|−3/2 . This estimate shows that g0t 2 is integrable with respect to t, such that the limits exist.

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Proof of Existence of the Limit (6.2). Since ∪j C0∞ (Gj ) is dense in L2 (B(ρc )) (see (4.6)), {φ(p)ep (k) : φ ∈ ∪j C0∞ (Gj )} is dense in {h(p)ep (k) : h ∈ L2 (B(ρc ))}. Fλ (p, λ◦ (p)) is smooth and strictly negative in B(r) for any r < ρc , it follows that it suﬃces to prove the existence of the limits, when fˆ1,0 ∈ C0∞ (Gj ) for some j. Using the fact that λ◦ (p) is the eigenvalue of H00 (p), it is easy to see that d itH e−itλ◦ (Dx ) f1,0 V e−itλ◦ (Dx ) f1,0 itH e = ie . e−ikx e−itλ◦ (Dx ) f1,1 V e−ikx e−itλ◦ (Dx ) f1,1 dt Thus, it suﬃces to show that both V e−itλ◦ (Dx ) f1,0 and V e−itλ◦ (Dx ) f1,1 are integrable functions of |t| ≥ 1. But we have seen in Lemma 4.1 that, |e−iλ◦ (Dx ) f1,0 (x)| ≤ C1 |t|−3/2

and |e−iλ◦ (Dx ) f1,1 (x, k)| ≤ C2 (k) |t|−3/2 .

and, moreover, if |x|/t ∈ [α, β], −N

|e−iλ◦ (Dx ) f1,0 (x)| ≤ C1 |t|

x

−N

−N

, |e−iλ◦ (Dx ) f1,1 (x, k)| ≤ C2 (k) |t|

x

−N

,

where C2 (k) is square integrable over R3 . Thus, we have 1/2 V e−iλ◦ (Dx ) f1,0 2 ≤

1/2 2

|V (tx)| dx α<|x|<β

+ Ct−N

R3

1/2 |V (x)|2 x−2N dx

and the right-hand side is integrable by the short range assumption on V . The integrability of V e−iλ◦ (Dx ) f1,1 may be proved similarly. This completes the proof of the theorem.

Acknowledgments Part of this work was carried out while AJ was visiting professor at the Graduate School of Mathematical Sciences, University of Tokyo. The hospitality of the department is gratefully acknowledged. KY thanks Michael Loss for helpful discussions and encouragement at an early stage of this work, and Herbert Spohn, who insisted that we should separate the center of mass motion ﬁrst. We thank Gian Michele Graf for constructive remarks on a preliminary version of the manuscript.

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References [1] A. Arai, A note on scattering theory in nonrelativistic quantum electrodynamics, J. Phys. A 16, 49–69 (1983). [2] H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry, Springer-Verlag, Berlin, 1987. [3] J. Derezi´ nski and C. G´erard, Asymptotic completeness in quantum ﬁeld theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11, 383–450 (1999). [4] M. Dimassi and J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Notes Series 268, Cambridge University Press, Cambridge 1999. [5] J. Fr¨ ohlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Rayleigh scattering, Ann. Henri Poincar´e 3, 107–170 (2002). [6] J. Fr¨ ohlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Compton scattering, Preprint 2001, mp arc 01-420, to appear in Comm. Math. Phys. [7] R. Froese and I. Herbst, A new proof of the Mourre estimate, Duke Math. J. 49, 1075–1085 (1982). [8] C. G´erard, Asymptotic completeness for the spin-boson model with a particle number cutoﬀ, Rev. Math. Phys. 8, 549–589 (1996). [9] C. G´erard, On the scattering theory of massless Nelson models, Preprint 2001, mp arc 01-103. [10] M. Griesemer, E.H. Lieb, and M. Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145, 557–595 (2001). [11] L. H¨ormander, The analysis of linear partial diﬀerential operators. I, Second Edition, Springer-Verlag, Berlin, 1990. [12] J.D. Jackson, Classical electrodynamics, John Wiley & Sons, Inc., New York, London, Sydney, 1962. [13] R.A. Minlos, H. Spohn, The three-body problem in radioactive decay: the case of one atom and at most two photons, Topics in statistical and theoretical physics, 159–193. Amer. Math. Soc. Transl. Ser. 2, 177. Amer. Math. Soc., Providence, RI, 1996. [14] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 78, 391–400 (1981).

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[15] E. Nelson, Interaction of nonrelativistic particles with a quantized scalar ﬁeld, J. Math. Phys. 5, 1190–1197 (1964). [16] M. Reed and B. Simon, Methods of modern mathematical physics. III: Scattering theory, Academic Press, New York, 1979. [17] M. Reed and B. Simon, Methods of modern mathematical physics. IV: Analysis of operators, Academic Press, New York, 1978. [18] H. Spohn, Asymptotic completeness for Rayleigh scattering, J. Math. Phys. 38, 2281–2296 (1997). [19] Yu.V. Zhukov and R. A. Minlos, The spectrum and scattering in the “spinboson” model with at most three photons, Teoret. Mat. Fiz. 103, 63–81 (1995); translation in Theoret. and Math. Phys. 103, 398–411 (1995). A. Galtbayar University Street 3 School of Mathematics and Computer Science National University of Mongolia P.O.Box 46/145, Ulaanbaatar Mongolia A. Jensen Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G DK-9220 Aalborg Ø Denmark and MaPhySto Centre for Mathematical Physics and Stochastics funded by the Danish National Research Foundation email: [email protected] K. Yajima 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 12/04/02, accepted 26/01/03

Ann. Henri Poincar´e 4 (2003) 275 – 299 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020275-25 DOI 10.1007/s00023-003-0130-z

Annales Henri Poincar´ e

Long-Range Scattering of Three-Body Quantum Systems, II Erik Skibsted

1 Introduction and results This paper is the second in a series of two papers on asymptotic completeness for (generalized) three-body quantum systems with long-range interaction. Asymptotic completeness is henceforth abbreviated AC; an “eﬀective” version of AC is the existence of the limits (1.4) and (1.5) given below. For a succinct account of the statement we refer the reader to our ﬁrst paper [S1]. The reader will ﬁnd excellent detailed expositions in [D], [DG], [GL] and [HuSi], in particular historical remarks and extensive lists of references. We shall here prove results in the regime “µ ∈ (0, 12 ]” with µ measuring the decay of the “pair potentials” at inﬁnity as in (1.2) (given below). While there are many papers in the literature on AC for two-body systems with arbitrary µ > 0, the case of many-body systems is opposite. In fact all examples on AC for manybody systems seem so far to be restricted to µ greater than one half. Moreover Yafaev ([Y]) constructed counter-examples to AC for any µ ∈ 0, 12 in systems of one-dimensional particles. We present results for two diﬀerent classes of potentials: 1) One-dimensional potentials with a negative upper bound near inﬁnity (like those considered in [S1]). 2) Potentials (in any dimension) with a positive lower bound near inﬁnity. We recall the basic model and a reduction scheme for AC; see [S1] for more details. We consider a ﬁnite family of subspaces {Xa |a ∈ F } of a ﬁnitedimensional Euclidean space X. By deﬁnition amin , amax ∈ F are given by Xamin = X and Xamax = {0}, respectively, and for a and b diﬀerent from amin the “threebody” condition Xa ∩Xb = {0} is imposed. The position and momentum operators on the basic Hilbert space H = L2 (X) are denoted by x and p, respectively. The orthogonal complement of Xa in X is denoted by X a . The corresponding components of x and p are denoted by xa , pa and xa , pa , respectively. The basic Hamiltonian on H is 1 H = p2 + V ; V (x) = V a (xa ), (1.1) 2 a∈F

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where each “pair potential” V a is assumed to be a real-valued smooth function on X a obeying for some µ > 0 (independent of a) and all multi-indices β (1.2) ∂xβa V a (xa ) = O |xa |−µ−|β| . We consider for a = amax , amin the propagator Ua (t) generated by Ha (t) = H a + 21 p2a +Ia (t, x) (i.e., i∂t Ua (t) = Ha (t)Ua (t) and Ua (1) = I ), where by deﬁnition H a = 12 (pa )2 + V a and Ia (t, x) = J xt Ia (x) with J an arbitrary C0∞ cut-oﬀ function supported in Ya = X \ ∪b⊂a Xb and with Ia (x) = V (x) − V a (xa ). (Here b ⊂ a ⇔ X b ⊂ X a . ) Clearly we have the bound (1.3) ∂xβ Ia (t, x) = O t−µ−|β| uniformly in x.

There exists the asymptotic energy H a+ = lim Ua (t) H a Ua (t) (understood in t→+∞

˜a (t) denote the propagator generated by H a + the strong resolvent sense). Let U 1 2 2 pa + Ia (t, xa ). The notation EΩ (D) denotes the spectral projection for a self-adjoint operator D corresponding to a Borel set Ω ⊆ R. Now, the “eﬀective” version of AC that we are going to address in this paper a+ ) there exists the limit is the following statement: For all φ+ a ∈ E{0} (H ∗ + ˜ φ˜+ a = lim Ua (t) Ua (t)φa ; t→+∞

(1.4)

a and for all φ˜+ a ∈ E{0} (H ) there exists the limit ∗˜ ˜+ φ+ a = lim Ua (t) Ua (t)φa . t→+∞

1.1

(1.5)

Negative potentials

In addition to (1.2) we shall need the following negativity condition of [S1] for a (ﬁxed) a = amax , amin : For some c, R > 0 V a (xa ) ≤ −c|xa |−µ , |xa | ≥ R.

(1.6)

Moreover we shall assume that the subspace X a is one-dimensional. In our last set of conditions (1.7)–(1.11) stated below we identify the part of x denoted by xa by a coordinate for this vector given by ﬁxing a basis vector for the subspace X a . Suppose (1.7) V a (xa ) = V1a (xa ) + V2a (xa ) + V3a (xa ), where V1a (xa ), V2a (xa ) and V3a (xa ) obey (1.2), − Cr |xa |−µ−2 ≤ V1a (xa ) ≤ −cr |xa |−µ−2 ; xa ≥ R, 2

Cr , cr > 0 and Cr < 2−1 (2 + µ) cr ,

(1.8)

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Long-Range Scattering of Three-Body Quantum Systems, II

− Cl |xa |−µ−2 ≤ V1a (xa ) ≤ −cl |xa |−µ−2 ; xa ≤ −R, Cl , cl > 0 and Cl < 2 and for some > 0

−1

277

(1.9)

2

(2 + µ) cl ,

V2a (xa ) = O |xa |−1−µ− ,

(1.10)

1+ V3a (xa ) = O |xa |− α .

(1.11)

and

In (1.11) and henceforth α = 2(2 + µ)−1 . Now, suppose the conditions (1.2), (1.3) and (1.6) for some a = amax , amin with dim X a = 1; all conditions with the same (ﬁxed) µ ∈ (0, 12 ]. Then one can introduce, cf. [S1], P a+ = Pra+ + Pla+ ,

∗ Pra+ = s − lim Ua (t) E[tα− ,tα+ ] (xa )Ua (t)E{0} H a+ , t→+∞ Pla+ = s − lim Ua (t)∗ E[−tα+ ,−tα− ] (xa )Ua (t)E{0} H a+ .

(1.12)

t→+∞

It was proven that these limits are independent of (small) > 0 and that P a+ = E{0} (H a+ ). We shall show the existence of (1.4) by proving that P a+ = 0, cf. [S1]. Our main result is the following. Theorem 1.1 Under the conditions (1.2), (1.3), (1.6)–(1.11) for a fixed a = amax , amin with dim X a = 1 and for µ ∈ (0, 12 ] E{0} H a+ = 0; (1.13) in particular the existence of (1.4) holds. The existence of (1.5) with these assumptions follows from the fact that in this case E{0} (H a ) = 0 , cf. [O, Theorem 2.2 p. 196]. Combined with (1.13), AC follows. Remark 1.2 For simplicity of presentation we shall prove Theorem 1.1 with the additional assumption that V2a = V3a = 0 . The general case may be treated in the following fashion: First we may assume V3a = 0 since V3a is “short-range”. Next we keep V2a in the analysis (of Sections 2–7). We deﬁne the classical orbit in (2.3) in terms of the “dominating” term V1a only. Keeping track of contributions from error terms coming from V2a yields a weaker localization than (2.4), but strong enough for the arguments of Section 7 ((7.2) needs to be replaced by a weaker estimate). As indicated in the above remark we devote Sections 2–7 to a proof of a slightly simpliﬁed version of Theorem 1.1. Our basic strategy is similar to one applied to a diﬀerent problem, although with common spirit, in [HeSk]. We compare

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a+ a+ + the evolution of a state φ+ a ∈ Pr H (or φa ∈ Pl H) with a simpliﬁed evolution in terms of a relative wave operator. Setting up this wave operator is the content of Sections 2–6. Our techniques at this point resemble at many points those applied in [HeSk]. In Section 7 we verify conditions of [S2] for the simpliﬁed evolution. Using the relevant result of [S2] we infer that indeed states propagated with this evolution cannot be localized to regions of the conﬁguration space that the projection Pra+ a priori prescribes. Consequently φ+ a = 0. Notice that this is a purely quantum statement, it does not have an analogue in classical mechanics. The technique of Section 7 diﬀers completely from the one applied at the similar step in [HeSk]. We remark that one may modify the latter technique as to provide another approach to our second step in Section 7. However it is more complicated. Moreover we remark that neither of these approaches seems to be optimal. For example we consider the conditions (1.8) and (1.9) to be “technical”; the condition (2.1) stated below should suﬃce. As another open problem motivated by the analysis of [S1] we mention AC for negative potentials in higher dimensions with spherical symmetry. In Appendix A we prove bounds for classical orbits of some one-dimensional quadratic Hamilton functions, that are needed at various points in Sections 5–7.

1.2

Positive potentials

In addition to (1.2)) for a µ ∈ (0, 12 ] we shall need the following positivity condition 2µ for a (ﬁxed) a = amax , amin : For some µ+ ∈ [µ, 1−µ ) and R > 0 +

V a (xa ) ≥ |xa |−µ , |xa | ≥ R.

(1.14)

Our main result is the following. Theorem 1.3 Under the conditions (1.2), (1.3), (1.14) for a fixed a = amax , amin and with µ ∈ (0, 12 ] , the limits (1.4) and (1.5) exist; in particular AC holds. The proof of Theorem 1.3 is given in Section 8. It is based on some energy bounds which may be viewed as modiﬁcations of results of [S1]. Conceptually and technically it is much simpler than the proof of AC for negative potentials due to the fact that there are no classical orbits (for the internal dynamics) at inﬁnity with zero energy. To put our result √ and method into perspective we also give a proof of Wang’s result for µ ∈ ( 12 , 3 − 1], [W], referred to in [S1]. Obviously Theorems 1.1 and 1.3 can be combined to obtain asymptotic completeness as deﬁned in [S1] for families of pair potentials of mixed type, each either negative or positive (at inﬁnity) with further properties as speciﬁed in the theorems.

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2 Preliminary estimates (negative potentials) In addition to (1.2), (1.3) and (1.6) with dim X a = 1 and µ ∈ (0, 12 ], we shall in Sections 2–6 need the concavity assumption V a (xa ) ≤ 0 for |xa | ≥ R.

(2.1)

Only in Section 7 the stronger conditions (1.8) and (1.9) are needed (to verify (7.7)). Recall that we put V2a = V3a = 0, cf. Remark 1.2. a+ + We aim at showing that for any given φ+ a ∈ Pr H indeed φa = 0 . Since our a+ proof can be adapted for Pl we then conclude (1.13). In this section we are going to use the freedom to change Ia (t, x) to the eﬀect a a x x Ia (t, x) = Ia (t, x)F+ α− + Ia (t, xa = 0, xa )F− α− (2.2) t t for > 0 chosen arbitrarily small, cf. [S1, Section 5]. (Here and henceforth we adapt the notation F+ and F− of [S1, Deﬁnitions 2.1].) The proof of [S1, Lemma 4.5] (with j = 1) yields the following improved (and classically “optimal”) localization (cf. the proof of [S1, (5.6)]). We remark that we are not going to use the full strength of the result; it is stated here only for completeness of presentation. Lemma 2.1 Let L be the solution of the initial value problem d L(t) = −2V a (L(t)), L(0) = R; t ≥ 0. dt a+ Then for all ε > 0 and φ+ a ∈ Pr H 0 ||F+ tµ−ε−1 |xa − L| φ+ a (t)|| = o t ,

(2.3)

(2.4)

+ where φ+ a (t) = Ua (t)φa .

Proof. Given ε > 0 we may assume (2.2) for a small > 0 obeying 3/2µ < ε . We modify the proof of [S1, (5.6)] by introducing for small σ > 0 obeying 3/2µ < 3σ < ε α0 = α +

σ , µ

3 µ µ α0 + σ = α + σ, 2 2 2 5 α1 = 1 − γ1 − σ = α − σ, 2 µ δ = µ + α − σ, 2 µ β1 = 1 − µ + α + 3σ. 2 γ1 =

(2.5)

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All requirements of the proof of [S1, (5.6)] including [S1, (4.36)–(4.38)] are fulﬁlled for the choice (2.5) with σ > 0 small enough yielding the following statement: β1 + ¯ φa (t)|| = 0, (2.6) lim ||φ+ a (t) − B t, t t→+∞

where for parameters given by (2.5) and K(xa ) being the inverse of L(t) ¯ t, tβ1 = F¯0 F1 F2 F3 F4 ; B F¯0 = F− t−β1 |t − K(xa )| , (2.7) a a x x F1 = F+ α1 , F2 = F− α0 , F3 = F+ (tγ1 pa ), F4 = F− |tδ H a | . t t (In the present context we do not need [S1, (4.37)] though. We can use the fact that a+ a+ )H and [S1, Lemma 2.3] to avoid a certain symmetrizing φ+ a ∈ Pr H ⊆ E{0} (H using [S1, (4.37)].) We shall need some operators rt , bt and ˜bt which are modeled after constructions in [D] and [HeSk]: 1 3 rt (xa ) = f r f −1 (xa − L) ; f = t 4 , r(y) = y = 1 + |y|2 2 . We compute its Heisenberg derivative (with D =

d dt

(2.8)

+ i[Ha (t), ·] )

1 −1 a r f (x − L) pa − L˙ + h.c. + f˙dt ; 2 d −1 ˙ dt = f rt , L = L(t). dt

bt = Drt =

(2.9)

Furthermore (with dots used again for time-derivatives and V (t) = V a (xa ) + Ia (t, x)) −1 a ∂ 2 −1 −3 ¨ ¨ V (t) + L ; Dbt = D rt = f ct + f dt − f et − r f (x − L) ∂xa 2 d f˙ r f −1 (xa − L) P, P = pa − L˙ − (xa − L), (2.10) ct = P ∗ a2 dx f 4 d et = 4−1 r f −1 (xa − L) . a4 dx 2

Clearly both bt and Dbt are bounded relatively to pa2 = (pa ) . Also we notice the non-negativity of ct and the uniform boundedness (with respect to t) of the terms dt and et . We may estimate using also the fact that 1 ∂ ∂ a ¨ V (t) + L = Ia (t, x) + (x − L) dsV a (L + s(xa − L)) (2.11) ∂xa ∂xa 0

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and (2.1), ∂ 1 Ia (t, x) + C1 f¨ ≥ −C2 t− min (µ, 4 )−1 . Dbt ≥ −r f −1 (xa − L) ∂xa

(2.12)

We introduce the regularizations ˜bt = N −1 bt N −1 (= O(tν )), c˜t = N −1 ct N −1 = O t2ν ; t t t t Nt = I + t

(2.13)

−2ν a2

p , ν > 0.

We also introduce for small σ > 0 g = tρ1 ; ρ1 =

µ α + 3σ, 2

h = t1−ρ2 = tα−2σ ; ρ2 =

(2.14) µ α + 2σ. 2

Obviously (for future reference) f = o(h) and ρ2 < ρ1 . Moreover C1 tα ≤ L ≤ C2 tα , cf. [S1, (5.8)]. To start out the analysis we shall use the following weaker localization than the one presented in Lemma 2.1: −1 + ||φ+ (2.15) rt φa (t)|| = o t0 . a (t) − F− h Notice that this localization is very weak as opposed to the “optimal” one of Lemma 2.1. Similarly the following localization result is very weak; the “optimal” bound follows readily from the proof: Lemma 2.2 Let φ+ a be given as in Lemma 2.1. Then for all small σ > 0 (and all ν > 0) 0 ||F+ g˜bt φ+ (2.16) a (t)|| = o t . Proof. As in the proof of Lemma 2.1 we may assume (2.2) with > 0 small. We use the proof of this lemma. By (2.6) it suﬃces to estimate 0 ¯ t, tβ1 φ+ (2.17) ||g˜bt B a (t)|| = o t . By commutation (2.17) will follow from ¯ t, tβ1 φ+ (t)|| = o t0 . ||g pa − L˙ B a

(2.18)

To show (2.18) we may insert F+ = F+ (4tγ1 pa ) to the left and then write −1 2(H a − (V a (xa ) − V a (L))). F+ pa − L˙ = F+ pa + L˙

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The contribution from H a may by commutation be shown to be O t3σ−δ+µα , 6σ−µ+ µ α 2 while the one from the second term to the right is O t due to the presence −α(1+µ) a a a ¯ (x − L). of the factor F0 and the formula V (x ) − V (L) = O t a+ For any given φ+ a ∈ Pr H we aim at proving the existence of the limit

˘a∗ (t)φ+ lim U a (t),

(2.20)

t→+∞

˘a (t) to be deﬁned in Section 4 and for which a for some comparison dynamics U result of [S2] can be applied (to conclude φ+ a = 0).

3 Integral estimates In order to prove the existence of (2.20) we need certain integral estimates for the full dynamics Ua (t), cf. [D] and [HeSk]. Henceforth we shall not use or assume (2.2). Nevertheless we are going to apply some of the parameters of (2.5). The δ is changed to δ = µ − σ. (3.1) Lemma 3.1 For all small enough σ, ν > 0 (depending only on µ) and with F (t) = F1 F2 F3 F4 , where the factors to the right are given by (2.7) with α0 , γ1 , and α1 given by (2.5) and δ by (3.1), and with the expectation value · φ given in the state + + a+ φ = φ+ a (t) = Ua (t)φa for any φa ∈ Pr H ∞

1 1 −F−2 2 g˜bt A(t) −F−2 2 g˜bt dt < ∞; (3.2) G(t)φ+ a (t) 1 a x ¨ N −1 , V a + L A(t) = gf −1 c˜t + t−1 I − gNt−1 r f −1 (xa − L) F+ 4 α1 t t −1 −1 G(t) = G+ (t) = F+ h rt Nt F (t) or G(t) = G− (t) = F− h−1 rt Nt−1 F (t), and 1

∞

F−2 g˜bt t−1−σ 2I − g˜bt + t−1 I

1 H(t) = H+ (t) = F+2 2 h−1 rt F (t) or 1 H(t) = H− (t) = −F−2 2 h−1 rt F (t).

H(t)φ+ a (t)

dt < ∞;

(3.3)

Proof. We notice that indeed the expression A(t) of (3.2) is a sum of non-negative terms, cf. (2.1) and (2.11). A similar remark is due for the integrand of (3.3).

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For (3.2) we claim that the estimate with G(t) = Nt−1 F (t) follows by considering the “propagation observable” (more precisely the uniformly bounded family of observables) ∗ Φ(t) = Nt−1 F (t) F−2 g˜bt Nt−1 F (t). Let us compute the “leading” term coming from diﬀerentiating the middle term. It suﬃces to diﬀerentiate w.r.t. the Heisenberg derivative

a 1 d x + i p2 + V a (xa )F+ 4 α1 + Ia (t, x), · ; Dt = dt 2 t the remainder is integrable. We compute 1 1 −F−2 2 g˜bt + R1 (t), Dt F−2 g˜bt = − −F−2 2 g˜bt Dt g˜bt ||R1 (t)|| ≤ C|| g˜bt , Dt g˜bt ||,

with

(3.6) (3.7)

cf. [DG, Appendix C]. Using (3.8) given below (and concrete expressions for the derivatives involved) we readily estimate the right-hand side by a constant times g 2 t2ν f −2 which is in L1 (dt) for ν > 0 small enough. Obviously Dt g˜bt = g˙ ˜bt + gNt−1 (Dt bt )Nt−1 + 2gRe Dt Nt−1 bt Nt−1 . (3.8) The contribution from the ﬁrst two terms on the right-hand side of (3.8) to the Heisenberg derivative of Φ(t) is T1 (t) + R2 (t); ∗

T1 (t) = −B(t)

1 g˙ ˜bt + gNt−1 (Dbt )Nt−1 B(t), B(t) = −F−2 2 g˜bt Nt−1 F (t),

||R2 (t)|| ∈ L1 (dt).

a a Here we used again that the functions F1 = F+ txα1 and F− 4 txα1 have disjoint support. Commutation picks up an integrable term. (In fact R2 (t) = O(t−∞ ) .) As for the contribution from the last term on the right-hand side of (3.8) we compute for a suitable real-valued Fc ∈ C0∞ (R) ∗ − 2gB(t) Re Dt Nt−1 bt Nt−1 B(t) = T2 (t) + R3 (t) + R4 (t); ∗ T2 (t) = −4νB(t) Fc g˜bt t−2ν−1 pa2 Nt−1 Fc g˜bt B(t), µ 7 ||R3 (t)|| ≤ C|| Fc g˜bt , t−2ν−1 pa2 Nt−1 || = O gf −1 t−1 = O t 2 α− 4 +3σ , a

x ||R4 (t)|| ≤ C|| V a (xa )F+2 4 α1 + Ia (t, x), Nt−1 Nt || = O t−ν−(1+µ)α1 . t Clearly it follows that ||R3 (t)||, ||R4 (t)|| ∈ L1 (dt).

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By (2.10), (2.11) and (2.12) ∗

T1 (t) + T2 (t) ≤ −CB(t) A(t)B(t) + R(t), where C > 0 and ||R(t)|| ∈ L1 (dt). Next we look at the contributions ∗ T3 (t) = Nt−1 F (t) F−2 g˜bt Dt Nt−1 F (t) + h.c.,

(3.9)

(3.10)

∗ T4 (t) = Nt−1 F (t) F−2 g˜bt Nt−1 DF (t) + h.c.,

and

to the Heisenberg derivative of Φ(t): The ﬁrst term on the right-hand side of the formula a

d −1 x −1 −1 a a 2 Dt Nt = Nt + i V (x )F+ 4 α1 + Ia (t, x), Nt dt t contributes to (3.10) by a term that is O t−1−2ν (since pa2 may be bounded by the factor F4 ). Obviously by the above bound for R4 (t) the second term is integrable. As for the term T4 (t) the derivatives of the factors of F (t) are readily handled (i.e., proven integrable) by using various estimates of [S1], cf. the proof of Lemma 2.1.Straightforward computations of commutators with the middle term Nt−1 F−2 g˜bt Nt−1 needed when symmetrizing expressions of the derivatives of the factors F1 and F2 show that those contribute by integrable terms. We refer to (5.10) and (5.11) for a similar issue. In combination with (3.9) we ﬁnally conclude the estimate ∞

1 1 dt < ∞. (3.11) −F−2 2 g˜bt A(t) −F−2 2 g˜bt −1 + Nt

1

F (t)φa (t)

To obtain (3.2) for G(t) = G− (t) it suﬃces by (3.11) to show the statement for G(t) = G+ (t). We show the latter and (3.3) for H(t) = H+ (t) in one stroke by considering the propagation observable ∗ Φ(t) = G+ (t) F−2 g˜bt G+ (t). We notice that 1 ˙ −2 rt + h.c. DF+ h−1 rt = F+ h−1 rt h−1 Drt − hh 2 h˙ = h−1 bt F+ h−1 rt − h−1 rt F+ h−1 rt + O h−2 , h

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tends to be negative when sandwiched by factors of F− g˜bt , cf. the proof of [D, Proposition 5.6]. Explicitly, by commutation cf. (5.11) given below, Nt−1 DF+ h−1 rt F−2 g˜bt F+ h−1 rt Nt−1 + h.c. h˙ −1 −1 = −(gh) A∗1 A1 − A∗2 A2 + O (gh) ; (3.12) 1h A1 = f1 g˜bt F+2 2 h−1 rt , A2 = f2 h−1 rt F− g˜bt Nt−1 , √ √ 1 f1 (x) = 2 − xF− (x), f2 (x) = x F+2 2 (x). The previous arguments for the contribution from Dt F−2 g˜bt applies again (this term contributes by another non-positive term). obtain (3.3) for H(t) = H− (t) we may diﬀerentiate Φ(t) = G− (t)∗ To F 2 tρ1 ˜bt G− (t) using of similar computations as for the ﬁrst estimate of (3.3), −

and we use (3.2) for G(t) = G− (t).

Remark 3.2 We are only going to use the bounds of Lemma 3.1 for the cases G(t) = G− (t) and H(t) = H− (t). In the ﬁrst case the contribution to the integral from the third term of the a expression of (3.2) simpliﬁes as follows: First we replace the factor F+ 4 txα1 −1 A(t) by F− 4 h−1 rt . This can be done up to an integrable error due to the presence of the factor F− h−1 rt in the product G− (t). Next we use a Taylor expansion (cf. Section 4) to write ¨ Nt−1 − gNt−1 r f −1 (xa − L) F− 4−1 h−1 rt V a + L = γ˜t∗ F− 4−1 h−1 rt γ˜t + O gf t−2 + O gt−(3+µ)α h2 ; 1

γ˜t = (−V a (L)grt ) 2 Nt−1 . Clearly the second term O gf t−2 is in L1 (dt), and since gt−(3+µ)α h2 = t−1−σ the same holds for the third term. Finally we may remove the factor F− h−1 rt and obtain consequently, that we can replace the third term of the expression A(t) by γ˜t∗ γ˜t for the estimate of (3.2) with G(t) = G− (t). We also notice that the functions F1 and F2 of Lemma 3.1 are one on a 1 neighborhood of the support of the functions F− h−1 rt and −F−2 2 h−1 rt of the products G− (t) and H− (t), respectively. Consequently we may remove the factors F1 and F2 in the applications given in the coming sections.

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4 A simplified comparison dynamics ˘a (t) by We introduce a comparison dynamics U d ˘ ˘ a (t)U ˘a (t), U ˘a (1) = I, Ua (t) = H dt

(4.1)

˘ a (t) = 1 p2a + H ˘ a (t) + R(t, ˘ x); H 2 ˘ a (t) = 1 pa2 + V a (L) + (xa − L)V a (L) + 2−1 V a (L)(xa − L)2 , H 2 ˘ x) = Ia (t, x) + R ˘ a (t, xa ), R(t, 1 2 d3 a (1 − s) 3 a ∗ a ˘ ˘ R = F (x − L) V (L + s(xa − L))dsF˘ , 2 dxa3 0 F˘ = F− 4−1 h−1 rt .

(4.2)

i and

Clearly for all k ∈ N ∪ {0}

˘ a = O t−(3+µ)α h3−k , ∂xka R

(4.3)

uniformly in x. ˘a (t) preserves the domain of p2 + x2 , see [S2, Section 4]. It is known that U

5 Further integral estimates We need further integral estimates for the full dynamics. To motivate those we con˘ a (t) of (4.2). Let U ˘ a (t) denote the corresponding sider the “model Hamiltonian” H a a a a ˘ (t) = H ˘ (t)U ˘ (t) and U ˘ (1) = I. propagator, i.e., i∂t U We shall introduce “radiation operators” for the generator. This is done in terms of two solutions α+ (t) and α− (t) to the Riccati equation α˙ = −V a (L) − α2 with the properties that for some C > 1 and all large t t−1 ≤ α+ (t) ≤ Ct−1 , 0 ≤ −α− (t) ≤ Ct−1 .

(5.2)

(See Appendix A for an elaboration.) We deﬁne η + = pa − L˙ − α+ (xa − L), η − = pa − L˙ − α− (xa − L), and notice that

˘ a η − = −α− η − , ˘ a η + = −α+ η + , D D

(5.4)

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˘ a refers to the Heisenberg derivative with respect to H ˘ a (t). where D Let 2 2 Nt+ = η + , Nt− = η − , and

G1 (t) = F− g˜bt F−2 h−1 rt Nt−1 F3 F4 , G2 (t) = F− g˜bt F−2 Nt− F−2 Nt+ Nt−1 , G3 (t) = F− h−1 rt Nt−1 F3 F4 ,

where F3 and F4 (and the other factors) are given as in Lemma 3.1. Lemma 5.1 With φ+ a (t) given as in Lemma 3.1 ∞ t−1 −F−2 Nt+ F− (g˜bt )G3 (t)φ+ (t) dt < ∞, a

1

and

1

∞

α− (t) F−2 Nt− F− (g˜bt )G3 (t)φ+ (t) dt < ∞. a

(5.6)

(5.7)

Proof. We consider for (5.6) the propagation observable ∗ Φ1 (t) = G3 (t) F+ Nt+ F−2 g˜bt F+ Nt+ G3 (t). To treat the contribution from the Heisenberg derivatives of the two factors ˘t = d +i H ˘ a (t), · . Due F+ Nt+ in Φ1 (t) we introduce the modiﬁed derivative D dt to the fact that the F˘ of (4.2) is one on a neighborhood of the support function −1 ˘ t F+ N + : of the function F− h rt of the product G3 (t) it suﬃces to consider D t We compute using (5.4) ˘ t F+ Nt+ = −2α+ Nt+ F+ Nt+ + i R(t, ˘ x), F+ Nt+ , D cf. (5.11) stated and (4.3) the second term on the right-hand side below. By (1.3) is O t−1−µ + O t−(3+µ)α h2 and hence in particular integrable. Next we compute − 2α+ Nt+ F+ Nt+ F−2 g˜bt F+ Nt+ + h.c. (5.10) = F− g˜bt −2α+ Nt+ F+2 Nt+ F− g˜bt + O t−1 gf −1 tν , by ﬁrst representing with B = F− g˜bt B, F Nt+ (5.11) 1 −1 −1 =− B, Nt+ Nt+ − w G− (t)dudv; w = u + iv, ∂¯F˜ (w) Nt+ − w π C

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cf. [DG, Appendix C.3], and then using the representation formula once again for the commutator to the right. The ﬁrst term on the right-hand side of (5.10) is non-positive; the second term is integrable. Next to treat the contribution from the Heisenberg derivative of the factor 2 ˜ F− g bt we notice that the proof of Lemma 3.1 yields the upper bound ∗ G3 (t) F+ Nt+ DF−2 g˜bt F+ Nt+ G3 (t) (5.12) 1 1 ≤ −CG3 (t)∗ F+ Nt+ −F−2 2 g˜bt A(t) −F−2 2 g˜bt F+ Nt+ G3 (t) + R(t), for a positive C and with R(t) ∈ L1 (dt). Notice that the ﬁrst term on the right-hand side of (5.12) is non-positive. Assuming for the moment that the contribution from the Heisenberg derivatives of the two factors G3 (t) is integrable we obtain from (5.10) and (5.12) that ∞

1 1 dt < ∞. (5.13) −F−2 2 g˜bt A(t) −F−2 2 g˜bt F+ (Nt+ )G3 (t)φ+ a (t) 1 Combining (5.13) and Lemma 3.1 leads to the estimate ∞

1 1 dt < ∞, −F−2 2 g˜bt A(t) −F−2 2 g˜bt F− (Nt+ )G3 (t)φ+ a (t) 1

(5.14)

cf. Remark 3.2. It remains to check the integrability of the contribution from the derivatives of the two factors G3 (t). The contributions from the derivatives of the factors of Nt−1 , F3 and F4 are integrable due to [S1, Lemmas 2.3 and 4.1], cf. the proof of Lemmas 2.1 and 3.1. As for the contribution from the derivatives of the two factors F− h−1 rt we compute for a positive C, cf. (3.12), Nt−1 DF− h−1 rt F+ Nt+ F−2 g˜bt F+ Nt+ F− h−1 rt Nt−1 + h.c. h˙ = (gh)−1 A∗1 F+2 Nt+ A1 + A∗2 F+2 Nt+ A2 + O t−1−σ h ˙ h −1 ≤ (gh) A∗1 A1 + A∗2 A2 + O t−1−σ ; h 1 −1 2 2 ˜ h rt , A2 = f2 h−1 rt F− g˜bt N −1 , A1 = f1 g bt −F −

(5.15)

t

√ 1 √ f1 (x) = 2 − xF− (x), f2 (x) = x −F−2 2 (x). Combining (5.15) and Lemma 3.1 (and Remark 3.2) leads to the wanted integrability and therefore (5.14).

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Next we consider the Heisenberg derivative of Φ2 (t) = G3 (t)∗ F− Nt+ F−2 g˜bt F− Nt+ G3 (t). We compute the main contribution from the derivatives of the factors F− Nt+ : ∗ (5.17) G3 (t) −2α+ Nt+ F− Nt+ F−2 g˜bt F− Nt+ G3 (t) + h.c. ∗ = G3 (t) F− g˜bt −2α+ Nt+ F−2 Nt+ F− g˜bt G3 (t) + O t−1 gf −1 t2ν , cf. (5.10).

The contribution from the derivative of the factor F−2 g˜bt is integrable by (5.14). The contribution from the derivatives of the two factors G3 (t) is integrable by the same arguments as used for the same factors in Φ1 (t). Finally we conclude (5.6) by integrating (5.17). The estimate (5.7) follows by considering the observable ∗ Φ(t) = G3 (t) F− Nt− F−2 g˜bt F− Nt− G3 (t); the computations and arguments are similar as for those given above for Φ1 (t). By (2.6), (2.15) and (2.16) 0 ∗ + ||φ+ a (t) − G2 (t) G1 (t)φa (t)|| = o t .

Therefore to show the existence of (2.20) it suﬃces to show the existence of ˘a (t)∗ G2 (t)∗ G1 (t)φ+ lim U a (t).

t→+∞

For future reference we notice that ∗ ||F+ 4−1 h−1 rt G2 (t) G1 (t)|| ∈ L1 (dt).

(5.20)

(5.21)

6 Integral estimates for the comparison dynamics ˘a (t), cf. Lemmas 3.1 and 5.1. (The We shall need the following estimates for U operator γ˜t in (6.3) is deﬁned in Remark 3.2.) Lemma 6.1 For all small enough σ, ν > 0 and all φ ∈ H the following estimates ˘a (t)φ : hold with φ˘a (t) = U ∞ t−1 −F−2 Nt+ φ˘a (t) dt ≤ C||φ||2 , (6.1)

1

1

∞

α− (t) F−2 Nt− φ˘

a (t)

dt ≤ C||φ||2 ,

(6.2)

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∞

1

1 1 −F−2 2 g˜bt B(t) −F−2 2 g˜bt

˘a (t) H1 (t)φ

Ann. Henri Poincar´e

dt ≤ C||φ||2 ;

(6.3)

B(t) = gf −1 c˜t + t−1 I + γ˜t∗ γ˜t , H1 (t) = Nt−1 F− Nt− F− Nt+ , 1

∞

F−2 g˜bt t−1−σ 2I − g˜bt + t−1 I

1 H2 (t) = −F−2 2 h−1 rt F− Nt− F− Nt+ .

˘a (t) H2 (t)φ

dt ≤ C||φ||2 ;

(6.4)

Proof. As for (6.1) and (6.2) we notice that the estimates follow from the proof of Lemma 5.1. For (6.3) we consider the observable ∗ Φ(t) = H1 (t) F−2 g˜bt H1 (t). We use the representation formula applied for (5.11), the proof of Lemma 3.1, (6.1) and (6.2), the bounds sup ||pa2 F− Nt− F− Nt+ || < ∞, (6.6) t≥1

|xa | − + F− Nt F− Nt || < ∞, (6.7) t t≥1 ˘ x)|| = O t−1−σ ∈ L1 (dt) . and that g||∂xa R(t, We notice that the bound (6.6) compensates for energy-localization. The bound (6.7) is used to bound (by commutation) certain expressions containing the 2 d −1 a a unbounded factor dxa 2 V (L)(x − L) . sup ||

For (6.4) we ﬁrst consider the observable ∗ Φ1 (t) = H1 (t) F+ h−1 rt F−2 g˜bt F+ h−1 rt H1 (t), cf. the scheme of the proof of Lemmas 3.1 and 5.1. The arguments above combined with (3.12) and (6.3) yield ∞

1 1 −F−2 2 g˜bt B(t) −F−2 2 g˜bt dt ≤ C||φ||2 . (6.9) 1

˘a (t) F− (h−1 rt )H1 (t)φ

Next we ﬁrst consider the observable ∗ Φ2 (t) = H1 (t) F− h−1 rt F−2 g˜bt F− h−1 rt H1 (t). Finally by combining previous arguments, (6.9) and a computation similar to (5.15) we conclude (6.4).

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7 Proof of φ+ a = 0 (negative potentials) a+ The ﬁrst step of the proof of the statement, that any given φ+ a ∈ Pr H indeed must vanish, is the following result.

Lemma 7.1 The limit (5.20) exists. Proof. We prove the existence of the limit using the integral estimates for Ua (t) ˘a (t) proven in Sections 3, 5 and 6. Due to the presence of the factor F− h−1 rt and U in the product G1 (t) it suﬃces to consider Dt (H(t))F3 F4 + H(t)(Dt (F3 )F4 + F3 DF4 ); H(t) = Nt−1 F−2 Nt+ F−2 Nt− F−2 g˜bt F−2 h−1 rt Nt−1 ,

(7.1)

where the derivative Dt is given as in the proof of Lemma 3.1, cf. (5.21): For the ﬁrst term on the right-hand side of (7.1) we compute the derivative of all factors (cf. the proof of Lemma 3.1 and Remark 3.2). After symmetrizing all “leading” terms (i.e., those that are not in L1 (dt)) we may invoke Lemmas 3.1 and 5.1 for Ua (t) ˘a (t). We notice that in some cases we use the estimates for and (6.1)–(6.4) for U some functions F− replaced by F−2 (which are just special cases). By construction 2 ˜ the contribution from the derivative of F g bt does not need to be commuted. −

In other cases that is needed; the errors are easily shown to be in L1 (dt). For the second term on the right-hand side of (7.1) we use [S1, Lemmas 2.3 and 4.1]. The second step is to invoke [S2, Theorem 1.2]. We consider a state φ˘+ a for which for all ε > 0 0 (7.2) F+ tµ−ε−1 |xa − L| φ˘+ a (t) = o t , ˘+ ˘ where φ˘+ a (t) = Ua (t)φa (cf. Lemma 2.1). We need to show that φ˘+ a = 0. For that we introduce the asymptotic velocity a ˘a (t)∗ x − L(t) , xa U ˘a (t), x˘+ = s − C∞ − lim U t→+∞ x+ (t) t

(7.3)

(7.4)

t where x+ (t) = exp 1 α+ dt with α+ given as in (5.2) (see also Appendix A). Suppose for the moment the conditions of [S2, Theorem 1.2] so that this asymptotic velocity is well deﬁned and absolutely continuous w.r.t. the Lebesgue measure on X (the latter by the conclusion of the theorem). Then since x+ (t) ≥ t we obviously get from (7.2) that φ˘+ x+ )H, whence we conclude from the stated a ∈ E{0}×Xa (˘ absolute continuity that indeed (7.3) holds.

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Now to verify the conditions of [S2, Theorem 1.2] we notice that ˘ x)| ∈ L1 (dt), sup |∂x R(t, x

cf. (4.3) and (1.3). (This bound is suﬃcient for the existence of x ˘+ .) As for the second order derivatives [S2, (1.6)] we use (4.3) and the bound (A.5) to estimate ˘ a (t, xa ) as follows: ∂x2a R ˘ a (t, xa )| ≤ O t−2−2σ ≤ x+ (t)−1 x− (t)−1 O t−1−2σ , |∂x2a R uniformly in x. The condition [S2, (1.6)] follows for this second order derivative since t−1−2σ ∈ L1 (dt). We treat ∂x2a I(t, x) and ∂x2a I(t, x) similarly. It remains to bound |∂xa ∂xa I(t, x)| ≤ x+ (t)

−1

h2 (t),

(7.7)

for some h2 ∈ L1 (dt). For that we notice that the lower bound, V a (L) ≥ −˜ µt−2 for some µ ˜ ≥ 0, implies the bound x+ (t) ≤ t

1+

(7.8)

√

1+4µ ˜ 2

;

(7.9)

see Appendix A. The following computations show that under assumption (1.8), (7.8) holds for some µ ˜ ≥ 0 with √ µ 1 + 1 + 4˜ < 1 + µ. (7.10) 2 Obviously we may replace (1.8) by: V a (xa ) ≤ −cr µ(µ + 1)|xa |−µ−2 ; xa ≥ R, cr > 0,

(7.11)

V a (xa ) ≥ −Cr µ(µ + 1)|xa |−µ−2 ; xa ≥ R, Cr > 0,

(7.12)

Cr < 2−1 (2 + µ)2 cr .

(7.13)

and

By integrating (7.11) twice (to inﬁnity) we get V a (xa ) ≤ −cr |xa |−µ . Thus by the formula t =

L

t≤ R

yielding

L R

(−2V a (x))

− 12

dx we can estimate

− 1 µ µ −1 −1 (2cr ) 2 L1+ 2 , 2cr x−µ 2 dx ≤ 1 + 2

1

(2cr ) 2 α−1 t

α

≤ L.

(7.16)

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Next we use (7.12) for xa = L and estimate the right-hand side by (7.16) yielding Cr 2 −2 . (7.17) V a (L) ≥ − µ(µ + 1) 2t cr (2 + µ) Finally by combining (7.13) and (7.17) we get (7.8) for some µ ˜ < µ(µ + 1).

(7.18)

We notice that (7.18) implies (7.10). By combining (7.9) and (7.10) we get that x+ (t) ≤ t1+µ− for some > 0. Therefore −1 |∂xa ∂xa I(t, x)| ≤ O t−2−µ ≤ x+ (t) O t−1− , yielding (7.7). We have veriﬁed the conditions of [S2, Theorem 1.2] and hence proved the absolute continuity of x ˘+ .

8 Positive potentials We shall prove Theorem 1.3. We start out somewhat more generally assuming (1.2) for an arbitrary µ > 0. Suppose in addition that a = amax , amin is given such that for some µ− > µ, −

V a (xa ) ≥ −|xa |−µ , |xa | ≥ R.

(8.1)

1 We deﬁne w = 1 + |w|2 2 ; w ∈ C . Then we have the following modiﬁcation of [S1, Lemma 2.2], which follows from the same method of proof. Lemma 8.1 Let α ˜ ≥ 0 and κ, t ≥ 1. Then (with the above assumptions) for all w∈C 12 − w 2 −αµ −1 t ˜ + t−2α˜ + κ−1 . ||F t−α˜ |xa | > 1 pa (κH a − w) || ≤ C |Imw| 1

(8.2)

We obtain the following modiﬁcation of [S1, Lemma 2.3]. Lemma 8.2 Suppose (8.1) and that for some positive α ˘<µ I(t, x) = I(t, xa ) for |xa | < 2tα˘ .

(8.3)

a+ + Suppose φ+ ). Then with φ+ a ∈ E{0} (H a (t) = U (t)φa and any positive δ and σ with − µ ,1 , (8.4) δ ≤ min 2(µ − 2σ), µ − 2σ + α ˘ min 2

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there holds the bound

Ann. Henri Poincar´e

−σ ||F+ |tδ H a | φ+ . a (t)|| ≤ Ct

(8.5)

Proof. Following the proof of [S1, Lemma 2.3] using now Lemma 8.1 we get F+ (|κH a |) φ+ a (t) ∞ 12 ˜ − ≤ κC1 s−(1+µ) s−αµ + s−2α˜ + κ−1 ds t − 1 − µ+α ˜ min µ2 ,1 + C3 κ 2 t−µ . ≤ C2 κt

Let κ = tδ . We give a new proof of the main result of [W], cf. [S1, Remark 3.4].

Theorem 8.3 ([W]) Under the √ conditions (1.2), (1.3) and (8.1) for a fixed a = amax , amin and with µ ∈ ( 12 , 3 − 1] and µ− > 2µ−1 (1 − µ), the limits (1.4) and (1.5) exist; in particular AC holds. Proof. We shall only prove the existence of the limit (1.4); the existence of (1.5) can be shown completely similarly. We may assume (8.3) for any ﬁxed α ˘ < µ, cf. [E]. Let α ˘ = µ − and δ = 2(1 − µ) + 5 for a small > 0. In addition we can assume (8.4) (for σ small enough) and − δ α ˘µ ,α ˘, ≥1+−α ˘. (8.6) min 2 2 By Lemma 8.2 and by mimicking the proof of [S1, Lemma 3.1] using of (8.6) and Lemma 8.1 we obtain δ a −α˘ a δ a + φ+ |x | F− |t H | φa (t) as t→ + ∞. (8.7) a (t) ≈ F− |t H | F− t Moreover,

1

∞

t−1 | F tδ |H a | ≈ 1 φ+ (t) |dt < ∞, a

cf. [S1, Lemma 2.4], and ∞ t−1 | F t−α˘ |xa | ≈ 1 F− (|tδ H a |)φ+ (t) |dt < ∞, a

1

cf. [S1, Lemma 3.3]. ˜ =U ˜a (t) read with φ(t) ˜a (t)φ˜ : The analogous statements for U ∞ ˜ 2 t−1 | F tδ |H a | ≈ 1 φ(t) ˜ |dt ≤ C||φ|| , 1

(8.8)

(8.9)

(8.10)

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Long-Range Scattering of Three-Body Quantum Systems, II

1

∞

˜ 2 t−1 | F t−α˘ |xa | ≈ 1 F− (|tδ H a |)φ(t) ˜ |dt ≤ C||φ|| .

295

(8.11)

Substituting (8.7) in the expression ˜a (t)∗ φ+ U a (t), and then diﬀerentiating yields integrable terms: The one with the potentials is bounded by ||(Ia (t, xa ) − Ia (t, x))F− t−α˘ |xa | || + || Ia (t, x), F− |tδ H a | || = O t−1−2σ ; notice that the ﬁrst term to the left vanishes by (8.3) and that the bound of the second term comes about by using the proof of Lemma 8.2. Clearly the bound is integrable. The contribution from the derivative of the factor F− (t−µ |xa |), 1 −α˘ xa α ˘ DF− t−α˘ |xa | = t F− (·) a · pa + h.c. − F− (·)(·), 2 |x | t is handled as follows: Using (8.6) and Lemma 8.1 again we infer that the ﬁrst term on the right-hand side is O t−1− . The second term is treated by (8.9) and (8.11). d The contributions from the two factors dt F− |tδ H a | are treated by (8.8) and (8.10) after commutations using (8.6) and Lemma 8.1. Now we consider the regime µ ∈ (0, 12 ]: Proof of Theorem 1.3. We shall only prove the existence of the limit (1.4); the existence of (1.5) can be shown completely similarly. We pick > 0 so small that (1 − µ + 2)µ+ ≤ 2µ − 3.

(8.15)

Let α ˘ = µ − and δ = 2α ˘ . We can assume (8.3). Clearly (8.4) holds for σ small enough. Proceeding exactly as in the beginning of the proof of Theorem 8.3 we conclude that µ−1−2 a δ a + φ+ |x | F− |t H | φa (t) as t→ + ∞. (8.16) a (t) ≈ F− t The next result does not have an analogue in the previous proof. Proceeding as the proof of [S1, Lemma 4.1] we introduce for ψ ∈ H ψ˜ = F1 F2 F3 ψ; F1 = F t−α˘ |xa | > 1 , F2 = F tµ−1−2 |xa | < 1 , F3 = F |tδ H a | < 1 .

296

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Suppose we know that ||F1 F2 F3 || = O(t−s ), then we shall show the bound with the right-hand side replaced by O t−s− 2 , leading inductively to the conclusion that (8.17) ||F1 F2 F3 || = O t−∞ . We estimate, cf. [S1, (4.11)],

2 2H a ψ˜ = H a F12 F22 + F12 F22 H a + t−α˘ F1 F2 + tµ−1−2 F1 F2 F3 ψ δ a −δ ˜ (8.18) ≤ 2t ||F1 F2 t H F3 ψ|| ||ψ|| + 2t−2α˘ ||F1 F2 F3 ψ||2 + 2t2(µ−1−2) ||F1 F2 F3 ψ||2 , and, using here (8.15), + ˜ 2 ≥ Ct−2µ+3 ||ψ|| ˜ 2. 2H a ψ ≥ Ct−(1−µ+2)µ ||ψ||

(8.19)

˜ ≤ Ct−s− 2 ||ψ||, whence we conclude Combining (8.18) and (8.19) leads to ||ψ|| (8.17). Combining (8.16) and (8.17) yields δ a −α˘ a δ a + |x | F− |t H | φa (t). φ+ a (t) ≈ F− |t H | F− t Following now the last part of the proof of Theorem 8.3 we diﬀerentiate the expression ˜a (t)∗ F− |tδ H a | F− t−α˘ |xa | F− |tδ H a | φ+ U a (t), and pick up integrable terms: The contribution with the potentials is estimated by || Ia (t, x), F− |tδ H a | || = O t−1−2σ ∈ L1 (dt), cf. the proof of Lemma 8.2. The one from the Heisenberg derivative of the factor F− t−α˘ |xa | is clearly integrable by (8.17). The same conclusion holds for the d contributions from the two terms with a factor dt F− |tδ H a | ; this follows by using the bounds (8.8) and (8.10) (with the present δ) after commutations using (8.17) again.

A

Classical orbits

In this Appendix we shall construct solutions α+ (t) and α− (t) to the Riccati equation 2 α˙ = q(t) − α2 , (A.1) where q(t)2 is a continuous (non-negative) function. For the example q(t)2 = −V a (L) of Section 5 (cf. (2.1)) we establish (5.2) using in this case the upper bound (A.2) q(t)2 ≤ Ct−2 , cf. [S1, (5.8)] (or (7.16)).

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First we construct two solutions α+ (t) and α− (t) satisfying the bounds t−1 ≤ α+ (t) and α− (t) ≤ 0. This will be done without using (A.2). To ﬁnd α+ (t) we notice that β(t) = t−1 is a solution to β˙ = −β 2 . Clearly we can solve (A.1) in a neighborhood of t = 1 with the initial condition α(1) = β(1) = 1. By the standard comparison theorem (see for example [BR, Theorem 1.8]) we conclude that α(t) ≥ t−1 for t ≥ 1. Using the equation (A.1) we readily continue α(t) to the whole half-axis. The obtained solution is denoted by α+ (t). It remains to construct α− (t) ≤ 0. For that we consider the Schr¨odinger equation (A.5) −x (t) − V a (L(t))x(t) = 0. t From the solution x+ (t) = exp 1 α+ dt we obtain another one, cf. [BR, Section 2.5], by the formula ∞

x− (t) = x+ (t)

x+ (t )

−2

dt .

t

Let us note the following bounds

∞ t + 0 < x (t) = x (t) e−2 t α dt dt t ∞ −1 −1 + −2 tt t1 dt ≤ x (t) e dt = x+ (t) t ≤ 1. −

+

−1

(A.7)

t

From (A.3) we get x− (t) ≥ 0. Consequently if x− (t) is positive for some t the solution x− (t) will grow at least linearly contradicting (A.4). Therefore x− (t) ≤ 0 and we conclude that x− (t) ≤ 0. α− (t) := − x (t) We shall now show the bounds α+ (t) ≤ C t−1 and − C t−1 ≤ α− (t) under the condition (A.2). To get the upper bound of α+ (t) we introduce the function √ 1 + 1 + 4C −1 β(t) = t 2 which satisﬁes β(1) ≥ α+ (1) = 1 and solves β˙ = Ct−2 − β 2 .

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By another comparison we conclude that α+ (t) ≤ β(t). To get the lower bound α− (t) ≥ −C t−1 we compute using the upper bound + α (t) ≤ C t−1 α− (t) = α+ (t) − ≥t

−1

−

∞

e−2

t t

t ∞

e

−2

t t

C t

dt

dt

α+ dt

−1

dt

−1

= −2(C − 1)t−1 .

t

The proof of the bounds is completed. In particular we have veriﬁed (5.2). For an application in Section 7 we notice that the formula x+ x− =

x+ x− 1 = + , x+ x− − x− x+ α − α−

and (5.2) yield the bounds −1

(2C)

t ≤ x+ (t)x− (t) ≤ t

(A.13)

with C given as in (5.2).

References [BR]

G. Birkhoﬀ, G.C. Rota, Ordinary diﬀerential equations, 4. edition, New York, John Wiley & Sons, 1989.

[D]

J. Derezi´ nski, Asymptotic completeness for N -particle long-range quantum systems, Ann. Math. 138, 427–476 (1993).

[DG]

J. Derezi´ nski, C. G´erard, Scattering theory of classical and quantum N particle systems, Texts and Monographs in Physics, Springer, Berlin Heidelberg New York, 1997.

[E]

V. Enss, Long-range scattering of two- and three-body quantum systems, Journ´ees Equations `a D´eriv´ees Partielles, Saint Jean de Monts, Juin, 1989, Publ. Ecole Polytechnique, Palaiseau, 1–31 (1989).

[G]

C. G´erard, Asymptotic completeness for 3–particles systems, Invent. Math. 114, 333–397 (1993).

[GL]

C. G´erard, I. Laba, Multiparticle quantum scattering in constant magnetic ﬁelds, Mathematical Surveys and Monographs, 90, American Mathematical Society, Providence, 2002.

[HeSk] I. Herbst, E. Skibsted, Quantum scattering for homogeneous of degree zero potentials: Absence of channels at local maxima and saddle points, MaPhySto preprint 24, (1999).

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Long-Range Scattering of Three-Body Quantum Systems, II

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[HuSi] W. Hunziker, I.M. Sigal, Time-dependent scattering theory of N -body quantum systems, Rev. Math. Phys. 12 no. 8, 1033–1084 (2000). [H]

L. H¨ ormander, The analysis of linear partial diﬀerential operators III, Berlin, Springer-Verlag, 1985.

[O]

F.W.J. Olver, Asymptotic and special functions, New York, Academic Press 1974.

[S1]

E. Skibsted, Long-range scattering of three-body quantum systems: Asymptotic completeness, Invent. Math. 151, 65–99 (2003).

[S2]

E. Skibsted, Asymptotic absolute continuity for perturbed time-dependent quadratic Hamiltonians, Proc. Indian Acad. Sci. (Math. Sci.) 112 no. 1, 209–228 (2002).

[W]

X.P. Wang, On the three-body long-range scattering problems, Reports Math. Phys. 25, 267–276 (1992).

[Y]

D. Yafaev, New channels of scattering for three-body quantum systems with long-range potentials, Duke Math. J. 82 no. 3, 553–584 (1996).

Erik Skibsted Institut for Matematiske Fag and MaPhySto1 Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark email: [email protected] Communicated by Gian Michele Graf submitted 4/06/02, accepted 30/01/03

To access this journal online: http://www.birkhauser.ch

1 Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation.

Ann. Henri Poincar´e 4 (2003) 301 – 341 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020301-41 DOI 10.1007/s00023-003-0131-y

Annales Henri Poincar´ e

On the Discrete Spectrum of a Pseudo-Relativistic Two-Body Pair Operator Semjon Vugalter and Timo Weidl Abstract. We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.

1 Introduction 1.1

Statement of the problem

In this paper we consider the behaviour of two particles with the given positive masses m+ and m− in the absence of external ﬁelds. The non-relativistic Hamiltonian of such a system is given by −

1 1 ∆+ − ∆− − V (x+ − x− ) on L2 (R2d ), 2m+ 2m−

(1)

where x+ , x− ∈ Rd denote the spatial coordinates and −V stands for the interaction between the particles. Due to translational invariance, this operator is unitary ⊕ equivalent to the direct integral Rd h(P )dP , where h(P ) = −

M p2 , ∆y − V (y) + 2m+ m− 2M

p = |P |,

acts on L2 (Rd ). The parameter M = m+ + m− is the total mass of the system and P ∈ Rd is the total momentum. The spectrum of (1) is the union of the spectra of the pair operators h(P ) for all P ∈ Rd . Notice that h(P ) depends on P only p2 by a shift of 2M , and the spectra of all h(P ) coincide modulo the respective shift. In other words, the fundamental properties of the pair operator do not depend on the choice of the inertial system of coordinates. On the other hand, if we consider the pseudo-relativistic Hamiltonian [H, LSV]

−∆− + m2− − V (x+ − x− ), ⊕ the corresponding decomposition into a direct integral Rd hrel (P )dP gives rise to the pair operators (2) hrel (P ) = |µ+ P − i∇y |2 + µ2+ M 2 + |µ− P + i∇y |2 + µ2− M 2 − V (y), −∆+ + m2+ +

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Ann. Henri Poincar´e

where µ± = m± M −1 ∈ (0, 1).1 Obviously these operators show a much more involved dependence on the total momentum P ∈ Rd . This implies a non-trivial behaviour of the spectra of hrel (P ) in P . For example, if −V is a smooth, compactly supported attractive well, the essential spectrum of hrel (P ) coincides with the interval [(p2 + M 2 )1/2 , ∞) and the discrete spectrum is ﬁnite. However, the distribution of the negative eigenvalues of qrel (P ) = hrel (P ) −

p2 + M 2 ,

p = |P |,

(3)

depends on P . Even if the attractive force −V is too weak to induce negative bound states for small p, eigenvalues will appear as p grows and their total number tends to inﬁnity as p → ∞. Operators with relativistic kinetic energy play an important role in mathematical physics and ﬁnd applications in astrophysics for the explanation of stellar collapses (see [LY1]) as well as in atomic physics (see [HS, LY, L, H]). Despite that to our knowledge the mathematical eﬀect of the non-invariance of the phase space volume of the symbol of hrel (P ) on the spectral properties of this operator has never been discussed. Our paper is devoted to the study of this phenomenon and to bring these observations to the attention of the research community, which is using pseudo-relativistic models in applications. It is natural to ask whether these mathematical results correspond to some real physical eﬀects, or whether they are just a trace of the imperfection of the model, which combines a relativistic kinetic energy with some non-relativistic potential. One way to approach this question is to study a more reﬁned model. This model should include the retardation corrections, see, e.g., [BS], which compensate for the fact, that due to the ﬁniteness of the speed of light a particle at the position x does not feel the pair potential V (x − y) with y being the position of the second particle at the same time, but a pair potential V (x− y˜) with some slightly retarded position y˜ = y˜(y; P ) of the second particle. Some ﬁrst and very rough estimates indicate, that such corrections should not compensate the increase of the number of eigenvalues of the operator (2) completely. The detailed mathematical analysis of the operator with retarded potentials is technically very involved. Therefore, in the present paper we do the ﬁrst step in this direction, studying this phenomenon for the operator (2) with no corrections of the potential. More precisely, we shall study the following quantities. First, for given P we chose the system of coordinates such that P = (p, 0, . . . , 0) and we stretch the spatial variables by the factor p−1 . Obviously p−1 qrel (P ) is unitary equivalent to the operator Qp (i∇, y) = Hp (i∇) − Vp (y), 1 Throughout

this paper we assume m± > 0 (and hence µ± > 0) to be fixed.

(4)

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where Vp (y) = p−1 V (yp−1 ) and 1 + M 2 p−2

Hp (ξ) = T+ (ξ) + T− (ξ) −

|(η ∓ µ± )|2 + |ζ|2 + µ2± M 2 p−2 ,

for T± (ξ) =

with ξ ∈ Rd , ξ = (η, ζ) for ξ1 = η ∈ R and (ξ2 , . . . , ξd ) = ζ ∈ Rd−1 , µ± ∈ (0, 1), p > 0. Throughout this paper we focus on the case of higher dimensions d ≥ 3. We will discuss the behaviour of the total number of negative eigenvalues (including multiplicities)2 Np (V ) = tr χ(−∞,0) (Qp (i∇, y)) and the sum of the absolute values of the negative eigenvalues3 Sp (V ) = tr (Qp (i∇, y))− of the operator Qp (i∇, y). In particular, we shall compare these spectral quantities with their classical counterparts Ξp = Ξp (V ) = (2π)−d dξdy, (5) Qp <0

Σp = Σp (V )

1.2

= (2π)−d

(Qp (ξ, y))− dξdy.

(6)

The classical picture

Already the initial analysis of the phase space averages (5) and (6) shows somewhat unexpected results. Put V ≥ 0. It is not diﬃcult to see, that Ξp is ﬁnite if and only d d if V ∈ L 2 (Rd ) ∩ Ld (Rd ), while Σp is ﬁnite if and only if V ∈ L 2 +1 (Rd ) ∩ Ld+1 (Rd ). However, within these classes of potentials the quantities Ξp and Σp show various asymptotical orders in p as p → ∞. Indeed, we have4 Ξp (V ) = Σp (V ) =

ωd p

d+1 2 (1+o(1)) 3d+1 2 2 πd

d−1 ωd p 2

(1+o(1))

3d−1 (d+1)2 2

πd

V

Rd

d−1 2

V

dy

d+1 2

dy

if V ∈ L

d−1 2

∩ Ld ,

(7)

if V ∈ L

d+1 2

∩ Ld+1

(8)

as p → ∞.5 On the other hand, consider the model potentials Vθ (y) = min{1, v|y|−d/θ }. 2 By

(9)

χ(0,∞) we denote the characteristic function of the negative semi-axes. real x we put 2x− = |x| − x. 4 Below ω is the volume of the d-dimensional unit ball. d 5 We point out that the powers of V in (7), (8) are typical for the phase space behaviour of Schr¨ odinger operators in the spatial dimension d − 1. 3 For

304

If

S. Vugalter, T. Weidl

d−1 2

d

< θ < d then Vθ ∈ Lθw ∩ Ld ⊂ (L 2 ∩ Ld )\L

d−1 2

and it holds

Ξp (Vθ ) = c1 (d, θ, µ± )pθ+1 v θ M d−1−2θ (1 + o(1)), as p → ∞, see also (104). Similarly, if d+1 d Lθw ∩ Ld+1 ⊂ (L 2 +1 ∩ Ld+1 )\L 2 and

d+1 2

Ann. Henri Poincar´e

d−1 < θ < d, 2

(10)

< θ < d + 1 then we have Vθ ∈

Σp (Vθ ) = c2 (d, θ, µ± )pθ−1 v θ M d+1−2θ (1 + o(1)),

d+1 < θ < d + 1, 2

(11)

as p → ∞, cf. (107). Obviously formulae (10) and (11) diﬀer from (7) and (8) not only in the leading order of p, but also in the character of the dependence of the asymptotic constants on V . For the beneﬁt of the reader we attach the calculation of these formulae in Appendix I. To discuss the diﬀerence in character of (7)–(8) and (10)–(11) it is useful to consider the massless limit case. Put ˜ ˜ p (ξ, y) = H(ξ) − Vp (y), Q

(12)

where ˜ H(ξ) = lim Hp (ξ) = |e+ − ξ| + |e− − ξ| − 1, M→0

e± = (±µ± , 0, . . . 0).

˜ p (V ) and Σ ˜ p (V ) be the analogs of (5) and (6), if we replace Qp by Q ˜ p . Then Let Ξ d−1 d+1 d ˜ p (V ) and Σ ˜ p (V ) are ﬁnite, if and only if V ∈ L 2 ∩ L or V ∈ L 2 ∩ Ld+1 , Ξ respectively. For these classes of potentials the asymptotics (7) and (8) can be carried over to the case M = 0 as well. For potentials (9), corresponding to the ˜ p (Vθ ) are inﬁnite for all p > 0. ˜ p (Vθ ) and Σ cases (10) or (11), the quantities Ξ

1.3

Estimates on the counting function

In Section 3 we start the spectral analysis of the operators (4) and develop CwikelLieb-Rosenbljum type bounds on the counting function Np (V ). The strong inhomogeneity of the symbol prevents us from using ready standard versions of the Cwikel inequality [C, BKS]. Instead we apply a modiﬁcation [W1, W2], where the estimate follows the phase space distribution as close as possible even for complicated symbols. In particular, we show that for p ≥ M > 0 and 3 (13) Np (V ) ≤ c p2 1 + ln pM −1 V L1 + V L3 , d = 3, d+1 d−1 d Np (V ) ≤ c p 2 V 2d−1 + V Ld , d ≥ 4, (14) L 2 θ d Np (V ) ≤ c p1+θ M d−1−2θ V θ,w + V Ld , d ≥ 3, (15)

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6 where d−1 2 < θ < d in (15) , whenever the respective right-hand side is ﬁnite. The leading terms in the bounds (14) and (15) reduplicate the correct asymptotic order in p in (7) and (10). The appearance of some mass dependence in (13) is natural, since one expects ˜ p has generically inﬁnite negative spectrum for d = 3 that the massless operator Q ˜ and all p > 0. Indeed, the massless kinetic energy H(ξ) vanishes on the interval between e+ and e− , the ﬁrst coordinate of the momentum will not contribute in this region and we experience practically a d − 1-dimensional kinetic behaviour. Hence, to establish (13) for d = 3 we have to deal with problems resembling spectral estimates for two-dimensional Schr¨ odinger operators. In the massless case virtual bound states will prevent any estimates on Np (V ). The inclusion of a ﬁnite mass suppresses this eﬀect to some extend, but leads with our method of proof to the additional factor (1 + ln pM −1 ) in (13) compared to (7). If the potential V has a repulsive tail at inﬁnity, the bound (13) can be complemented by the estimate

Np ≤ c(V )p2 ,

p ≥ M > 0,

d = 3.

This is carried out in Theorem 7 in Appendix II. Moreover, combining the techniques of Appendix II and inequality (13) it is possible to show that N = o(p2 ln pM −1 ) as p → ∞ for arbitrary V ∈ L1 (R3 ) ∩ L3 (R3 ). Nevertheless it remains an open problem, up to what extend the logarithmic increase in p can be removed from (13) in general.

1.4

Estimates on the eigenvalue moments

In Section 4 we integrate the estimates (13)–(15) according to the AizenmanLieb trick [AL] to obtain Lieb-Thirring type bounds on the sums of the negative eigenvalues and ﬁnd that for p ≥ M > 0 4 (16) Sp (V ) ≤ c p 1 + ln pM −1 V L2 + p−1 V L4 , d = 3, d−1 d+1 d+1 (17) Sp (V ) ≤ c p 2 V 2d+1 + p−1 V Ld+1 , d ≥ 4, L 2 θ d+1 Sp (V ) ≤ c p1−θ M d+1−2θ V θ,w + p−1 V Ld+1 , d ≥ 3, (18) where d+1 2 < θ < d in (18). The bounds (16) and (17) are immediate consequences of (13) and (14), respectively. The estimate (16) carries again an additional logarithmic factor. Since eigenvalue moments behave usually more regular than counting functions, the question on the essence of this term stands even more pressing in this situation. The derivation of (18) from (15) is somehow more involved, because bounds with Lorentz norms cannot be handled in the same way as in [AL]. 6 Here

·θ,w stands for the “weak” norm of the Lorentz space Lθw .

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Ann. Henri Poincar´e

Spectral asymptotics and coherent states

In Section 5 we state in Theorems 5 and 6 the main asymptotic results of this paper. In a ﬁrst step we obtain the formula Sp (V ) = (1 + o(1))Σp (V ) as

p → ∞,

(19)

if for d = 3 the potential V has uniformly bounded, continuous second derivatives d+1 and V ∈ Lθ (R3 )∩L4 (R3 ) for some θ < 2; or if V ∈ L 2 (Rd )∩Ld+1 (Rd ) for d ≥ 4. This result, which is obtained by means of coherent states, corresponds essentially to the case of the phase space asymptotics (8) and relates to the bounds (16), (17). In Section 5 we provide the necessary background information on BerezinLieb inequalities. In Sections 6 and 7 we implement these methods for the speciﬁc symbol at hand. The proof of Theorem 5 is ﬁnally given in Section 8. We point out that our methods do not avail for spectral asymptotics in the case (11). While the coherent state method works well for traces of convex functions of the operator, such as Sp (V ), the application to counting functions is more subtle. Essentially one has to diﬀerentiate the asymptotic formula (19), what requires special attention. In Section 9 we avail to the extend, that we can give asymptotics of the local spectral density. Assume that U, V ≥ 0, U, V ∈ Lθ ∩ Ld+1 for some and that U and V possess uniformly bounded second derivatives. Put θ < d+1 2 U (y; p) = U (p−1 y). Then d−1 ωd − d+1 2 tr U (y; p)χ0 (Qp (i∇, y)) = 3d+1 lim p U (x)V 2 (x)dx. p→∞ d 2 2 π The function U has to decay at inﬁnity and one cannot put U = 1 and deduce an asymptotic for Np (V ) itself. However, it is clear that d+1 d−1 ωd lim inf p− 2 Np (V ) ≥ 3d+1 V 2 dx. p→∞ 2 2 πd This sharp lower bound complements the estimates from above (13) and (14). We follow an approach similar to [ELSS]. Our methods do not provide sharp asymptotics in the setting of (10).

2 Notation Let Lp (Rd ) be the space of p-integrable functions with respect to the Lebesgue measure ν = dx on Rd equipped with the standard norm · Lp (Rd ) . We shall omit the spaces from our notation where possible. If f is a real-valued function on Rd and measurable with respect to the Lebesgue measure ν, then put f± (x)

=

νf (s) = f ∗ (t) =

(|f (x)| ± f (x))/2, ν ({|f (x)| > s}) , inf s, t > 0. νf (s)≤t

(20) s > 0,

(21) (22)

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On the Discrete Spectrum of a Two-Body Pair Operator

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Note that |f |q dν = (f ∗ )q dt and that |f1 (x)| ≥ |f2 (x)| for a.e. x ∈ Rd implies f1∗ (t) ≥ f2∗ (t) for all t > 0. We say that f ∈ Lqw (Rd ) if

f q,w = sup t−q

−1

f ∗ (t)

t>0

is ﬁnite. Beside the quasi-norm · q,w we shall also use the asymptotical functionals −1

δq (f ) =

lim inf t−q

∆q (f ) =

lim sup t−q

t→∞

t→∞

fν∗ (t),

−1

fν∗ (t),

which are continuous on Lqw (Rd ). The function χM will denote the characteristic function of the set M . If M = (−∞, t) ⊂ R we write in shorthand χt = χ(−∞,t) . Let ωd stand for the volume of the unit ball in Rd . Finally, by c or cj.k we denote various constants where we do not keep track of their exact values. In particular, the same notion c in diﬀerent equations does not imply that these constants coincide.

3 Uniform Estimates on the Number of Negative Eigenvalues: Cwikel’s Inequality Revised 3.1

Statement of the result

In this section we discuss a priori bounds on the counting function of the discrete spectrum of the operator Qp (i∇, y) = Hp (i∇) − Vp (y). Our goal is to ﬁnd estimates, which reproduce the behaviour of the phase space −d Ξp = Ξp (V ) = (2π) dξdy Qp <0

in general, and the asymptotics of Ξp for p → ∞ in particular, as closely as possible. In particular, we shall obtain the following two statements. d−1

Theorem 1 Assume that V ≥ 0, V ∈ L 2 (Rd ) ∩ Ld (Rd ) and p ≥ M > 0 . Then there exists a finite constant c = c(d), which is independent on p, M and V , such that 3 (23) Np (V ) ≤ c p2 1 + ln pM −1 V L1 + V L3 , d = 3, d+1 d−1 (24) Np (V ) ≤ c p 2 V 2d−1 + V dLd , d ≥ 4. L

2

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Remark 1 Note that for d = 3 in contrast to the asymptotical behaviour of the phase space volume Ξp p2 V L1 as p → ∞ for V ∈ L1 (R3 ) ∩ L3 (R3 ), the bound (23) contains an additional logarithmic factor. This underlines, that formula (23) has in fact a two-dimensional character, see [W2]. Remark 2 We point out, that in the case M = 0 in the dimension d = 3 one expects infinite many negative eigenvalues for any non-trivial attractive potential V ≥ 0. In contrast to that in higher dimensions the bound (24) holds true in the massless case as well. d Theorem 2 Assume that d ≥ 3, V ≥ 0 and V ∈ Lθw (Rd )∩Ld (Rd ) for d−1 2 < θ < 2. Then there exist finite constants c1 (d, θ) and c2 (d, θ) independent on p, M and V , such that θ d Np (V ) ≤ c1 p1+θ M d−1−2θ V θ,w + c2 V Ld (25)

for all 0 < M ≤ p. Remark 3 The corresponding asymptotics shows that for large p the right-hand side of (25) is of the same order in p as Ξp (V ), if the potential V satisfies δθ (V ) = ∆θ (V ) = v > 0. The remaining part of this section is devoted to the proof of Theorem 1 and Theorem 2.

3.2

A modification of Cwikel’s inequality

Let QA,B be an operator of the type QA,B = B(i∇) − A(x) on L2 (Rd ), where A = a2 and B = b−2 with a, b ≥ 0. Assume that the operator Ea,b = a(x)b(i∇) is compact in L2 (Rd ) and let {sn (Ea,b )}n≥1 be the non-increasing sequence of the singular values (approximation numbers) of Ea,b . According to the BirmanSchwinger principle [B, S] the total multiplicity of the negative spectrum of QA,B equals to the number of singular values sn (Ea,b ) exceeding one, that is NA,B := tr χ0 (QA,B ) = card {n : sn (Ea,b ) > 1} . Hence, spectral estimates on the operators QA,B can be found in terms of estimates on the sequence {sn (Ea,b )}n≥1 . In particular, if a and b satisfy a ∈ Lr (Rd ) and b ∈ Lrw (Rd ) for some 2 < r < ∞, then according to [C] Ea,b ∈ S∞ (L2 (Rd )) and sn (Ea,b ) ≤ c3.1 (r, d)n−1/r a Lr b r,w

for all n ∈ N.

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The bound (26) is of particular interest if b(ξ) = |ξ|−d/r ∈ Lrw (Rd ), since r then the factor a Lr is proportional to the volume of the portion of the classical phase space given by {(x, ξ) ∈ Rd × Rd |a(x)b(ξ) > 1}. For functions b(ξ) which are not “optimal” members of the weak class Lrw (Rd ), the right-hand side of (26) does not capture the respective phase space volumina. We are therefore in need for a suitable generalization of (26), which is applicable to a suﬃciently wide class of symbols b and which reﬂects the phase space character of the estimate even for non-homogeneous symbols. Corresponding results can be found in [W1, W2]. For the problem at hand we shall use the following statement from [W2]. Consider the function q(x, ξ) = a(x)b(ξ) on Rd × Rd and assume that q ∈ 2 2d 2d ∞ 2d L (R ) + L∞ 0 (R ). Here L0 (R ) stands for the subspace of bounded functions q satisfying q(x, ξ) → 0 as |x|+|ξ| → ∞. Let q ∗ be the non-increasing rearrangement of q, see (22) and put q (tˆ) =

tˆ−1

tˆ

1/2

(q ∗ (t))2 dt

,

(27)

0

which is ﬁnite for any tˆ > 0. If ν = dxdξ is the Lebesgue measure on R2d and the distribution function νq is deﬁned according to (21), then using integration by parts the quantity (27) can also be rewritten as follows q (tˆ) =

2 (q (tˆ)) + tˆ ∗

2

1/2

∞

q∗ (tˆ)

sνq (s)ds

,

tˆ > 0.

(28)

The following proposition holds true: 2d Proposition 1 ([W2]) Assume that q(x, ξ) = a(x)b(ξ) ∈ L2 (R2d ) + L∞ 0 (R ). Then 2 d Ea,b ∈ S∞ (L (R )) and the inequality

sn (Ea,b ) ≤ 5 q ((2π)d n)

(29)

holds true for all n ∈ N. Remark 4 In conjunction with the Birman-Schwinger principle the bound (29) implies 1 ≤ q (2π)d NA,B . (30) 5

310

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Cwikels inequality for the operator Hp (ξ) − Vp (y). Preliminary estimates

Now we apply Proposition 1 to the particular symbol qp (x, ξ) = ap (x)bp (ξ) with Ap (x) = a2p (x) = Vp (x) ≥ 0 and Bp (ξ) = b−2 p (ξ) = Hp (ξ). We start with some basic observations. Obviously it holds νqp (s) = ν (x, ξ) ∈ Rd × Rd |qp (x, ξ) > s = Ξp (s−2 V ), s > 0. The behaviour of the quantity Ξp is analyzed in Appendix I. We establish there that according to (98) and (100) for p ≥ M the two-sided bound νqp (s) νqp,1 (s) + νqp,2 (s) + νqp,3 (s)

(31)

holds true, where νqp,1 (s) =

p 2 +1 sd M

νqp,2 (s) =

p 2 sd−1

νqp,3 (s) =

s−2d

d

d+1

d

V 2 dx

(32)

d−1 2

(33)

Ω1 (p,s)

V

dx,

Ω2 (p,s)

V d dx,

(34)

Ω3 (p,s)

and Ω1 (p, s) =

{x|V (x) ≤ s2 M 2 p−1 },

(35)

Ω2 (p, s) = Ω3 (p, s) =

{x|s2 M 2 p−1 < V (x) ≤ s2 p}, {x|V (x) > s2 p}.

(36) (37)

Moreover, note that from (31) and (35), (36) one concludes pd+1 d −2d νqp (s) ≥ c3.2 2d d+1 V dx + c3.3 s V d dx, s M Ω1 (p,s) Ω2 (p,s)∪Ω3 (p,s)

s > 0.

Since we assume p ≥ M , the bound νqp (s) ≥ c3.4 s−2d V dLd holds true. Hence, for the inverse qp∗ of νqp we have 1

1/2

qp∗ (t) ≥ c3.5 t− 2d V Ld ,

3.4

Potentials V ∈ L

d−1 2

t > 0.

(Rd ) ∩ Ld (Rd )

For this class of potentials (31) and (35) imply

d−1 d+1 d νqp (s) ≤ c3.6 max s1−d p 2 V 2d−1 , s−2d V Ld , L

2

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311

d+1 1 1 1/2 1/2 qp∗ (t) ≤ c3.7 max t− d−1 p 2(d−1) V d−1 , t− 2d V Ld .

or

L

2

(39)

Assume now that d ≥ 4. Then (39), (27) and (30) imply 2

1 ≤ c3.8 (Np (V ))− d−1 p d−1 V

1

d+1

L

+ c3.9 (Np (V ))− d V Ld .

d−1 2

(40)

The analogous bound for the case d = 3 requires some more attention. For this we insert each of the three summands (32)–(34) in (31) into the integral in (28) and obtain ∞ 1 ∞ 3 −1 −1 52 ˆ 2 sν1,qp (s)ds ≤ c3.10 t M p dxV (x) s−2 ds √ tˆ qp∗ (tˆ) M −1 pV (x) R3 ≤

1 tˆ

∞

qp∗ (tˆ)

sν2,qp (s)ds

c3.11 tˆ−1 p2 V L1 ,

ˆ−1 2

(41)

M −1

≤

c3.12 t

≤

c3.13 V L1 tˆ−1 p2 ln pM −1 ,

p

R

√

pV (x)

dxV (x) √ 3

s−1 ds

p−1 V (x)

(42)

as well as 1 tˆ

∞

qp∗ (tˆ)

sν3,qp (s)ds

3

∞

≤

c3.14 tˆ−1

≤

c3.15 tˆ−1 (qp∗ (τ ))−4 V 3L3 .

dxV (x) R3

qp∗ (tˆ)

By (38) the last bound implies 1 ∞ sν3,qp (s)ds ≤ c3.16 tˆ−1/3 V L3 . tˆ qp∗ (tˆ)

s−5 ds

(43)

If we insert (39)–(43) into (28) and (30) we arrive at

1 ≤ c3.17 max (Np (V ))−1 V L1 p2 1 + ln pM −1 , (Np (V ))−1/3 V L3 . (44) The relations (40) and (44) imply Theorem 1.

3.5

Potentials V ∈ Lθw (Rd ) ∩ Ld (Rd ),

d−1 2

<θ<

d 2

First observe, that (34) implies ν3,qp (s) ≤ c3.18 s−2d V Ld , d

s > 0.

(45)

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Furthermore, by (32) and (33) we have ν1,qp (s) + ν2,qp (s) d d+1 p 2 +1 d p 2 2 V (x), d−1 V min ≤ c3.19 sd M s Rd

d−1 2

(x) dx.

(46)

1

Assume now V Lθ ≤ v, that is V ∗ (t) ≤ vt− θ for all t > 0. Passing from integraw tion in space to integration of rearrangements (46) turns into d ∞ d+1 p 2 +1 d − d p 2 d−1 − d−1 2 2θ 2 2θ ν1,qp (s) + ν2,qp (s) ≤ c3.20 v t min , d−1 v t dt sd M s 0 d p 2 +1 d 1− 2θ p 2 v 2 1− d−1 v 2 tc + c3.22 (θ) tc 2θ d s M sd−1 d+1

d

≤

c3.21 (θ)

d−1

with tc = M −2θ s−2θ v θ pθ , and ν1,qp (s) + ν2,qp (s) ≤ c3.23 (θ)v θ M d−1−2θ s−2θ p1+θ . Together with (45) this gives

νqp (s) ≤ c3.24 (θ) max p1+θ M d−1−2θ s−2θ V θθ,w , s−2d V dLd , and

1 1 1 1 d−1 1 1 2 t− 2θ , V L2 d t− 2d , qp∗ (t) ≤ c3.25 (θ) max p 2 + 2θ M 2θ −1 V θ,w

s > 0,

t > 0.

From (30) we conclude Theorem 2.

4 Uniform estimates on the Eigenvalue Moments: Lieb-Thirring Inequalities Revised 4.1

Statement of the results

Alongside with estimates on the number of negative eigenvalues we shall make use of estimates on the moments of eigenvalues. Given a bound on the counting function Np (V ), estimates on eigenvalue sums can be deduced from the identity ∞ Np (V − pu)du. (47) Sp (V ) = 0

We shall obtain the following estimates. d+1

Theorem 3 Assume that V ≥ 0, V ∈ L 2 (Rd ) ∩ Ld+1 (Rd ) and 0 < M ≤ p. Then there exist finite constants c = c(d) independent on V , M and p, such that 2 4 (48) Sp (V ) ≤ c p 1 + ln pM −1 V L2 + p−1 V L4 , d = 3, d−1 d+1 d ≥ 4. (49) Sp (V ) ≤ c p 2 V 2d+1 + p−1 V d+1 Ld+1 , L

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Remark 5 The respective asymptotics in Section 4 show that the right-hand side of (49) captures the correct asymptotical order of the phase space average Σp (V ) as p → ∞, while (48) carries an additional logarithmic factor similar to (23). Theorem 4 Assume that d ≥ 3, V ≥ 0 and V ∈ Lθw (Rd ) ∩ Ld (Rd ) for d+1 2 < θ < d + 1. Then there exist finite constants c (d, θ) and c (d, θ) independent on p, M 1 2 2 and V , such that Sp (V ) ≤ c1 pθ−1 M d+1−2θ V θθ,w + c2 V d+1 Ld+1

(50)

for all 0 < M ≤ p. Remark 6 The asymptotics in Section 4 show that the right-hand side of the second estimate has the same asymptotical order in p as Σp (V ) for p → ∞, if the potential V ≥ 0 satisfies δθ (V ) = ∆θ (V ) = v > 0.

4.2

Potentials V ∈ L

d+1 2

(Rd ) ∩ Ld+1 (Rd )

First put d = 3. Standard variational arguments and the Aizenman-Lieb integration [AL] of the bound (23) give ∞ −1 Sp (V ) ≤ c4.1 p 1 + ln pM du (V − pu)+ dx Rd ∞ 0 du (V − pu)3+ dx, +c4.2 0

Rd

which implies (48). In higher dimensions a similar integration of (24) implies (49).

4.3

Potentials V+ ∈ Lθw (Rd ) ∩ Ld+1 (Rd ) with

d+1 2

<θ<

d 2

+1

The inequality (25) contains a term with a weak Lθw -norm. In contrast to the usual Lp -norms, these weak norms in the bound for the counting function cannot be carried over a respective weak norm in the Lieb-Thirring inequality via the Aizenman-Lieb trick. In fact, for the proof of our results below it shows to be necessary to reﬁne (25) for potentials V = (W − pu)+ . Using the same notation as in the previous section in analogy to (45) we ﬁrst ﬁnd that −2d νqp ,3 (s) ≤ c4.3 s (W (x) − pu)d+ dx. (51) Rd

On the other hand, in analogy to (46) passing to the integration of rearrangements we ﬁnd d d+1 d−1 d p 2 p 2 +1 2 2 min (W − pu)+ , d−1 (W − pu)+ dx νqp ,1 + νqp ,2 ≤ c4.4 M sd s Ω1 ∪Ω2 d ∞ d+1 d−1 d p 2 p 2 +1 ≤ c4.5 min (W ∗ − pu)+2 , d−1 (W ∗ − pu)+2 dt. M sd s 0

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1

Put W ∈ Lθw and W w,θ ≤ v, that is W ∗ (t) ≤ vt− θ for t > 0. Then we see that

p 2 +1 ≤ c4.6 M sd d

νqp ,1 (s) + νqp ,2 (s)

∞

tc

d+1

+c4.7

d

1

(vt− θ − pu)+2 dt

p 2 sd−1

tc

0

1

d−1

(vt− θ − pu)+2 dt,

where tc = v θ (pu + p−1 s2 M 2 )−θ . The later integral transforms into

v θ p 2 +1 c4.8 M sd d

νqp ,1 (s) + νqp ,2 (s) ≤

0 θ

+c4.9 Notice that for

d+1 2 a

<θ<

d 2

d

(t + u ˜)−θ−1 t

d+1

v p 2 sd−1

(t + pu)−θ−1 t 2 dt

∞

d

(t + pu)−θ−1 t

d−1 2

dt. (52)

p−1 s2 M 2

+ 1 we have

(t + u˜)−θ−1 t 2 dt

0 ∞

p−1 s2 M 2

d−1 2

dt

≤

d d c4.10 min a 2 +1 u˜−θ−1 , u ˜ 2 −θ ,

(53)

≤

d−1 d−1 c4.11 min u ˜ 2 −θ , a 2 −θ ,

(54)

a

where the minimum is taken for the ﬁrst elements of the respective sets if 0 < a ≤ u ˜, and for the second elements if 0 < u ˜ ≤ a. From (52) and (53), (54) we conclude that d−1 νqp ,1 (s) + νqp ,2 (s) ≤ c4.12 v θ s2 M d+1 p−θ−1 u−θ−1 + s1−d pd−θ u 2 −θ ≤ c4.13 v θ s1−d pd−θ u

d−1 2 −θ

if s2 M 2 p−2 ≤ u, and

d νqp ,1 (s) + νqp ,2 (s) ≤ c4.14 v θ pd+1−θ M −1 s−d u 2 −θ + p1+θ M d−1−2θ s−2θ

for s2 M 2 p−2 ≥ u. These two bounds in conjunction with (51) give

d−1 d νqp (s) ≤ c4.15 s−2d (W − pu)+ Ld + c4.16 v θ min s1−d pd−θ u 2 −θ , d pd+1−θ M −1 s−d u 2 −θ + p1+θ M d−1−2θ s−2θ , s > 0. The inverse qp∗ of νqp satisﬁes then the bound qp∗ (t) ≤

1 d−θ θ 1 θ 1 c4.17 t− 2d (W − pu)+ dLd + c4.18 min t 1−d v d−1 p d−1 u 2 − d−1 , 1−θ 1+θ d−1 1 θ 1 1 θ 1 1 t− d v d p1+ d M − d u 2 − d + t− 2θ v 2 p 2θ M 2θ −1

Vol. 4, 2003

On the Discrete Spectrum of a Two-Body Pair Operator

for all t > 0. Hence, if d ≥ 4 we get qp (t)

≤ c4.19 t

1 − 2d

(W −

v d p1+ θ

1−θ d

1

pu)+ dLd

+ c4.20 min

1

1

u2−d θ

1

td M d

+

v2p

1+θ 2θ

M

qp (t)

≤ c4.21 t θ

+

1

θ

θ

3−θ

θ

1

,

1

(W − pu)+ L3 + c4.22 min

4−θ 3

1

t d−1 u d−1 − 2

t 2θ

1 2

u2−3 v 2 p 2 u , 1 1 1 t3 M 3 t2

v3p

d−θ

θ

v d−1 p d−1

d−1 2θ −1

for all t > 0, while for the dimension d = 3 we obtain − 16

315

1−θ 2

1

1

1

v 2 p 2θ + 2

1 1 + t 2θ M 1− θ θ θ−1 tu p 1 + ln+ vθ M 2

as t > 0. In view of (30) we conclude, that it holds either in higher dimensions

d−1 d Np (W − pu) ≤ c4.23 (W − pu)+ Ld + c4.24 v θ min pd−θ u 2 −θ , d pd+1−θ M −1 u 2 −θ + p1+θ M d−1−2θ , d ≥ 4, (55) or

3 d Np (V − pu) ≤ c4.25 (W − pu)+ Ld + c4.26 v θ min M −1 p4−θ u 2 −θ + √ if d = 3, (56) +M 2−2θ p1+θ , u−θ M 2 p1−θ f (C upM −1 )

where C is some ﬁxed ﬁnite positive constant and y = f (x) is the inverse function √ to x = y/(1 + ln+ Cy) on R+ . Inserting (55) into (47) we obtain immediately √ (50) for d ≥ 4. To settle the case d = 3 we ﬁrst note that f (x) ≤ cx2 (1 + ln+ Cx)2 if c is chosen such that −1 √ c ≥ (1 + ln+ t) 1 + ln+ 1+lnt + t for all t > 0. Hence, the bound (56) can be developed as follows 3 M2 for u ≤ Cp v θ M −1 p4−θ u 2 −θ + v θ M 2−2θ p1+θ 2 √ Np (V −pu) ≤ c4.27 . (57) M2 v θ p3−θ u1−θ (1 + ln+ ( CupM −1 ))2 for u > Cp 2 For θ > 2, the following identity holds true ∞ √ u1−θ (1 + ln(a u))2 du −2 a 1 1 1 + = + a2θ−4 θ − 2 (θ − 2)2 2(θ − 2)3 for any a > 0. If we integrate (57) in u for 2 < θ < we arrive at (50).

5 2

(58)

and take (58) into account,

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5 Asymptotics of the Eigenvalue Moments and the Counting Function 5.1

Statement of the main results

We turn now to the calculation of the asymptotical behaviour of Σp (V ) and Np (V ) for certain cases. In particular, we shall obtain the following two formulae: Theorem 5 Assume that V ∈ Lθ (R3 ) ∩ L4 (R3 ) for some θ < 2 and that V has d+1 uniformly bounded, continuous second derivatives if d = 3, or that V ∈ L 2 (Rd )∩ Ld+1 (Rd ) if d ≥ 4. Then the asymptotical formula Sp (V ) = (1 + o(1))Σp (V ) =

(1 + o(1))p (d + 1)2

d−1 2

3d−1 2

ωd

πd

d+1

Rd

V+ 2 (y)dy

(59)

holds true as p → ∞. Remark 7 For d = 3 the assumptions on the potential V in Theorem 5 are more restrictive than the natural one V ∈ L2 ∩ L4 . The additional logarithmic factor in (48) prevents one to use this bound to close formula (59) to the natural class of potentials. It remains an open problem, whether (59) holds actually for all V ∈ L2 ∩ L4 if d = 3. Theorem 6 Assume that U, V ≥ 0, U, V ∈ Lθ ∩ Ld+1 for some θ < d+1 2 and that U and V possess uniformly bounded second derivatives. Put U (y; p) = U (p−1 y). Then d+1 d−1 ωd lim p− 2 tr U (y; p)χ0 (Qp (i∇, y)) = 3d+1 (60) U (x)V 2 (x)dx. p→∞ 2 2 πd We mention the following obvious consequence of Theorem 6: Corollary 1 If V ≥ 0 has uniformly bounded second derivatives and if V ∈L

d−1 2

∩ Ld+1

then lim inf p− p→∞

d+1 2

Np (V ) ≥

ωd 2

3d+1 2

πd

V

d−1 2

dx.

The remaining part of this paper is devoted to the proof of Theorem 5 and Theorem 6. Our approach is based on the methods of coherent states. Therefore we ﬁrst give a short survey of the necessary general material from this subject.

Vol. 4, 2003

5.2

On the Discrete Spectrum of a Two-Body Pair Operator

317

Coherent States and Berezin-Lieb Inequalities: Preliminaries

Fix some spherically symmetric, smooth, non-negative function f with compact support in Rd , such that f L2 (Rd ) = 1. Put f (x) = d/2 f (x) where > 0. For given γ = {y, ξ} with y, ξ ∈ Rd we deﬁne the coherent states Πγ (x) = e−iξx f (x − y). For any ﬁxed γ and it holds Πγ L2 (Rd ) = 1.

(61)

Let J be a non-negative, locally integrable function on Rd with not more than polynomial growth at inﬁnity. We deﬁne the operator J(i∇) = Φ∗ JΦ in the usual way with Φ being the unitary Fourier transformation. Put fˆ = Φf . In view of our choice of coherent states it is associated with the symbol function j (γ) = j (ξ) = (J(i∇x )Πγ (x), Πγ (x))L2 (Rd ,dx) = (J |fˆ |2 )(ξ).

(62)

The operator of multiplication by a locally integrable real-valued function W on R3 corresponds to the symbol w (γ) = w (y) = (W (x)Πγ (x), Πγ (x))L2 (Rd ,dx) = (W f2 )(y), Here (·, ·)L2 (Rd ,dx) is the scalar product in L2 (Rd ) with respect to the variable x and u v denotes the convolution (u v)(x) = u(x − x )v(x )dx . If now W = W1 +W2 , where W1 is uniformly bounded and W2 is form compact with respect to J(i∇), the operator sum J(i∇)+W (x) can be deﬁned in the form sense. Let ψ be some non-negative convex function on R, such that ψ(J(i∇) + W (x)) is trace class. Then the Berezin-Lieb inequality states that ([Be], see also [LS]) ψ(j (ξ) + w (y))dγ ≤ tr ψ(J(i∇) + W (x)). (63) R2d

Moreover, if the average of ψ(J(ξ) + W (y)) in R2d with respect to dγ is ﬁnite, then ψ(j (i∇) + w (x)) is trace class and ψ(J(ξ) + W (y))dγ (64) tr ψ(j (i∇) + w (x)) ≤ R2d

Let us ﬁnally assume that in addition to this J or W are twice continuously diﬀerentiable with the following uniform bounds on the matrix norms of the respective Hessians ∂ 2 J d ∂ 2 W d ϑ(J) = max and ϑ(W ) = max . ξ∈Rd ∂ξl ∂ξk k,l=1 ξ∈Rd ∂ξl ∂ξk k,l=1

Put ψ(x) = x− =

−x 0

for for

x<0 . x≥0

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We also recall that χ−t is the characteristic function of the interval (−∞, −t). Under the above conditions we have Lemma 1 The two-sided bound (J(ξ) + W (y) + κ)− dγ

≤

tr (J(i∇) + W (x))−

≤

holds true, where κ = 2 ϑ(J)ϑ(W ) xf (x) ∇f

tr (J(i∇) + W (x))− , (J(ξ) + W (y))− dγ + Θκ

and

Θκ =

(65) (66)

κ

tr χ−t (J(i∇) + W (x))dt.

0

Proof. Indeed, by Taylors formula we have J(ξ − ξ ) = J(ξ) − ξ · ∇J(ξ) +

∂ 2 J(ξ(ξ, ˜ ξ )) ∂ξk ∂ξl

k,l

ξk ξl ,

where ξ˜ is some point on the line segment connecting ξ and ξ . Inserting this into ˆ the integral expression for (62), because of f 2 d = 1 one ﬁnds that j (ξ) − J(ξ) = −∇J(ξ) ·

L (R )

ξ |fˆ (ξ )|2 dξ +

∂ 2 J(ξ) ˜ ξ ξ |fˆ (ξ )|2 dξ . ∂ξk ∂ξl k l k,l

Since fˆ is spherically symmetric, the ﬁrst integral on the right-hand side vanishes and |j (ξ) − J(ξ)| ≤ ϑ(J) |ξ |2 |fˆ (ξ )|2L2 (Rd ) dξ ≤

2

ϑ(J)2 ∇f L2 (Rd ) .

(67)

In a similar way we get |w (y) − W (y)| ≤ ϑ(W )−2 xf (x) 2L2 (Rd ) .

(68)

Now (63), (67) and (68) for the optimal choice of give the ﬁrst inequality of Lemma 1. On the other hand (67) and (68) imply J(i∇) + W (x) + κ ≥ j0 (i∇) + w0 (x) and tr (J(i∇) + W (x))− ≤ tr (j0 (i∇) + w0 (x))− + tr gκ (J(i∇) + W (x)) with gκ (x) = min {κ, −x} for x < 0 and gκ (x) = 0 for x ≥ 0. Since κ χ−t (x)dt, gκ (x) = 0

the bound (64) implies the second statement of the Lemma.

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6 Moments of negative Eigenvalues. An Estimate from Below 6.1

Summary

We turn here to the study of the asymptotics of eigenvalue moments S(p) = tr (Qp (i∇, y))− ,

Qp (i∇, y) = Hp (ξ) − Vp (y).

Because of the divergence of the second derivatives of Hp (ξ) near the points e± = (±µ± , 0, . . . , 0) ∈ Rd as p → ∞, a straightforward application of the bound (65) in Lemma 1 will not lead to the desired results. Therefore we have to implement a suitable smoothing procedure of the symbol ﬁrst. In this section we consider the bound from below.

6.2

Basic properties of the symbol Hp (ξ)

Consider the functions T± (ξ) =

(η ∓ µ± )2 + |ζ|2 + µ2± M 2 p−2 ,

with ξ ∈ Rd , ξ = (η, ζ) for ξ1 = η ∈ R and (ξ2 , . . . , ξd ) = ζ ∈ Rd−1 , M = m+ +m− , µ± = m± M −1 . Here m± and p are positive parameters. We have Hp (ξ) = T+ (ξ) + T− (ξ) − 1 + M 2 p−2 . This is a convex non-negative function, which is rotational symmetric with respect to the η-axes. It achieves a unique, non-degenerate minimum at the point ξ = 0 where Hp (0) = 0. The gradient and the Hessian of T± calculate as follows t ∇T± (ξ) = T±−1 (ξ) η ∓ µ± , ζ t , (∇∇t )T± = T±−1 I − (∇T± )(∇T± )t . Hence,

∂Hp (ξ) ∂ξk ≤ 2 and

2 ∂ Hp (ξ) −1 −1 ∂ξk ∂ξl ≤ T+ (ξ) + T− (ξ)

(69)

for all ξ ∈ Rd , p, M > 0 and l, k = 1, . . . , d.

6.3

Smoothing of the symbol

Let g be a smooth, sphericallysymmetric non-negative function on Rd supported within the unit ball, such that g(x)dx = 1. If σ > 0 we put gσ (x) = σ −d g(σ −1 x), for σ = 0 we set g0 (x) = δ(· − x) and deﬁne Hp (ξ − y)gσ(ξ) (y)dy (70) Hp,σ (ξ) = d R = Hp (ξ − σ(ξ)t)g(t)dt. Rd

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It holds Lemma 2 The functions Hp (ξ) and Hp,σ (ξ) satisfy the pointwise estimate Hp (ξ) ≤ Hp,σ (ξ),

ξ ∈ Rd .

(71)

Proof. Note that Hp is convex and the spherically symmetric weight gσ has the total mass 1. If we represent in (70) the term Hp (ξ − y) in a Taylor series at the point ξ of order one with a positive quadratic form as remainder term, the inequality (71) follows immediately (Jensen’s inequality). Put τ± = τ± (ξ) = |ξ − e± |. Below we chose  if ξ ∈ Br+ ∪ Br−  0 ς− (ξ,r) σ(ξ) = σr (ξ) = re if ξ ∈ Br− ,  ς+ (ξ,r) re if ξ ∈ Br+

(72)

where 0 < r < min{µ+ , µ− }/2, Br± = {ξ : τ± (ξ) < r} and ς± (ξ, r) =

−1 2 (ξ) . 1 − r−2 τ±

Lemma 3 One can find an appropriate finite constant c, which is independent on p, M, r > 0, ξ ∈ Rd and k, l = 1, . . . d, such that ∂Hp,σ (73) ∂ξk ≤ c, 2 ∂ Hp,σ −1 (74) ∂ξk ∂ξl ≤ c(1 + r ). Proof. Obviously it holds ∂Hp,σ (ξ) = ∂ξk and ∂ 2 Hp,σ (ξ) = ∂ξk ∂ξl Since

d ∂νj ∂Hp (ν) g(t)dt, ∂ξk ∂νj j=1

νj = ξj − σ(ξ)tj ,

   d d ∂ 2 νj ∂Hp (ν) ∂νj ∂νi ∂ 2 Hp (ν)  + g(t)dt.  ∂ξk ∂ξl ∂νj ∂ξk ∂ξl ∂νj ∂νi  j=1 j,i=1 ∂σr ∂ξk ≤ c6.1

and

2 ∂ σr −1 ∂ξk ∂ξl ≤ c6.2 r ,

from (75) and the ﬁrst estimates in (69), (77) we conclude (73).

(75)

(76)

(77)

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To estimate the second derivatives we note , that by (69) and (77) the ﬁrst part of the integral on the right-hand side of (76) can be estimated by c6.3 (1+r−1 ), while the second term in (76) does not exceed c6.4 (T+−1 (ν) + T−−1 (ν))g(t)dt. Note that T± (ν) ≥ |τ± − σr t| and because g is bounded and of compact support we have π τσr d−1 τ± ± g(t)dt td−1 dt √ ≤ c6.5 d dφ dθ sin θ σr Sd−2 1 + t2 − 2t cos θ 0 Rd T± (ν) 0 d−1 τσr τ ± ≤ c6.6 ±d td−2 (t + 1 − |t − 1|)dt σr 0 ≤

−1 , σr−1 }. c6.7 min{τ±

For 0 ≤ τ± ≤ r/2 the function σr can be estimated by σr ≥ e−4/3 r. Hence, g(t)dt ≤ c(1 + r−1 ) T± (ν)

and we conclude (74).

6.4

The estimate from below

We are now in the position to obtain the main result of this section. Put Qp,σr (ξ, y) = Hp,σr (ξ) − Vp (y), Qp,σr (i∇, y) = Hp,σr (i∇) − Vp (y). By Lemma 2 we ﬁnd that tr (Qp (i∇, y))− ≥ tr (Qp,σr (i∇, y))− .

(78)

Next we apply the ﬁrst part of Lemma 1 with J = Hp,σr and W = Vp to this bound. By (74) we have ϑ(Hp,σr ) ≤ cr−1 for 0 < r < min{µ+ , µ− }, while ϑ(Vp ) ≤ p−3 ϑ(V ). Then (65) implies that tr (Qp,σr (i∇, y))− ≥ (Qp,σr (ξ, y) + κ)− dγ, (79) where κ ≤ c6.8

ϑ(V )r−1 p−3 . From (78) and (79) we ﬁnally conclude

Lemma 4 The inequality

S(p) ≥

(Qp,σr (ξ, y) + κ)− dγ

(80)

holds true for some κ ≤ c ϑ(V )r−1 p−3 , where the constant c in the estimate for κ can be chosen to be independent on V , p, M and r, 0 < r < min{µ+ , µ− }.

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7 Moments of Negative Eigenvalues: An Estimate from Above 7.1

Summary

We shall now accompany Lemma 4 by a corresponding estimate from above. As in the previous section we smooth the symbol before applying (66) from Lemma 1. But in the absence of a replacement of Lemma 2 we have to modify the symbol additionally.

7.2

Modification of the symbol

We put δ ∈ (0, 1/2), ξ = (η, ζ) with ξ1 = η ∈ R and (ξ2 , . . . , ξd ) = ζ ∈ Rd−1 , and set Gp,δ (ξ)

=

Gp,δ,σ (ξ)

= =

Hp ((1 − δ)η, ζ), Gp,δ (ξ − y)gσ(ξ) (y)dy d R Gp,δ (ξ − σ(ξ)t)g(t)dt.

(81)

Rd

In analogy to (72) let the function σ(ξ) = σr,δ (ξ)  if  0 reς−,δ (ξ,r) if σ(ξ) = σr,δ (ξ) =  ς+,δ (ξ,r) re if

be given by + − ∪ Br,δ ξ∈ Br,δ − ξ ∈ Br,δ , + ξ ∈ Br,δ

(82)

± where 0 < r < min{µ− , µ+ }, Br,δ = {ξ : |ξ − e±,δ | < r}, e±,δ = (1 − δ)−1 e± and

ς±,δ (ξ, r) =

−1 . 1 − r−2 |ξ − e±,δ |2

Similar to the proof of Lemma 3 one can show that the derivatives of Gp,δ,σ (ξ) satisfy the bounds ∂Gp,δ,σ (83) ∂ξk ≤ c, 2 ∂ Gp,δ,σ −1 (84) ∂ξk ∂ξl ≤ c(1 + r ). The constant c in (83), (84) can be chosen to be independent on p, M , r, ξ, k, l, and δ ∈ (0, 1/2) as well. Lemma 5 There exists a finite positive constant C = C(µ+ , µ− ), such that the bound (85) Gp,δ,σr (ξ) ≤ Hp (ξ), ξ ∈ Rd , holds true for all r ≤ min{µ− , µ+ , Cδ}, 0 < δ < 1/2 and all p ≥ M > 0.

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Proof. Let r ≤ min{µ− , µ+ }. Since Gp,δ,σr (ξ) = Hp ((1 − δ)η, ζ)

+ − if ξ ∈ Br,δ ∪ Br,δ ,

the bound (85) for that case is an obvious consequence of the local monotonicity of Hp (η, ζ) in η for ﬁxed p, M and ζ. On the other hand, by (69) it holds |∂Hp /∂η| ≤ 2 and |∂Gp,δ /∂η| ≤ 2. Hence, if r ≤ r(δ) = (Hp (e±,δ ) − Gp (e±,δ ))/4 we have

min Hp (ξ ) ≥ max Gp,δ (ξ ).

± ξ ∈Br,δ

± ξ ∈Br,δ

(86)

± and t ∈ Rd , |t| ≤ 1 it holds For any ξ ∈ Br,δ

|(ξ − tσr,δ (ξ)) − e±,δ | ≤ |ξ − e±,δ | + σr,δ (ξ) ≤ r. 2 −1

The later inequality follows from the fact that x+ e−(1−x ) ≤ 1 for all 0 ≤ x < 1. ± on the Thus, the argument ξ = ξ − tσr,δ (ξ) of Gp,δ in (81) satisﬁes ξ ∈ Br,δ support of g, and we conclude (85) from (86) and the normalization of g. It remains to estimate r(δ) from below. Note that M p−1 ≤ 1 and 0 < δ < 1/2. Then 4r(δ)

= ≥ ≥

Hp (e±,δ ) − Hp (e± ) ∂ δµ± min Hp (η, 0, 0) ∓ 1 − δ µ± ≤η≤µ± (1−δ)−1 ∂η 1 δ ≥ C(µ+ , µ− )δ. 1 − δ µ2 (1 + (1 − δ)−1 )2 + 1 ∓

This completes the proof.

7.3

The estimate from above

We put now Qp,δ,σ (ξ, y)

= Gp,δ,σ (ξ) − Vp (y),

Qp,δ,σ (i∇, y) = Gp,δ,σ (i∇) − Vp (y). From (85) it follows that for σ = σr,δ tr(Qp (i∇, y))− ≤ tr(Qp,δ,σr,δ (i∇, y))− if r ≤ min{µ− , µ+ , C(µ+ , µ− )δ}. For the eigenvalue sum on the right-hand side we can apply (66) in Lemma 1 and we conclude

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Lemma 6 Assume that 0 < r ≤ min{µ− , µ+ , C(µ+ , µ− )δ} and 0 < δ < 1/2. Then the inequality κ S(p) ≤ (Qp,δ,σr,δ (ξ, y))− dγ + tr χ−t (Qp,δ,σr,δ (i∇, y))dt (87) 0

holds true for some κ ≤ c ϑ(V )r−1 p−3 , where the constant c in the estimate for κ can be chosen to be independent on V , p, M , r and δ.

8 The Proof of Theorem 5 We are now in the position to complete the proof of formula (59). In the beginning we shall assume that V has uniformly bounded second derivatives and that V ∈ Lθ (Rd ) ∩ Ld+1 (Rd ) for some θ < d+1 2 and d ≥ 3.

8.1

The estimate from above

First note that Gp,δ is convex and consequently Gp,δ (ξ) ≤ Gp,δ,σr (ξ) for all ξ ∈ Rd . Thus, (Qp,δ,σr,δ (ξ, y))− dγ ≤ (Gp,δ (ξ) − Vp (x))− dγ 1 Σp (V ). ≤ (1 − δ)−1 (Qp (ξ, x))− dγ = 1−δ Simultaneously we have tr χ−t (Qp,δ,σr,δ (i∇, y)) ≤ tr χ−t (Gp,δ (i∇) − Vp (y)) ≤ Np ((V ((1 − δ)x1 , x2 , x3 ) − tp)+ ) for all t ≥ 0. Hence, relations (87), (23) and (24) imply that c8.2 p

V+ 4L4 (1 − δ)Sp (V ) ≤ Σp (V ) + c8.1 p(1 + ln ) min V+2 , κV+ dx + M p in the dimension d = 3, or

d+1 d−1 c8.4 d+1

V+ Ld+1 min V+ 2 , κV+ 2 dx + p if d ≥ 4, hold true for all p ≥ M with κ = c ϑ(V )r−1 p−3 . Since V+∗ (t) ≤ −1 c8.5 V+ Lθ t−θ we ﬁnd that ∞

d+1 d−1 d+1 d−1 2 2 dx = min (V+∗ ) 2 , κ(V+∗ ) 2 dt min V+ , κV+ (1 − δ)Sp (V ) ≤ Σp (V ) + c8.3 p

d−1 2

0

≤

θ

c8.6 V+ Lθ κ

d+1 2 −θ

,

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and consequently β p 4 ϑ (V ) V+ θLθ p 1 + ln M

V+ L4 Σp (V ) + c8.7 + c Sp (V ) ≤ 8.8 1−δ (1 − δ)p3β rβ (1 − δ)p as 0 < M ≤ p with β =

2−θ 2

(88)

> 0 if d = 3 and d−1

Sp (V ) ≤

p 2 ϑβ (V ) V+ θLθ

V+ d+1 Σp (V ) Ld+1 + c8.9 + c 8.10 1−δ (1 − δ)p3β rβ (1 − δ)p

(89)

θ as 0 < M ≤ p with β = d+1 4 − 2 > 0 if d ≥ 4. − Pick now δ(p) = p and r = r(p) = min{µ+ , µ− , C(µ+ , µ− )δ} with d−1 0 < < 3. Since Σp (V ) is of order p 2 for large p, we claim

lim sup p−1 Sp (V ) ≤ lim p−1 Σp (V ). p→∞

p→∞

8.2

The estimate from below

On the other hand, from (80) and from the identity Hp (ξ) = Hp,σr (ξ) for ξ ∈ Rd \(Br+ ∪ Br− ) it follows that (Qp,σr + κ)− dγ Sp (V ) ≥ ≥ Σp (V − pκ) − (Qp + κ)− dγ. y∈Br+ ∪Br−

Next note that at least

µ± −r 2r 2

disjoint balls of radius r can be placed into the

domains [r − µ− , 0] × (−r, r) and [0, µ+ − r] × (−r, r)2 , respectively. Because of Hp (η, ζ) ≥ Hp (η , ζ) for all |η | ≤ |η| we can conclude that ! µ± − r (Qp + κ)− dγ ≤ (Qp + κ)− dγ 2r y∈Br± ξ∈[r−µ− ,1−µ+ ]×(−r,r)2 ≤ Σp (V − pκ) and





Sp (V ) ≥ 1 −

1 µ− −r 2r

−

1 µ+ −r 2r

 Σp (V − pκ).

(90)

Put now r = r(p) = p−α with 0 < α < 1. Then r → 0 and simultaneously pκ = c8.11 ϑ1/2 (V )r−1/2 p−1/2 → 0 as p → ∞. Thus, it holds Σp (V − pκ) ≥ Σp (V − δ)

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for arbitrary δ > 0 if p is large enough. Because of the given class of potentials this means lim inf Sp (V ) ≥ p→∞

≥ Since V ∈ L

8.3

d+1 2

lim Σp (V − δ)

p→∞

ωd (d + 1)2

3d−1 2

πd

Rd

d+1

(V (x) − δ)+2 dx.

we can pass to the limit δ → 0.

The closure of the asymptotical formula

If d ≥ 4, we ﬁnally apply inequality (49) in a standard manner to close asymptotics d+1 (59) to all potentials V+ ∈ L 2 ∩ Ld+1 . However, for d = 3 the appropriate Liebp , which prevents Thirring inequality (48) contains the logarithmic factor 1 + ln M us from carrying out the same procedure in that case.

9 The Proof of Theorem 6 For the proof of Theorem 6 we follow the main strategy of [ELSS] and apply the bounds (88), (89)and (90) of the previous section in a more subtle way. For the shortness of notation we shall write Yp = U (y; p)χ0 (Qp (i∇, y)) = pUp χ0 (Qp (i∇, y)), where in agreement with our previous notation Up (y) = p−1 U (yp−1 ).

9.1

The estimate from above

Let {ψp,n } be an o.n. system of eigenfunctions corresponding to the negative part of Qp (i∇, y). Then for any ∈ (0, 1) it holds tr p−1 Yp = Up (x)|ψp,n (x)|2 dx n

≤

1 (tr (Qp (i∇, y) − Up )− − tr (Qp (i∇, y))− ) .

Here we make use of the variational property tr (Qp (i∇, y) − Up )− ≥ tr D(Up − Qp (i∇, y)) for any operator 0 ≤ D ≤ 1. Put V = V + U . Then (88)–(90) imply that tr p−1 Yp ≤

1 (Σp (V ) − Σp (V − pκ) + R(p, , µ± , δ, V, U )) ,

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for all 0 < M ≤ p and ∈ (0, 1), where d+1

R(p, , µ± , δ, V, U ) =

V Ld+1 δ Σp (V ) + c9.1 1−δ (1 − δ)p +c9.2

p

d−1 2

zd (p)ϑβ (V ) V θLθ + c9.3 rΣp (V − pκ) (1 − δ)p3β rβ

p θ with β = d+1 4 − 2 , z3 (p) = 1 + ln M and zd (p) = 1 for d ≥ 4. Pick now δ(p) = p−α and r = r(p) = min{µ− , µ+ , C(µ− , µ+ )δ(p)} with 0 < α < 1. Fix ∈ (0, 1). Then r → 0, pκ → 0 and the limits d+1 d−1 ωd V 2 dx, p− 2 Σp (V ) → 3d−1 (d + 1)2 2 π d d+1 ωd − d−1 2 p Σp (V ) → V 2 dx, 3d−1 d (d + 1)2 2 π d−1 d+1 ωd − 2 p Σp (V − pκ) → V 2 dx 3d−1 d (d + 1)2 2 π

hold true as p → ∞. From this we conclude that d+1 d+1 −1 ωd 2 lim sup p− 2 tr Y p ≤ dx − V V 3d−1 p→∞ (d + 1)2 2 π d Note that for non-negative U, V ∈ L −1 |V

d+1 2

−V

d+1 2

d+1 2

d+1 2

dx .

and all ∈ (0, 1) we have

|≤

d−1 d+1 U (V + U ) 2 , 2

where the function on the right-hand side is integrable. Hence, by Lebesgues majorization theorem we can pass to the limit → +0 and ﬁnd d+1 d−1 ωd lim sup p− 2 Yp ≤ 3d+1 U V 2 dx. d p→∞ 2 2 π

9.2

The estimate from below

Reversely, it holds tr p−1 Yp ≥

1 (tr (Qp (i∇, y))− − tr (Qp (i∇, y) + Up )− ) .

Let V− = V − U with ∈ (0, 1). Then tr p−1 Yp ≥

1 ˜ , µ± , δ, V, U ) , Σp (V ) − Σp (V− − pκ) + R(p,

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for all 0 < M ≤ p and ∈ (0, 1), where

V d+1 δ Ld+1 Σp (V ) + c9.4 1−δ (1 − δ)p

˜ , µ± , δ, V, U ) = R(p,

+c9.5

p

d−1 2

θ

zd (p)ϑβ (V ) V Lθ + c9.6 rΣp (V− − pκ) (1 − δ)p3β rβ

p θ with β = d+1 4 − 2 , z3 (p) = 1 + ln M and zd (p) = 1 for d ≥ 4. Passing to p → ∞ as above we obtain −1 d+1 d+1 ω d − d+1 p 2 lim inf p 2 tr Y ≥ V 2 dx − (V− )+ dx 3d−1 p→∞ (d + 1)2 2 π d

and for → +0 by a majorized convergence argument this turns into d+1 d−1 ωd lim inf p− 2 Yp ≥ 3d+1 U V 2 dx. p→∞ 2 2 πd

10 Appendix I: Phase Space Estimates for the Symbol Qp (ξ, y) = Hp (ξ) − Vp (y) 10.1

Preliminaries

Let V be a real function on Rd . Set Vp (y) = p−1 V (p−1 y) and Qp (ξ, y) = Hp (ξ) − Vp (y), where

1 + M 2 p−2

Hp (ξ) = T+ (ξ) + T− (ξ) − for T± (ξ) =

|(η ∓ µ± )2 + |ζ|2 + µ2± M 2 p−2 ,

with ξ ∈ Rd , ξ = (η, ζ) for ξ1 = η ∈ R and (ξ2 , . . . , ξd ) = ζ ∈ Rd−1 , M = m+ +m− , µ± = m± M −1 > 0, p > 0. Below we shall study properties of the phase space averages Σp = Σp (V ) = (2π)−d (91) (Qp (ξ, y))− dξdy, Ξp = Ξp (V ) = (2π)−d dξdy. (92) Qp <0

Set −d

Λp (y; V ) = (2π)

dξ. Qp<0

(93)

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329

Lemma 7 Assume that τ = M p−1 ≤ 1. Then for any y ∈ Rd it holds Λp (y) =

d d d ωd W 2 (W + υ) (W + 2υ) 2 W 2 + 2W υ + τ 2 (1 − 4˜ µ2 ) 2 (4π)d (W 2 + 2W υ + τ 2 )

where W = W (y) = (Vp (y))+ and υ =

d+1 2

,

(94)

√ 1 + τ 2.

Proof. Fix some point y ∈ Rd . Since Hp (ξ) ≥ 0 we have Qp (ξ, y) ≥ 0 if Vp (y) ≤ 0, ˜ = (µ− − what settles the statement in that case. Assume now Vp (y) ≥ 0. Put µ µ+ )/2 and η˜ = η + µ ˜. Then Qp (ξ, y) < 0 is equivalent to & 2 1 1 ˜2 + η−µ ˜τ 2 )2 2 |ζ|2 + η˜2 + + µ τ 2 − (˜ 4 4 1 1 2 2 2 2 ˜ + A − 2 |ζ| + η˜ + + µ τ2 , 4 4

< where A = Vp +

(95)

√ 1 + τ 2 . Thus, in particular, the condition 2

2

|ζ| + η˜ < B,

A2 1 1 2 − µ ˜ + B= τ2 − 2 4 4

(96)

has to be satisﬁed. The bound (95) transforms into A2 − 1 |ζ| + A2 2

η˜ +

µ ˜τ 2 A2 − 1

2
µ ˜2 τ 4 A2 + 2 , 4 A −1

(97)

√ subject to the additional condition (96). For Vp (y) ≥ 0 we have A ≥ 1 + τ 2 and inequality (97) describes an ellipsoid with symmetry semi-axes of the length l1 l2 = · · · = ld

' µ ˜2 τ 4 A A2 + 2 = √ B− 2 4 A −1 A −1 ' 2 2 4 µ ˜ τ A + 2 . = B− 4 A −1

It is not diﬃcult to see that lj2 ≤ B for j = 2, . . . , d and τ ≤ 1, while l1 ≤ 2

˜ B 1/2 − Aµτ 2 −1 . Thus the ellipsoid given by (97) is a subset of the sphere (96), and the volume of all admissible ξ is given by ωd l1 . . . ld , what by

B−

µ ˜2 τ 4 (A2 − 1 − τ 2 )(A2 − 1 − 4˜ µ2 τ 2 ) A2 + 2 = 4 A −1 4(A2 − 1)

implies the second statement of the Lemma 7.

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Let us now assume that τ = M p−1 ≤ 1 and m± > 0. Then µ ˜2 < 1/4 and by (94) the quantity Λp permits the following two-sided estimate  d/2 −1  on Ω1 = {y|Vp (y) ≤ τ 2 }  τ (Vp (y))+ d−1 Λp (y; V ) (98) on Ω2 = {y|τ 2 ≤ Vp (y) ≤ 1} , (Vp (y))+2   d (Vp (y))+ on Ω3 = {y|Vp (y) ≥ 1} or equivalently,

d−1 d + (Vp (y))d+ , Λp (y; V ) min τ −1 (Vp (y))+2 , (Vp (y))+2

(99)

which for ﬁxed µ ˜ is uniform for all p and M satisfying τ ≤ 1. Hence, V+ ∈ Ld/2 (Rd ) ∩ Ld (Rd ) is suﬃcient and necessary for (100) Ξp (V ) = Λp (y; V )dy = pd Λp (px; V )dx to be ﬁnite.

10.2

Potentials V+ ∈ L

d−1 2

(Rd ) ∩ Ld (Rd ) d−1

For this class of potentials by (98) the function p 2 Λp (p·) has an integrable majorant, and by Lebesgues limit theorem it holds d+1 d−1 d−1 ωd (101) p 2 Λp (py)dy = 3d+1 (V+ (y)) 2 dy. lim p− 2 Ξp = lim p→∞ p→∞ d 2 2 π

10.3

Potentials V+ ∈ L

d+1 2

(Rd ) ∩ Ld+1 (Rd )

We ﬁnd that the integrand on the right-hand side of ∞ ∞ 1−d 1−d d−1 p 2 Σp (V ) = p 2 Ξp (V − sp)ds = p 2 Λp (py; V − t)d3 ydt 0

0

Rd

is for ﬁxed µ ˜ bounded by a uniform multiple of d−1

max{(V (y) − t)+2 , (V (y) − t)d+ }, d+1

which is integrable on [0, ∞) × Rd for V+ ∈ L 2 (Rd ) ∩ Ld+1 (Rd ). Thus, ∞ d−1 d−1 lim p− 2 Σp (V ) = dt lim p 2 Λp (py; V − t) dy p→∞ 0 Rd p→∞ ∞ d−1 ωd = dt (V (y) − t)+2 dy 1+3d d 2 2 πd 0 R d+1 ωd = V+ 2 (y)dy. 3d−1 d (d + 1)2 2 π Rd

(102)

Vol. 4, 2003

10.4

On the Discrete Spectrum of a Two-Body Pair Operator

Potentials V+ ∈ Lθw (Rd ) ∩ Ld (Rd ) with

d−1 2

<θ<

331 d 2

d−1

d

For potentials V where V+ is “strictly between” L 2 (Rd ) ∩ Ld (Rd ) and L 2 (Rd ) ∩ Ld (Rd ) the phase space volume shows a diﬀerent behaviour in p. Let us study the model potential V (y) = min{1, v|y|−d/θ },

(103)

d θ d where d−1 2 < θ < 2 . Then V = V+ ∈ Lw (R ) and V θ,w = c(θ, d)v. The preliminary estimate (98) shows that

Ξp

p

1−d 2

θ

θ

θ

p1− d v d ≤|y|≤pv d

+v

d−1 2

p

(d−1)(d−θ) 2θ

θ pv d d2

+v 2 p1− 2 + 2θ M −1 d

d

dy

θ ≤|y|≤p1+ d

θ vd

2θ M− d

|y|−

d(d−1) 2θ

dy

d2

θ |y|≥p1+ d

θ vd

2θ M− d

|y|− 2θ dy

pθ+1 v θ M d−1−2θ (1 + o(1)) as p → ∞. After one has established the order of Ξp in p, the same estimate now shows that θ Λp (y)dy, 0 < c < , Ξp = (1 + o(1)) d |y|>p1+c as p → ∞. Hence, for

d−1 2

<θ<

d 2

it holds

Ξp

= (1 + o(1)) |y|>p1+c

= (1 + o(1))

ωd (4π)d

Λp (y)dy

|y|>p1+c

2 2 (2W + τ 2 µ ˆ) 2 d

d

W2

d

(2W + τ 2 )

d+1 2

dy,

d d where µ ˆ = 1 − 4˜ µ2 ∈ (0, 1] and W (y) = p−1 min 1, vp θ |y|− θ as p → ∞. This implies

Ξp = (1 + o(1))

d 2

d+1

2 ωd p (4π)d M

d2 vp d2 −d θ + µ 2 |x| ˆ 2 2 M v d |x|− 2θ dx d+1 p c 2 |x|>p vp −d θ 2M |x| + 1 2

or Ξp = (1 + o(1))

2θ θωd2 θ d−1−2θ v M L(d, θ, µ ˆ)pθ+1 (4π)d

as p → ∞

(104)

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Ann. Henri Poincar´e

with

∞

L(d, θ, µ ˆ) =

(t + µ ˆ) 2 t 2 −θ−1 dt d

d

d−1

(t + 1) 2 d d d−1 d+1 1 d−θ − θ, θ − − θ, ; ,1 − µ ˆ , µ ˆ B 2 F1 2 2 2 2 2 0

=

where 2 F1 is Gauss’ hypergeometric function ([PBM] 2.2.6.24 p. 303).This result d can be generalized to all potentials V with V+ ∈ Lθw ∩ Ld for d−1 2 < θ < 2 and ∆θ (V+ ) = δθ (V+ ) = c(θ, d)v.

10.5

Potentials V+ ∈ Lθw (Rd ) ∩ Ld (Rd ) with

d+1 2

<θ<

d+2 2

A similar calculation can be carried out for the average Sp (V ) if the potential d+1 d 2 ∩ Ld and (103) satisﬁes d+1 2 < θ < 2 + 1 and is therefore “strictly” between L d L 2 +1 ∩ Ld . First, from (98) one concludes in general that ∞ Σp = Λp (y; V − sp)dyds (105) 0 Rd Θp (y; V )dy, (106) Rd

for suﬃcient large p, where d

+ τ d+1 χΩ2 ∪Ω3 (y) + τ −1 (Vp (y))+2 Θp (y; V ) = (Vp (y))d+1 +

+1

χΩ1 (y)

with χ being the characteristic functions of (unions of) the respective sets Ωj d deﬁned in (98). For the potential V (y) = min{1, v|y|− θ } at hand this gives the preliminary estimate Σp pθ−1 M d+1−2θ v θ

as p → ∞.

Moreover, the integration in (106) and therefore in (105) can be reduced to |y| > p1+c , 0 < c < θd , without changing the asymptotical behaviour of the integrals. d d −d 2 Hence, if d+1 ωd as p → ∞ and a = 1 + cdθ−1 , 2 < θ < 2 + 1, φ = (1 + o(1))2 (4π) we have Σp

=

∞

d

(Vp − s)+2 (2(Vp − s)+ + τ 2 µ ˆ ) 2 dyds

φ

=

d d

∞

∞

φdω p

0 d θ d−θ

∞

d

d

(2( vp r− θ − s)+ + τ 2 ) d

φθω v p

0

d

d+1 2

( vp r− θ − s)+2 (2( vp r− θ − s)+ + τ 2 µ ˆ) 2 rd−1 drds

pc

=

(2(Vp − s)+ + τ 2 )

|y|≥p1+c

0

d

0

vp−a

d

d

d+1 2

(t − s)+2 (2(t − s)+ + τ 2 µ ˆ ) 2 t−θ−1 dtds d

(2(t − s)+ + τ 2 )

d+1 2

.

Vol. 4, 2003

On the Discrete Spectrum of a Two-Body Pair Operator

333

The later integral can be simpliﬁed as follows Σp

=

d θ d−θ

vp−a

φθω v p

dtt

−θ−1

=

x 2 (2x + τ 2 µ ˆ) 2 d

t

d

dx d+1 (2x + τ 2 ) 2 vp−a 2tτ −2 d d u 2 (u + µ ˆ) 2 − d+3 d θ d+1 d−θ −θ−1 2 θω v τ p dtt φ2 d+1 du (u + 1) 2 0 0 0

0

=

φ2θ−

d+3 2

× For a < 2 and θ >

d+1 2

θω d v θ τ d+1−2θ pd−θ × 2vM −2 p2−a

dww−θ−1

w

(u + 1)

0

0

d

d

u 2 (u + µ ˆ) 2 d+1 2

du.

we ﬁnally claim 3

Σp = (1 + o(1))

2θ− 2 θωd2 θ d+1−2θ θ−1 v M p K(d, θ, µ ˆ ), (4π)d

(107)

where K(d, θ, µ ˆ) denotes the ﬁnite positive constant

∞

K(d, θ, µ ˆ) =

dww

−θ−1

0

0

w

d

d

u 2 (u + µ ˆ) 2 (u + 1)

d+1 2

In fact, this asymptotics holds true for all V+ ∈ Lθw ∩ Ld , ∆θ (V+ ) = δθ (V+ ) = c(θ, d)v.

du.

d+1 2

<θ<

d 2

+ 1, with

11 Appendix II: An Estimate Np (V ) ≤ c(V )p2 in the Dimension d = 3 11.1

Statement of the result

In this appendix we show, that for certain short-range potentials with some repulsive tail at inﬁnity the counting function Np (V ) in the dimension d = 3 is bounded by a multiple of p2 . This complements the estimate (13). As above we concentrate on the case of positive masses m± > 0. Theorem 7 Assume that d = 3, m± > 0 and that the bounded potential V satisfies the condition (108) V (x) ≤ −a(1 + |x| − b)−γ , x ∈ R3 , |x| ≥ b, for appropriate positive finite constants a, b and γ. Then Np (V ) ≤ C(b + 1)3 p2 ,

p ≥ M,

where C = C(a, γ, V L∞ ) does not depend on p and b.

(109)

334

S. Vugalter, T. Weidl

11.2

Ann. Henri Poincar´e

A localization estimate in spatial coordinates

Consider the operator T =

√ −∆ + 1

on L2 (R3 ).

Let (·, ·) and · be the scalar product and the norm in L2 (R3 ). For positive b and γ set ςγ,b (x) = (1 + |x| − b)−γ/2 , x ∈ R3 . The proof of Theorem 7 is based on the following improved localization estimate: Lemma 8 For any given positive number b one can find spherically symmetric functions χ1 , χ2 ∈ C 2 (R3 ), which are monotone w.r.t. the radial variable and satisfy χ1 (x) = 1

if

|x| ≤ b,

χ1 (x) = 0

if

|x| ≥ b + 1,

χ21 + χ22 = 1,

such that for any > 0 and γ > 0 the estimate 2 2 2 (T u, u) − (T uχ , uχ ) j j ≤ c uχ1 + uχ2 ςγ,b

j=1

(110)

(111)

holds true for all u ∈ C0∞ (R3 ) with some appropriate finite constant c = c(γ, ). Proof. For given b > 0 we can obviously chose spherically symmetric cut-oﬀ functions χ1 , χ2 ∈ C 2 (R3 ), which are monotone in the radial variable and satisfy (110) as well as χ1 (x)χ2 (x) > 0 for b < |x| < b + 1. (112) According to formula (3.8) in [LY] the localization error of the operator T is given as follows 2 (T u, u) − (T uχj , uχj ) = (Lu, u), (113) j=1

where L is an integral operator with the kernel L(x, y) =

K2 (|x − y|)

)2

j=1 (χj (x) (2π)2 |x − y|2

− χj (y))2

.

Here K2 stands for the modiﬁed Bessel function and satisﬁes the estimate |K2 (|x − y|)| ≤ α|x − y|−2 e−κ|x−y|

(114)

for appropriate α, κ > 0. We shall now estimate the quadratic form on the right-hand side of (113). Because of symmetry it suﬃces to estimate the respective integrals over the region

Vol. 4, 2003

On the Discrete Spectrum of a Two-Body Pair Operator

335

|x| ≤ |y| only. Let δ ∈ (0, 1/2) be a positive number, which will be speciﬁed later. Put bδ = b + 1 − δ and deﬁne O1

=

{(x, y)| |x| ≤ |y| ≤ bδ },

O2 O3

= =

{(x, y)| |x| ≤ b2δ , |y| ≥ bδ }, {(x, y)| |x| ≤ |y|, |x| ≥ b2δ , (x, y) ∈ O1 ∪ O2 }.

Then (Lu, u)L2 (R3 ) = 2Re(I1 + I2 + I3 ),

(115)

where Ik =

L(x, y)u(y)¯ u(x)dxdy,

k = 1, 2, 3.

Ok

To estimate I1 we notice that for all x, y ∈ R3 .

|χj (x) − χj (y)| ≤ min {1, c11.1 |x − y|}

(116)

From (114) and (116) we conclude |L(x, y)| ≤ c11.2 |x − y|−2 min{1, |x − y|−2 } for all (x, y) ∈ O1 . Hence, it holds |I|1

≤ 2

−1

≤ c11.3

(|u(x)|2 + |u(y)|2 )|L(x, y)|dxdy |u(x)|2 dx |x − y|−2 min{1, |x − y|−2 }dy.

(x,y)∈O1

|x|≤bδ

This gives

R3

|I1 | ≤ c11.4

2

|x|≤bδ

|u(x)|2 dx ≤ c11.5 (δ) uχ1 L2 (R3 ) .

(117)

In the last step we used that χ1 (x) ≥ c11.6 (δ) > 0 for all |x| ≤ bδ , what on its turn follows from (112) and the radial monotonicity of χ1 . We study now the integral I2 and observe that |L(x, y)| ≤ c11.7 δ −2 e−κ|x−y| In view of

for (x, y) ∈ O2 .

2 2 |u(y)u(x)| ≤ 4−1 −1 1 |u(x)| + 1 |u(y)| ,

(118)

1 > 0,

we ﬁnd from (118) that for any given γ > 0 and 1 > 0 the bound −2 −1 2 |I2 | ≤ c11.8 δ 1 dx|u(x)| e−κ|x−y|dy 1 +c11.9 2 δ

|x|≤b2δ

|y|≥bδ

dy|u(y)|2 κ e 2 (|y|−b2δ )

|y|≥bδ

|x−y|≥δ

e− 2 |x−y|dx κ

(119)

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S. Vugalter, T. Weidl

Ann. Henri Poincar´e

holds true. By (112) and by the radial monotonicity of χ1 and χ2 we have χ1 (x) ≥ c11.10 (δ) > 0

for

|x| ≤ b2δ ,

χ2 (y) ≥ c11.11 (δ) > 0

for

|y| ≥ bδ > b.

Moreover, it holds e− 4 (|y|−b2δ ) ≤ c11.12 (γ, δ)ςγ,b (y), κ

|y| ≥ bδ .

Hence, the inequality (119) implies 2

2

|I2 | ≤ c11.13 (δ, 1 ) uχ1 L2 (R3 ) + c11.14 (γ, δ)1 uχ2 ςγ,b L2 (R3 ) .

(120)

Estimating I3 we recall that χ1 (x) ≡ 0

and χ2 (x) ≡ 1 for all |x| ≥ b + 1.

(121)

Since χ1 , χ2 ∈ C 2 (R2 ), for any given 2 > 0 we can ﬁnd an appropriate δ = δ(2 ) ∈ (0, 1/2) such that |∇χ1 (x)|2 + |∇χ2 (x)|2 ≤ 2 ,

b2δ ≤ |x| ≤ b + 1.

(122)

With this value of δ the relations (121) and (122) imply 2

(χj (x) − χj (y))2 ≤ 2 min 4δ 2 , |x − y|2 ,

b2δ ≤ |x| ≤ |y|.

(123)

j=1

Moreover, from (118), (121) and (123) we conclude that

|L(x, y)| ≤ 2 c11.15 |x − y|−2 min 4δ 2 |x − y|−2 , 1 min e−κ(|y|−b−1) , 1 for b2δ ≤ |x| ≤ |y| and L(x, y) = 0 for b + 1 ≤ |x| ≤ |y| . Therefore it holds |I3 | ≤ 2−1 (|u(x)|2 + |u(y)|2 )|L(x, y)|dxdy (x,y)∈O3 |u(x)|2 dx |x − y|−2 min{4δ 2 |x − y|−2 , 1}dy ≤ 2 c11.16 R3

|x|≤b+1

+2 c11.17

|y|≥b2δ

2

|u(y)| dy eκ(|y|−b−1)

R3

min{4δ 2 |x − y|−2 , 1} dx. |x − y|2

Since e− 2 (|y|−b−1) ≤ c11.18 (γ, δ)ςγ,b (y) for |y| ≥ b2δ and δ ∈ (0, 1/2), we conclude that 2 2 (124) |I3 | ≤ 2 c11.19 (γ, δ)( uχ1 L2 (R3 ) + uχ2 ςγ,b L2 (R3 ) ). κ

We proceed now as follows. For given > 0 chose 2 > 0 such that the total constant in front of the bracket in (124) for given b and γ does not exceed /4.

Vol. 4, 2003

On the Discrete Spectrum of a Two-Body Pair Operator

337

Fix the corresponding δ(2 ) > 0 for (122) and subsequently (124) to be satisﬁed. 2 Finally, ﬁx 1 > 0 such that the total constant in front of the term uχ2 ςγ,b L2 (R3 ) in (120) for given b, γ and δ(2 ) does not exceed /4. Then (113) together with (116), as well as (117), (120) and (124) yield (111). Remark 8 Let trel = trel (P ) be the regularized kinetic part of the operator (2) on L2 (R3 ), that is (125) trel = |µ+ P − i∇|2 + m2+ + |µ− P + i∇|2 + m2− − p2 + M 2 , where M > 0, µ± = m± M −1 > 0, and P ∈ R3 , p = |P |. As an immediate consequence of (110) in Lemma 8 we find that for arbitrary positive and γ it holds 2 2 2 (trel u, u) − (t uχ , uχ ) (126) rel j j ≤ c(γ, , µj , M ) uχ1 + uχ2 ςγ,b . j=1 The constant c(γ, , µj , M ) can be chosen to be independent on P and b.

11.3

A local estimate in momentum space

Let trel (ξ, P ) =

|µ+ P − ξ|2 + m2+ + |µ− P + ξ|2 + m2+ − p2 + M 2

be the symbol of the operator (125) where M > 0, µ± = m± M −1 > 0 and P, ξ ∈ R3 . Put ξ = (η, ζ) with η ∈ R and ζ ∈ R2 . We recall that P = (p, 0, 0) and µ+ + µ− = 1. Lemma 9 Assume that p ≥ ν ≥ M and that ξ = (η, ζ) satisfies

Then

ξ ∈ W (ν, p) = {ξ|(|η| ≥ 3p)} ∪ {ξ|(|ζ|2 ≥ νp)}.

(127)

trel (ξ, P ) ≥ 2−1 3−1/2 ν.

(128)

Proof. Assume ﬁrst |η| ≥ 3p ≥ 3ν ≥ 3M . Then √ √ trel (ξ, P ) ≥ 4p2 + M 2 − p2 + M 2 ≥ 3( 5 + 2)−1 ν.

(129)

2

If instead |ζ| ≥ νp from ν ≥ M it follows that trel (ξ, P ) ≥ p2 + |ζ|2 + M 2 − p2 + M 2 ≥ ≥

2−1 |ζ|2 (p2 + |ζ|2 + M 2 )−1/2 2−1 ν(1 + νp−1 + ν 2 p−2 )−1/2 .

Since p ≥ ν we conclude trel (ξ, P ) ≥ 2−1 3−1/2 ν. Together with (129) this completes the proof.

338

11.4

S. Vugalter, T. Weidl

Ann. Henri Poincar´e

The proof of Theorem 7

Let qrel (P ) = trel (P ) − V (y),

P = (p, 0, 0),

be the operator (3) for d = 3. Obviously the total multiplicity of the negative eigenvalues of this operator coincides with Np (V ). To verify (109) it suﬃces to construct a subspace G in L2 (R3 ) of ﬁnite dimension dim G ≤ Cb3 p2 such that (qrel (P )u, u)L2 (R3 ) ≥ 0

for all u ∈ G⊥ 0,

(130)

⊥ 2 3 where G⊥ 0 is a qrel (P )-form dense subset of G = L (R ) G. For given b construct the cut-oﬀ functions χ1 , χ2 from Lemma 8. Set n = (n1 , n2 , n3 ) for nj ∈ N+ and x = (x1 , x2 , x3 ) with xj ∈ R, j = 1, 2, 3. Let b = b+1. We deﬁne − 3 *3 x for |xj | ≤ b , j = 1, 2, 3, b 2 j=1 sin πnj 12 + bj un (x) = 0 otherwise.

˜ = G(τ ˜ 1 , τ⊥ ) be the linear span of all un where Let G n1 ≤ τ1 b p,

n2,3 ≤ τ⊥ b p1/2 ,

and the positive real numbers τ1 , τ⊥ will be speciﬁed below. We put ˜ G = G(τ1 , τ⊥ ) = {u|u = u ˜ χ1 , u ˜ ∈ G}. Obviously we have

2 3 2 ˜ ≤ τ1 τ⊥ b p . dim G = dim G

To verify (130) we ﬁrst notice, that from the boundedness of V , (108) and (126) it follows that 2

(qrel (P )u, u)L2 (R3 ) ≥ (trel (P )uχ1 , uχ1 )L2 (R3 ) − c˜ uχ1 L2 (R3 )

(131)

for all u ∈ C0∞ (R3 ). For ﬁxed χ1 the constant c˜ = c˜(a, γ, V L∞ ) does not depend on p, b or u. Let W (ν, p) be the set deﬁned in (127) of Lemma 9 for the choice ν = 23 31/2 c˜. Below we shall show, that for appropriate constants τ1 = τ1 (˜ c) and τ⊥ = τ⊥ (˜ c), which do not depend on p, the bound χ1 L2 (R3 ) ,

uχ +1 W (ν,p) ≥ 2−1 u+

u⊥G(τ1 , τ⊥ ),

holds true. From (132) and (128) we conclude that (trel (P )uχ1 , uχ1 )L2 (R3 )

≥ (trel (ξ, P )+ uχ1 (ξ), uχ +1 (ξ))L2 (W (ν,p)) 2

≥ 4˜ c + uχ1 L2 (W (ν,p)) 2

≥ c˜ uχ1 L2 (R3 ) ,

(132)

Vol. 4, 2003

On the Discrete Spectrum of a Two-Body Pair Operator

339

where u⊥G(τ1 , τ⊥ ) and u ∈ C0∞ (R3 ). Together with (130) and (131) the later bound settles the proof. In the remaining part of this section we establish (132). Consider some ˜ and consequently uχ1 = )3 σj , where σj = function u⊥G. Then uχ1 ⊥G j=1 ) n∈Υj cn un and Υ1

=

{n|n1 ≥ τ1 b p},

Υ2 Υ3

= =

{n|(n1 < τ1 b p)} ∩ {n|(n2 ≥ τ⊥ b p1/2 )}, {n|(n1 < τ1 b p)} ∩ {n|(n2 < τ⊥ b p1/2 )} ∩ {n|(n3 ≥ τ⊥ b p1/2 )}.

˜ = R3 \W (ν, p). Since Put W

uχ +1 L2 (W ˜) ≤

3

ˆ σj L2 (W ˜),

j=1

for (132) it suﬃces to show that −1

ˆ σj L2 (W

uχ +1 L2 (R3 ) , ˜) ≤6

j = 1, 2, 3.

(133)

We shall verify (133) ) for j = 1. The proof for the cases j = 2, 3 is similar. ˆn where Obviously we have σ ˆ1 = n∈Υ1 cn u 3

u ˆ1 (ξ) = b− 2

3 ,

1

eiπ(nj + 2 )

j=1

Since 2

ˆ σj L2 (W ˜)

πnj sin(ξj b − 2 j ) . π 2 n2 b ξj2 − 4b2j πn

≤

|ξ1 |<3p

|ˆ σ1 (ξ)|2 dξ,

after integration in ζ = (ξ2 , ξ3 ) and using the notation η = ξ1 , we ﬁnd that π2 b3

2

ˆ σj L2 (W ˜) ≤

|η|<3p

n 2 , n 3 ∈ N+ n1 , n1 ≥ τ1 b p

|c(n1 ,n2 ,n3 ) c(n1 ,n2 ,n3 ) |n1 n1 dη . 2 π2 n21 2 π2 n2 η − 4b2 η − 4b21

Let us assume that τ12 ≥ 72π −2 . Then we have both 2

π n2 1 8b2

π 2 n21 8b2

(134)

≥ 9p2 ≥ η 2 and

≥ 9p2 ≥ η 2 in the denominator in the previous sum and thus |η|≤3p

2 η −

π 2 n21 4b2

dη 2 η −

π 2 n2 1 4b2

8b2 8b2 ≤ · · 6p. 2 π 2 n1 π 2 n2 1

(135)

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Applying the Schwarz inequality in the summations over n1 and n1 together with (135) to (134) we obtain   384b p  2 

ˆ σj L2 (W n−2 |cn |2 . ˜) ≤ 1 2 π 3 n1 ≥τ1 b p

Since

) n1 ≥τ1 bp

n−2 ≤ 2(τ1 b p)−1 and 1

)

n∈N3+

n∈N+

2

|cn |2 = u+ χ1 L2 (R3 ) a choice of

τ1 = 2 · 36 · 384 · π −2 ≥ 72 · π −2 will yield (133) for j = 1.

Acknowledgments The authors acknowledge the ﬁnancial support from the Swedish Institute, the DAAD and the EU TMR Network ERBFMRXCT960001. We wish also to express our gratitude to H. Siedentop for the numerous fruitful discussions on this material. The ﬁrst author acknowledges the kind support of ESI Vienna.

References [AL]

M. Aizenman, E. Lieb, On semi-classical bounds for eigenvalues of Schr¨ odinger operators, Phys. Lett. 66A, 427–429 (1978).

[Be]

F.A. Berezin, Wick and anti-Wick symbols of operators, (Russian) Mat. Sb. (N.S.) 86 (128), 578–610 (1971).

[BS]

H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One and Two Electron Atoms, Springer, NY, 1977.

[B]

M.S. Birman, The spectrum of singular boundary problems, (Russian) Mat. Sb. (N.S.) 55 (97), 125–174 (1961). (English) Amer. Math. Soc. Transl. 53, 23–80 (1966).

[BKS] M.Sh. Birman, G.E. Karadzhov, M.Z. Solomyak, Boundedness conditions and spectrum estimates for operators b(x)a(D) and their analogues, Adv. in Sov. Math. 7, 85–106 (1991). [C]

M. Cwikel, Weak type estimates for singular values and the number of bound states of Schr¨ odinger operators, Trans. AMS 224, 93–100 (1977).

[ELSS] W.D. Evans, R.T. Lewis, H. Siedentop, J.Ph. Solovej, Counting eigenvalues using coherent states with an application to Dirac and Schr¨ odinger operators in the semi-classical limit, Ark. Mat. 34 (2), 265–283 (1996). [HS]

G. Hardekopf, J. Sucher, Critical coupling constant for relativistic equations and vacuum breakdown in quantum electrodynamics, Phys. Rev A 31, 2020–2029 (1985).

Vol. 4, 2003

[H]

On the Discrete Spectrum of a Two-Body Pair Operator

341

I.W. Herbst, Spectral theory of the operator H = (p2 + m2 )1/2 − Ze2 /r, Comm. Math. Phys. 53 (3), 285–294 (1977).

[LSV] R.T. Lewis, H. Siedentop, S. Vugalter, The essential spectrum of relativistic multi-particle operators, Ann. Inst. H. Poincar´e Phys. Theor. 67 (1), 1–28 (1997). [L]

E. Lieb, The stability of matter from atoms to stars, 1989 Gibbs Lecture, in selecta of E. Lieb, Springer, 1977.

[LS]

E. H. Lieb, J.Ph. Solovej, Quantum coherent operators: a generalization of coherent states, Lett. Math. Phys. 22 (2), 145–154 (1991).

[LY]

E. H. Lieb, H.-T. Yau, The Stability and Instability of Relativistic Matter, Comm. Math. Phys. 118, 177–213 (1988).

[LY1]

E.H. Lieb, H.-T. Yau, A rigorous examination of the Chandrasekhar theory of stellar collapse, Astrophys. J. 323, 140–144 (1987).

[PBM] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series. Elementary functions, (Russian) Nauka, Moscow, 1981 799pp. [S]

Y. Schwinger, On the bound states for a given potential, Proc. Nat. Acad. Sci. USA 47, 122–129 (1961).

[W1]

T. Weidl, Cwikel type estimates in nonpower ideals, Math. Nachrichten 176, 315–334 (1995).

[W2]

T. Weidl, Another look at Cwikel’s inequality, in “Diﬀerential Operators and Spectral Theory. M.Sh. Birman’s 70th Anniversary Collection”, AMS Translations Series 2 189, 247–254 (1999).

Semjon Vugalter Mathematisches Institut der LMU Theresienstrasse 39 D-80333 Muenchen Germany email: [email protected] Timo Weidl Universit¨ at Stuttgart Fakult¨ at Mathematik Pfaﬀenwaldring 57 D-70569 Stuttgart Germany email: [email protected] Communicated by Volker Bach submitted 8/02/02, accepted 3/01/03

Ann. Henri Poincar´e 4 (2003) 343 – 368 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020343-26 DOI 10.1007/s00023-003-0132-x

Annales Henri Poincar´ e

Lp Norms of Eigenfunctions in the Completely Integrable Case John A. Toth and Steve Zelditch Abstract. The eigenfunctions eiλ,x of the Laplacian on a flat torus have uniformly bounded Lp norms. In this article, we prove that for every other quantum integrable Laplacian, the Lp norms of the joint eigenfunctions blow up at least at the p−2

−

rate ϕk Lp ≥ C()λk4p when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in Lp unless (M, g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.

0 Introduction This paper, a companion to [TZ1], is concerned with the growth rate of the Lp norms of L2 -normalized ∆-eigenfunctions ∆ϕj = λ2j ϕj ,

ϕj , ϕk = δjk

on compact Riemannian manifolds (M, g) with completely integrable geodesic ﬂow Gt on S ∗ M . The motivating problem is to relate sizes of eigenfunctions to dynamical properties of its geodesic ﬂow Gt on S ∗ M . In general this is an intractable problem, but much can be understood by studying it in the special case of integrable systems. To be precise, we assume that ∆ is quantum completely integrable or QCI in the sense that there exist n = dim M ﬁrst-order pseudo-diﬀerential operators P1 , . . . , Pn such that √ (1) P1 = ∆, [Pi , Pj ] = 0 and whose symbols (p1 , . . . , pn ) satisfy the independence condition dp1 ∧ dp2 ∧ · · · ∧ dpn = 0 on a dense open set Ω ⊂ T ∗ M − 0. Since {pi , pj } = 0, their joint Hamiltonian ﬂow Φt (x, ξ) := exp(t1 Xp1 ) ◦ · · · ◦ exp(t1 Xpn )(x, ξ) (where Xp denotes the Hamiltonian vector ﬁeld of p) deﬁnes a Hamiltonian Rn action with R+ -homogeneous moment map P = (p1 , . . . , pn ) : T ∗ M − 0 → Rn .

(2)

Throughout, we will assume that the orbits of this action are non-degenerate in the sense of Eliasson (see [El] and Deﬁnitions 2 and 3). The main result of [TZ1] was that the L∞ -norms of the L2 -normalized joint eigenfunctions {ϕλ } of (P1 , . . . , Pn ) are unbounded unless (M, g) is ﬂat. The rough

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idea was to prove that orbits of the Rn action which had singular projection to M (under the natural projection π : T ∗ M → M ) caused sup-norm blow-up in associated sequences of eigenfunctions. In this paper, we make use of singular orbits rather than just singular projections of possibly regular orbits. The nice feature of singular orbits is that the associated modes blow up at high rates. To use this approach, we ﬁrst prove that singular orbits must occur unless (M, g) is a ﬂat torus, and we relate blow-up rates of Lp -norms of sequences ||ϕλ ||Lp of eigenfunctions to the dimensions of the singular orbits. For the deﬁnition of singular orbits and Eliasson non-degeneracy, see Deﬁnition (1). Theorem 1 Suppose that (M, g) is a compact Riemannian manifold whose Laplacian ∆ is quantum completely integrable as in (1) and suppose that the Hamiltonian Rn action deﬁned by (2) satisﬁes Eliasson’s non-degeneracy condition (Definition 2). Then, unless (M, g) is a ﬂat torus, this action must have a singular orbit of dimension < n. If the minimal dimension of the singular orbits is , then for every > 0, there exists a sequence of eigenfunctions satisfying:  n− −  .  ϕk L∞ ≥ C()λk 4  

(n−)(p−2) − 4p

ϕk Lp ≥ C()λk

, 2 < p < ∞.

The proof does not determine the minimal dimension . By taking products of lower-dimensional manifolds, it is easy to construct examples with any value of = 1, . . . , n − 1. But, as will be discussed in §1, it is quite plausible that onedimensional orbits ‘often’ occur for Hamiltonian Rn actions on cotangent bundles. Such leaves correspond to closed geodesics which are invariant under the Rn action. Simple examples are given by the ‘equatorial geodesics’ on convex surfaces of revolution, i.e., geodesics invariant the S 1 action. In such cases, the eigenfunction n−1

−

blow-up estimate becomes the optimal rate ϕk L∞ ≥ C()λk 4 . In the case of the sphere or other convex surfaces of revolution, the blow-up rate is achieved by ‘highest weight’ eigenfunctions (corresponding to joint spectral points on the boundary of the image of the moment map P). Regarding the sharpness of the result, based on the case of the quantum Euler n−

top [T1], it seems reasonable to conjecture that ϕλj ∞ ≥ Cλj 8 n− 4

in the elliptic

case and ϕλj ∞ ≥ Cλj (log λj )−α for some α > 0 in the hyperbolic case. The method of proof of Theorem 1 is partly based on a study of trace formulae, as in [TZ1], and partly on a study of Birkhoﬀ normal forms of quantum integrable systems and their modes and quasi-modes near singular torus orbits. We derive Lp estimates from studying matrix elements Aϕµ , ϕµ of pseudodiﬀerential operators relative to the joint eigenfunctions ϕµ . In the case of multiplications functions of domains Ω ⊂ M , such maA = 1Ω by (smoothed out) characteristic trix elements measure the L2 - mass Ω |ϕµ (x)|2 dV of the eigenfunction in Ω. To obtain Lp norm information from L2 -mass estimates, we study small scale eigen-

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function mass, that is, the mass in shrinking tubes. The main estimate is given in Lemma 8. To our knowledge, these and the results of [TZ1, TZ2] are the ﬁrst lower bounds on eigenfunctions in the completely integrable case. They were contained in the original posted version (arXiv (math-ph/0002038)) of our article [TZ1]. In revising that paper for publication, we decided to separate the results into the qualitative one of [TZ1] and the present quantitative ones. A recent paper of Donnelly [D] has obtained a similar maximal blow up rate n−1

λk 4 for L∞ norms of eigenfunctions on compact Riemannian manifolds (M, g) with warped product metrics or isometric S 1 actions under certain non-degeneracy assumptions. In the S 1 -action case, the assumption is analogous to the existence of a singular Eliasson non-degenerate one-dimensional orbit. Donnelly’s setting has two commuting operators of a very special type (i.e., S 1 acts on the base), whereas our QCI setting has n commuting operators of a more general type. Comparison of the results suggests a common generalization to partially integrable Laplacians √ P1 = ∆, P2 , . . . , Pm , m ≤ n, [Pi , Pj ] = 0. If the associated Rm action has a one-dimensional orbit satisfying an analogue of Eliasson non-degeneracy, then it is reasonable to expect the existence of a sequence of joint eigenfunctions which blow up at the maximal rate. In another paper [TZ2], we give a proof of the eigenfunction blow-up result of [TZ1] which in some ways is closer in spirit to the approach of this paper than that of [TZ1]. The method is to relate norms of modes to norms of “quasi-modes”, i.e., approximate eigenfunctions associated to Bohr-Sommerfeld leaves of the Liouville foliation. That paper also gives a number of detailed examples of quantum completely integrable systems such as Liouville tori, surfaces of revolution, ellipsoids, tops and so on. We close by stating a conjecture on a much larger class of Riemannian manifolds which is suggested by the recent results: Conjecture 2 Any compact (M, g) with a stable elliptic orbit has a sequence of n−1

eigenfunctions whose L∞ norms blow up at the rate λk 4 . Such orbits occur in the KAM setting of perturbations of integrable systems. There surely exist quasi-modes with this property, and the diﬃculty is to show that this implies the existence of modes with this blow-up rate.

1 Geometry of completely integrable systems This section is devoted to the geometric aspects of our problem. We begin with some preliminary background on completely integrable systems. In particular, we describe Eliasson’s normal form theorem for completely integrable Hamiltonians near non-degenerate singular orbits. We then prove the existence of singular orbits of Hamiltonian Rn actions commuting with geodesic ﬂows on manifolds other than ﬂat tori.

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A completely integrable system is deﬁned by an Abelian subalgebra p = R{p1 , . . . , pn } ⊂ (C ∞ (T ∗ M − 0), {, }).

(3)

Here, {, } is the standard Poisson bracket. We assemble the generators into the moment map (4) P = (p1 , . . . , pn ) : T ∗ M → B ⊂ Rn . The Hamiltonians pj generate the Rn -action Φt = exp t1 Ξp1 ◦ exp t2 Ξp2 · · · ◦ exp tn Ξpn . We denote Φt -orbits by Rn ·(x, ξ). By the Liouville-Arnold theorem [AM], the orbits of the joint ﬂow Φt are diﬀeomorphic to Rk × T m for some (k, m), k + m ≤ n. We now consider level sets P −1 (b) of the moment map and their decompositions into orbits. First, we suppose that b is a regular value. Since P is proper, a regular level has the form (5) P −1 (b) = Λ(1) (b) ∪ · · · ∪ Λ(mcl ) (b), (b ∈ Breg ) where each Λ(l) (b) T n is an n-dimensional Lagrangian torus. Here, mcl (b) = #P −1 (b) is the number of orbits on the level set P −1 (b). In suﬃciently small neighbourhoods Ω(l) (b) of each component torus, Λ(l) (b), the Liouville-Arnold theorem (l) (l) (l) (l) also gives the existence of local action-angle variables (I1 , . . . , In , θ1 , . . . , θn ) in terms of which the joint ﬂow of Ξp1 , . . . , Ξpn is linearized [AM]. Now let us consider singular levels of the moment map and singular orbits of the Rn -action. We use the following notations: Deﬁnition 1 • A point (x, ξ) is called a singular point of P if dp1 ∧ · · · ∧ dpn (x, ξ) = 0. • A level set P −1 (c) of the moment map is called a singular level if it contains a singular point (x, ξ) ∈ P −1 (c). (We then say c is a singular value and write c ∈ Bsing .) • A connected component of P −1 (c) is a singular component if it contains a singular point. • An orbit Rn ·(x, ξ) of Φt is singular if it is non-Lagrangian, i.e., has dimension < n; Suppose that c ∈ Bsing . We ﬁrst decompose the singular level (j)

P −1 (c) = ∪rj=1 Γsing (c)

(6)

(j)

into connected components Γsing (b) and then decompose (j)

Γsing (c) = ∪pk=1 Rn · (xk , ξk )

(7)

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each component into orbits. Both decompositions can take a variety of forms. The (j) regular components Γsing (b) must be Lagrangian tori. Under a non-degeneracy assumption (see Deﬁnition 1.1 ), the singular component consists of ﬁnitely many orbits. The orbit Rn · (x, ξ) of a singular point is necessarily singular, hence it has the form Rk × T m for some (k, m) with k + m < n. Regular points may of course also occur on a singular component; their orbits are Lagrangian and can take any one of the forms Rk × T m for some (k, m) with k + m = n. Now let v ∈ P −1 (c) and assume the orbit Rn · (v) := {exp t1 Ξp1 ◦ · · · ◦ exp tk Ξpk (v); t = (t1 , . . . , tk ) ∈ Rn } is compact and of rank k in the sense that rank (dp1 , . . . , dpn )|v = rank (dp1 , . . . , dpk ) = k < n.

(8)

Following [El] (see p. 9), we observe that the Hessians d2v pj determine an Abelian subalgebra (9) d2v p ⊂ S 2 (K/L, ωv )∗ of quadratic forms on the reduced symplectic subspace K/L, where we put K=

n

ker dpi (v),

L = span {Ξp1 (v), . . . , Ξpn (v)}.

i=1

Deﬁnition 2 [El] The orbit Rn · v is said to be non-degenerate of rank k if d2v p is a Cartan subalgebra of S 2 (K/L, ωv )∗ . A Cartan subalgebra is a maximal Abelian subalgebra generated by semisimple elements. The above deﬁnition is (superﬁcially) more general than the one in [El] (p. 6), since Eliasson assumes through most of [El] that the subalgebra is elliptic (in a sense we describe below). However, most of Eliasson’s ideas apply to generic integrable systems where the Cartan subalgebra is of mixed type, with real or complex hyperbolic generators as well as elliptic ones, as discussed in the last section of [El] and in [El2]. Also, our assumption that (9) is a CSA is somewhat stronger than in [El]. The deﬁnition can be rephrased in terms of reduced Hamiltonian systems, as follows. First, there is a singular Liouville-Arnold theorem (cf. [N]) which produces action variables conjugate to the angle variables on the singular orbit. As in (8), we choose indices so that dp1 , . . . , dpk are linearly independent everywhere on Rn ·(v0 ). The singular Liouville-Arnold theorem [AM] states that there exists local canonical transformation ψ = ψ(I, θ, x, y) : R2n → T ∗ M − 0, where I = (I1 , . . . , Ik ), θ = (θ1 , . . . , θk ) ∈ Rk , x = (x1 , . . . , xn−k ), y = (y1 , . . . , yn−k ) ∈ Rn−k m

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deﬁned in an invariant neighbourhood of Rn · (v) such that pi ◦ ψ = Ii (i = 1, . . . , k),

(10)

and such that the symplectic form ω on T ∗ M takes the form ψ∗ ω =

k

dIj ∧ dθj +

j=1

n−k

dxj ∧ dyj .

(11)

j=1

As for the remaining Hamiltonians pj , there exist constants cij with i = k+1, . . . , n and j = 1, . . . , k, such that at each point of the orbit, Rn · (v), dpi =

k

cij dpj .

(12)

j=1

Since dp1 , . . . , dpk are linearly independent in a suﬃciently neighbourhood U of v ∈ P −1 (c), the action of the ﬂows corresponding to the Hamilton vector ﬁelds, Ξp1 , . . . , Ξpk generates a symplectic Rk action on P −1 (c0 ) ∩ U . We reduce U with respect to the partial moment map I := (I1 , . . . , Ik )(= (p1 , . . . , pk )), i.e., we take {I = 0} and divide by the Hamiltonian ﬂow. This produces a 2(n − k)-dimensional symplectic manifold, (13) Σk := P −1 (c0 ) ∩ U/Rk , with the induced symplectic form, σ. We will denote the canonical projection map by: πk : P −1 (c0 ) ∩ U −→ Σk . Since {pi , pj } = 0 for all i, j = 1, . . . , n, it follows that pk+1 , . . . , pn induce C ∞ functions on Σk , which we will, with some abuse of notation, continue to write as pk+1 , . . . , pn . From (12), it follows that dpi (πk (v)) = 0; i = k + 1, . . . , n. Here, we denote the single point πk (Rn · (v)) by πk (v). We thus obtain an Abelian subalgebra pred = R{pk+1 , . . . , pn } of (C ∞ (Σk ), {, }) equipped with the Poisson bracket deﬁned by σ, consisting of functions with a critical point at πk (v). Equivalent to Deﬁnition 2 is: Deﬁnition 3 The orbit Rn · v is non-degenerate of rank k if d2v pred is a Cartan subalgebra of the Lie subalgebra of quadratic forms in (C ∞ (Σk ), {, }).

1.1

Normal forms of integrable Hamiltonians near non-degenerate singular orbits

Eliasson’s normal form theorem for completely integrable systems near a compact non-degenerate singular orbit Λ ⊂ P −1 (c) of rank k expresses the Hamiltonians pj

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in terms of the linear action variables Ik of (10) and of additional action variables in the symplectic transversal (or reduced space). Before stating the normal form theorem, we recall the deﬁnitions of the action variables. Let Q(2m) denote the Lie algebra of quadratic forms on R2m equipped with its standard Poisson bracket. It contains the following special elements (action variables): (i) Real hyperbolic: Iih = xi ξi ; (ii) Elliptic: Iie = x2i + ξi2 ; √ (iii) Complex hyperbolic: Iich = xi ξi+1 − xi+1 ξi + −1(xi ξi + xi+1 ξi+1 ). Let us call the reduced (or transversal) Hamiltonian system around the equilibrium point (or singular orbit) non-degenerate elliptic, if it is non-degenerate in the sense of Deﬁnitions 2 and 3 and if the generators of d2v pred are elliptic as in (ii). Elliason’s elliptic normal form theorem states that in this non-degenerate elliptic case, there exists a local symplectic diﬀeomorphism κ : V → U, κ(Tk × 0) = Rn · v from a neighbourhood V of Tk × 0 in T ∗ (Tk × Rn−k ) to a neighborhood U of the orbit and a locally deﬁned C ∞ function f : (Rn , 0) → (R, 0) such that e pi ◦ κ−1 − ci = f (I1e , . . . , In−k , I1 , . . . , Ik ).

(14)

There is a corresponding normal form theorem in the hyperbolic case or in the case of mixed elliptic-hyperbolic systems. The statement and proof are alluded to in [El] and discussed in detail in [El2]. We let 2m = 2(n − k) = dim K/L as above. By our assumption, the sub-algebra d2v p is a Cartan subalgebra of Q(2m). By simultaneously diagonalizing the quadratic forms, we can ﬁnd a basis of d2v p consisting of generators of the above types. The normal form theorem on the reduced (or transversal) space now states that there exists a locally-deﬁned canonical mapping, κ : U → U0 , from a small neighbourhood, U , of πk (v) ∈ Σk to a neighbourhood, U0 , of 0 ∈ R2m , with the property that: ∀i, j {pi ◦ κ−1 , Ije } = {pi ◦ κ−1 , Ijh } = {pi ◦ κ−1 , Ijch } = 0.

(15)

Here, pj are actually the functions induced by pk+1 , . . . , pn on Σk . By making a second-order Taylor expansion about I e = I ch = I h = 0, it follows from (15) that ∀i = 1, . . . , n, there locally exist Mij ∈ C ∞ (U0 ) with {Iie , Mij } = {Iih , Mij } = {Iich , Mij } = 0 such that: pi ◦ κ−1 − ci =

H j=1

Mij · Ijh +

H+L+1

n

Mij · Ijch +

j=H+1

j=H+L+1

Non-degeneracy is easily seen to be equivalent to: (Mij )(0) ∈ Gl(n; R).

Mij · Ije .

(16)

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The corresponding result for the original integrable system near a singular compact orbit (Theorem C of [El]) is a parameter-dependent version of the reduced normal form theorem, generalizing (14). With the same assumptions, there exists a neighbourhood, Ω, of the orbit, Rn · (v), and a canonical map κ : Ω → T ∗ (Tk ) × D with the property that, for all i = 1, . . . , n, pi ◦κ−1 −ci =

H j=1

Mij ·Ijh +

H+L+1

Mij ·Ijch +

j=H+1

n−k

Mij ·Ije +

j=H+L+1

n

Mij ·In+1−j .

j=n−k+1

(17) h ch ch e e ), I ch := (IH+1 , . . . , IH+L+1 ) and I e := (IH+L+2 , . . . , In−k ) Here I h := (I1h , . . . , IH denote the elements deﬁned above and I := (I1 , . . . , Ik ) are momentum coordinates of T ∗ (Tk ). The Mij Poisson-commute with all the action functions. As mentioned above, the proofs of (16) and (17) are similar to the elliptic case in [El]; for discussion of how the results can be extended to mixed elliptichyperbolic systems we refer to [El, El2, CP, VN, VN2].

1.2

Existence of singular orbits

In this section, we prove the ﬁrst part of Theorem 1: Lemma 3 Let (M, g) be a compact Riemannian manifold whose geodesic ﬂow commutes with a Hamiltonian Rn action. Then, unless (M, g) is a ﬂat torus, the action must have a singular orbit of dimension < n. Proof. The hypothesis that the Liouville foliation is non-singular has two immediate geometric consequences: (i) By Mane’s theorem [M] , (M, g) is a manifold without conjugate points; (ii) The (homogeneous) moment map P : T ∗ M − 0 → Rn − 0 is a torus ﬁbration by T n . Statement (ii) follows from the Liouville-Arnold theorem. On a non-singular leaf, we must have dp1 ∧ · · · ∧ dpn = 0. Since this holds everywhere, P is a submersion; and since it is proper, it is a ﬁbration. The ﬁber must be T n , again by the LiouvilleArnold theorem. Since P is homogeneous, the image P(T ∗ M − 0) = R+ · P(S ∗ M ). Since P is a submersion, the image is a smooth compact submanifold of S n−1 hence must be all of S n−1 . ˜ /Γ where (M ˜ , g˜) is the universal Riemannian By (i) it follows that M = M cover of (M, g) (diﬀeomorphic to Rn ), and where Γ ∼ π1 (M ) is the group of covering transformations. We claim that (ii) implies π1 (M ) = Zn .

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Indeed, T ∗ M − 0 is a double ﬁbration (with indicated ﬁbers): T ∗M − 0 π (Rn − 0)

(T n ) P

M

. Rn − 0

By the homotopy sequence of a ﬁbration π : E → B, · · · πq (F ) → πq (E) → πq (B) → πq−1 (F ) · · · → π0 (E) → π0 (B) → 0 and using that π2 (T n ) = 1 and that π2 (M ) = 1 by (i), we obtain 1 → π1 (S n−1 ) → π1 (S ∗ M ) → π1 (M ) → π0 (S n−1 ) → π0 (S ∗ M ) → π0 (M ) → 1 1 → π2 (S ∗ M ) → π2 (S n−1 ) → π1 (T n ) → π1 (S ∗ M ) → π1 (S n−1 ) → → π0 (T n ) → π0 (S ∗ M ) → π0 (S n−1 ) → 1

(18)

˜ × ˜ ) (where N ˜ is the universal cover), and since S ∗˜M = S n−1 Since π2 (N ) = π2 (N Rn , we have π2 (S ∗ M ) = π2 (S n−1 ). From its deﬁnition, we see that the homomorphism π2 (S ∗ M ) → π2 (S n−1 ) is an isomorphism, hence the second sequence simpliﬁes to 1 → π1 (T n ) → π1 (S ∗ M ) → π1 (S n−1 ) → π0 (T n ) → π0 (S ∗ M ) → π0 (S n−1 → 1 (19) Let us ﬁrst assume that n ≥ 3. Then π1 (S ∗ M ) = π1 (M ) and π1 (S n−1 ) = 1, so we get 1 → π1 (T n ) → π1 (M ) → 1, i.e., π1 (T n ) ∼ = π1 (M ). We now consider dimension 2. By (i), the genus g ≥ 1. The unit tangent bundles of surfaces of genus g ≥ 2 do not ﬁber over a circle, so we can disqualify them and conclude g = 1. (In fact, it follows by a classic result of Kozlov [K] (see also [TAI]) that the only surfaces that can possibly have a completely integrable geodesic ﬂow (even with singularities) are M = S 2 , T 2 .) It follows then that M is diﬀeomorphic to Rn /Zn , i.e., M is a torus. Since g has no conjugate points, the proof is concluded by Burago-Ivanov’s theorem that metrics on tori without conjugate points are ﬂat [BI]. 1.2.1 Problem on homogeneous moment maps As mentioned above, the existence proof of singular orbits does not give any information on the dimension of such orbits. Do there exist one-dimensional orbits in ‘generic’ cases?

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In the case of Hamiltonian torus T n actions, one-dimensional orbits are detected by the image C of T ∗ M −0 under the moment map (I1 , . . . , In ) : T ∗ M −0 → Rn . By assumption, the Ij generate 2π-periodic Hamiltonian ﬂows. In the case of the ﬂat torus, C = RN − 0, but in other cases C is a convex polyhedral cone with boundary. The boundary faces correspond to singular orbits and, in particular, an edge (a one-dimensional boundary face) corresponds to a periodic orbit. The moment polyhedra of toric moment maps have been studied by Lerman and Lerman-Shirokova [LS, L2]. Lerman has proved that if no subtorus acts freely on T ∗ M − 0, then C always has an edge [L3]. The problem we would like to pose is to formulate and prove an analogous result for homogeneous Rn actions on cotangent bundles. Can one detect singular torus orbits of dimension one from C and must C have an edge? By taking products with ﬂat tori one can see that not all torus actions have a one-dimensional orbit, but perhaps ‘generic’ examples do. To our knowledge, the cone C has not been studied to date for general homogeneous Rn actions. Clearly it depends on a choice of generators of the Rn action. In the toric case, the special generators gave rise to a polyhedron C from which one could read oﬀ the existence of singular orbits. Is there a good replacement in the case of general Rn actions, one which does not already presuppose a knowledge of the singular orbits?

2 Quantum integrable systems and Birkhoﬀ normal forms Our purpose in this section is to construct a microlocal Birkhoﬀ normal form for a QCI (quantum completely integrable) system near a singular orbit. We ﬁrst set up some notation for the joint spectrum of quantum integrable systems. Throughout, we follow the notation and terminology of [TZ1] and refer there for background on QCI systems.

2.1

Quantum integrable systems

In this article, the only Hamiltonians we consider are Laplacians ∆ on compact Riemannian manifolds (M, g), although many of the methods and results extend √ to Schr¨ odinger operators as in [TZ1]. We therefore assume that P1 = ∆, and that the other commuting operators P2 , . . . , Pn are classical pseudodiﬀerential operators of order one. Although our operators are homogeneous pseudodiﬀerential operators, it is convenient to use the notation and methods of semiclassical microlocal analysis. We therefore rescale our operators to obtain 0,1 ). Qj := Pj ∈ Op(Scl

In forming microlocal models, we will need to introduce more general kinds of semiclassical quantum integrable systems, so we pause to deﬁne the class.

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m,k Deﬁnition 4 We say that the operators Qj ∈ Op (Scl ); j = 1, . . . , n, generate a semiclassical quantum completely integrable system if

[Qi , Qj ] = 0; ∀1 ≤ i, j ≤ n, and the respective semiclassical principal symbols q1 , . . . , qn generate a classical integrable system with dq1 ∧ dq2 ∧ · · · ∧ dqn = 0 on a dense open set Ω ⊂ T ∗ M − 0. Here, we use the notation of Dimassi-Sj¨ ostrand [DSj] for operator classes: Given an open U ⊂ Rn , we say that a(x, ξ; ) ∈ C ∞ (U × Rn ) is in the symbol class S m,k (U × Rn ), provided |∂xα ∂ξβ a(x, ξ; )| ≤ Cαβ −m (1 + |ξ|)k−|β| . m,k We say that a ∈ Scl (U × Rn ) provided there exists an asymptotic expansion:

a(x, ξ; ) ∼ −m

∞

aj (x, ξ)j ,

j=0

valid for |ξ| ≥ C1 > 0 with aj (x, ξ) ∈ S 0,k−j (U × Rn ) on this set. The associated Kohn-Nirenberg quantization is given by Op (a)(x, y) = (2π)−n

Rn

ei(x−y)ξ/ a(x, ξ; ) dξ.

As is well known, the deﬁnition can be globalized to M using a partition of unity. We denote this class by Op (S m,k )(T ∗ M ). The symbol of the composition is given by the usual formula: Given a ∈ S m1 ,k1 and b ∈ S m2 ,k2 , the composition Op (a) ◦ Op (b) = Op (c) + O(∞ ) in L2 (M ) where locally, c(x, ξ; ) ∼ −(m1 +m2 )

∞ |α| (−i) (∂ξα a) · (∂xα b). α!

|α|=0

For further details, we refer to [DSj]. For a general QCI system, we denote by {ϕµ } an orthonormal basis of joint eigenfunctions, Qj ϕµ = µj ()ϕµ , ϕµ , ϕµ = δµ,µ ,

(20)

and the joint spectrum by Σ() := {µ() := (µ1 (), . . . , µn ())}.

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Model cases

Quantum Birkhoﬀ normal forms are microlocal expressions of a given QCI system in terms of certain model system. Model quantum completely integrable systems are direct sums of the quadratic Hamiltonians: • Iˆh := (Dy y + yDy ) (hyperbolic Hamiltonian), • Iˆe := 2 Dy2 + y 2 , (elliptic Hamiltonian), √ • Iˆch := [(y1 Dy1 + y2 Dy2 ) + −1(y1 D2 − y2 Dy1 )] (complex hyperbolic Hamiltonian), • Iˆ := Dθ , (regular Hamiltonian). The corresponding model eigenfunctions are: • uh (y; λ, ) = | log |−1/2 [c+ ()Y (y)|y|−1/2+iλ()/ + c− ()Y (−y)|y|−1/2+iλ()/ ]; |c− ()|2 + |c+ ()|2 = 1; λ() ∈ R. • ue (y; n, ) = −1/4 exp(−y 2 /) Φn (−1/2 y); n ∈ N. • uch (r, θ; t1 , t2 , ) = | log |−1/2 r(−1+it1 ())/ eit2 ()θ/ ; t1 (), t2 () ∈ R. • ureg (θ; m, ) = eimθ ; m ∈ Z. Here, Y (x) denotes the Heaviside function, Φn (y) the nth Hermite polynomial and (r, θ) polar variables in the (y1 , y2 ) complex hyperbolic plane. The important part of a model eigenfunctions is its microlocalization to a neighborhood of x = ξ = 0, so we put: ψ(x; ) := Op ( χ(x)χ(y)χ(ξ) ) · u(y; ), where > 0 and χ ∈ C0∞ ([−, ]). In the hyperbolic, complex hyperbolic, elliptic and regular cases, we write ψh (y; ), ψch (y; ), ψe (y; ) and ψreg (y; ) respectively. A straightforward computation [T2] shows that when t1 (), t2 (), n, m = O() the model quasimodes are L2 -normalized; that is Op (χ(x)χ(y)χ(ξ) ) u(y; ) L2 ∼ 1

(22)

as → 0. Note that, although the model eigenfunctions above are not in general smooth functions, the microlocalizations are C ∞ and supported near the origin.

2.3

Ladders of eigenfunctions

Semiclassical limits are taken along ladders in the joint spectrum. For ﬁxed b = (b1 , b2 , . . . , bn ) ∈ Rn , we√deﬁne a ladder of joint eigenvalues of the original homogeneous problem P1 = ∆, P2 , . . . , Pn by: {(λ1k , . . . , λnk ) ∈ Spec(P1 , . . . , Pn ); ∀j = 1, . . . , n, lim

k→∞

where |λk | :=

λ21k + · · · + λ2nk .

λjk = bj }, |λk |

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In the corresponding -scaled system, the ladder will be denoted by: Σb () := {µ() := (µ1 (), . . . , µn ()) ∈ Spec(Q1 , . . . , Qn ); |µj () − bj | ≤ C, j = 1, . . . , n}. (24) We deﬁne the joint eigenspace corresponding to Σb () as follows: For b ∈ P(T ∗ M − 0), deﬁne Vb () := {ϕµ ; ϕµ L2 = 1 with µ() ∈ Σb ()}.

2.4

(25)

Microlocal solution space

Following [CP] Deﬁnition 2, we call a family of distributions u ; ∈ (0, 0 ] admissible if there exist constants Nj ; j = 1, 2, 3 such that for any ϕ ∈ S(R),

ϕ(x)u (x)dx ≤ −N1 ||(1 + x2 )N2 ϕ||C N3 .

We now deﬁne the microlocal (quasi-)eigenvalue problem in Ω. We denote -microlocal equivalence on an open set Ω by =Ω [CP]. We say that νk () is a quasi-classical eigenvalue if there exists a non-trivial, admissible solution of the eigenvalue problem (26) Qj ψν =Ω νj ()ψν ; j = 1, . . . , n. The set of quasi-classical eigenvalues around c is thus: QΣc () := {ν() : (26) holds, with |ν() − c| ≤ C, j = 1, . . . , n}.

(27)

We deﬁne the corresponding microlocal solution space as the span of QVc () := {ψν ; ψν L2 = 1, (26) holds with µ() ∈ QΣb ()}.

(28)

The solution space (28) can be characterized uniquely (up to a C()-multiple) in terms of the model quasimodes ψe , ψh , ψch and ψreg . In the following, we k use the abbreviation (ue (y; n, ) · uh (y; λk (), ) · uch (y; t1,k (), t2,k (), ) j=1 eimj θj ) for the expression H+L H+L+E ΠH j=1 ψhj (yj ; ) ⊗ Πj=H+1 ψchj (yj ; ) ⊗ Πj=H+L+1 ψej (yj ; )⊗

Πnj=L+H+E+1 ψrj (yj ; ) in which the (m, n, λ, t1 , t2 ) parameters are put in. Proposition 4 For any ψν() ∈ QVc (), there exist t1 (), λ() ∈ R and t2 (), n(), m() ∈ Z and an -dependent constant c() such that ψν =Ω c() F (ue (y; n, ) · uh (y; λ(), ) · uch (y; t1 (), t2 (), )

k

eimj θj ),

j=1

where F is the microlocally unitary -Fourier integral operator in Lemma (5).

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Proof. The microlocal solutions of the model eigenfunction equations are unique up to C()-multiples. This was proved for the strictly hyperbolic case in [CP] and for the complex hyperbolic case in [VN2]. The elliptic case follows from the local normal form. An application of Lemma (5) then gives the result.

2.5

Singular Birkhoﬀ normal form

In this section we introduce Birkhoﬀ normal forms for a quantum completely integrable system near a singular orbit. The main result is the following quantum analogue (see also [VN]) of the classical Eliasson normal form in (17). In the following lemma, e,h,ch,reg () are each classical symbols of order 1 in ∈ (0, 0 ]; that is,

∞ ∈ C; j = 1, 2, . . . such that e,h,ch,reg () ∼ j=1 e,h,ch,reg j there exist e,h,ch,reg j j as → 0+ . Lemma 5 Let c ∈ Rn be a singular value of the moment map P and Rn ·v be a rankk Eliasson non-degenerate orbit of the joint ﬂow. Then, there exists a microlocally elliptic -Fourier integral operator, F , and a microlocally invertible n × n matrix of -pseudodiﬀerential operators, Mij , with [F−1 Mij F , Qk ] =Ω 0; k = 1, . . . , n and satisfying: F−1 (Q1 − c1 , . . . , Qn − cn ) F =Ω M · ( Iˆh − h (), Iˆch − ch (), Iˆe − e (), Dθ − reg () ) + O(∞ ). (29) Here, Iˆjh = (Dyj yj + yj Dyj ), Iˆje = 2 Dy2j + yj2 , Iˆch = [(yj Dyj + yj+1 Dyj+1 ) + √ −1(yj Dj+1 − yj+1 Dyj )] and Iˆj = Dθj . Proof. The proof is essentially the same as in ([VN] Theorem 3.6). The only complication here is that since Rn · (v) is a k < n-dimensional torus and not a point, Dθ1 , . . . , Dθk must be added to the space of model operators. The proof can be reduced to that in [VN] by making Fourier series decompositions in the (θ1 , . . . , θk ) variables (see, for instance [T2] Theorem 3).

3 Blow-up of eigenfunctions attached to singular orbits of the Lagrangian ﬁbration The purpose of this section is to prove Theorem 1. We break up the proof into a sequence of three Lemmas, each concerned with estimates of matrix elements relative to the eigenfunctions. They culminate in an estimate in Lemma 8 of the small scale L2 mass of certain eigenfunctions ϕµ ∈ Vc () near any singular orbit Λ ⊂ Γsing (c). The end result is that for each ∈ (0, 0 ], there exist ϕµ ∈ Vc () with (30) ( Op (χδ1 (x; ))ϕµ , ϕµ ) | log |−m for some m > 0 and where χδ1 (x; ) := χ1 (−δ x) with χ1 (x) a cut-oﬀ function supported near π(Λ).

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In the proof, we will need to use additional pseudo-diﬀerential operators belonging to a more reﬁned semi-classical calculus, containing cut-oﬀs such as χ1 (−δ x), which involve length scales ∼ δ with 0 < δ < 1/2.

3.1

Eigenfunction mass near non-degenerate, singular orbits

In this section, we give a lower bound for the microlocal mass on ‘large’ length scales of joint eigenfunctions with joint eigenvalues µ() ∈ Σc () near any compact, singular orbit Λ ⊂ Σc () satisfying Eliasson’s non-degeneracy condition. Note that each connected component Γsing (c) of a singular level always contains a k < ndimensional compact orbit Λ (see Proposition 1.3 of [TZ1]). Our lower bound is analogous to that of Lemma 2.8 of [TZ1] in the regular case, which proves that each component torus of a regular level carries an amount of eigenfunction mass bounded below by a constant independent of . In the singular case, the same kind of proof gives a weaker result in which the mass can decrease but at most logarithmically. It would be interesting to know if in fact there exists a uniform lower bound for the mass as in the regular case, but we do not need such a strong result for our application and do not pursue the matter here. In the following lemma, we write f () g() if there exists a constant C0 > 0 such that for suﬃciently small, |f ()| ≥ C10 |g()|. Lemma 6 Let Λ := Rn · (v) ⊂ Γsing (c) be an Eliasson non-degenerate orbit, and let χΛ (x, ξ) ∈ C0∞ (Ω; [0, 1]) be a cut-oﬀ function supported in an invariant neigh˜ of Λ. bourhood Ω of Λ and identically equal to one on a smaller neighbourhood Ω Then for each ∈ (0, 0 ] there exist ϕµ ∈ Vc () and a constant m ≥ 0 such that, (Op (χΛ )ϕµ , ϕµ ) | log |−m . Proof. Let Λj ; j = 1, . . . , K denote all the compact Rn -orbits contained in the singular component, Γsing (c) and χj (x, ξ) ∈ C0∞ ; j = 1, . . . , K cut-oﬀ functions with χj = 1 near Λj . We choose the indices so that Λ1 = Λ and put χ := χ1 . Let F ∈ S(R) with F ≥ 0, F (0) = 1 and Fˇ ∈ C0∞ (R) with suﬃciently small support near 0 ∈ R. We deﬁne fˆ(t1 , . . . , tn ) := n Fˆ (tj ). We have j=1

Op (χ)ϕµ , ϕµ f (−1 (µ() − c))

µ∈Σ()

= Rn

fˇ(t)T rOp (χ)ei

n j=1

tj [Qj ()−cj ]/

n

dt. (31)

We calculate the trace using the microlocal conjugation of ei j=1 tj [Qj ()−cj ] to its normal form given in Lemma 5. Then, by a well-known parametrix construction jˆ n ([BPU] Lemma 2.1) for Op (χ ◦ κ) · ei i=1 ti Mi Ij / for |t| small, we have that

358

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fˇ(t)T rOp (χ)ei

T := (2π)−n

J.A. Toth and S. Zelditch

n j=1 tj [Qj ()−cj ]/

Ann. Henri Poincar´e

dt is locally a sum of integrals of the form

e−iM(x,ξ)·(Ie ,Ih ,Ich ,I),t/ fˇ(t) χ ◦ κ(x, ξ) a(t, x, x, ξ) dtdxdξ + O(),

(32)

where a ∈ C0∞ with a(0, x, x, ξ) = 1. Here, Ih (x, ξ) := (Ih (x1 , ξ1 ), . . . , Ih (xH , ξH )), Ich (x, ξ) = (Ich (xH+1 , ξH+1 ), . . . , Ih (xH+L+1 , ξH+L+1 )), Ie (x, ξ) = (Ie (xL+H+2 , ξL+H+2 ), . . . , Ie (xn−k , ξn−k )) and I(x, ξ) = (In−k+1 ), . . . , In ). Next, make the change of variables (t1 , . . . , tn ) →t M · (t1 , . . . , tn ) in (32) and get that T := (2π)−n

e−i(Ie ,Ih ,Ich ,I),s/ b(s, x, x, ξ) dsdxdξ + O(),

(33)

where, b(s, x, x, ξ) = fˇ(t M −1 (x, ξ) · s) χ ◦ κ(x, ξ) a(t M −1 (x, ξ)s, x, x, ξ) | det M (x, ξ)|−1 . Since M (0) ∈ GLn (R), by choosing supp χ suﬃciently small it follows that b ∈ C0∞ . Then, by carrying out the iterated (si , xi , ξi )-integrals in (33), we are reduced to computing the asymptotics of the integrals: Treg := (2π)−1 Te := (2π)−1 Th := (2π)−1

∞ 2π ∞ −∞ 0

−∞

∞ ∞ ∞

eis(x

−∞ −∞ −∞

∞ ∞ ∞

Tch := (2π)−2

eiξs/ Fˆ (s)c(s, x, ξ)dsdxdξ ∼ creg ,

−∞ −∞ −∞ 0

0

+ξ 2 )/

c(s, x, ξ)Fˆ (s)dsdxdξ ∼ ce

eisxξ/ c(s, x, ξ)Fˆ (s)dsdxdξ ∼ ch | log |,

2π 2π ∞ ∞ 0

2

0

eirρ[s1 cos θ−s2 sin α]/ c(s, r, ρ, θ, α)

×Fˆ (s1 )Fˆ (s2 )ds1 ds2 rdrρdρdθdα ∼ cch | log |. where, c ∈ C0∞ with c(0) = 0 and creg , ce , ch , cch denote non-zero constants. The estimate for Treg follows by stationary phase in the (s, ξ)-variables, whereas for Te , Th , Tch the asymptotics follow from Proposition 3.4 and Theorem 3.5 in [BPU]. It follows that for some non-zero constant C0 f (−1 (µ() − c))Op (χΛ )ϕµ , ϕµ ∼ C0 | log |m1 , (35) µ∈Σ()

where, m1 denote the total number of complex and real hyperbolic summands in Λ1 . The same estimate holds for all singular tori. Put (36) m = max {mj } − m1 . j=1,...,K

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We now argue by contradiction that the estimate in the Lemma is correct. If not, then for all ϕµ ’s the matrix element (Op (χΛ )ϕµ , ϕµ ) = o(| log |−m ), and therefore f (−1 (µj () − c))Op (χΛ )ϕµ , ϕµ = o(| log |−m ) f (−1 (µ() − c)). µ∈Σ()

µ∈Σ()

(37) Since the Λj ; j = 1, . . . , K are ω-limit sets for the joint ﬂow on Γsing (c), it follows by the semiclassical Egorov theorem and the G˚ arding inequality ([T2] Proposition 1) that for any joint eigenfunction ϕµ with |µ() − c| = o(1), K ( (Op (χi )ϕµ , ϕµ )) >> 1.

(38)

i=1

Thus,

µ∈Σ()

f (−1 (µ() − c))

µ∈Σ()

K i=1

K i=1 (Op (χi )ϕµ , ϕµ )

f (−1 (µ() − c))

| log |mi . (39)

In the last line, we used (35) in each term. Combining (35) and (39), we get the contradiction | log |

m1

−m

≤ o(| log |

)

K

| log |mi = o(| log |m1 ),

(40)

i=1

by the choice of m in (36).

3.2

Localization on singular orbits

Let Λ be any Eliasson non-degenerate compact orbit. We claim that the joint eigenfunctions ϕµ ∈ V () satisfying the estimate in Lemma 6 must blow up along π(Λ). The ﬁrst way of quantifying this blowup involves computing the asymptotics for the expected values (Op (q) · Op (χΛ )ϕµ , Op (χΛ )ϕµ ) where q ∈ C0∞ (T ∗ M ). Lemma 7 Let ϕµ ∈ V () satisfy the bound in Lemma 6. Then: 2

(Op (q) · Op (χΛ )ϕµ , Op (χΛ )ϕµ ) = |c()|

Rn ·(v)

again with |c()| | log |−m for some m ≥ 0.

−1/2

q dµ + O(| log |

) , (41)

Proof. Since ϕµ solves the equation (26) exactly (and a fortiori microlocally on Ω), we may express it by Proposition (4) in the form: ϕµ =Ω c() F uµ ,

(42)

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for some constant c(). Here, uµ = (ue (y; n, ) · uh (y; λ(), ) · uch (y; t1 (), t2 (), )

k

eimj θj )

(43)

j=1

Here, by applying the operators on both sides of the QBNF in Lemma (5) to the model eigendistributions uµ and using the uniqueness result in Lemma (4), it follows that for some n × n matrix M with M (0) ∈ GLn , M (m, n, λ(), t1 (), t2 ()) · (m, n, λ(), t1 (), t2 ()) = µ(). By the inverse function theorem, the (m, n, λ(), t1 (), t2 ()) are uniquely determined (modulo O(∞ ) ) by the joint eigenvalues µ() and moreover, when µ() ∈ Σ() it follows that m, n, λ(), t1 (), t2 () = O(). By Lemma (6), by (22) and by (42), it follows that for ∈ (0, 0 ], | log |−m (Op (χΛ )ϕµ , ϕµ ) = |c()|2 (F ∗ Op (χΛ )F uµ , uµ ) ≤ |c()|2 .

(44)

Granted this lower bound on |c()|, the Lemma reduces to estimating matrix elements of model eigenfunctions. We now evaluate the matrix elements case by case. The most interesting case is where the orbit Λ is strictly real or complex hyperbolic. We use (42) to conjugate to the model setting. The function q goes to q ◦ χ where χ is the canonical transformation underlying F . The model Rn - action locally reduces to a compact torus T k -action, so we can average the function q ◦ χ over the action to obtain a smooth invariant function. We then Taylor expand this averaged function in the directions (y, η) transverse to the action. We obtain: (Op (q) ◦ Op (χΛ )ϕµ , Op (χΛ )ϕµ ) 2 q dµ + (Op (rh )uµ , uµ ) + (Op (rch )uµ , uµ ) + O() . (45) = |c()| Rn ·(v)

where, rh , rch ∈ C0∞ (Ω) are the Taylor remainders with rh , rch = O(|y| + |η|). A direct computation for the model distributions, uµ (see [T2] Lemma 5 and Proposition 3) shows that: (Op (rh )uµ , uµ ) = O(| log |−1/2 ), (Op (rch )uµ , uµ ) = O(| log |−1/2 ).

(46)

The remaining cases are where elliptic (i.e., Hermite factors). Each such factor satisﬁes (Op (re )uµ , uµ ) = O() so is better than what is claimed.

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Mass concentration on small length scales

Let Λ := Rn · v be a compact, k-dimensional singular orbit of the Hamiltonian Rn -action generated by (p1 , . . . , pn ). In this section, we study mass concentration of modes in shrinking tubes of radius ∼ δ for 0 < δ < 1/2 around π(Λ) in M , where π : T ∗ M −→ M denotes the canonical projection map. Such small scale concentration of mass estimates quickly lead to sup-norm estimates. For the sake of simplicity we will assume in this section that π:Λ→M is an embedding. This seems to be a reasonable assumption since dim Λ = k < dim M , and it is satisﬁed in most (if not all) the examples we know. As we explain in §3.5, the proof extends with only minor modiﬁcations to the general case. We denote by T (π(Λ)) the set of points of distance < from π(Λ). For 0 < δ < 1/2, we introduce a cut-oﬀ χδ1 (x; ) ∈ C0∞ (M ) with 0 ≤ χδ1 ≤ 1, satisfying (i) supp χδ1 ⊂ Tδ (π(Λ)) (ii) χδ1 = 1 on T3/4δ (π(Λ)). (iii) |∂xα χδ1 (x; )| ≤ Cα h−δ|α| . Under the assumption that Λ is an embedded submanifold of M , the functions χδ1 (x; ) = ζ1 (−2δ d2 (x, π(Λ)))

(48)

are smooth on T (π(Λ)) and satisfy the conditions. Here, d(., .) is the Riemannian distance function. Also, ζ1 ∈ C0∞ (R) with 0 ≤ ζ1 ≤ 1, ζ1 (x) = 1 for |x| ≤ 3/4 and supp ζ1 ⊂ (−1, 1). Lemma 8 Let ϕµ ∈ Vc () satisfy the bounds in Lemma 6. Then for any 0 ≤ δ < 1/2, (Op (χδ1 )ϕµ , ϕµ ) | log |−m . 3.3.1 Outline of proof The proof uses the somewhat technical properties of small scale pseudodiﬀerential operators. We ﬁrst sketch the proof without these technicalities. Let χδ2 (x, ξ; ) ∈ C0∞ (T ∗ M ; [0, 1]) be a second cut-oﬀ supported in a radius δ tube, Ω(), around Λ with Ω() ⊂ supp χδ1

(49)

χδ1 = 1 on supp χδ2 .

(50)

χδ1 (x, ξ) ≥ χδ2 (x, ξ),

(51)

and such that Then, clearly

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for any (x, ξ) ∈ T ∗ M . Modulo small errors (see (57)), inequality (51) implies the corresponding operator bound for the matrix elements: ( Op (χδ1 )ϕµ , ϕµ ) ( Op (χδ2 )ϕµ , ϕµ ).

(52)

Now, take ϕµ ∈ Vc () satisfying the bounds in Lemma 6. We basically use (42) and (44) to estimate the matrix elements on the RHS of (52) from below in terms of k the masses of the model distributions u(y, θ; ) = j=1 eimj θj ue (y) · uh (y) · uch (y). But to obtain the ﬁne estimate in the lemma, we need to compute these masses on shrinking neighbourhoods of diameter δ centered around a singular < ndimensional orbit, Λ. Therefore we need to introduce appropriate classes of small scale pseudodiﬀerential operators. By estimating matrix elements of such operators, we show that in shrinking neighbourhoods of diameter δ , the model distributions have ﬁnite mass bounded from below by a positive constant independent of ∈ (0, 0 ] provided we choose 0 ≤ δ < 1/2, giving the estimate stated in the Lemma. 3.3.2 Small scale semiclassical pseudo-diﬀerential calculus The more reﬁned symbols are deﬁned as follows: Given an open set U ∈ Rn and 0 ≤ δ < 12 , we say that a(x, ξ; ) ∈ Sδm (U × Rn ) if |∂xα ∂ξβ a(x, ξ; )| ≤ Cαβ −δ(|α|+|β|).

(53)

Model symbols include cut-oﬀs of the form χ(h−δ x, h−δ ξ) with χ ∈ C0∞ (R2n ). There is a pseudo-diﬀerential calculus Op Sδm (U × Rn ) associated with such symbols with the usual symbolic composition formula and Calderon-Vaillancourt L2 boundedness theorem [Sj]. Composition with operators in our original class Op S m,0 (U × Rn ) preserves Op Sδm (U × Rn ). We can now give the proof. Proof. We need to deﬁne shrinking cut-oﬀs around Λ ⊂ T ∗ M , and therefore introduce a Riemannian distance function on T ∗ M . A natural choice is to use the Riemannian metric induced by the Riemannian connection of (M, g) on T ∗ M × T ∗ M : the connection induces a splitting T (T ∗ M ) = H ⊕ V into horizontal and vertical sub-bundles. We deﬁne a metric by requiring that H ⊥ V ; on H we lift the metric g under π; for V we use the Euclidean metrics; g induces on the vertical spaces. ˜ .) be the associated distance function between points of We then let d(., ∗ T (M ). For > 0 suﬃciently small we deﬁne ˜ ξ), Λ) ≤ }. A := {(x, ξ) ∈ T ∗ (M ); d((x,

(54)

We then choose χδ2 (x, ξ; ) ∈ C0∞ (T ∗ M with 0 ≤ χδ2 ≤ 1 so that: supp χ2 ⊂ A 34 δ , χ2 = 1 on A 12 δ , and so that χδ2 (x, ξ; ) ∈ Sδ0 (T ∗ M ). (55)

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For example, we can deﬁne χδ2 (x, ξ; ) := ζ2 (−2δ d˜2 ((x, ξ); Λ)),

(56)

where ζ2 ∈ C0∞ (R), ≤ ζ2 ≤ 1 with ζ2 (x) = 1 for |x| ≤ 1/2 and supp ζ2 ⊂ (−3/4, 3/4). We choose the cut-oﬀ χδ1 as deﬁned in (48). Clearly, χδj ∈ Sδ0 (T ∗ M ); j = 1, 2, and χδ1 (x, ξ) ≥ χδ2 (x, ξ), ∀(x, ξ) ∈ T ∗ M. By the G˚ arding inequality, there exists a constant C1 > 0 such that: (Op (χδ1 )ϕµ , ϕµ) ) ≥ (Op (χδ2 )ϕµ , ϕµ ) − C1 1−2δ .

(57)

We now conjugate the right side to the model by the -Fourier integral operator F of Lemma (5). Since F is a microlocally elliptic -Fourier integral operator associated to a canonical transformation κ, it follows by Egorov’s theorem (Op (χδ2 )ϕµ , ϕµ ) = |c()|2 (Op (χδ2 ◦ κ)uµ , uµ ) − C3 1−2δ

(58)

where c()uµ (y, θ; ) is the microlocal normal form (43) for the eigenfunction ϕµ . Since ϕµ ∈ Vc () satisﬁes the bounds in Lemma (6) it follows that |c()|2 | log |−m and from (58) we are left with estimating the matrix elements (Op (χδ2 ◦ κ)uµ , uµ ) from below. To simplify the calculation, deﬁne the product-type cut-oﬀ function

χ2δ ◦ κ(y, η, I; ) =

L+N

χ(−δ yj ) χ(−δ ηj ) ·

L+M+N +1

χ(−δ ρj )χ(−δ αj )·

j=L+N +1

j=1

n

χ(−δ In+1−j ). (59)

j=n−l+1

Here (rj , αj ) denote radial variables in the jth complex hyperbolic summand and χ(x) ∈ C0∞ (R) is a cut-oﬀ equal to one near zero. Since y = η = I = 0 on (κ−1 )∗ Λ, it follows that for χ(x) with suﬃciently small support,

χδ2 ◦ κ(y, η, I; ) ≥ χ2δ ◦ κ(y, η, I; ).

(60)

Thus it suﬃces to estimate χ2δ ◦ κ(y, η, I; ).from below. To simplify the notation a little, we will write χδ (x; ) := χ(−δ x) below. Now, (Op (χδ 2 ◦ κ)u, u) consists of products of four types of terms. The ﬁrst three are: ∞ δ u (η; )|2 dη, χδ (η; ) |χ Me = e −∞

Mh =

∞ −∞

δ u (η; )|2 dη, χδ (η; ) |χ h

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and ﬁnally, Mch =

∞

−∞

∞

−∞

δ u (η , η ; )|2 dη dη . χδ (η1 , η2 ; ) |χ ch 1 2 1 2

To estimate Me , we note that, since ϕµ ∈ V (), and F (e−|y| it follows that,

2

/

Me =

Φn (y−1/2 ))(η) = e−|η|

∞

−∞

e−2|η|

2

/

2

/

Φn (η−1/2 ),

|Φn (−1/2 η)|2 dη + O(∞ )

and so for ∈ (0, 0 ], Me () ∼ 1. To estimate Mh , we recall that λ() = O() and that by [CP] Section 4.3 it suﬃces to estimate:

∞

2 ∞

dξ 1 δ

−ix −1/2+iλ/ δ . (61) χ(ξ/ )

e x χ(x/ ξ)dx

log ξ 0 0 The integral in (61) equals: (log )−1

δ−1

0

dξ ξ

δ

2

ξ

e−ix x−1/2+iλ/ dx + O(| log |−1 ).

0

(62)

To estimate this last integral, assume ﬁrst that ξ ∈ [0, −δ ]. Then,

δ ξ

0

e−ix x−1/2+iλ/ dx = O(|δ ξ|1/2 )

and so, (61) equals | log |−1

δ−1

−δ

dξ ξ

δ

2

ξ

e−ix x−1/2+iλ/ dx + O(| log |−1 ).

0

(63)

From (63), it follows that: Mh = C(δ) + O(| log |−1 )

(64)

where C(δ) > 0 when 0 < δ < 12 . Finally, we are left with the integral Mch corresponding to a loxodromic subspace. Since |Jk (ρ)| ≤ 1 for all k ∈ Z and ρ ∈ R, and k() = O(M ), t() = O(), it follows that:

2 δ−1

δ α

dα

+ O(| log |−1 ). Jk (ρ)ρit/ dρ

(65) Mch = | log |−1

α −δ 0

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Here, Jk (ρ) denotes the kth integral Bessel function of the ﬁrst kind [AS]. For α ≥ −δ ,

δ

2it/ Γ k+1+it/ )

α

it/ 2 Jk (ρ)ρ dρ =

+O(|δ α|−1/2 ) = 1+O(|δ α|−1/2 ), (66)

Γ( k+1−it/ )

0 2

and so, Mch = | log |−1

δ−1

−δ

dα + O(| log |−1 ) = 1 − 2δ + O(| log |−1 ). α

(67)

Consequently, given δ = 1/2 − it again follows that Mch = C() > 0 uniformly for ∈ (0, 0 ()]. The ﬁnal step involves estimating (Op (χδ (I))eimθ , eimθ ). An integration by parts in the I1 , . . . , I variables shows that: (Op (χδ (I))eimθ , eimθ ) = 1 + O(1−δ ).

(68)

As a consequence of the estimates above for Mh , Mch , Me and the estimates in (57),(58) and (60), it follows that for any > 0 and δ = 1/2 − , there exists a constant C() > 0 such that for all ϕµ ∈ Vc () satisfying the bounds in Lemma 6, (Op (χδ1 )ϕµ , ϕµ ) ≥ C()| log |−m .

3.4 Since M

(69)

Completion of the proof of Theorem 1 |ϕµ (x)|2 χδ1 (x; ) dvol(x)

≤

supx∈Th2δ (π(Λ)) |ϕµ (x)|2

≤ ϕµ 2L∞ ·

M

M

χδ1 (x; ) dvol(x)

χδ1 (x; ) dvol(x) (70)

it follows from Lemma 8 or (69) that χδ1 (x; ) dvol(x) ≥ C()| log |−m , ϕµ 2L∞ ·

(71)

uniformly for ∈ (0, 0 ()]. Since χδ1 (x; ) dvol(x) = O(δ(n− ) ),

(72)

M

M

the lower bound coming from (71) is: 1

Since we take −1

ϕµ 2L∞ ≥ C()− 2 (n− )+ | log |−m . √ ∈ {λj ; λj ∈ Spec − ∆}, this gives: n−

ϕλj L∞ ≥ C()λj 4

−

.

The lower Lp bounds when 2 < p < ∞ follow by applying the H¨ older inequality in the estimate (69).

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We doubt that much is lost in the second inequality in (70), since we expect ϕµ (x) to take its supremum at or near π(Λ) and to have a roughly constant size on this set. This completes the proof of Theorem 1 when π(Λ) is embedded. We now brieﬂy explain how to modify the proof in the general case.

3.5

General π(Λ)

The only problem is that the function d2 (x, π(Λ)) is not generally smooth if π(Λ) has singularities (such as self-intersections). We therefore need to use diﬀerent cut-oﬀs from ζ1 (h−2δ d2 (x, π(Λ)). It turns out to be suﬃcient just to localize our argument as follows. For any C ∞ map such as π : Λ → M , there exists a relatively open dense set U on which π : Λ → π(Λ) attains its maximum rank. By the implicit function theorem, there further exists a relatively open set V ⊂ U on which π : V → π(V ) is one to one. For small , the image of the normal bundle of radius along the relative interior of π(V ) exponentiates to a product tubular neighborhood of the form π(V ) × Dn−k where Dr is the r-dimensional ball of radius . We use the corresponding Fermi normal product coordinates (y, v). We pick a function FV (y), compactly supported in the interior of π(V ) and equal to one on a somewhat smaller open subset V ⊂ π(V ) and with the same δ as above we deﬁne the cut-oﬀ χδ1 (x; ) = FV (y)ζ1 (−2δ |v|2 ) .

(73)

Due to FV there is no singularity at the boundary of F (V ). The cut-oﬀ is smooth and satisﬁes (i) supp χδ1 ⊂ Tδ (π(V )) n−r (ii) χδ1 = 1 on a product neighborhood of the form V × D3/4h δ; (iii) |∂xα χδ1 (x; )| ≤ Cα h−δ|α| . Correspondingly, we deﬁne χδ2 (x, ξ; ) := FV (θ)ζ2 (−2δ d˜2 ((x, ξ); V )), θ ∈ V, π(θ) = y.

(74)

Here, we are using the singular action-angle coordinates (I, θ) near Λ. The argument is then the same as in the embedded case, except that we need to multiply by the additional cut-oﬀ factor FV (θ) in the angle coordinate. If π : V → π(V ) has rank k, this modiﬁcation therefore does not change the remaining calculations in any essential way, since the eigenfunctions corresponding to the singular action variables, i.e., the exponentials eik,θ , have modulus one. So the dθ integral simply becomes the integral of FV over Λ rather than the integral of 1, and this merely changes the lower bounds in (68)–(69) by an -independent factor. If on the other hand, π : V → π(V ) has rank < k, then the estimates actually improve because the tube volume in (72) decays to a higher power in . We leave the details of this to the reader.

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References [AM]

R. Abraham and J.E. Marsden, Foundations of mechanics, second edition, Benjamin/Cummings (1978).

[AS]

M. Abramovitz and R. Stegun, Handbook of mathematical functions, Dover (1970).

[BPU] R. Brummelhuis, T. Paul, A. Uribe, Spectral estimates around a critical level, Duke Math. J. 78 no. 3, 477–530 (1995). [BI]

D. Burago, and S. Ivanov, Riemannian tori without conjugate points are ﬂat, Geom. Funct. Anal. 4 no. 3, 259–269 (1994).

[CP]

Y. Colin de Verdiere and B. Parisse, Equilibre instable en regime semiclassique I: concentration microlocale, Comm. in P.D.E. 19, 1535–1563 (1994).

[DSj]

M. Dimassi and J. Sj¨ ostrand, Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999.

[D]

H. Donnelly, Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds, J. Funct. Anal. 187 no. 1, 247–261 (2001).

[El]

L.H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Comment. Math. Helv. 65, 4–35 (1990).

[El2]

L.H. Eliasson, Hamiltonian systems with Poisson commuting integrals. PhD thesis, University of Stockholm (1984).

[K]

V.V. Kozlov, Topological obstacles to the integrability of natural mechanical systems, (Russian) Dokl. Akad. Nauk SSSR 249, no. 6, 1299–1302 (1979).

[LS]

E. Lerman and N. Shirokova, Toric integrable geodesic ﬂows, Math. Res. Lett. 9, no. 1, 105–115 (2002) (arXiv math.DG/0011139).

[L2]

E. Lerman, Geodesic ﬂows and contact toric manifolds, preprint xxx.lanl.gov, math.SG/0201230.

[L3]

E. Lerman, personal communication.

[M]

R. Mane´e, On a theorem of Klingenberg. Dynamical systems and bifurcation theory (Rio de Janeiro, 1985), 319–345, Pitman Res. Notes Math. Ser., 160, Longman Sci. Tech., Harlow, 1987.

[N]

N.T. Zung, Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities, Comp. Math. 101 no. 2, 179–215 (1996).

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J. Sj¨ ostrand, Semi-excited states in nondegenerate potential wells, Asymp. Anal. 6, 29–43 (1992).

[TAI] Ta˘ımanov, I. A. Topology of Riemannian manifolds with integrable geodesic ﬂows, Proc. Steklov Inst. Math., no. 4 (205), 139–150 (1995). [T1]

J.A. Toth, Eigenfunction localization in the quantized rigid body, J. Differential Geom. 43, no. 4, 844–858 (1996).

[T2]

J.A. Toth, On the quantum expected values of integrable metric forms, J. Diﬀerential Geom. 52, no. 2, 327–374 (1999).

[TZ1] J.A. Toth and S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111(1), 97–132 (2002). [TZ2] J.A. Toth and S. Zelditch, Norms of modes and quasi-modes revisited, to appear Proceedings of the Mt. Holyoke Conference on Harmonic Analysis, Eds. W. Beckner, A. Nagel, A. Seeger, and H. Smith, AMS Contemporary Math. Series. [VN]

S. Vu Ngoc, Formes normales semi-classiques des syst`emes compl`etement int´egrables au voisinage d’un point critique de l’application moment, Asymptot. Anal. 24, no. 3–4, 319–342 (2000).

[VN2] S. Vu Ngoc, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure and Applied Math. 53 no. 2, 143–217 (2000). John A. Toth1 Department of Mathematics and Statistics McGill University Montreal Canada, H3A-2K6 email: [email protected] Steve Zelditch2 Department of Mathematics Johns Hopkins University Baltimore, MD 21218 USA email: [email protected] Communicated by Bernard Helﬀer submitted 13/06/02, accepted 5/02/03 1 Research partially supported by an Alfred P. Sloan Research Fellowship and NSERC grant #OGP0170280 2 Research partially supported by NSF grant #DMS-0071358

Ann. Henri Poincar´e 4 (2003) 369 – 383 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020369-15 DOI 10.1007/s00023-003-0133-9

Annales Henri Poincar´ e

On the Topology of Vacuum Spacetimes James Isenberg, Rafe Mazzeo and Daniel Pollack Abstract. We prove that there are no restrictions on the spatial topology of asymptotically ﬂat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Σn to an asymptotically Euclidean solution of the constraints on Ên . For any Σn which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically ﬂat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.

1 Introduction A basic question in general relativity is whether there are any restrictions on the topology of the spacetime manifold M n+1 of a physically reasonable solution of Einstein’s equations. If we restrict attention to globally hyperbolic solutions, so that M n+1 = Σn × R, and appeal to well-known results on the local well-posedness of the Einstein equations [6], then this question reduces to whether there are any restrictions on the topology of manifolds Σn which carry physically realistic solutions of the Einstein constraint equations. The initial data on Σ is a pair of symmetric 2-tensors (γ, Π), where γ is a Riemannian metric and Π represents the second fundamental form in a Lorentzian development. The vacuum constraint equations are the compatibility conditions on these initial data sets arising from the putative embedding in a Ricci-ﬂat Lorentz manifold. They take the form div Π − ∇tr Π = 0 2

2

R − |Π| + (tr Π)

= 0

(1) (2)

All geometric quantities, norms and operators here are computed with respect to γ, and in particular R is the scalar curvature of this metric. We write τ = tr Π and call this the mean curvature function of the initial data set. Of particular interest and simplicity are the data sets with τ constant, and these are called constant mean curvature, or CMC. For Σ compact, there are always CMC data sets on Σ which satisfy the constraints: Proposition 1 If Σn is compact, then it admits solutions of the vacuum constraint equations (1) and (2) with constant mean curvature τ = 0.

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In fact, any compact Σ admits a metric γ of constant scalar curvature R = −n(n − 1). Setting Π = γ, then it is straightforward to check that (γ, γ) solves both (1) and (2) with τ = n. By rescaling we obtain solutions with τ an arbitrary positive constant. Related to this result is the work of Witt [24]. When Σ is compact, any smooth function which is negative somewhere is the scalar curvature function for some metric on Σ [18], and he uses this to produce non-vacuum ‘dust’ solutions to the constraints on Σ with arbitrarily prescribed non-negative energy density. Witt also addresses the question of whether there are topological restrictions for asymptotically Euclidean solutions of the constraints. His results in this setting only hold for nonvacuum solutions, however. More speciﬁcally, he describes a procedure for gluing a solution of the constraints on an arbitrary manifold Σ with nonvanishing energy density to a set of time symmetric initial data for a Schwarzschild solution. This relies crucially on the nonvanishing of the energy density; hence his construction shows that there are asymptotically Euclidean solutions of the non-vacuum constraints on Σ \ {p} for any compact manifold Σ, but it says nothing about solutions of the vacuum constraints. In fact, in his construction, the mass of the exterior Schwarzschild solution depends on the energy density of the interior, with the Schwarzschild mass equal to zero if the energy density vanishes. So if Witt’s construction could be extended to the vacuum (non-ﬂat) case, it would produce nonﬂat solutions of the vacuum constraints which would be exactly Euclidean outside a compact set. This would violate the positive mass theorem [23]. The recent work of Corvino [11] shows that one can glue an exact exterior Schwarzschild metric (with non-zero mass) to compact subsets of fairly general time symmetric, asymptotically Euclidean initial data sets (with vanishing energy density). We emphasize, though, that Corvino’s construction begins with a pre-existing asymptotically Euclidean solution of the vacuum constraint equations. Hence, Corvino’s work does not bear on the issue of topological restrictions for asymptotically Euclidean solutions. Are there any restrictions on the topology of asymptotically Euclidean vacuum initial data sets? We shall prove that this is not the case. Theorem 1 Let Σ be any closed n-dimensional manifold, and p ∈ Σ arbitrary. Then Σ \ {p} admits an asymptotically Euclidean initial data set satisfying the vacuum constraint equations. We are not claiming that this solution is CMC. In fact, a CMC asymptotically Euclidean initial data set necessarily has τ = 0, so that (2) becomes R = |Π|2 ≥ 0. In addition, an asymptotically Euclidean metric on Σ \ {p} with non-negative scalar curvature is conformally equivalent to (the restriction of) a metric on Σ with positive scalar curvature. This limits the possibilities for the topology of Σ dramatically, cf. [22], [12] and [13]. For example, when n = 3, this implies that Σ is

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the connected sum of manifolds with ﬁnite fundamental group (i.e., the quotient of a homotopy sphere with positive scalar curvature by a ﬁnite group of isometries) and a ﬁnite number of copies of S 2 × S 1 . The paper [20] surveys some of what is known in higher dimensions. Suppose we apply this construction when Σ is a compact manifold which admits no metric of positive scalar curvature. We show in §6 that, subject to an extra hypothesis which appears in the statement of Theorem 4, the maximal spacetime development of this data, which is asymptotically ﬂat, admits no maximal slices. The existence of spacetimes with this property was heretofore unknown. Theorem 1 is proved by an analytical gluing method closely related to our earlier work [16]. More speciﬁcally, we produce solutions on Σ \ {p} by joining together a CMC solution on Σ with a non-CMC solution on Rn . The main result of [16] is that arbitrary nondegenerate (in a sense we explain below) solutions of the constraints on manifolds which are CMC and either compact, asymptotically Euclidean or asymptotically hyperbolic may be glued together. In particular, the method of [16] shows that if Σ is any compact 3-manifold, then Σ \ {p} admits an asymptotically hyperbolic solution of the vacuum constraint equations. The gluing construction in the present paper closely follows that earlier work, but the new feature here is the use of non-CMC solutions on the asymptotically Euclidean summand. As we explain in the next section, this complicates the analysis slightly because the linearizations of (1), (2) uncouple when τ is constant, but not otherwise. In the next section we review the conformal method for solving the vacuum constraints. In §3 we use the implicit function theorem to ﬁnd appropriate nonCMC asymptotically Euclidean solutions on Rn . The gluing is done in two steps: a family of approximate solutions is produced on Σ#Rn , and then these are perturbed using a contraction mapping argument to exact solutions. These steps are reviewed in §4 and §5, respectively. In §6 we discuss the existence of asymptotically ﬂat vacuum spacetimes with no maximal slices. A crucial idea in this analysis is the notion of nondegeneracy of a solution, which concerns the surjectivity of the linearized operator. We explain this concept in the next section. Theorem 1 is a special case of a more general gluing theorem for non-CMC initial data sets. Theorem 2 Let (Σj , γj , Πj ), j = 1, 2, be two initial data sets which solve the vacuum constraint equations; these may be either compact, asymptotically Euclidean or asymptotically hyperbolic. Suppose that both solutions are nondegenerate with respect to the appropriate function spaces (which contain functions weighted at inﬁnity if either factor is noncompact). If the mean curvature functions τj are both equal to the same constant τ0 in a neighborhood of the points pj ∈ Σj , j = 1, 2, then the manifold Σ1 #Σ2 obtained by forming a connected sum based at these two points again carries a one-parameter family of solutions of the vacuum constraint equations. Moreover, for large values of the parameter, these solutions are small

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perturbations of the original initial data sets (γj , Πj ) outside of small balls around the points pj . This more general result is proved in almost exactly the same way as the more specialized Theorem 1, and so we provide details only in the special case. Note that in Theorem 2 one can also take the points pj ∈ Σ, j = 1, 2 as lying in the same connected manifold. In this case, rather than forming the connected sum, the theorem produces a family of solutions on the manifold obtained from Σ by adding a handle diﬀeomorphic to S n−1 × R. The authors wish to thank the American Institute of Mathematics, the National Science Foundation and the Stanford Mathematics Department for funding the extended Workshop on General Relativity in the Spring of 2002, during which this work was begun.

2 The conformal method A very useful tool for the construction and enumeration of solutions to the vacuum constraint equations is the conformal method of Lichnerowicz, Choquet-Bruhat and York, and many of the basic existence results for these equations, e.g., [14], [17], [5], [9], [10], [7], [8], [1], rely on it. This method is most successful when dealing with CMC data because in this case the equations decouple. It can also handle non-CMC data, as shown in some of these references, and as our work shows here, although results to date suggest that one place restrictions on the size of the gradient of the mean curvature. Solutions are constructed via the conformal method as follows. One begins by ﬁxing a background metric γ (representing a given conformal structure), and a symmetric (0, 2) tensor Π which decomposes into trace-free and pure trace parts as µ + nτ γ. The mean curvature function τ is speciﬁed through the second term on the right. (Note that one often makes the additional demand that µ is transversetraceless, i.e., also divergence-free. This is useful in parametrizing the set of solutions to the vacuum constraint equations but is misleading for our current purposes.) We then modify this data by a conformal factor and a ‘gauging’ term by setting 4 4 = φ−2 (µ + DW ) + τ φ n−2 Π γ, (3) γ˜ = φ n−2 γ, n where φ and W are a positive function and a vector ﬁeld, respectively. Note that ˜ The operator D, which maps vector the mean curvature is preserved, τ = trγ˜ (Π). ﬁelds to trace-free symmetric (0, 2) tensors, is the conformal Killing operator, and is given in local coordinates by the formula 1 1 1 1 LX γ − (divX)γ, (DX)jk = (Xj;k + Xk;j ) − div (X) γjk . 2 n 2 n We have DX = 0 if and only if X is a conformal Killing ﬁeld. The formal adjoint of D on trace-free tensors is D∗ = −div and the operator L ≡ D∗ ◦ D is formally self-adjoint, nonnegative and elliptic. DX =

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The modiﬁed data (3) satisﬁes the vacuum Einstein constraint equations (1) and (2) if and only if ∆γ φ −

2 −3n+2 n+2 n−2 n − 2 n − 2 2 n−2 Rγ φ + µ + DW φ n−2 − τ φ 4(n − 1) 4(n − 1) 4n 2n n − 1 n−2 φ LW − (div µ − ∇τ ) n

=

0 (4)

=

0 (5)

The ﬁrst of these is usually called the Lichnerowicz equation. We write this coupled system as N (φ, W, τ ) = 0. The mean curvature τ is emphasized in this notation; however, the dependence of N on γ and µ is suppressed. The linearization L of N in the directions (φ, W ) (but not τ ) is central to our construction. Definition 1 A solution to the constraint equations N (φ, W, τ ) = 0 is nondegenerate with respect to Banach spaces X and Y provided L : X → Y is an isomorphism. Remark 1 It might seem more natural to require only that L be surjective. However, in the main cases of interest, when Σ is compact, asymptotically Euclidean or asymptotically hyperbolic, these are equivalent (provided we use spaces of functions which decay at inﬁnity). Nondegeneracy conditions like this one are crucial to any gluing construction. The main result of [16] is that any two nondegenerate solutions of the vacuum constraint equations with the same constant mean curvature τ can be glued. For compact CMC solutions, nondegeneracy is equivalent to Π ≡ 0 together with the absence of conformal Killing ﬁelds. On the other hand, asymptotically Euclidean or asymptotically hyperbolic CMC solutions are always nondegenerate (cf. §7 of [16]). While the paper [16] only treats the case n = 3, the generalization to higher dimensions is not diﬃcult; this is discussed in [15], which also considers the extension to various types of non-vacuum solutions. Suppose (φ, W, τ ) solves (4) and (5) with background data (γ, Π); then we ˜ of the constraints (1) and (2), deﬁned can choose the resulting solution (˜ γ , Π) by (3), as the new background data. For the moment, all objects with tildes are associated to this new data. Let us determine the solution of N˜ (·, ·) = 0 associated to this new data. Obviously the conformal factor φ˜ = 1, but we need to ﬁnd the ˜ is nondegenerate, then we new vector ﬁeld. If we assume that our solution (˜ γ , Π) ˜ ˜ ˜ ˜ ˜ in the appropriate may uniquely solve LW = (div µ − ∇τ ) (since φ = 1) for W ˜ summand of X. Thus N (1, W ) = 0. We drop the tildes henceforth.

3 Non-CMC asymptotically Euclidean initial data on Rn A metric γ on Rn is said to be asymptotically Euclidean, or AE, if γ decays to the Euclidean metric at some rate. More precisely, we assume that there is a ν > 0 so that in Euclidean coordinates z, |γij (z) − δij | ≤ C|z|−ν , along with appropriate decay of the derivatives, as z → ∞. To formulate this precisely, we make the

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k,α Definition 2 The family of weighted H¨ older spaces C−ν (Rn ) is deﬁned as follows:

C00,α (Rn ) =

  

0,α u ∈ Cloc (Rn ) ∩ L∞ (Rn ) : sup

R≥1

sup

z=z R≤|z|,|z |≤2R

  |u(z) − u(z )|Rα < ∞  |z − z |α

C0k,α (Rn ) = {u : (1 + |z|2 )|β|/2 ∂zβ u ∈ C00,α (Rn ),

for |β| ≤ k},

k,α C−ν (Rn ) = {u = (1 + |z|2 )−ν/2 v : v ∈ C0k,α (Rn )}. k,α (Rn ) provided it is in the ordinary H¨ older space C k,α on any comThus u ∈ C−ν pact set and also satisﬁes |∂zβ u| ≤ Cβ |z|−ν−|β| as |z| → ∞, for |β| ≤ k, with an appropriate decay condition for the α H¨ older seminorm on the k th derivatives of u. We deﬁne k,α n Mk,α −ν = {metrics γ on R : γij − δij ∈ C−ν }. k,α , then its scalar curvature function Rγ is in C−ν−2 . Thus Note that if γ ∈ Mk+2,α −ν (4) and (5) suggest that we should assume that k+1,α W ∈ C−ν/2 (Rn , T Rn ),

and

k,α τ ∈ C−ν/2−1 (Rn ).

We now assume that µ ≡ 0 and look for solutions of the two equations (4) and (5) with (φ, W ) close to (1, 0) in appropriate weighted H¨ older spaces. Set

k+2,α k+1,α k,α X = (φ, W, τ ) : φ ∈ C−ν (Rn ), W ∈ C−ν/2 (Rn , T Rn ), τ ∈ C−ν/2−1 (Rn ) and k,α k−1,α (Rn ) × C−ν/2−2 (Rn , T Rn ). Y = C−ν−2

Then it is obvious that X (φ, W, τ ) −→ N (φ, W, τ ) ∈ Y is a C 1 mapping in some neighbourhood U of (1, 0, 0) ∈ X. Let L denote the linearization D12 N of N in the (φ, W ) directions, evaluated at the solution (1, 0, 0). Then L(ψ, Z) = (∆ψ, LZ), and clearly k+2,α k+1,α (Rn ) × C−ν/2 (Rn , T Rn ) −→ Y. L : C−ν

(6)

The fact that this linearization decouples reﬂects that we are linearizing about a CMC solution.

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Theorem 3 Let N denote the vacuum constraint operator as above evaluated at the ﬂat metric on Rn with vanishing extrinsic curvature. Fix 0 < ν < n − 2. Then for some δ > 0, there exists a C 1 mapping k+2,α k+1,α F : {τ : ||τ ||k,α,−ν/2−1 < δ} −→ C−ν (Rn ) × C−ν/2 (Rn , T Rn )

such that N (F (τ ), τ ) ≡ 0 (and all solutions near (1, 0, 0) are of this form). In particular, we may solve the vacuum constraint equations for any speciﬁed mean k,α (Rn ). curvature function τ with suﬃciently small norm in C−ν/2−1 Proof. This follows immediately from the implicit function theorem once we recognize that for ν in this range, L is an isomorphism, cf. [9], [7] for the analogous statement on weighted Sobolev spaces. k,α We shall apply this theorem by choosing a function τ ∈ C−ν/2−1 (Rn ) with suﬃciently small norm, and which is equal to a nonzero constant on a ball around the origin. Writing F (τ ) = (φ, W ), then (φ, W, τ ) is a non-CMC solution of the vacuum constraints. Proposition 2 The linearization L = D12 N of the operator N in the ﬁrst two slots, evaluated at this special solution (φ, W, τ ), is an isomorphism in (6) when δ is suﬃciently small. Hence (φ, W, τ ) is a nondegenerate solution. This follows simply because the invertibility is an open condition, hence is stable under perturbations of small norm. Finally, notice that we can scale this solution by dilations on Rn so that τ is equal to any desired constant on a (possibly smaller) ball around the origin. We assume that this has been done so that τ = n on B2R (0) for some R > 0. These dilated solutions are still nondegenerate. We summarize this as Proposition 3 There exist nondegenerate, asymptotically Euclidean, non-CMC solutions of the vacuum constraint equations (γ, Π) on Rn which satisfy τ ≡ n on B2R (0) ⊂ Rn for some R > 0. These solutions are given as (1, W, τ ) (or equivalently as (1, µ, τ ) with µ = DW ) relative to the metric φ4/(n−2) δ with (φ, W, τ ) ∈ X.

4 The approximate solutions We now sketch the construction of the family of approximate solutions. This proceeds exactly as in [16] and so we refer to §2 of that paper for details. We include a brief description here both to set notation for what follows and so that the geometry of the construction can be readily understood without reference to [16]. Our two summands are the manifolds Σ and Rn . These each have solutions of the vacuum constraints, which we write as (γj , φj , µj , τj ), j = 1, 2 (with j = 1 corresponding to Σ and j = 2 to Rn ). The metric γ1 is provided by Proposition 1 and corresponds to the solution with φ1 ≡ 1, µ1 ≡ 0, and τ1 ≡ n. Note that if

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(Σ, γ1 ) has non-trivial conformal Killing ﬁelds, hence is degenerate, small perturbations of the conformal class of γ1 generically have none. We then appeal to [14] to ﬁnd the corresponding solutions of the constraints, which by construction are now nondegenerate. The solution (γ2 , φ2 , µ2 , τ2 ) on Rn is provided by Proposition 3 with φ2 ≡ 1, τ2 ≡ n in B2R (0) ⊂ Rn . Note that we are including the functions φj ≡ 1, for j = 1, 2, to emphasize that, relative to the other pieces of data, these are solutions to the Lichnerowicz equation (4) on each summand. We begin by removing a small ball of radius R around the point p ∈ Σ and also around the origin in Rn . (These points will be written p1 and p2 .) The remaining manifolds both have boundaries diﬀeomorphic to S n−1 . The usual connected sum construction proceeds by identifying these copies of S n−1 , and we denote the = Σ#Rn . The mean curvature functions τ1 and τ2 have an resulting manifold Σ obvious smooth extension, which we denote by τ , to all of Σ. We now construct a one-parameter family of metrics γT and symmetric 2tensors ΠT which serve as background data (in the conformal method) for a family a family of metrics of approximate solutions. More speciﬁcally, we construct on Σ n γT , functions ψT (which are equal to 1 on (Σ \ BR (p)) ∪ (R \ BR (0))), and tracefree (0, 2) tensors µT (which are equal to 0 on Σ \ BR (p) and µ2 = on Rn \ BR (0), respectively), such that N (ψT , 0, τ ) = ET , where the error term ET is exponentially small in T as T → ∞ (see [16] for a geometric description of the parameter T ). Note that here the constraint operator, N (·, ·, τ ), is computed with respect to γT and µT . This part of the construction is completely localized on the ‘neck region’ bridging the two summands. These approximate solutions are perturbed in the next section to a family of exact solutions of the constraints, and it is only at this last step that we introduce a global correction term, which is exponentially small in T . To construct these approximate solutions we ﬁrst choose conformal factors ψj on each of the summands which are identically one outside the balls B3R/2 (pj ) and which are equal to (dist (·, pj ))(n−2)/2 in BR (pj ). We then deﬁne 4 − n−2

(γj )c = ψj

γj .

These are complete metrics with asymptotically cylindrical ends in place of the punctured balls BR (pj ). There is a decomposition of the Πj into trace-free and pure-trace parts, τ1 τ2 Π2 = µ2 + γ2 , Π1 = γ1 , n n (Notice that µ2 is not transverse-traceless: it has non-vanishing divergence since ∇τ2 ≡ 0.) Let (µ2 )c = ψ22 µ2 and set (µ1 )c = µ1 = 0. The discussion from §2 shows that ((γj )c , ψj , (µj )c , τj ) are solutions (with complete metrics) of the constraint equations on Σ \ {p} and Rn \ {0}, respectively. If rj = distγj (·, pj ), then tj = − log rj is a natural linear coordinate on each of these cylindrical ends. Let T be a large parameter and A = − log R. Truncate the

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cylindrical ends of each of these manifolds by omitting the regions where tj > A+T . is obtained by identifying the two ﬁnite A smooth manifold diﬀeomorphic to Σ cylindrical segments {(tj , θ) : A ≤ tj ≤ A + T } via the map (t1 , θ) → (T − t1 , −θ). We call this new manifold ΣT , and denote the long cylindrical tube it contains by CT . It is convenient to use s = t1 − A − T /2 = −t2 + A + T /2 as a linear coordinate on CT . This parametrizes CT via the chart (s, θ) ∈ [−T /2, T /2] × S n−1 . We next deﬁne the family of metrics γT and trace-free tensors µT on ΣT . Choose cut-oﬀ functions χ1 , χ2 so that χ1 = 1 on all of Σ and vanishes in the ball Be−T /2 R (p1 ) and similarly for χ2 , and moreover such that when these functions are transferred to ΣT , then χ2 = 1 − χ1 and the supports of dχj are contained in the region Q = [−1, 1]s × Sθn−1 ⊂ CT . Now deﬁne γT = χ1 (γ1 )c + χ2 (γ2 )c ,

µT = χ2 (µ2 )c .

Notice that γT = (γj )c and µT = (µj )c , where j = 1 when s ≤ −1 and on the rest of Σ and j = 2 when s ≥ 1 and on the rest of R3 . The key point to emphasize here is that (µ2 )c is very close to zero in the region where we have cut it oﬀ to be exactly zero, so this introduces only a small error. This is described in more detail in §2 of [16]. The conformal factors ψj on either summand must be joined together somewhat diﬀerently, by cutting them oﬀ at the far ends of the cylinder (relative to their domain of deﬁnition), as follows. We choose nonnegative cut-oﬀ functions χ ˜1 on Σ \ {p} and χ ˜2 on Rn \ {0} which are identically one for tj ≤ A + T − 1 and which vanish when tj ≥ A + T . Now set ˜1 (ψ1 )c + χ ˜2 (ψ2 )c . ψT = χ This is deﬁned on ΣT , is identically 1 away from CT , and equals ψ1 + ψ2 on most of the cylinder except at the ends. The estimates for the error term ET = N (ψT , 0, τ ) now follow readily from the estimates in §3.4, §4 and §6 of [16]. In particular, writing ET = (ET1 , ET2 ) ∈ Y (corresponding to (4) and (5), respectively) we have that ET1 k,α,−ν−2 + ET2 k−1,α,−ν/2−2 ≤ Ce−T /4 ,

(7)

and furthermore, the components ET1 and ET2 of ET are supported on all of CT and Q ⊂ CT , respectively. In establishing this estimate it is important to note that, since ∇τ ≡ 0 on CT , the vector constraint equation (5) does not introduce any new error terms beyond those previously encountered in [16].

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5 The perturbation result We now sketch the argument to perturb the approximate solution 4 4 T = ψ −2 µT + τ ψ n−2 γT Π γ T = ψTn−2 γT , T n T ¯ T ) of the vacuum on ΣT , to an exact, asymptotically Euclidean solution (¯ γT , Π constraints, corresponding to a solution (φT , ZT ) to N (φT , ZT , τ ) = 0 on ΣT (where N (·, ·, τ ) is computed relative to γT and µT ), when T is suﬃciently large. Here φT = ψT +φ and the pair (φ, ZT ) are small in the appropriate function space. Definition 3 Let wT be an everywhere positive smooth function on ΣT which equals e−T /4 cosh(s/2) on CT and which equals 1 outside both balls B2R (pj ). For any δ ∈ R, and any φ ∈ Cνk,α (ΣT ), set ||φ||k,α,ν ,δ = ||wT−δ φ||k,α,ν ; the corresponding space is denoted Cνk,α ,δ (ΣT ). Thus elements of this function space not only have restricted growth or decay at inﬁnity, but their norms are also measured in CT with an extra weighting factor. We let

k+2,α k+1,α (ΣT ), Z ∈ C−ν/2 (ΣT ; T ΣT ) XT,δ = (φ, Z) : φ ∈ C−ν,δ and

k,α k−1,α (ΣT ) × C−ν/2−2 (ΣT ; T ΣT ). YT,δ = C−ν−2,δ

Notice that we are only including the weight δ on the ﬁrst component of this space, but do not measure the vector ﬁeld Z with a weighted norm in the neck. We use the obvious product norm on XT,δ , but the less obvious one ||(f, Y )||YT ,δ = ||f ||k,α,−ν−2,δ + T −3 ||Y ||k−1,α,−ν/2−2 on YT,δ . We also assume that the weight ν is always in the range (0, n − 2), so that the conclusion of Proposition 2 is valid. The mapping ˜ (φ, Z) ≡ N (ψT + φ, Z, τ ) ∈ YT,δ XT,δ (φ, Z) −→ N is C 1 in some small neighbourhood U of the origin in XT,δ for each T . The only subtlety here is that even when it has small norm, the function φ may be rather large in a pointwise sense in CT . However, ∇τ = 0 in CT and so this does not ˜ at (φ, Z) = (0, 0) aﬀect (5) there. Furthermore, if we write the linearization of N as LT , then LT : XT,δ −→ YT,δ is bounded as well. We state the two fundamental results, which are essentially obtained by combining Propositions 7 and 8 from §5 and Corollary 1 from §3.3 in [16].

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Proposition 4 Fix any δ ∈ R. For T suﬃciently large, the mapping LT : XT,δ −→ YT,δ is an isomorphism. Let GT denote the inverse of LT provided by this proposition. Thus GT : YT,δ −→ XT,δ

(8)

and LT GT = GT LT = I. Of course GT also depends on δ, but we suppress this in the notation. Proposition 5 If 0 < δ < 1, then the norm of GT is uniformly bounded as T → ∞. Although we refer to [16] for the proofs, let us make a few comments. Proposition 4 reﬂects the fact that the solutions we are joining are nondegenerate on their respective summands. One way to prove this result is to patch together the (pseudodiﬀerential) inverses on each piece with the inverse on CT (which is constructed rather explicitly in [16]). The resulting parametrix GT satisﬁes LT GT = I − RT , where RT has very small norm. As for Proposition 5, the reason we have added the T −3 factor in the second component in the deﬁnition of YT,δ is because the inverse of the vector Laplacian L = D∗ D localized to the neck region CT has norm bounded by T 3 (cf. §3.3 of [16]). This proposition is most handily proved by assuming this uniform bound fails and arguing to a contradiction. From here, the rest of the proof of Theorem 1 is straightforward. The system we wish to solve is written as N (ψT + φ, Z, τ ) = 0. We write this as L(φ, Z) = FT (φ, Z) where FT depends on all of the approximate data and consists of the error term ET together with a nonlinear operator which is quadratically small. Using Proposition 4 this may in turn be written as (φ, Z) = GT (FT (φ, Z)) The existence of a ﬁxed point for this map then follows from an application of the contraction mapping principle using Proposition 5 and the error estimate (7). This is explained carefully in §6 of [16]. This completes the proof of Theorem 1. We emphasize that the reason no substantial changes need to be made to any of these arguments is that their most diﬃcult aspects involve the explicit analysis of the linearizations of the Lichnerowicz operator (4) and the vector Laplacian (5) in CT and the estimates of the approximate solutions in this same region. Because

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we are always assuming that ∇τ = 0 there, these parts of the arguments carry through completely unchanged. The remaining more global parts are quite general and it can be readily veriﬁed that they do not ‘notice’ the fact that these operators are now coupled. Very few modiﬁcations are required to prove the more general Theorem 2. In fact, we need only note that now each of the initial solutions may have a transverse-traceless part, but these are easily incorporated into all the arguments.

6 Asymptotically flat vacuum spacetimes with no maximal slices It has been commonly believed that every physically reasonable asymptotically ﬂat solution of the Einstein equations should admit a foliation by asymptotically Euclidean, maximal (τ = 0) slices. Such foliations are unique when they exist, and they are useful because they have desirable “singularity avoidance” properties. Roughly 20 years ago, Brill showed [4] that there exist asymptotically ﬂat spacetimes which are dust solutions of the Einstein equations and which admit no maximal spacelike hypersurfaces. These are constructed by explicitly gluing a Friedman-Robertson-Walker spacetime to a Schwarzschild spacetime. The presence of dust is crucial to this argument. Our analysis here shows that there exist asymptotically ﬂat vacuum spacetimes which admit no maximal Cauchy slices (satisfying certain decay conditions). Indeed, if (M, g) is the maximal development of any initial data set constructed as in Theorem 1, with Σ admitting no metric of positive scalar curvature, then (M, g) admits no such maximal slice. As mentioned earlier, such manifolds Σ are quite abundant; for example, any closed hyperbolic 3-manifold has this property. The precise result is as follows: Theorem 4 Suppose that Σ is a closed n-manifold which admits no metric of positive scalar curvature. For any p ∈ Σ, let (γ, Π) be an asymptotically Euclidean solution of the vacuum constraint equations on Σ \ {p} provided by Theorem 1, and let (M, g) be the maximal development of this data. Then there exists no in M for which the maximal (τ = 0) asymptotically Euclidean Cauchy surface Σ k,α n−2 induced metric γˆ ∈ M−ν for some ν > 2 , and for which the scalar curvature R(ˆ γ ) ∈ L1 (Σ). γˆ ) has nonnegative Proof. We have already observed that any maximal slice (Σ, scalar curvature. Schoen and Yau have proved, cf. [19] or [21], that an asymptotically Euclidean manifold with nonnegative scalar curvature may be perturbed to an asymptotically Euclidean manifold which is scalar ﬂat and in addition conformally ﬂat near inﬁnity. Let γ denote this new metric. Then there exists a compact which is diﬀeomorphic to an exterior region in Rn such that in the asset K ⊂ Σ 4 sociated Euclidean coordinates, γ = u(x) n−2 δ with u(x) → 1 as |x| → ∞ (δ is the Euclidean metric). Since both δ and γ are scalar ﬂat, u(x) is harmonic and thus

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has an expansion u(x) = 1 + E |x|2−n + O(|x|)1−n ,

|x| → ∞;

here E is the total energy of γ , see [21]. The Kelvin transform u ¯(x) = |x|2−n −2 γ ) u(|x| x) of u is harmonic on some punctured ball B(0, ρ) \ {0}. Hence (Σ, may be conformally compactiﬁed as follows. Choose a positive smooth function G 4 so that G(x)−1 = u on Σ ¯(x) when |x| > 2R and G ≡ 1 on K. Then γ¯ = G− n−2 γ ∪ {∞}. Regard G as a function on Σ \ {p}. The extends smoothly to Σ ∼ = Σ facts that it is positive, diverges near p like dist(·, p)2−n , and is in the nullspace of the conformal Laplacian away from p (since γ is scalar ﬂat) shows that it is a constant multiple of Green’s function for the conformal Laplacian (for γ) on Σ. The positivity of Green’s function is equivalent to the positivity of the ﬁrst eigenvalue of the conformal Laplacian for γ, which in turn implies that this metric is conformally equivalent to a metric on Σ with positive scalar curvature. This is a contradiction and so the proof is ﬁnished. Remark 2 This equivalence between asymptotically Euclidean metrics of nonnegative scalar curvature and metrics of positive scalar curvature on ‘stereographic compactiﬁcations’ is well known, although the proof does not seem to be readily available in the literature. Note that by choosing mean curvature functions τ on Rn with suﬃciently fast decay one can use Theorem 1 to produce initial data sets which satisfy the asymptotic conditions required in Theorem 4. It is not at all clear, however, and may be quite subtle to prove, that any other asymptotically Euclidean Cauchy surface in the resulting maximal development must also satisfy these same decay conditions.

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[21] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, “Topics in Calculus of Variations” LNM 1365, Springer-Verlag, M. Giaquinta ed. (1987) 120–154. [22] R. Schoen and S.T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28, 159–183 (1979). [23] R. Schoen and S.T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65, 45–76 (1979). [24] D. Witt, Vacuum space-times that admit no maximal slice, Phys. Rev. Let. 75 No. 12, 1386–1389 (1986). James Isenberg1 University of Oregon Department of Mathematics Eugene, OR 97403-1221 USA email: [email protected] Rafe Mazzeo2 Stanford University Department of Mathematics Stanford, CA 94305 USA email: [email protected] Daniel Pollack University of Washington Mathematics Department Box 354350 Seattle, WA 98195-4350 USA email: [email protected] Communicated by Piotr Chrusciel and Sergiu Klainerman submitted 11/07/02, accepted 15/11/02

ics

1 Supported

by the NSF under Grant PHY-0099373 and the American Institute of Mathemat-

2 Supported

by the NSF under Grant DMS-9971975

Ann. Henri Poincar´e 4 (2003) 385 – 411 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/020385-27 DOI 10.1007/s00023-003-0134-8

Annales Henri Poincar´ e

Probl`emes de Cauchy sur un Cono¨ıde Caract´eristique pour les Equations d’Einstein (Conformes) du Vide et pour les Equations de Yang-Mills-Higgs Marcel Dossa Abstract. Existence and uniqueness results established in [7], [8] for the hyperbolic quasilinear Cauchy problem of second order, with initial data given on a characteristic cone are applied to solve partially or completely the Cauchy problem, with initial data prescribed on a characteristic cone, for some classical physical equations of gauge theories and gravitation. R´esum´e. Les r´ esultats d’existence et d’unicit´e ´ etablis dans [7], [8] pour le probl`eme de Cauchy quasi-lin´eaire hyperbolique du second ordre avec donn´ee initiale port´ee par un cono¨ıde caract´eristique, sont appliqu´es ` a la r´ esolution partielle ou compl`ete du probl`eme de Cauchy, avec donn´ee initiale sur un cono¨ıde caract´eristique, pour certaines ´equations physiques classiques des th´eories des champs de jauge et de la gravitation.

Introduction Dans des travaux pr´ec´edents (cf. [7], [8], [11]), pour des donn´ees initiales port´ees par un cono¨ıde caract´eristique C, le probl`eme de Cauchy pour des syst`emes d’´equations hyper-quasilin´eaires hyperboliques du second ordre a ´et´e r´esolu dans le domaine Y int´erieur `a C, sous des hypoth`eses de diﬀ´erentiabilit´e minimale sur les donn´ees. L’objet du pr´esent travail est d’appliquer les r´esultats d’existence et d’unicit´e ainsi obtenus a` la r´esolution du probl`eme de Cauchy, avec donn´ee initiale sur un cono¨ıde caract´eristique pour certaines ´equations physiques classiques des th´eories des champs et de la gravitation: les Equations de Yang-MillsHiggs, les Equations d’Einstein du vide, le syst`eme conforme r´egulier des Equations d’Einstein de [6]. Malheureusement cette application ne s’op`ere pas sans diﬃcult´es. En eﬀet, consid´erons dans un espace-temps M , un syst`eme d’´equations des champs relativistes ou des th´eories de jauge, avec des conditions initiales port´ees par une hypersurface S spatiale ou caract´eristique. Le probl`eme de Cauchy ainsi obtenu est, en g´en´eral, mal pos´e car les Equations des Champs exprim´ees dans un syst`eme de coordonn´ees quelconque ne sont pas, en g´en´eral, un syst`eme d’´evolution (hyperbolique ou parabolique). Pour r´esoudre cette diﬃcult´e, on est amen´e `a adjoindre au syst`eme des Equations des champs ´etudi´es une condition suppl´ementaire appel´ee “condition de jauge” qui, compte tenu de la structure profonde des Equations ´etudi´ees, doit poss´eder les propri´et´es suivantes:

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(i) lorsque cette condition de jauge est partout v´eriﬁ´ee dans l’Espace-temps, le syst`eme des Equations des Champs ´etudi´e se r´eduit a` un syst`eme d’´evolution. (ii) lorsque le syst`eme d’´evolution associ´e est partout v´eriﬁ´e et que la condition de jauge est v´eriﬁ´ee sur l’hypersurface initiale S, alors cette condition est partout v´eriﬁ´ee. Il en d´ecoule que lorsqu’une condition de jauge est choisie, le probl`eme de Cauchy pour les Equations des champs se scinde en deux sous-probl`emes: le probl`eme des contraintes initiales et le probl`eme de l’´evolution. Le probl`eme des contraintes initiales consiste `a construire, a` partir du choix arbitraire sur S de certaines composantes des champs appel´ees “donn´ees ind´ependantes”, des donn´ees initiales compl`etes telles que la solution du probl`eme de l’Evolution associ´e `a ces donn´ees initiales v´eriﬁe sur S la condition de jauge. Le probl`eme de l’Evolution consiste `a r´esoudre dans l’Espace-temps le syst`eme d’Evolution (appel´e encore Equations r´eduites) obtenu pour des donn´ees initiales compl`etes, solution du probl`eme des contraintes initiales. Ces remarques permettent de pr´eciser les r´esultats, en fait, partiels obtenus de la fa¸con suivante; nous avons r´esolu: 1) de fa¸con compl`ete, pour la jauge de Lorentz, le probl`eme de Cauchy pour les Equations de Yang-Mills-Higgs, en utilisant outre [7], [8], les r´esultats de [34] o` u Rendall a trait´e, sous des hypoth`eses de classe C ∞ sur les donn´ees, le probl`eme des contraintes initiales sur un cono¨ıde caract´eristique, 2) les probl`emes de l’Evolution associ´es, en jauge harmonique, respectivement aux Equations d’Einstein du vide et au syst`eme conforme r´egulier des Equations d’Einstein de Choquet Bruhat et Novello ([6]). Avant de consacrer la suite de ce travail a` l’´etude strictement math´ematique des probl`emes annonc´es, il convient de dire quelques mots sur les motivations physiques du probl`eme de Cauchy caract´eristique en Relativit´e g´en´erale. Le probl`eme de Cauchy caract´eristique intervient naturellement dans une grande vari´et´e de probl`emes de la Th´eorie de la Relativit´e g´en´erale. Plusieurs sp´ecialistes de cette th´eorie (cf. [1], [29], [31]) pensent que ce probl`eme survient beaucoup plus naturellement dans les questions relativistes que le probl`eme de Cauchy traditionnel a` donn´ees initiales port´ees par une hypersurface spatiale; ceci est dˆ u au fait qu’en Relativit´e g´en´erale, les directions isotropes et les hypersurfaces isotropes (qui sont des hypersurfaces caract´eristiques des Equations d’Einstein) de l’espace-temps relativiste que sont les horizons, l’inﬁni isotrope pass´e ou futur, les cˆ ones isotropes pass´e et futur d’un observateur de l’espacetemps, semblent jouer un rˆ ole plus fondamental que les hypersurfaces spatiales et les directions temporelles. Ainsi, par exemple, de nombreux probl`emes d’existence et d’unicit´e de propri´et´es physiquement int´eressantes ou bien pathologiques des espaces-temps relativistes se r´eduisent a` des probl`emes de Cauchy sur des horizons, l’inﬁni isotrope pass´e ou futur, le cˆ one isotrope pass´e ou futur d’un point, etc. . . (cf. Hajicek [23], [24], Newman et Unti [30], Christodoulou [4], [5], Moncrief

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et Isenberg [28], H. Friedrich [15], [19], Kannar [26], Balean [1], Balean et Bartnik [2]). Plusieurs relativistes ont soulign´e l’importance du probl`eme de Cauchy sur un cono¨ıde caract´eristique dans les questions de cosmologie; dans [12], on a montr´e que les observations astronomiques id´eales, `a grande ´echelle, de l’Univers sont ´equivalentes a` des donn´ees port´ees par le cˆone isotrope pass´e (ou futur) d’un seul point (ici et maintenant). Le probl`eme de Cauchy caract´eristique pour les Equations d’Einstein a des applications int´eressantes en Relativit´e num´erique (cf. [25], [35], [36]); ce succ`es relatif est dˆ u au fait que le probl`eme des contraintes initiales associ´e au probl`eme de Cauchy caract´eristique se r´eduit, lorsque le choix des donn´ees ind´ependantes est convenablement op´er´e, `a une hi´erarchie d’´equations diﬀ´erentielles ou d’´equations alg´ebriques plus faciles `a r´esoudre que les contraintes elliptiques du probl`eme de Cauchy traditionnel. L’un des probl`emes les plus importants en Relativit´e g´en´erale est l’estimation de la radiation gravitationnelle ´emise par les syst`emes isol´es; le probl`eme de Cauchy caract´eristique pour les Equations d’Einstein avec donn´ee initiale port´ee par le cˆone isotrope futur d’un point situ´e au voisinage des sources, constitue un moyen tr`es eﬃcace pour ´etudier ce probl`eme. Les espaces-temps relativistes repr´esentant la radiation gravitationnelle pure qui vient de l’inﬁni et interragit avec elle-mˆeme, peuvent se caract´eriser comme les solutions des Equations d’Einstein du vide poss´edant une structure r´eguli`ere `a l’inﬁni isotrope pass´e (qui constitue le cˆ one isotrope de sommet l’inﬁni temporel pass´e, avec des g´en´eratrices compl`etes (cf. [15])). Dans une s´erie de travaux [14], [15], [16], [17], [18], [19], H. Friedrich a r´eussi, en utilisant le formalisme spinoriel de Newman-Penrose et des m´ethodes de transformation conforme, `a r´eduire des probl`emes d’existence globale ou semi-globale, de comportement asymptotique (existence d’espaces-temps relativistes asymptotiquement plats au sens de Penrose), des solutions des Equations d’Einstein, a` des probl`emes de Cauchy locaux avec donn´ee sur l’inﬁni isotrope.

1 Rappels des r´esultats de [8] 1.1

Cadre g´eom´etrique et espaces fonctionnels utilis´es

Soient: C le demi-cono¨ıde de sommet 0 dans R4 d’´equation: • xo = s, s = (x1 )2 + (x2 )2 + (x3 )2 • Y l’int´erieur de C: Y= (xo , x1 , x2 , x3 ) ∈ R4 / s ≤ xo o ≤ T }, CT = C ∩ {xo ≤ T }, ΣT = C ∩ {xo = T }, • YT = Y ∩ {x 0 GT = Y ∩ x = T si T est un r´eel > 0.

Pour β = (β1 , β2 , β3 ) ∈ N3 , on note: ∂β =

∂ |β| (∂x1 )β1 (∂x2 )β2 (∂x3 )β3

.

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Pour v = (vr ), d´eﬁnie dans un domaine de C et pour p ∈ N, on pose:  1/2 β 2  ∂ vr dσ(Σt ) (si le second membre existe), vH p (Σt ,C) =   |β|≤p r

Σt

dσ(Σt ) ´etant la mesure induite sur Σt par dx1 dx2 dx3 . Pour α = (αo , α1 , α2 , α3 ) ∈ N4 , on note: Dα =

∂ |α| . (∂xo )αo (∂x1 )α1 (∂x2 )α2 (∂x3 )α3

Pour X = (Xr ) d´eﬁnie dans un domaine de Y et k ∈ N, on pose: k 2 2 D X = |Dα Xr | , |α|≤k

r

XH p (Gt ,Y) =

p k=0

k 2 1 2 3 D X dx dx dx

1/2

Gt

(si les seconds membres de ces ´egalit´es existent, les d´eriv´ees ´etant prises au sens des distributions). • C ∞ (Yt ) est l’espace des restrictions `a Yt des fonctions num´eriques de classe C ∞ sur R4 ∞ C∞ (Yt ) est le sous espace de C ∞ (Yt ) form´e des fonctions dont les d´eriv´ees de tous ordres sont nulles en 0. F p (Yt ) est l’espace des fonctions num´eriques X d´eﬁni par la norme: XF p (Yt ) = Ess sup τ −3/2 XH p (Gτ ,Y) . τ ∈]0,t]

F p (Ct ) est l’espace des fonctions num´eriques X d´eﬁni par la norme: XF p (Ct ) =

2p−1 I

r=0

Ess sup τ −2p+r XH p (Στ ,C) . τ ∈]0,t]

F p (Ct ) est la fermeture dans Fp (Ct ) de l’espace des restrictions `a Ct des ∞ (Yt ). fonctions de C∞ p F (Yt ) est le sous espace de F p (Yt ) d´eﬁni par la norme: XF p (Yt ) = XF p (Yt ) +

p−1 k [∂ X] o

k=o

o` u [ ] indique la restriction a` Ct et ∂ok =

∂k . (∂xo )k

F p−k (Ct )

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∞ (Yt ) dans F p (Yt ). F p (Yt ) est la fermeture de C∞ P k (Yt ) est l’espace des restrictions `a Yt des fonctions polynˆ omes sur R4 de degr´e ≤ k.

1.2

Enonc´es des r´esultats de [8]

On consid`ere le syst`eme d’´equations du second ordre: (E) Aλµ (xα , u)Dλµ u + f (xα , u, Du) = 0 2 I ∂ u ∂uI u = (uI ), Du = ( ν ), Dλµ u = , I = 1, . . . , N, ν, λ, µ = 0, 1, 2, 3. ∂x ∂xλ ∂xµ Hypoth`ese Hm • m est un entier ≥ 3 • Aλµ (xα , (uI )) est de classe C 2m−1 sur U × W, o` u U est un ouvert de R4 N contenant l’origine 0, W est un ouvert de R . Pour tout (xα , uI ) ∈ U × W , Aλµ (xα , uI ) d´eﬁnit une forme quadratique d´eﬁnie de signature + − avec Aoo > 0, Aij Xi Xj d´eﬁnie n´egative (i, j = 1, 2 ou 3). Il existe (aI ) ∈ W tel que Aλµ (0, aI ) = η λµ (m´etrique de Minkowski) • f (xα , uI , Dλ uI ) est de classe C 2m−3 dans U × W × W o` u W est un ouvert 4N de R . Si m = 3, les d´eriv´ees DuI f et DDν uI f sont de classe C 2m−3 . • ϕ = (ϕI ) est une fonction num´erique continue sur CT ; ϕ(0) = aI et CT est caract´eristique pour (E): Aoo (s, xl ; ϕI (xl ))−2

3 k=1

Aok (s, xl ; ϕI (xl ))

3 xk ij xi xj + A (s, xl ; ϕI (xl )) 2 = 0 s i,j=1 s

∀(s, xl ) ∈ CT , ϕ peut s’´ecrire ϕ = ϕ|CT + ϕ1 , o` u ϕ1 ∈ F m (CT ) et ϕ = (ϕI ) est une fonction vectorielle `a composantes polynomiales de degr´e ≤ 2(m − 1) telle que: ∂ϕI I I ϕ (0) = a , Dϕ(0) ≡ ( λ )(0) ∈ W . ∂x Th´eor`eme 1 Soit le probl`eme de Cauchy (∗): u solution de (E) dans Y et u = ϕ sur CT . Sous l’hypoth`ese Hm avec m ≥ 3, on a: 1) la donn´ee initiale ϕ du probl`eme de Cauchy (∗) peut se red´ecomposer en: ϕ = u|CT + ϕ 1 avec: • u fonction polynˆ ome de degr´e ≤ 2(m − 1) v´eriﬁant au point 0 l’´equation (E) et les ´equations d´eriv´ees jusqu’` a l’ordre 2(m − 2). m • ϕ 1 ∈ F (CT ) 2) ∃To ∈]0, T ] tel que le probl`eme de Cauchy (∗) admet, dans le domaine YTo , une solution unique u = (uI ) ∈ P 2(m−1) (YTo ) + Fm (YTo ).

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Cette solution se d´ecompose en: u = u + u1 ,

avec

u1 ∈ F m (YTo ), u1 (0) = 0, Du1 (0) = 0

3) Si les Aλµ sont de classe C ∞ et sont ind´ependants de u et si ϕ1 ∈ F m (CT ), alors u1 ∈ F m (YTo ). 4) Si f (xα , uI , Dλ uI ) est lin´eaire en Du, on peut remplacer la condition m ≥ 3 par m ≥ 2. 5) Si (E) est lin´eaire, on peut supposer m ≥ 2 et prendre To = T . Th´eor`eme 2 (Existence Globale dans YT ) Sous les mˆemes hypoth`eses et notations que le Th´eor`eme 1, on pose: ϕm,T = |u|m + ϕ 1 F m (CT ) avec |u|m = max Dα uI (0) . |α|≤2(m−1) I

On suppose en plus: f(xα , 0, 0) = 0 et les d´eriv´ees partielles d’ordre 2 de f par rapport aux variables u et Du sont de classe de C 2m−3 . Alors il existe un r´eel d > 0 tel que, si ϕm,T ≤ d la solution u = u + u1 est globale, c’est-` a-dire To = T .

2 Formulation des principaux r´esultats obtenus 2.1

Probl`eme de l’Evolution pour les Equations d’Einstein du vide en coordonn´ees harmoniques; Th´eor`eme 3

Les Equations d’Einstein du vide s’´ecrivent en coordonn´ees quelconques: Rαβ (g) = 0

(2.1.1)

o` u les Rαβ d´esignent les composantes du tenseur de Ricci du tenseur m´etrique hyperbolique inconnu g, de composantes gαβ . On sait (cf. [3]) que le tenseur de Ricci se d´ecompose sous la forme suivante: 1 gαλ ∇β F λ + gβλ ∇α F λ 2

(2.1.2)

1 2 Rαβ (g) ≡ − g λµ ∂λµ gαβ + hαβ (xν , gµν , ∂δ gµν ) 2

(2.1.3)

Rαβ (g) ≡(h) Rαβ (g) − avec: (h)

F λ (g) ≡ g αβ Γλαβ

(2.1.4)

la matrice de composantes g αβ ´etant l’inverse de celle de composantes gαβ , les Γλαβ ´etant les symboles de Christoﬀel de g. Les coordonn´ees sont dites harmoniques si on a: F λ (g) ≡ 0 partout.

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En coordonn´ees harmoniques, les Equations d’Einstein du vide se r´eduisent, vu (2.1.2) et (2.1.3), au syst`eme hyperbolique: (h)

Rαβ (g) = 0 .

Th´eor`eme 3 A Soit m un entier ≥ 3. Soit T un r´eel > 0. Soit donn´ee sur CT une m´etrique h = (hαβ ) v´eriﬁant: 1) hαβ = hαβ C + hαβ o` u hαβ ∈ P 2(m−1) (R4 ) et hαβ ∈ F m (CT ) T

2) hαβ (0) = hαβ (0) = ηαβ (m´etrique de Minkowski) a la m´etrique (hαβ ). 3) CT est isotrope relativement ` Alors: ∗ a) la donn´ee h peut se red´ecomposer sous la forme hαβ = g αβ C + hαβ o` u: T

• g αβ est une m´etrique sur R4 , g αβ ∈ P 2(m−1) (R4 ) et g αβ v´eriﬁe au point a l’ordre 2(m − 2). 0 (h) Rαβ (g) = 0 et les ´equations d´eriv´ees jusqu’` ∗ m • hαβ ∈ F (CT )

b) ∃To ∈]0, T ] et une unique m´etrique hyperbolique (gαβ ) = g telle que: • gαβ ∈ P 2(m−1) (YTo ) + F m (YTo ), (h) Rαβ (g) = 0 dans YTo et gαβ |CT = o hαβ Cette solution peut s’´ecrire: gαβ = g αβ + gαβ avec gαβ ∈ F m (YTo ) . B On pose: hm,T =

M ax

α,β 1≤|ν|≤2(m−1)

∗ ν D g αβ (0) + hαβ

F m (CT )

.

Alors il existe un r´eel d > 0 tel que si hm,T ≤ d la solution (gαβ ) est globale: To = T. C Si on suppose en plus des hypoth`eses de A ou B que les donn´ees de Cauchy (hαβ ) v´eriﬁent les contraintes suivantes: Il existe des fonctions γ = (γαβ ) d´eﬁnies sur CT telles que: αβ o` u γ αβ ∈ P 2m−3 (R4 ) et γαβ ∈ F m−1 (CT ) 4) γαβ = γ αβ C + γ T 5) γ = (γαβ ) v´eriﬁe les relations d´eduites des ´equations F λ (g) C = 0 et T (h) Rαβ (g) CT = 0 en rempla¸cant dans ces derni`eres [gαβ ], [∂o gαβ ], [∂i gαβ ] xi (i = 1, 2, 3) respectivement par hαβ , γαβ et ∂i [hαβ ] − γαβ ([ ] d´esigne la s restriction a ` C). Alors la solution g = (gαβ ) du probl`eme de Cauchy (h) Rαβ (g) = 0 dans Y et gαβ |C = hαβ est aussi solution des Equations d’Einstein du vide.

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Probl`eme de Cauchy sur un cono¨ıde caract´eristique pour les Equations de Yang-Mills-Higgs en jauge de Lorentz; Th´eor`eme 4

Soit (M, g) un espace-temps de dimension 4, de classe C ∞ , muni d’une m´etrique hyperbolique g de classe C ∞ . On ﬁxe sur un ouvert U de (M, g) un syst`eme de coordonn´ees g´eod´esiques normales (xα ) centr´e au point p. Soit C le demi-cono¨ıde de sommet p de U , isotrope par rapport a` g et d’´equation: xo = s avec s = (x1 )2 + (x2 )2 + (x3 )2 Soit Y l’int´erieur de C. On consid`ere dans l’espace-temps (Y, g) les Equations de Yang-Mills-Higgs usuelles de potentiel de Yang-Mills (Aα ) a` valeurs dans G 4 (alg`ebre de Lie de N × N matrices antihermitiennes) et de champ scalaire u = (uI ) a valeurs dans RN . ` Le champ de Yang-Mills associ´e `a A = (Aα ) est F = (Fαβ ) d´eﬁni par: Fαβ = ∇α Aβ − ∇β Aα + [Aα , Aβ ]

(2.2.1)

∇ d´esignant ici la d´erivation covariante dans la m´etrique g. Fαβ est donc une N × N matrice de composantes: I I K = ∇α AIβL − ∇β AIαL + (AIαK AK FαβL βL − AβK AαL ) .

(2.2.2)

Les Equations de Yang-Mills-Higgs s’´ecrivent alors: α F αβ = J β ∇

(2.2.3)

α∇ α uK = RK (uL , u∗L ) ∇

(2.2.4) K

o` u le courant J engendr´e par le champ de Higgs u = (u ) est donn´e par: β uI β u∗ uI + u∗ ∇ JLβI = ∇ L L

(2.2.5)

est la d´erivation covariante de jauge o` u u∗ d´esigne le conjugu´e hermitien de u, ∇ et riemannienne pour g: α uI = ∂α uI + AI uL (action sur les fonctions scalaires) ∇ αL

(2.2.6)

α F αβ = ∇α F αβ + [Aα , F αβ ] (action sur les fonctions a` valeurs dans G) ∇ (2.2.7) Les RK sont des fonctions C ∞ de leurs arguments qui, compte tenu de: β∇ α F αβ ≡ 0 ∇

(2.2.8)

doivent v´eriﬁer, vu (2.2.3) et (2.2.4): βJβ ≡ 0 . ∇

(2.2.9)

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On a:

α F αβ ≡ LAβ + ∇β (∇α Aα ) ∇

393

(2.2.10)

o` u L est l’op´erateur diﬀ´erentiel hyperbolique d´eﬁni par: β α LAβ ≡ ∇α ∇α Aβ − Rα A + ∇α [Aα , Aβ ] + [Aα , F αβ ] .

(2.2.11)

Si A v´eriﬁe la condition de Lorentz: ∇λ Aλ = 0

(2.2.12)

alors vu (2.2.10), (2.2.3) se r´eduit a` LAβ = J β

(cf. [27]) .

(2.2.13)

Th´eor`eme 4 Soit m un entier > 2. Soit T > 0 tel que YT ⊂ U. Soient donn´ees sur YT b = (bo , b1 , b2 , b3 ) ` a valeurs dans G 4 et φ = (φ1 , . . . , φN ) ` a valeurs dans N R et v´eriﬁant: (i) xα bα = 0 dans YT et bα = bα + bα ∈ P 2(m+1) (YT ) + F m+2 (YT ) (ii) φk = φk + φk ∈ P 2m (YT ) + Fm+1 (YT ). Alors il existe To ∈]0, T ] et une unique solution (A,u) des Equations de YangMills-Higgs (2.2.3), (2.2.4) surYTo telle que: • A = (Aα ) ∈ P 2(m−1) (YTo ) + F m (YTo ); y α (Aα − bα )| = 0 pour tout champ CTo

a CTo de vecteurs (y α ) tangent ` • u = (uI ) ∈ P 2(m−1) (YTo ) + F m (YTo ); u|CTo = φ|CTo .

Remarque 1 Ce probl`eme a ´et´e r´esolu dans [34] pour les Equations de Yang-Mills pures sous des hypoth`eses de classe C ∞ sur les donn´ees. Comme dans les Th´eor`emes 2 et 3-B, si les normes des donn´ees b|CT et φ|CT sont assez petites, alors la solution (A, u) est globale: T = To .

2.3

Probl`eme de l’´evolution pour le syst`eme conforme r´egulier des Equations d’Einstein (cf. [6]) en coordonn´ees harmoniques

Pr´eliminaires Dans [6], Y. Choquet-Bruhat et M. Novello ont obtenu un syst`eme hyperbolique au sens de Leray, v´eriﬁ´e par une m´etrique g conforme `a une m´etrique d’Einstein G = ω −2 g et le facteur de conformit´e ω. Ce syst`eme est dit r´egulier parce que ses coeﬃcients sont r´eguliers en ce sens qu’ils ne comportent aucune puissance n´egative de ω. Il semble que l’´etude de ce syst`eme et de sa variante d´eduite auparavent par Friedrich H. (cf. [15], [19], [20]) pourrait fournir des renseignements int´eressants sur l’existence globale et le comportement asymptotique du champ gravitationnel d´eﬁni par la m´etrique d’EinsteinG = ω −2 g; dans tous les cas, elle permettrait de

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mettre en ´evidence la compatibilit´e de la notion d’espace-temps asymptotiquement plat au sens de Penrose avec les Equations d’Einstein. Pour ´ecrire le “syst`eme conforme r´egulier des Equations d’Einstein”, pla¸cons nous dans un syst`eme de coordonn´ees harmoniques d’une vari´et´e hyperbolique (M, g), de classe C ∞ , de dimension 4. Notons: • (gαβ ) les composantes de la m´etrique hyperbolique g. • (dβγλµ ) les composantes du tenseur de Weyl de g, multipli´e par ω −1 , ω ´etant une fonction C ∞ sur M ne s’annulant en aucun point de d’une sous-vari´et´e de M telle que ω = 0 et dω = 0 sur ∂ M , ouverte M • rαβ ) les composantes du tenseur de Ricci de g, • σ ≡ ∇λ ∂λ ω, ∇λ ´etant la d´erivation covariante dans la m´etrique g. • f la courbure scalaire de g. Alors Y. Choquet-Bruhat et M. Novello ont montr´e que si G = ω −2 g est solution . des Equations d’Einstein, RiccG = 0 dans M β Les fonctions (gαβ ), (dγλµ ), (rαβ ), ω, σ v´eriﬁent, pour f donn´ee, le syst`eme suivant, appel´e “syst`eme conforme r´egulier des Equations d’Einstein”: (1)

(h)

(2)

1 λ 1 λ α 1 α 1 1 λ α λ 2 ∇ ∇λ rγµ − 2 Rµγ rαλ − 2 Rµ rγα − 6 ∇µ ∂γ f − 12 gγµ ∇ ∂λ f −dγλµ ∇ ∂α ω

Rαβ = rαβ =0

αρ β αβ ρ αρ β (3) ∇α ∇α dβγλµ + Rµρ dαγλ + Rµγ dαλ + Rµλ dγαρ − λ ←→ µ = 0

(4) 6∇α ∇α σ − f σ + 12rαβ ∇α ∂ β ω + 6∂α f ∂ α ω + ω∇α ∂α f = 0 (5) ∇α ∂α ω = σ o` u: λ α • (Rµγ ) d´esigne la fonction de g, ∂g, ∂ 2 g, qu’on obtient quand on exprime le tenseur de Riemann de g en fonction de g, ∂g, ∂ 2 g, • (Rµλ ) d´esigne la fonction de g, ∂g, ∂ 2 g, qu’on obtient quand on exprime le tenseur de Ricci de g en fonction de g, ∂g, ∂ 2 g, (h) Rαβ d´esigne la fonction de g, ∂g, ∂ 2 g, qu’on obtient quand on exprime • le tenseur de Ricci de g en coordonn´ees harmoniques en fonction de g, ∂g, ∂ 2 g.

On a alors le r´esultat suivant: Th´eor`eme 5 1) Soit m un entier ≥ 3, soit T un r´eel > 0. Soit f = f + f1 avec f fonction polynˆ ome sur R4 dedegr´e ≤ 2m et f1 ∈ Fm+1 (YT ). Soient donn´ees sur le cono¨ıde C, d’´equation xo = (x1 )2 + (x2 )2 + (x3 )2 , 0 ≤ xo ≤ T, une m´etrique hyperbolique g = ( gαβ ) et des fonctions (dβγλµ ), ( rαβ ), ω , σ v´eriﬁant:

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(i) les conditions de structure:  gαβ = g αβ C + gαβ  T    dβ = dβ + dβ   γλµ γλµ γλµ CT (6) r = r | + r αβ αβ αβ  CT    = ω|CT + ω   ω σ = σ|CT + σ o` u:  g αβ , ω sont des fonctions polynˆ omes sur R4 de degr´e ≤ 2m    β  r αβ , dγλµ , σ sont des fonctions polynˆ omes sur R4 de degr´e (6 )  sont des ´el´ements de F m+1 (CT ) ≤ 2(m − 1) gαβ , ω    β rαβ , dγλµ , σ sont des ´el´ements de Fm (CT ) (ii) gαβ (0) = gαβ (0) = ηαβ , ∀α, β = 0, 1, 2, 3; les ηαβ d´esignent les composantes de la m´etrique de Minkowski: ηoo = 1, ηoi = 0, ηij = −δij i, j = 1, 2, 3. (iii) CT est isotrope relativement ` a la m´etrique ( gαβ ), c’est-` a-dire: X ≡ 0 sur VT = {(x1 , x2 , x3 ) ∈ R3 /(s, x1 , x2 , x3 ) ∈ CT } avec X(x1 , x2 , x3 ) ≡ goo (s, x1 , x2 , x3 ) − 2

3

g oi (s, x1 , x2 , x3 )

i=1

+

xi s

gij (s, x1 , x2 , x3 )

1≤i,j≤3

xi xj s2

( g αβ ) est la matrice inverse de la matrice ( gαβ ). Alors: a) les donn´ees initiales (( gαβ ), (dβγλµ ),( rαβ ), ω , σ ) peuvent se red´ecomposer sous la forme:  ∗  gαβ = g αβ C + g αβ   T   β ∗β   β    dγλµ = dγλµ + dγλµ CT ∗ (7) rαβ = r αβ CT + r αβ    ∗    ω = ω CT + ω    ∗  σ = σ C + σ T

β

` composantes polyo` u ((g αβ ), (dγλµ ), (r αβ ), (ω), (σ)) est une fonction vectorielle a nomiales sur R4 de degr´e inf´erieur ou ´egal respectivement ` a 2m, 2(m−1), 2(m−1), ∗

∗

∗

∗

∗

2m, 2(m − 1) et v´eriﬁant au point 0 le syst`eme non lin´eaire (1), (2), (3), (4), (5)

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(d´eduit du syst`eme (1), (2), (3), (4), (5) en rempla¸cant f par f ) et les ´equations ∗

∗

a l’ordre 2(m − 2) pour d´eriv´ees jusqu’` a l’ordre 2(m − 1) pour (1) et (5) et jusqu’` ∗β ∗ ∗ ∗ ∗ ∗ ∗ ∗ m+1 m (CT ), d , r αβ , σ∈ F (CT ). (2), (3) et (4), gαβ , ω ∈ F γλµ

b) ∃To ∈]0, T ] et une unique m´etrique hyperbolique (gαβ ) et une unique fonction vectorielle (dβγλµ , rαβ , ω, σ) telles que: gαβ , ω ∈ P 2m (YTo ) + F m+1 (YTo ) (8) rαβ , dβγλµ , σ ∈ P 2(m−1) (YTo ) + F m (YTo ) et ((gαβ ), (dβγλµ ), (rαβ ), ω, σ) est solution dans le domaine YTo du “syst`eme conforme r´egulier des Equations d’Einstein” (1), (2), (3), (4), (5) et v´eriﬁe la condition initiale:  β  gαβ | = g , d = dβγλµ , rαβ |CT = rαβ αβ γλµ CTo o CTo (9)  ω| = ω , σ| = σ CTo CTo Cette solution ((gαβ ), (dβγλµ ), (rαβ ), ω, σ) peut s’´ecrire sous la forme suivante:  gαβ = gαβ + gαβ avec gαβ ∈ Fm+1 (YTo )    β  β  β   dβγλµ = dγλµ + dγλµ avec dγλµ ∈ Fm (YTo ) (10) rαβ = r αβ + rαβ avec rαβ ∈ Fm (YTo )    m+1 avec ω ∈ F  ω =ω+ω (YTo )    m σ =σ+σ avec σ ∈ F (YTo ) 2) Les notations et hypoth`eses ´etant celles de 1), posons: • g m+1,T = max gαβ m+1,T αβ ∗ avec gαβ m+1,T = max Dν gαβ (0) + gαβ m+1 1≤|ν|≤2m

• ω m+1,T • d

m,T

|ν|≤2m

= max dβγλµ β,γ,λ,µ

avec dβγλµ

m,T

avec rαβ m,T =

m+1,T

F m+1 (CT )

β ν β ∗ = max D dγλµ (0) + dγλµ |ν|≤2(m−1)

F m (CT )

α,β

• σ m,T =

(CT )

m,T

• r m,T = max rαβ m,T

• f

F

∗ = max Dν ω(0) + ω

max

|ν|≤2(m−1)

max

|ν|≤2(m−1)

∗ ν D rαβ (0) + r αβ

∗ ν D σ(0) + σ

F m (CT )

= max Dν f (0) + f1 F m+1 (YT ) . |ν|≤2m

F m (CT )

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Alors il existe un r´eel p > 0 tel que si

|f |m+1,T + g m+1,T + ω m+1,T + d

m,T

+ r m,T + σ m,T ≤ p,

la solution ((gαβ ), (dβγλµ ), (rαβ ), ω, σ) du probl`eme de Cauchy (1), (2), (3), (4), (5), (9), (10) est globale: To = T. = Y, Remarque 2 Le th´eor`eme s’applique au cas physiquement int´eressant M ∂ M = C et la donn´ee initiale ω = 0.

3 Preuve du Th´eor`eme 3 Preuve de la partie A du Th´eor`eme 3 L’op´erateur (h) Rαβ (g) ´etant, vu (2.1.3), hyperbolique, la partie A du Th´eor`eme 3 est une cons´equence imm´ediate des Th´eor`emes 1 et 2,

Preuve de la partie B du Th´eor`eme 3 Soit (gαβ ) la solution du probl`eme de Cauchy: ∼m

(1) gαβ ∈ P 2(m−1) (YTo ) + F (YT0 ) (2) (h) Rαβ (g) = o dans YTo (3) gαβ |CTo = hαβ ∀α, β = 0, 1, 2, 3 Pour montrer que (gαβ ) est aussi solution des Equations d’Einstein du vide: (4) Rαβ (g) = 0 dans YTo ∀α, β = 0, 1, 2, 3 il suﬃt, compte tenu de (2.1.2) , de montrer que: (5) F λ (g) = o dans YTo . ∀λ = 0, 1, 2, 3. Or, (gαβ ) ´etant solution des Equations r´eduites (2), les [∂o gαβ ] v´eriﬁent les relations: xj 1 oi ij xj g − g ∂i [∂o gαβ ] + 12 g ij ∂i [∂o gαβ ] (6) (h) Rαβ (g) CT ≡ − o 2 s s ij i 1 − 2 g ∂ij [gαβ ] + Kαβ x , [gµν ] , [∂o gµν ] , ∂i [gµν ] = o (µ, ν, α, β = 0, 1, 2, 3; i, j = 1, 2, 3). D’autre part, on sait que, par hypoth`ese, ces relations demeurent vraies si on y remplace les [∂0 gαβ ] par les γαβ . On en d´eduit, vu la forme des ´equations (6), que les [∂o gαβ ]−γαβ sont solution d’un syst`eme lin´eaire d’inconnu X = (Xαβ ) de la forme: x xj j µν ∂i Xαβ − g ij ∂i Xαβ + Bαβ (7) g oi − g ij Xµν = 0 s s µν ∈ R + F 4 (CT o ) avec: R = l’espace des restrictions `a CTo des polynˆ omes avec: Bαβ 4 i sur R , ´eventuellement multipl´ees par une fonction fraction rationnelle en x , s de degr´e 0, de d´enominateur s ( ∈ N) .

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Or, des calculs similaires (mais beaucoup plus faciles) a` ceux de la preuve du Th´eor`eme 8.4 de [11] permettent de montrer que toute solution X = (Xαβ ) ∈ omes sur R4 ) v´eriﬁe une P + F1 (CTo ) (P = espace des restrictions `a CT o des polynˆ in´egalit´e ´energ´etique de la forme suivante: t (8) ||X||L2 ( ) ≤ C 0 ||X||2L2 ( ) dτ ∀t ∈ ]0, To ] t

τ

ce qui entraˆıne d’apr`es le lemme de Gronwall que X ≡ (Xαβ ) = 0 sur CT o , c’esta-dire: ` ∀α, β = 0, 1, 2, 3. (9) [∂o gαβ ] = γαβ Il d´ecoule alors de (9) et de la deuxi`eme condition int´egro-diﬀ´erentielle sur les γαβ que: (10) F λ (g) CT o = 0. Comme cons´equence classique des Equations r´eduites (2) et des identit´es de Bianchi, les F λ (g) v´eriﬁent dans le domaine YTo un syst`eme quasi-diagonal, lin´eaire, hyperbolique, homog`ene, du second ordre, de partie principale g αβ ∂αβ F λ (g) . D’apr`es le r´esultat d’unicit´e contenu dans le Th´eor`eme 1, un tel syst`eme n’admet, dans le domaine YTo et dans l’espace fonctionnel ∼m−1

P 2(m−2) (YTo ) + F

(YTo ) , (m − 1 ≥ 2)

que la seule solution nulle qui v´eriﬁe la condition initiale (10) ; donc: F λ (g) = 0 dans YTo . Remarque 3. Probl`eme des contraintes initiales sur un cono¨ıde caract´eristique Pour pouvoir r´esoudre de fa¸con compl`ete le probl`eme de Cauchy, avec donn´ee initiale sur un cono¨ıde caract´eristique, pour les Equations d’Einstein du vide, il serait tr`es important de pouvoir discuter le probl`eme du choix des donn´ees ind´ependantes a partir desquelles on pourrait g´en´erer, `a l’aide de certaines ´equations de propa` gation port´ees par les g´eod´esiques isotropes issues du point 0 et g´en´eratrices du demi-cono¨ıde C de sommet 0 les donn´ees de Cauchy (hαβ ) v´eriﬁant les conditions int´egrodiﬀ´erentielles impos´ees dans l’´enonc´e du Th´eor`eme 3. Ce probl`eme qui a ´et´e r´esolu par M¨ uller Zum Hagen et Hans J¨ urgen Seifert [29] d’une part et A.D. Rendall [33] d’autre part, dans le cas o` u l’hypersurface initiale est constitu´ee de deux hypersurfaces r´eguli`eres, caract´eristiques et s´ecantes suivant une 2-surface spatiale, demeure encore a` notre connaissance, largement ouvert dans le cas d’un cono¨ıde caract´eristique.

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4 Preuve du Th´eor`eme 4 4.1

D´ebut de la preuve 2

Soit σ = − (xo ) +

3 2 ! xi . Posons: i=1

(1) o aα = bα + a∇α σ,

α = 0, 1, 2, 3.

Supposons que dans YT − {0}, Aα et u admettent les d´eveloppements limit´es respectifs: ∼

(2) Aα =o aα +1 aα σ +2 aα σ 2 + a α ∼

(3) u = φ +1 φσ+ u avec

(4) ∂ λ aα

CT −{0}

= 0,

∼ ∂ µ u

CT

−{0}

=0

si |λ| ≤ 2, |µ| ≤ 1 .

Alors, des calculs presque identiques `a ceux de [34], montrent que: (5) g αδ ∇δ σ (LAα − Jα ) − g βγ ∇β ∇γ σ − 4 ∇λ Aλ |CT −{0} = 0 si, sur CT , on a:  αδ βγ  g g ∇δ σ(∇β ∇oγ aα − ∇γ ∇oα aβ − ∇α ∇oβ aγ − [o aβ , ∇oα aγ ]) −(g ετ ∇ε ∇τ σ − 4)∇oβ aγ g βγ = M ≡ (MLI ) (6)  avec MLI ≡ 2xα (∇α φ∗L φI + φ∗L ∇α φI )

.

L’´equation (6) se r´eduit a` un syst`eme d’´equations aux d´eriv´ees partielles du 1er ordre d’inconnu (a, c) , dans le domaine , de la forme:  α  x ∂α a =c xβ ∂β c + 34 g αβ ∇α ∇β σ − 2 c + 18(g αβ ∇α ∇β σ g γδ ∇γ ∇δ σ − 4 (7) b,φ  +2xγ g αβ ∇γ ∇β σ∇α σ)a = ρ la fonction ρ ´etant une fonction C∞ des seuls arguments bα , ∂bα , ∂ 2 bα , φ, ∂φ. La restriction a` CT de (7)b,φ se r´eduit a` un syst`eme d’´equations aux d´eriv´ees partielles du 1er ordre d’inconnu ([a] , [c]) de la forme:  ! i x ∂i [a] − [c] = 0  i=1,2,3 ! (8) xi ∂i [c] + 4[a] + 4[c] + sH[a] + sG[c] = [ρ]  i=1,2,3

avec: [X] x1 , x2 , x3 = X s, x1 , x2 , x3 , s · H et s · G ´etant les restrictions respectives `a CT des fonctions C ∞ dans et nulles en 1 αβ 3 αβ g ∇α ∇β σ − 6, g ∇α ∇β σ g γδ ∇γ ∇δ σ − 4 + 2xγ g αβ ∇γ ∇β σ∇α σ − 4 . 4 8 H, G sont des fonctions born´ees de x1 , x2 , x3 sur CT . 0:

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Dans la suite de cette preuve, on se propose de montrer: • dans un premier temps, que lorsque les donn´ees b et φ sont de classe C ∞ dans Y, alors le syst`eme (7)b,φ poss`ede une solution unique de classe C ∞ dans , • dans un second temps, que le syst`eme (8) qui est hyperbolique admet des in´egalit´es ´energ´etiques v´eriﬁ´ees par [a] et [c] dans l’espace fonctionnel ∼m

P 2(m−1) (CT ) + F

(CT ) ;

en utilisant ensuite ces in´egalit´es ´energ´etiques dans des approximations par des fonctions C ∞ , ainsi que le r´esultat d’existence C ∞ associ´e `a (7)b,φ pour b et φ de classe C ∞ dans , on r´esout le syst`eme (8). L’extraction des composantes polynomiales de [a] et [c] se fera au moyen du syst`eme (7)− − . b ,φ

4.2

R´esolution du syst`eme (7)b,φ d’inconnu (a, c) dans l’espace C ∞ (Y) lorsque les donn´ees b et φ sont de classe C ∞ dans Y

On aura besoin de la formule suivante (cf. [13], page 132): 1

(9) g αβ ∇β ∇γ σ = 8 + 2xα ∂α log |g| 2 avec: |g| = det (gαβ ). En substituant (9) dans (7)b,φ , on peut mettre le syst`eme (7)b,φ sous la forme suivante: " α x ∂α a − b = 0 (10) xα ∂α b + 4a + 4b + xα Hα (x) a + xα Kα (x) b = e o` u les Kα et Hα sont des fonctions C ∞ dans . Posons: 0 −1 0 0 , Mα (x) = (11) V = (ab ) , N = , R= 0 . 4 4 Hα (x) Kα (x) Le syst`eme (10) peut alors se mettre sous la forme suivante: (12) xα ∂α ∨ +N. ∨ +xα Mα (x) ∨ = R. La matrice N a une seule valeur propre 2 de multiplicit´e g´eom´etrique 1. Soit P la matrice inversible telle que: 2 0 −1 . (13) P .N.P = D ≡ 1 2 Faisons le changement de fonction inconnue: (14) W = P −1 .∨ on obtient: α (x) W = R (15) xα ∂α W + DW + xα M avec: = P −1 R α (x) = P −1 .Mα (x) .P , R (16) M Posons maintenant: a mα (x) nα (x) 1 = α (x) = ,R . (17) W = , M pα (x) qα (x) 2 b

Vol. 4, 2003

Probl` emes de Cauchy sur un Cono¨ıde Caract´eristique

Alors le syst`eme (15) s’´ecrit: ∼ ∼ ∼ xα ∂α a +2 a +xα mα (x) a +xα nα (x) .b = ρ1 (18) ∼ ∼ ∼ ∼ xα ∂α b +2 b + a +xα pα (x) a +xα qα (x) .b = ρ2

401

.

Consid´erons l’´equation: (19) xα ∂α a1 (x) + xα mα (x) a1 (x) = 0 si on y remplace x par rx avec 0 ≤ r ≤ 1, alors on obtient: (20) xα ∂α a1 (rx) + xα mα (rx) a1 (rx) = 0 on en d´eduit que l’unique solution de (19) telle que a1 (o) = 1 est la fonction de classe C ∞ : (21) a1 (x) = e−

1 o

xα mα (sx)ds

.

De mˆeme l’´equation: (22) xα ∂α b1 (x) + xα qα (x) b1 (x) = 0 admet comme unique solution telle que b1 (o) = 1, la fonction C ∞ : (23) b1 (x) = e−

1 o

xα qα (sx)ds

.

Les fonctions a1 et b1 permettent de r´eexprimer le syst`eme (18) sous la forme: 

b  a a ρ1  α α    x ∂α a1 + 2 a1 = −x nα (x) a1 + a1

(24) . b b  a a ρ2  α α    x ∂α b1 + 2 b1 = −x pα (x) b1 − b1 + b1 Si on remplace dans (24) x par rx pour r ∈ [0, 1], on obtient:  # " b(rx) ρ1 (rx) a (rx)  2 2 α    ∂r r a (rx) = −r x nα (rx) a (rx) + r a (rx) 1 1 1 $ (25) b(rx) a (rx) ρ2 (rx) a (rx)    ∂r r2 −r +r = −r2 xα pα (rx)  b1 (rx) b1 (rx) b1 (rx) b1 (rx)

.

Le syst` eme (24) est donc ´equivalent $ au syst`eme int´egral: b(sx)  ρ (sx) 1  1 2  a (x) = −a1 (x) o s xα nα (sx) −s ds  a1 (sx) a1 (sx) (26) . " #  a (sx) ρ2 (sx) a (sx) 1  2 α  +s −s ds  b (x) = −b1 (x) o s x pα (sx) b1 (sx) b1 (sx) b1 (sx) On va montrer dans la suite comment on r´esout le syst`eme (26) en utilisant une m´ethode de point ﬁxe. Il d´ecoule d’abord de (26) en faisant x = 0: ∼ 1 1 1 (27) a (o) = 1 (0), b (o) = 2 (0) − 1 (0). 2 2 4 Pla¸cons nous sur la droite D (courbe g´eod´esique passant par l’origine et le point ω de la sph`ere-unit´e (centr´ee `a l’origine) de Rn+1 . Alors, sur la droite

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D, le syst`eme (26) se r´eduit au syst`eme diﬀ´erentiel d’inconnu la fonction r −→ a (rω) , b (rω) :  $ b(srω)  ρ (srω) 1  1 2  a (rω) = −a1 (rω) o s rω α nα (srω) −s ds  a1 (srω) a1 (srω) (28) " #  a (srω) ρ2 (srω) a (srω) 1   +s −s ds b(rω) = −b1 (rω) o s2 rω α pα (srω) b1 (srω) b1 (srω) b1 (srω) Faisons le changement de variable d’int´egration: (29) u = sr et posons: (30) a (rω) = aω (r) ,

b (rω) = bω (r) .

On obtient:  " # a1(rω) r 2 α bω (u) ρ1 (uω)   aω (r) = − 2 u ω n (uω) − u du  α r" 0 a1 (uω) a1 (uω) # (31) b1(rω) r aω (u) ρ2 (uω) aω (u)   +u −u du  bω (r) = − 2 0 u2 ω α pα (uω) r b1 (uω) b1 (uω) b1 (uω) qu’on va noter sommairement: (32) Xω (r) = F (Xω (r) , r, ω) en posant: aω (r) . (33) Xω (r) = bω (r) x x 2 = max {|x| , |y|} . Si ∈ R , on note y y Soit L un r´eel > 0 quelconque; soit k un r´eel > 0 dont la valeur sera ﬁx´ee ult´erieurement. Soit: " # XL = X ∈ C [0, L] , R2 | sup e−kr ||X (r)|| < +∞ 0≤r≤L

XL est un espace de Banach pour la norme: XXL = sup e−kr ||X (r)|| 0≤r≤L

Soit A : XL −→ XL d´eﬁnie par (AX) (r) = F (X (r) , r, ω). Il existe une constante M > 0 ne d´ependant que des bornes des co´eﬃcients du syst`eme int´ egral lin´eaire (31) dans le compact (u, ω) ∈ [0, L] × Sn Sn sph`ere-unit´e de Rn+1 telle que: (34) ||AX − AY ||XL ≤

M ||X − Y ||XL k

∀X, Y ∈ XL .

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Pour k > M , A admet un point ﬁxe quiest la solution de (32) dans le aω domaine r ∈ [0, L] . Cette solution Xω = d´epend continˆ ument de ω. En bω posant:   x X ||x|| (||x||) si x = 0       1 ρ (0) . (35) X(x) = 2 1  1 si x = o    1   ρ (0) − (0) ρ 1 2 2 4 On d´eﬁnit une fonction continue dans qui est la solution du syst`eme int´egral (26) dans . On peut ensuite montrer par r´ecurrence que X est de classe C ∞ dans et que sa d´eriv´ee d’ordre α ∈ N4 s’obtient comme l’unique solution dans l’espace des fonctions continues dans du syst`eme (26) d´eriv´e `a l ordre α. On a ainsi montr´e que le syst`eme (7)b,φ admet une unique solution (ab ) de classe C ∞ dans .

4.3

R´esolution du syst`eme 8 sous conditions

Nous r´esolvons le syst`eme (8) dans l’espace fonctionnel (

)2 P 2(m−1) (CT ) + Fm (CT )

lorsque les donn´ees b et φ v´eriﬁent les hypoth`eses de r´egularit´e suivantes: bα = bα + bα avec: bα ∈ P 2(m+1) (YT ) , bα ∈ F m+2 (YT ), φk = φk + φk avec : φk ∈ P 2m (YT ) , φk ∈ F m+1 (YT ). ∞ (YT ) Comme bα ∈ F m+2 (YT ) , il existe une suite bα,n d’´el´ements de C∞ m+2 m+1 (YT ) ; de mˆeme comme φk ∈ F (YT ) , il existe qui converge versbα dans F ∞ d’´el´ements de C (YT ) qui converge vers φ dans F m+1 (YT ) . une suite φ k,m

∞

k

Posons: " bα,n = bα + bα,n , b(n) = (bα,n ) (36) φk,n = φk + φ˜k,n ,φ(n) = (φk,n ) Notons: a, c) la solution d´eﬁnie et de classe C ∞ dans du syst`eme (12) b , φ ( b = bα , φ = φk .

avec

Notons: a le polynˆome de Taylor d’ordre 2(m − 1) en 0 de a, c le polynˆome de Taylor d’ordre 2(m − 1) en 0 de c. ∀n ∈ N, notons (an , cn ) la solution d´eﬁnie et de classe C ∞ dans du syst`eme (7)b(n) ,φ(n) .

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Alors il est ´evident que: $ ∞ an = a + a1,n avec a1,n ∈ C2(m−1) (YT ) (37) ∞ avec c1,n ∈ C2(m−1) (YT ) cn = c + c1,n ∞ (ici C2(m−1) (YT ) d´esigne l’espace des fonctions num´eriques de classe C ∞ dans YT dont toutes les d´eriv´ees jusqu’`a l’ordre 2 (m − 1) sont nulles en 0).

∀n, q ∈ N le couple (a1,n − a1,q , c1,n − c1,q ) v´eriﬁe le syst`eme diﬀ´erentiel:  α x ∂α (a1,n − a1,q ) −(c1,n − c1,q ) = 0     xα ∂α (c1,n − c1,q ) + 3/4g αβ ∇α ∇β σ − 2 (c1,n − c1,q ) 1 (38)  + (g αβ ∇α ∇β σ g γδ ∇γ ∇δ σ − 4 +   8  2xγ g αβ ∇γ ∇β σ∇α σ) (a1,n − a1,q ) = ρn − ρq Avec:

(39) ρi = ρ b(i) , ∂b(i) , ∂ 2 b(i) , φ(i) , ∂φ(i)

i = n ou q.

La restriction du syst`eme (38) `a CT s’´ecrit:  i  x ∂i ([a1,n ] − [a1,q ]) − ([c1,n ] − [c1,q ]) = 0 xi ∂i ([c (40) ]) + 4 ([c1,n ] − [c1,q ]) 1,n ] − [c1,q ]) + 4 ([a1,n ] − [a1,q  +sH xi ([a1,n ] − [a1,q ]) + sG xi ([c1,n ] − [c1,q ]) = [ρn ] − [ρq ] Il d´ecoule de calculs similaires (mais beaucoup plus faciles) a` ceux de [8] (pages 70–78) qu’il existe une constante C > o, ind´ependante de n et m, telle que: (41) ||[[a1,n ] − [a1,q ]]||Fm (CT ) + ||[c1,n ] − [c1,q ]||Fm (CT ) ≤ C ||ρn − ρq ||Fm (CT ) Or par d´eﬁnition des i (cf. (39)) et compte tenu des in´egalit´es de substitution des Th´eor`emes 2.1.4, 2.1.5, 2.1.6 et 2.1.7 de [8] , il existe une constante K > 0, ind´ependante de n et q, telle que: + * (42) ||ρn − ρq ||Fm (CT ) ≤ K b(n) − b(q) Fm+2 (Y ) + φ(n) − φ(q) Fm+1 (Y ) . T

T

Les in´egalit´es (41) et (42) entraˆınent que les suites ([a1,n ]), ([c1,n ]) sont de Cauchy dans l’espace de Banach Fm (CT ) donc convergent dans cet espace respectivement vers des ´el´ements [a1 ] , [c1 ]. Posons: [a] = [a] + [a1 ] [c] = [c] + [c1 ]. Alors le couple ([a] , [c]) est l’unique solution du syst`eme (8) dans l’espace * +2 fonctionnel P 2(m−1) (CT ) + F m (CT ) .

4.4

Fin de la preuve du Th´eor`eme 4

Consid´erons maintenant dans YT le probl`eme (2.2.4) , (2.2.13) avec les conditions initiales: Aα |CT = bα + [a] ∇α σ, u|CT = φ|CT . D’apr`es le Th´eor`eme 1, et vu (2.2.11) ∃T1 ∈ ]0, T ] tel que ce probl`eme admet dans YT1 une solution unique (A, u) telle que: A, u ∈ P 2(m−1) (YT1 ) + F m (YT1 ) . Comme dans YT1 − {0} , (A, u) peut

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s’´ecrire sous la forme (2), (3), on en d´eduit vu le choix de [a] et vu (5) qu il existe T0 ∈ ]0, T1 ] tel que (∗): ∇λ Aλ |CT0 = 0. D’autre part on sait que, toute β J β = 0 dans YT0 , donc aussi, puisque solution (A, u) de (2.2.4) , (2.2.13) v´eriﬁe ∇ αβ ∇β ∇α F ≡ 0 dans YT0 : (∗∗) ∇β ∇β (∇α Aα ) + Aβ , ∇β (∇α Aα ) = 0 dans YT0 . Or, d’apr`es le r´esultat d’unicit´e contenu dans le Th´eor`eme 1, le probl`eme de Cauchy lin´eaire, caract´eristique (∗), (∗∗) d’inconnu ∇α Aα admet dans l’espace fonctionnel P 2(m−2) (YT0 )+ F m−1 (YT0 ) l’unique solution: ∇α Aα = 0 dans YT0 . (A, u) est donc ﬁnalement solution des Equations de Yang-Mills-Higgs (2.2.3) , (2.2.4) et v´eriﬁe les conditions de structure et les conditions initiales annonc´ees dans l’´enonc´e du Th´eor`eme 4; de plus pour tout champ de vecteurs (y α ) tangent a` CT0 , on a vu (1): y α (Aα − bα )|CT = [a] y α ∇α σ|CT = 0. 0

0

5 Preuve du Th´eor`eme 5 Preuve de 1b) si on admet 1a) Notons sommairement: • g = gαβ , g1 = une d´eriv´ee premi`ere des gαβ , • ∂g1 = une d´eriv´ee premi`ere de g1 , • ω1 = une d´eriv´ee premi`ere de ω, • ∂ω1 = une d´eriv´ee premi`ere de ω1 , • d = dβγλµ , r = rαβ , f1 = une d´eriv´ee premi`ere de f, • ∂f1 = une d´eriv´ee premi`ere de f1 , • ∂r = une d´eriv´ee premi`ere de r, etc. . . . Alors le syst`eme conforme r´egulier des Equations d’Einstein (1), (2), (3), (4), (5) du Paragraphe 2.3 peut se noter sommairement: (1 ) γ λµ (g)Dλµ g + h1 (g, g1 , r) = 0 (2 ) γ λµ (g)Dλµ r + h2 (g, g1 , ∂g1 , r, f, f1 , ∂f1 ) = 0 (3 ) γ λµ (g)Dλµ d + h3 (g, g1 , ∂g1 , d) = 0 (4 ) γ λµ (g)Dλµ σ + h4 (g, g1 , r, σ, ω, ω1 , ∂ω1 , f, f1 , ∂f1 ) = 0 (5 ) γ λµ (g)Dλµ ω + h5 (g, g1 , σ) = 0 . Comme les seconds membres h2 , h3 , h4 des ´equations (2 ), (3 ), (4 ) contiennent les d´eriv´ees secondes de g ou de ω, on doit aussi consid´erer les ´equations d´eriv´ees de (1 ) et (5 ) qu’on notera respectivement (1 ) et (5 ). Ces ´equations peuvent s’´ecrire sommairement sous la forme suivante: (1 ) γ λµ (g)Dλµ g1 + h6 (g, g1 , ∂g1 , r, ∂r) = 0 (5 ) γ λµ (g)Dλµ ω1 + h7 (g, g1 , ∂g1 , σ, ∂σ) = 0 .

406

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Ann. Henri Poincar´e

Les conditions initiales (9) du Th´eor`eme 5 (Paragraphe 2.3) induisent sur les g1 , ω1 , des conditions initiales de la forme: g1 , ω1 |C = ω 1 , (11) g1 |C = C

C

1 d´esignent respectivement les d´eriv´ees premi`eres des fonctions g et o` u les g1 et ω ω qui sont respectivement les solutions des probl`emes de Cauchy caract´eristiques non lin´eaires: λµ g + h1 ( γ ( g)Dλµ g, g1 , r) = 0 dans Y (12) g = g

(13)

C λµ

γ ( + h5 ( g)Dλµ ω g, g1 , σ ) = 0 dans Y ω = ω |C C

avec: rαβ = r αβ + r1,αβ , σ =σ+σ 1 o` u r αβ et σ sont les polynˆ omes intervenant dans les ´egalit´es (7) du Th´eor`eme 5 du Paragraphe 2.3, ∗ ∗ 1 ∈ F m (YT ) avec r1,αβ =rαβ , σ 1 =σ r1,αβ , σ CT

CT

∗ ∗ (r αβ , σ ´etant les ´el´ements de Fm (CT ) qui interviennent dans les ´egalit´es (7) du Paragraphe 2.3). D’apr`es le Th´eor`eme 1 et vu les relations (7) et (8) du Paragraphe 2.3 (car a) est admis), il existe T1 ∈ ]0, T ] tel que:

g ∈ {g} + Fm+1 (YT1 ),

ω ∈ {ω} + F m+1 (YT1 ).

D’apr`es le Th´eor`eme 1 et vu les relations (7), et (8) du Paragraphe 2.3, il existe To ∈]0, T1 ] tel que le syst`eme (1 ), (2 ), (3 ), (4 ), (5 ), (1 ), (5 ) admet alors, dans le domaine YTo , sous les conditions initiales (10) du Paragraphe 2.3 et (11), (12), (13), une solution unique (g, g1 , r, d, σ, ω, ω1 ) ∈ {(g, g1 , r, d, σ, ω, ω 1 )} + [F m (YTo )]N . On montre enﬁn facilement que (g1 − ∂g, ω1 − ∂ω) v´eriﬁe dans le domaine YTo , un syst`eme quasi-diagonal lin´eaire, hyperbolique du second ordre de parties principales respectives γ λµ (g)∂λµ (g1 -∂g), γ λµ (g)∂λµ (ω1 −∂ω) et de seconds membres nuls. Comme g1 − ∂g|CTo = 0 et ω1 − ∂ω|CTo = 0, on a ﬁnalement d’apr`es le r´esultat d’unicit´e contenu dans le Th´eor`eme 1: g1 = ∂g

et ω1 = ∂ω

dans

YTo .

On en d´eduit que: g ∈ {g} + F m+1 (YTo ),

ω ∈ {ω} + F m+1 (YTo )

(g, , r, d, σ, ω, ) est alors l’unique solution du probl`eme de Cauchy (1), (2), (3), (4), (5), (6), (6 ), (9), (10) du Paragraphe 2.3.

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Preuve du 2) sous la condition que 1.a) est vraie Elle est identique `a celle de la preuve pr´ec´edente `a condition d’appliquer une variante ´evidente du Th´eor`eme 2 `a la place du Th´eor`eme 1. Les preuves de 1.b) et 2) seront compl`etes si on d´emontre 1.a). Preuve de 1)a) En reprenant les notations pr´ec´edentes reconsid´erons les ´equations: (1 ) γ λµ (g) Dλµ g + h1 (g, g1 , r) = 0 (2 ) γ λµ (g) Dλµ r + h2 g, g1 , ∂g1 , r, f , f 1 , ∂f 1 = 0 (3 ) γ λµ (g) Dλµ d + h3 (g, g1 , ∂g1 , d) = 0 (4 ) γ λµ (g) Dλµ σ + h4 g, g1 , r, σ, ω, ω1 , ∂ω1 , f , f 1 , ∂f 1 = 0 (5 ) γ λµ (g) Dλµ ω + h5 (g, g1 , σ) = 0 (1 ) γ λµ (g) Dλµ g1 + h6 (g, g1 , ∂g1 , r, ∂r) = 0 (5 ) γ λµ (g) Dλµ ω1 + h7 (g, g1 , ∂g1 , σ, ∂σ) = 0 . (Les ´equations (2 ), (4 ) ´etant respectivement d´eduites de (2 ), (4 ) en rempla¸cant f par f , f1 par f 1 et ∂f par ∂f 1 ). On munit le syst`eme (1 ), (2 ), (3 ), (4 ), (5 ), (1 ), (5 ) des conditions initiales: " g|C = g|C ; r|C = r|C ; d|C = d C (14) ω|C = ω|C ; σ|C = σ|C o` u C est le demi-cono¨ıde isotrope de sommet 0, orient´e vers les xo ≥ 0, d´etermin´e par la m´etrique γ λµ (g(xα )) qui est hyperbolique dans un voisinage U de xα = 0 dans R4 . D’apr`es [11] (cf. le Paragraphe III du Chapitre III de la 1`ere partie de [11]), C admet une repr´esentation param´etrique de la forme: xo = S(x1 , x2 , x3 ) o` u S est une fonction d´eﬁnie dans un voisinage V de 0 dans R3 ; S est positive et continue dans V ; S est de classe C ∞ dans V \{0}; S admet un d´eveloppement limit´e en fractions rationnelles homog`enes en x1 , x2 , x3 , s = (x1 )2 + (x2 )2 + (x3 )2 de la forme: (15) S(x1 , x2 , x3 ) = s + O(s2m+1 ). Notons: Y = {(xo , x1 , x2 , x3 ) ∈ U/ S(x1 , x2 , x3 ) ≤ xo < +∞} Yt = Y ∩ {xo < t}. Les conditions initiales (14) induisent sur les g1 , ω1 des conditions initiales de la forme: (16) g1 |C = G1 |C ; ω1 |C = Ω1 |C

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o` u G1 et Ω1 d´esignent respectivement les d´eriv´ees premi`eres des fonctions G, Ω qui sont respectivement les solutions des probl`emes de Cauchy caract´eristiques, non lin´eaires, `a donn´ees C ∞ : " γ λµ (G)Dλµ G + h1 (G, G1 , r) = 0, dans Y (17) G|C = g|C " γ λµ (G)Dλµ Ω + h5 (G, G1 , σ) = 0, dans Y (18) . Ω|C = ω|C D’apr`es le Th´eor`eme 2 de [7], il existe T2 ∈]0, T ] tel que le probl`eme de Cauchy caract´eristique (1 ), (2 ), (3 ), (4 ), (5 ), (1 ), (5 ), (14), (16) admet dans le o o o o o o o domaine YT2 , une solution unique (g , g 1 , r, d, σ, ω, ω1 ) ∈ C ∞ (YT2 ). On montre o o o o enﬁn facilement que (g 1 -∂ g, ω 1 -∂ ω) v´eriﬁe, dans le domaine YT2 , un syst`eme quasi-diagonal lin´eaire, hyperbolique, homog`ene, du second ordre de parties prino o o o o o cipales γ λµ (g)Dλµ (g 1 −∂ g); γ λµ (g)Dλµ (ω 1 −∂ ω). Comme de plus, g 1 −∂ g = CT2 o o 0, ω1 −∂ ω = 0, on a enﬁn, d’apr`es le r´esultat d’unicit´e contenu dans le CT2

Th´eor`eme 1:

o

o

o

o

g1= ∂ g; ω1= ∂ ω o

dans YTo . ∗

o o o o

∗

∗

(g , , r, d, σ, ω, ) est alors l’unique solution du probl`eme de Cauchy (1), (2), (3), ∗ ∗ (4), (5) dans le domaine YTo (cf. l’´enonc´e du Th´eor`eme 5 pour la d´eﬁnition des ´equations “´etoil´ees”). On choisit: o

g = polynˆ ome de Taylor d’ordre 2m de g au point xα = 0 o

ω = polynˆ ome de Taylor d’ordre 2m de ω au point xα = 0 o

r = polynˆ ome de Taylor d’ordre 2(m − 1) de r au point xα = 0 o

d = polynˆ ome de Taylor d’ordre 2(m − 1) de d au point xα = 0 o

σ = polynˆ ome de Taylor d’ordre 2(m − 1) de σ au point xα = 0. ∗

∗

∗

∗

∗

Alors (g, ω, r, d, σ) v´eriﬁe au point xα = 0 le syst`eme (1), (2), (3), (4), (5) et ∗ ∗ a l’ordre 2(m − 1) les ´equations d´eriv´ees jusqu’`a l’ordre 2m pour (1), (5) et jusqu’` ∗

∗

∗

pour (2), (3) et (4). On montre enﬁn, comme dans la preuve du Th´eor`eme 2 de [8], en utilisant r , (14), (15), que les donn´ees initiales ( g ,d, ω, σ ) associ´ees au syst`eme conforme r´egulier des Equations d’Einstein (1), (2), (3), (4), (5) peuvent se red´ecomposer comme dans les ´egalit´es (7) du Th´eor`eme 5. On a ainsi achev´e la preuve de 1) a) et compl´et´e celles de 1) b) et 2).

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Remerciements Je voudrais remercier l’I.C.T.P. que j’ai visit´e, en 2000 et o` u a d´emarr´e la r´edaction de ce travail. Mes remerciements vont aussi au C.D.E. (I.M.U.) qui a ﬁnanc´e mon voyage Yaound´e-Trieste.

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[29] H. M¨ uller zum Hagen et F.H.J. Seifert, “On characteristic initial value and mixed problems”, Gen. Rel. Grav. 8, 259–301 (1977). [30] E.T. Newman, T. Unti, J. Math. Phys. 3, 891 (1962). [31] R. Penrose, “Null hypersurface initial data for classical ﬁelds of arbitrary spin and for General Relativity” in Aerospace Research Laboratories Report 63–56 (1963); reprinted in General Rel. Grav. 12, 225–264 (1980). [32] A.D. Rendall, “Local and global existence theorems for the Einstein Equations” www. living reviews.org/Articles/vol m 3/2000-1rendall, Living Reviews in Relativity. [33] A.D. Rendall, “Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations”, Proc. R. Soc. London, Ser. A, 427, 221–239 (1990). [34] A.D. Rendall, “The characteristic initial value problem for the Einstein Equations” Nonlinear hyperbolic equations and ﬁeld theory (Lake Como 1991) Pitman, Res. Notes, Maths-ser., 253, Longman Sci. Harlow, 1992, 154–163. [35] R.K. Sachs, “Characteristic initial value problem for gravitational theory”, in Infeld, L., ed., Relativistic Theories of Gravitation, 93–105 (Pergamon Press, Oxford, 1964). [36] J.M. Stewart dans Classical General Relativity (ed. W.B. Bonnor J.N. Islam et M.A.H. Mac Callum) Cambridge University Press. Marcel Dossa D´epartement de Math´ematiques Facult´e des Sciences Universit´e de Yaound´e I B.P. 812 Yaound´e Cameroun email: [email protected] Communicated by Vincent Rivasseau submitted 23/05/02, accepted 19/09/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 413 – 438 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030413-26 DOI 10.1007/s00023-003-0135-7

Annales Henri Poincar´ e

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates Steven E. Golowich Abstract. The Green’s function for the nearest-neighbor self-avoiding walk on a hypercubic lattice in d > 2 dimensions is constructed and shown to be analytic for values of the killing rate a ∈ C satisfying |a| > , | arg a| < 3π/4 − b with > 0 and 0 > 0 in order to use the killing rate as an infrared cutoﬀ, which allows us to construct Green’s function using a single scale cluster expansion. The presence of non-real killing introduces complications that we resolve through the use of an appropriate choice of decoupling scheme and a subsidiary expansion. Our methods can be used to control a single momemtum slice in a phase-space expansion.

1 Introduction The self-avoiding walk (SAW) has attracted a great deal of interest both as a widely used mathematical model for systems arising in physics, chemistry, and biology, as well as for its intrinsic mathematical appeal. In this paper, we study the weakly self-avoiding nearest-neighbor walk in continuous time on a discrete, hypercubic lattice in d > 2 dimensions. This model is deﬁned as a perturbation of a simple random walk, which we take to be the continuous time Markov jump (or L´evy) process generated by the lattice Laplacian ∆. For this process, the probability of travelling from x to y in time T is (1.1) P (x, y, T ) = eT ∆ x,y if the jumping rate is chosen to be 2d. We will denote sample paths by ω(t), and expectations for this process, conditioned on the walk starting at the origin, by E0 (·). In order to deﬁne the self-avoiding walk, we ﬁrst note that a measure of the self-intersections of a walk ω of length T is 1 1 2 2 τ = dx τx,T = ds dt 1ω(s)=ω(t) , (1.2) 2 T 2 0≤s≤t≤T where

τx,T =

T

ds 1ω(s)=x

(1.3)

0

is the local time spent at site x. We use this to discourage walks from intersecting themselves by deﬁning the expectation for self-avoiding walks that live for time T

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to be E0T,λ (·) =

1 NT,λ

1 E0 (·)e− 2 λ

2 dx τx,T

Ann. Henri Poincar´ e

,

(1.4)

where NT,λ is the normalization. It is also of interest to study the ensemble of walks of all lengths, and allow the underlying free process to be killed at rate a (the killing rate), leading to the deﬁnition ∞ 2 1 1 (1.5) dT e−aT E0 (·)e− 2 λ dx τx,T . E0a,λ (·) = Na,λ 0 Clearly, the two ensembles (1.4) and (1.5) are related by the Laplace transform. A fundamental question is the behavior of the mean-square end-to-end distance of the walk as its length increases. It was longstanding conjecture [12] that d = 4 should be the upper critical dimension of the SAW; that is, in d > 4 dimensions the self-avoiding walk should behave diﬀusively, but for d < 4 the excluded volume eﬀect should qualitatively modify its behavior. The behavior for d = 4 was conjectured by de Gennes [9] to be diﬀusive, but with a logarithmic correction: E0T,λ ω(T )2 ∼ T log1/4 T. (1.6) The calculations leading to (1.6), based on the renormalization group, are not rigorous, but have provided a useful starting point for recent progress toward a proof. This progress began with the foundational work of Brydges and Spencer [5], who introduced the lace expansion to prove that the weakly SAW behaves diﬀusively for d > 4. This expansion was revisited several times in the context of the SAW [15, 23, 24, 25], and a version of it was used by Hara and Slade [18, 17] to remove the restriction of weak self-avoidance and prove that the strictly SAW also behaves diﬀusively. The inﬂuence of the self-avoidance becomes much more subtle in d = 4 dimensions, but recently much progress has been made toward proving (1.6). Brydges and Imbrie [6, 22, 7, 8] proved the result for walks on a type of hierarchical lattice chosen to simplify the renormalization group analysis. The result is obtained by constructing the Green’s function ∞ 2 1 dT e−aT E0 1ω(T )=x e− 2 λ dx τx,T (1.7) Gλ (x, a) = 0

for killing rates a in a large enough region in the complex plane to invert the Laplace transform and thereby extract the asymptotic behavior for ﬁxed T . The renormalization group analysis is much more involved when the hierarchical simpliﬁcation is removed. However, Iagolnitzer and Magnen [21] constructed the Green’s function for a non-hierarchical Edwards model [10] with ﬁxed ultraviolet cutoﬀ at the critical point by means of a novel phase-space expansion. A

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possible path to a proof of (1.6), therefore, would be to combine the ideas for inverting the Laplace transform from [7, 8] with the phase-space expansion of [21]. In the present paper, we take a step toward implementing this program by constructing the Green’s function of a nearest-neighbor self-avoiding walk on a hypercubic lattice in the region of the complex plane {a ∈ C : |a| > , | arg a| < 3π/4 − b} with > 0 and 0 < b < π/4. The region is chosen, as in [8], to enable the inversion of the Laplace transform to recover information about walks of ﬁxed length. The construction is an adaptation of the expansion of [21] to enable the use of a complex killing rate. We have performed a cluster expansion with the lattice providing a UV cutoﬀ and the killing rate providing the IR cutoﬀ. In order to use the killing rate in this way, we must restrict it to be bounded away from the critical point by a distance which may be small, but not too small. Our results could be used to study walks of ﬁxed length T ∼ 1/, where T is large compared to the waiting time at a site of the underlying free process, though not the limit T → ∞. However, the main interest of our result is as a step toward proving (1.6) by providing a construction of a slice in a phase-space expansion: instead of getting the ultraviolet cutoﬀ from the lattice and the infrared cutoﬀ from the killing rate, we could have considered the case of an arbitrary momentum slice in a phase space expansion without substantially modifying our analysis. For this reason we expect that our methods can be used to extend the full phase space expansion developed in [21] to complex killing rates in a region extending all the way to the critical point, and thereby lead to a proof of (1.6). A related set of questions arises when studying the ensemble of walks of all lengths (1.5) for values of the killing rate that are real, but less than the critical value. In this case, with careful handling of the inﬁnite-volume limit, the walk ﬁlls all of space to some density that depends on the strength of the interaction. The walk is said to be in the dense phase, and the density exhibits critical behavior as the killing rate approaches the critical value that, like (1.6), is expected to depart from that of a mean-ﬁeld model for d ≤ 4. This behavior was rigorously shown for the hierarchical weakly SAW in four dimensions in [16], where the dense phase was shown to be associated with a broken supersymmetry in a ﬁeld-theoretic representation. It is natural to hope that progress on the weakly SAW on a nonhierarchical lattice in the unbroken phase will translate into progress on the critical behavior of the dense phase for this lattice, as happened in the hierarchical case. A complication is the presence of Goldstone modes, which would considerably complicate the analysis in the non-hierarchical case, though considerable progress on such issues has been made by Balaban [2, 3, 4] in the context of low temperature bosonic models. Much of the progress on SAW, starting with (1.6), has been enabled by its close relationship to φ4 ﬁeld theories and spin systems. These latter models have been extensively studied with rigorous renormalization group techniques in four dimensions [11, 13, 19]. Another related model is the SAW on a Sierpi` nski gasket, which was studied in [20], again using renormalization group ideas.

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We now present a brief overview of the rest of the paper. Our ﬁrst step is to follow [21] and rewrite the interacting Green’s function Gλ (x, a) in a form of a noninteracting walk in an imaginary random potential, which is a lattice version of an Edwards model. We adapt the phase space expansion in [21] to our model beginning in Section 3. Because we do not attempt to study the limit of long walks, we have both a ﬁxed UV cutoﬀ, provided by the lattice, and a ﬁxed IR cutoﬀ, provided by the killing rate. If the killing rate is taken to be real and positive, a single cluster expansion of the type developed in [21] suﬃces. This cluster expansion has the property that only one cluster contributes to the Green’s function (we refer to this property as “factorization,” for reasons to be explained later). This is in contrast to the usual case in statistical mechanics or ﬁeld theory where the clusters containing external ﬁelds interact with vacuum clusters. It is a reﬂection of the fact that the underlying model is a random walk that interacts only with itself. The situation becomes more complicated in the presence of a complex killing rate, which we will need to invert the Laplace transform. These diﬃculties are detailed in Section 3. Very roughly, the issues are similar to those of preserving positivity when decoupling the covariance in an ordinary cluster expansion. In our case, the analogous condition is that a certain operator must be small enough in norm. Care must be taken that this norm condition is not violated by the interpolating scheme that is used to generate the expansion. The scheme that we developed is described in Section 3.2. It preserves the norm condition but tends to destroy the factorization property alluded to above. To save both we performed a preliminary expansion in Section 3.1. The cluster expansion is described in Section 3.3, and the norm and decoupling properties are proven in Section 3.4. Finally, convergence of the expansion is proven in Section 4.

2 The model In this section we will precisely describe the noninteracting (free) random walk, which we take to be a L´evy process on Zd , and explain how to add interactions to deﬁne the self-avoiding walk. The free process ω(t) leaves a site x ∈ Zd at a constant rate r, with probability Qx,y of jumping from x to y, given a jump. We work with a nearest-neighbor walk; i.e., 1 . Qx,y = 1 2d |x−y|=1 We wish to initially work in ﬁnite volume and take the inﬁnite-volume limit later, so we allow the process to be killed on ﬁrst exit from some set Λ ⊂ Zd , which we take to be a hypercube with sides of length L, centered at the origin (we assume L to be odd for convenience). If we set the jumping rate r to be 2d, then the probability PΛ (x, t) of ﬁnding the walk at site x at time t, given that it started at the origin, is given by PΛ (x, t) = E0,Λ 1ω(T )=x = et∆Λ 0,x . (2.1)

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Here ∆Λ is the lattice Laplacian with Dirichlet boundary conditions [∆Λ ]x,y ≡ Λ(x)∆x,y Λ(y) where ∆x,y = −2d 1x=y +1|x−y|=1 is the lattice Laplacian, and Λ(x) is the indicator function for the set Λ. In analogy with the free case, we deﬁne 2 1 Pλ,Λ (x, T ) ≡ E0,Λ 1ω(t)=x e− 2 λτT (Λ) even though it needs further normalization to be interpreted as a probability density. Our ultimate goal is to control Pλ ≡ Pλ,Zd well enough to calculate quantities such as the mean square end-to-end distance of a self-avoiding walk. We will study Pλ through the Laplace transform ∞ Gλ,Λ (x, a) = dT e−aT Pλ,Λ (x, T ) (2.2) 0 ∞ 1 2 = dT E0,Λ 1ω(t)=x exp − dx a τx,T + λ τx,T . 2 0 Λ When a ≥ 0, we obtain by monotone convergence that Gλ (x, a) = lim Gλ,Λ (x, a). Λ↑Zd

(2.3)

Because Gλ and Gλ,Λ are Laplace transforms, they are analytic in a for a in a domain containing a right half-plane. One expects that Gλ has a singularity in a at ac = O(−λ). Since we are not attempting to study this singularity, we will restrict attention to the domain 3π 3π + b < arg a < −b (2.4) D = a ∈ C : |a| > and − 4 4 with 0 0 chosen small, but not too small. We will see below that the Gλ,Λ can be continued analytically to all a ∈ D, and that they are uniformly bounded on each compact subset of D. Hence, by Theorem 15 of [1], they form a normal family, and the convergence (2.3) extends to all a ∈ D [8]. The fact that we are avoiding the singularity at ac means that, when we invert the Laplace transform, we will only be able to make statements about relatively short walks, those of length T −1 . To study the limit T → ∞ requires letting → 0, which will require the more involved multi-slice expansion. We now rewrite (2.2) in a form more amenable to study. We use the identity √ 2 1 e− 2 λ Λ dx τx,T = dµ(σ) ei λ Λ dx σx τx,T (2.5) where dµ(σ) is Gaussian measure with covariance equal to the identity matrix. In order to continue to values of a with a ≤ 0 it will prove necessary to rotate the

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contour in the σ integrals. Similar issues arose in a somewhat diﬀerent context in [8]. We deﬁne σ ˜x = eiθ σx and integrate over σ ˜ real. In order to preserve positivity we must have |θ| < π/4. The choice of θ depends on arg a, and will be made below. Inserting (2.5) into (2.2) and taking a > 0 to enable use of Fubini’s theorem, we obtain ∞ √ −iθ Λ λ˜ σx −a) . d˜ µ (˜ σ) dT E0,Λ 1ω(t)=x e Λ dx τx,T (ie Gλ,Λ (x, a) = 0

Here d˜ µΛ (˜ σ ) is the Gaussian measure with covariance e2iθ I. Applying the Feynman-Kac formula ([6], Proposition 3.2) and performing the integral over T , √ −1 d˜ µΛ (˜ σ ) −∆Λ + a − ie−iθ λ˜ σ , Gλ,Λ (x, a) = 0,x

, and where by σ ˜ inside the inverse we mean the diagonal matrix with√ eigenvalues σ ˜x√ by a we mean a times the identity matrix. If we deﬁne g˜ = e−iθ λ, A = B = CΛ , with CΛ this becomes

Gλ,Λ (x, a)

=

= (−∆Λ + a)−1 , −1

d˜ µ (˜ σ ) B 1 − i˜ g A˜ σB A Λ

.

(2.6)

0,x

Note that CΛ exists as long as a is not in the spectrum of ∆Λ , which is contained in the interval [−4d, 0] ⊂ R. In this case CΛ also has a square root, since −∆Λ is real symmetric and a is a multiple of the identity. Details about the matrices used in this paper can be found in Appendix A. We are now in a position to state our main result. Theorem 2.1 Fix d > 2. The Green’s function for the weakly self-avoiding walk on Zd is analytic in the region a ∈ C satisfying |a| > , | arg a| < 3π/4 − b, for > 0 and 0 < b < π/4, with λ < λ0 suﬃciently small, depending on . In this region it can be written Gλ (x, a) =

G0 (x, a) + Grem (x, a)

where G0 (x, a) = (−∆ + a)−1 0,x and the remainder satisﬁes |Grem (x, a)|

≤

K1 λ −νa |x| e 1 + |a|

where K1 = K1 (), and νa = K2 ln(1 + |a|1/2 ) with K2 = O(1). The proof is contained in Sections 3–4. Throughout this paper the symbol K will be used to denote ﬁxed O(1) constants, whose values can change from usage to usage.

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3 The expansions We ultimately wish to study Gλ,Λ by means of a cluster expansion, which we call the h-expansion (Section 3.3). It will involve paving Λ with a family of expansion cells and decoupling between diﬀerent cells by perturbing the operators A and B (Section 3.2). It is necessary to use care in choosing the decoupling scheme so as not to destroy any needed properties of the operators involved. In our case, we gA˜ σ B)−1 to satisfy a norm condition, which will force us to use a need D−1 ≡ (1−i˜ decoupling scheme that only decouples points that are well separated. However, we also require D−1 to satisfy a certain factorization property, which this decoupling scheme tends to destroy. Hence, we are led to do a preliminary expansion, called the s-expansion (Section 3.1), which restores the desired factorization properties while preserving the norm condition (Section 3.4).

3.1

The s-expansion

We begin by paving Λ with hypercubes ∆ with sides of length α, which is an integer chosen large enough below (so L is chosen to be a multiple of α, for simplicity). We will use the symbol ∆ to denote the hypercube itself, the indicator function for the hypercube, and the operator on 2 (Λ) consisting of multiplication by the indicator function. We introduce the following dependence on the set of variables {s∆ }∆⊂Λ , all of which take values in [0, 1]:

d˜ µΛ (˜ σ ) BD(s)−1 A 0,x Gλ,Λ (x, a; s) ≡ where

D(s) ≡

1 − i˜ gA˜ σ

s∆ ∆ B .

(3.1)

∆⊂Λ

Note that when s∆ = 1 for all ∆ ⊂ Λ this reduces to (2.6). The s-expansion is eﬀected by a ﬁrst-order Taylor expansion in each of the s∆ . By the fundamental theorem of calculus, we have Gλ,Λ (x, a; 1) = (I∆ + R∆ )Gλ,Λ (x, a; s) (3.2) ∆⊂Λ

I∆ (·)

≡

R∆ (·)

≡

(·) s∆ =0 1 d ds∆ (·). ds ∆ 0

The derivatives in the operators R∆ act only on the D(s)−1 , with the result d −1 D(s) = dz i˜ g D(s)−1 A(·, z)˜ σ (z)B(z, ·)D(s)−1 . ds∆ ∆

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That is, the derivative creates two links A and B, both with one end localized in the center cube ∆. The product in (3.2) breaks into a sum of terms, each of which is uniquely associated with a set S of ∆’s in which the corresponding s∆ has been diﬀerentiated. We call such ∆’s painted. So we can write GS (x, a; s) Gλ,Λ (x, a; 1) = S

with i˜ g GS (x, a; s) = ∆∈S

×

permutations π of {1,...,|S|}

1

Λ

ds∆

σ) d˜ µ (˜

0

|S| j=1

BD(s)−1 Vπ1 (zπ1 )D(s)−1 · · ·

dzj σ ˜ (zj )

(3.3)

Λ

· · · D(s)−1 Vπ|S| (zπ|S| )D(s)−1 A

x,y s∆ =0

∆ ∈S /

if

and Vn (zn ) ≡

A(·, zn )∆n (zn )B(zn , ·) .

Here the ordered set of painted boxes {∆1 , . . . , ∆|S| } consists of the boxes comprising S, and the sum over permutations comes from the Leibnitz rule. Each of the GS (x, a; s) serves as an input to the h-expansion.

3.2

Decoupling

We are given a set of boxes S ⊂ Λ, and an associated contribution GS (x, a; s). We now deﬁne the set of cells that will be decoupled, which √ depends on the set S. We say two boxes ∆ and ∆ are neighbors if d(∆, ∆ ) ≤ d. Deﬁne any two painted ∆’s that are neighbors to be connected, and from this rule form maximally connected sets of painted blocks. An expansion cell δ is either • one such maximally connected region, or • an unpainted cell. Deﬁne DS to be the set of all such δ’s for some choice of S. We have, for any S, δ ∩ δ δ

=

∅

=

Λ.

if δ = δ

δ∈DS

In order to do the h-expansion, we need to deﬁne a perturbation of the matrices A = B that decouples points that are widely separated, and also a scheme of

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interpolating between the decoupled matrices and the original ones. We introduce a set of variables {hδδ }, one for each distinct non-ordered pair of expansion cells (δ, δ ), that take values in [0, 1]. We deﬁne the interpolated version of A by A(x, y; h) = hδδ δ(x)δ (y) + δ(y)δ (x) A(x, y) + (δ,δ ) not neighbors

δ(x)δ (y) + δ(y)δ (x) A(x, y) + δ(x)δ(y)A(x, y),

(δ,δ ) neighbors

δ

where the sums are on non-ordered pairs of √distinct cells or on cells in DS , and two cells δ and δ are neighbors if d(δ, δ ) ≤ d. (An analogous formula holds for B, with the same set of hδδ ’s.) We rewrite this as = A(I − G(h)) = (I − H(h))A

A(h)

where the products are operator products, and we have deﬁned G(h) = (1 − hδδ )A−1 δAδ δ,δ not neighbors

H(s) =

(1 − hδδ )δAδ A−1 .

(3.4)

δ,δ not neighbors

Here the sum is over ordered pairs of cells. When some hδδ = 0, the cells δ and δ are decoupled as far as the operators A(h) and B(h) are concerned. We will show that, for all values of all of the hδδ ’s, the operators A(h) and B(h) are small perturbations of A and B in the sense that G and H are small in norm. This fact will be used in Section 3.4. Lemma 3.1 Let G and H be deﬁned by (3.4). Then G, H ≤ Kα , with α ≡ 2 e− a α/2 (1 + K−d a ) . Proof. We bound G; the case the operator by of H is analogous. We bound max(C1 , C2 ), where C1 = sup Λ dy |G(x, y)| and C2 = sup Λ dx |G(x, y)|. So C1 ≤ sup x∈Λ

δ,δ not neighbors

x∈Λ

x∈Λ

≤ Ke−

a 2

δ,δ not neighbors

α

sup x∈Λ

dy dz A−1 (x, z)δ(z)A(z, y)δ (y)

δ,δ not neighbors

≤ K sup

y∈Λ

Λ

≤ K sup

x∈Λ

dy dz δ (y)δ(z)e− a (|x−z|+|z−y|)

Λ

a a dy δ (y) sup e− 2 |z−y| dz δ(z)e− 2 (|x−z|+|z−y|) z∈δ

Λ

δ,δ unrestricted

Λ

dy dz δ (y)δ(z)e−

Λ a 2

(|x−z|+|z−y|)

.

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Using the exponential decay to do the integrals, and noting that the estimate of C2 is similar, we obtain the stated result.

3.3

The h-expansion

Next, according to the scheme developed in Section 3.2, we introduce a dependence on the set of variables {hδδ } in all of the A’s and B’s in (3.3), except for the initial A and ﬁnal B. As with the s-expansion, we apply the fundamental theorem of calculus to generate the h-expansion: (Iδδ + Rδδ ) GS (x, a; s; h) GS (x, a; s) = pairs (δ,δ )

≡ (·) hδδ =0 1 d ≡ dhδδ (·). dhδδ 0

Iδδ (·) Rδδ (·)

Expanding the product, we obtain a sum of terms, each associated with a set of distinct pairs of cells (δ, δ ): 1 1 (3.5) ds∆ dhδδ GS (x, a; s) = sets of ∆∈S pairs P

(i˜ g)k

k=kmin

×

0

(δ,δ )∈P

|S|+|P |

×

0

σ) d˜ µΛ (˜

k m=1

dzj σ ˜ (zj )

Λ

BD(s, h)−1 V1 (z1 )D(s, h)−1 · · · · · · D(s, h)−1 Vk (zk )D(s, h)−1 A

x,y

s∆ =0 if ∆ ∈S / ¯ δ¯ )∈P hδ¯δ¯ =0 if (δ, /

.

Here kmin = max(|S|, |P |/2), because the |P | derivatives in the h-expansion can act on the |S| numerators Vn that result from the s-expansion before creating any new Vn ’s, and up to two h-derivatives can act on each Vn . The unlabeled sum in (3.5) is over all choices for the Vn ’s that could have been produced by the expansion. The possible forms of the Vn ’s here are Vn (zn )     A(·, zn ; h) B(zn , ·; h) ∆ s∆ ∆(zn )    or or or = δn A(·, zn )δn ∆n (zn ) δn B(zn , ·)δn

. (3.6) s∆ =0 if ∆ ∈S / ¯ δ¯ )∈P hδ¯δ¯ =0 if (δ, /

The lower of each of the possibilities occurs in the case of a derivative in the corresponding variable. Hence the only one of the eight possibilities that is excluded

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is the A(h)( s∆ ∆)B(h), since at least one s- or h-derivative is necessary to produce a Vn at this stage. The set of Vn summed over depends on k; for instance if k = kmin and |P |/2 = |S| then none of the Vn will have any h or s dependence. In addition, due to the Leibnitz rule, for any choice of {Vn } there will be k! distinct terms in the expansion consisting of the permutations {Vπn }. At this point each term contains a product of k random ﬁelds σ ˜ (zj ). We use integration by parts to remove them; since the covariance matrix for σ ˜ is e2iθ times ˜ (zj ). These derivatives can the identity, this means each σ ˜ (zj ) turns into e2iθ ∂/∂ σ either act on another σ ˜ (zi ), yielding δ(zi , zj ), or on a D(s, h)−1 , yielding ∂ D(s, h)−1 (3.7) ∂σ ˜ (zj ) −1 dz δ(zj , z )D(s, h) A(·, z ; h) s∆ ∆(z ) B(z , ·; h)D(s, h)−1 . = Λ

∆

This has the form of the sole type of Vn in (3.6) that was excluded before. So the expansion after removal of σ ˜ ’s is similar to (3.5), with the insertion of some δ(zi , zj ) factors, and now the sum over k has an upper bound of 2(|S| + |P |). It is convenient to write the expansion in a slightly diﬀerent form. We choose to expand all of the δ’s in all of the Vn as δ = ∆⊂δ ∆, which gives rise to the expression GS (x, a; s) =

sets of ∆∈S pairs P

×

d˜ µΛ (˜ σ)

1 0

sets of triplets tn (∆n ,∆n ,∆ n)

ds∆

dhδδ

(δ,δ )∈P

Θ

k m=1

1

0

dzj ∆j

2(|S|+|P |) (i˜ g )k

(3.8)

k=kmin

σ−pairings (α,β)

 

 δ(zα − zβ )

α,β

x,y

× BD(s, h)−1 V1 (z1 )D(s, h)−1 · · · D(s, h)−1 Vk (zk )D(s, h)−1 A

.

s∆ =0 if ∆ ∈S / ¯ δ¯ )∈P hδ¯δ¯ =0 if (δ, /

Here Θ is the product of phases that arose in the σ ˜ removal process. The sum over sets of triplets runs over all sets that can be produced by the expansion. The sets consist of k triples (∆n , ∆n , ∆n ) of boxes in Λ, along with an ordering, and each triplet has a type tn ∈ {1, . . . , 8}. The eight types correspond to the eight possibilities in Vn (s, h; zn )

   hδn δn s∆ n hδn δn = ∆n A(·, zn )  or   or   or  B(zn , ·)∆n 1 1 1 

. (3.9) s∆ =0 if ∆ ∈S / ¯ δ¯ )∈P hδ¯δ¯ =0 if (δ, /

424

Steven E. Golowich

Ann. Henri Poincar´ e

Note that the speciﬁcation of a set of triplets along with a type uniquely speciﬁes a term in the h-expansion. With a further speciﬁcation of σ ˜ -pairings, a term resulting from the removal of σ ˜ ’s is uniquely speciﬁed. In (3.9), hδδ refers to the cells δ ⊃ ∆ and δ ⊃ ∆ . At this point, for every non-ordered set of k triplets we have k! distinct terms in the expansion. We combine all of these contributions via the Cauchy formula (· · ·)BD−1 V1 (z1 ) · · · Vk (zk )D−1 A (3.10) ordered sets of triplets tn (∆n ,∆n ,∆ n)

=

non−ordered sets of triplets tn (∆n ,∆n ,∆ n)

=

non−ordered sets of triplets tn (∆n ,∆n ,∆ n)

k ∂ (· · ·) B ∂γ j j=1

1 (· · ·) 2πi

|γi |=ri

−1 D+ γi Vi A i=1 k

dγ1 · · · dγk 2 B i γi

D+

γi =0 ∀ i

k

−1 γi Vi

A.

i=1

We will comment on the choice of ri below. First, we notice that the integrand in (3.8) has no dependence on σ ˜ (x) for all x ∈ S, so we do those integrals. The ﬁnal expression for the expansion is Gλ,Λ (x, a; 1) =

S

sets of ∆∈S pairs P

×

d˜ µS (˜ σ)

Θ

|γi |=ri

1 0

k

ds∆

m=1

non−ordered sets of triplets tn (∆n ,∆n ,∆ n)

1 × 2πi

dhδδ

(δ,δ )∈P

2(|S|+|P |) (i˜ g )k

0

k=kmin

   δ(zα − zβ )

dzj ∆j

1

σ−pairings (α,β)

 −1  k dγ1 · · · dγk  2 γi Vi A B D+ i γi i=1

α,β

. (3.11)

s∆ =0 if ∆ ∈S / ¯ δ¯ )∈P 0,x hδ¯δ¯ =0 if (δ, /

The contours γi are circles of radius ri , chosen to be ri

=

χ

1 + |a| (1−η) a (d(∆i ,∆i )+d(∆i ,∆i )) e 1 + n(∆i )

where χ is a constant chosen small enough below, and 0 < η < 1. The deﬁnition of n(∆) depends on the set of triplets. In each such set there are 2k pairs of ∆’s, two per triplet. Of these, exactly |P | pairs (∆, ∆ ) are associated with h-derivatives; that is, there has been a derivative with respect to hδδ with δ ⊃ ∆ and δ ⊃ ∆ , which is reﬂected in the type t of the associated triplet.

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We deﬁne n(∆) = number of such derived pairs that contain ∆ in this set of triplets. So n(∆) is bounded above by the number of pairs of cells in P that contain δ ⊃ ∆, but can be much smaller if δ is large. Lemma 3.2 The number of triplets in our set with center box ∆j = ∆, for any given ∆, is bounded above by 2(n(∆) + 1). Proof. Before the removal of σ ˜ ’s, all factors Vn were of types 1 . . . 7; that is, they were acted on by at least one of the three possible derivatives with respect to s, h, and h again. There was at most a single Vn without an h-derivative that had ∆ as a center box, since there is just one s∆ per box. So at this point, after expansion into triplets, there were at most 1 + n(∆) triplets with center box ∆. Now consider the eﬀects of removal of σ ˜ ’s. The number of factors Vn can potentially be doubled, with all newly created Vn ’s of type 8, but the center point zn of each new Vn so created is tied by a δ-function to the center point of an existing Vn (see (3.7)). Also, each existing Vn can be tied to at most one such new Vn , so after expansion into triplets we have the stated bound. The choice of rj was to enable the following result, which will in turn validate the use of the Cauchy formula (3.10) in our estimates. Lemma 3.3 k " " " " γj Vj " ≤ "

O(χ) .

(3.12)

j=1

Proof. We bound the norm of the operator by max(C1 , C2 ), as Lemma 3.1. So C1 ≤ sup dy rj |Vj (x, y)| x∈Λ

≤

sup x∈Λ

Λ

j

j

χ

1 + |a| 1 + n(∆j )

K 1 + |a|

∆ j

dy e−η a (d(∆x ,∆j )+d(∆j ,∆j )) ,

where we have used Lemma A.3 to bound A and B, and written ∆x for the box containing x. Performing the integral over y, we ﬁnd C1

≤ χKαd sup

x∈Λ ∆ j:∆ =∆ j

1 e−η d(∆x,∆) 1 + n(∆)

≤ χKαd where we have used Lemma 3.2. The same bound applies to C2 .

426

3.4

Steven E. Golowich

Ann. Henri Poincar´ e

Properties of D −1

We prove two lemmas regarding norm and factorization properties of D−1 . Lemma 3.4 If α is chosen large enough, then D(s, h)−1

≤

O(1)

uniformly in {s∆ }, {hδδ }. Proof. We ﬁrst note that A(h) = B(h) are invertible since A = B are and, from Lemma 3.1 we have A(h) = A(1 − G), with G ≤ O(α ). Write M ≡ ∆ s∆ ∆, a multiplication operator. Then D = D(s, h) = 1 + i˜ g A(h)˜ σ MB(h). Choose any ϕ ∈ 2 (Λ), ϕ = 0, and by the Cauchy-Schwarz inequality, (A(h)−1 )† B(h)ϕ, Dϕ Dϕ ≥ . (A(h)−1 )† B(h)ϕ

(3.13)

(3.14)

First concentrate on the denominator on the right-hand side. We have (A(h)−1 )† B(h) = =

((I − H)−1 )† (A−1 )† A(I − G) (I + H )(A−1 )† A(I − G).

(3.15)

with H satisfying H ≤ O(α ), which is true because H ≤ O(α ). A is unitarily diagonalizable, so (A−1 )† A = U , a unitary matrix with eigenvalues in the arc of the unit circle deﬁned by |z| = 1, arg z ∈ [0, − arg a]. We expand the product (3.15) into (A(h)−1 )† B(h) = U + H U − U G − H U G ≡ U + R

(3.16)

where R ≤ O(α ). So (A(h)−1 )† B(h)ϕ

=

ϕ(1 + O(α )),

(3.17)

and we have an upper (as well as lower) bound on the denominator in (3.14). Next we ﬁnd a lower bound on the numerator. Consider = (A(h)−1 )† B(h)ϕ, ϕ (A(h)−1 )† B(h)ϕ, Dϕ σ MB(h)ϕ + i˜ g (A(h)−1 )† B(h)ϕ, A(h)˜ ≡

X + Y.

Recall that σ ˜ ∈ R. Hence Y = i˜ g(B(h)ϕ, σ˜ MB(h)ϕ) satisﬁes arg Y = −θ + π/2, since σ ˜ M is a multiplication operator by a real-valued function. Next apply (3.16) to X, ﬁnding X

= (U ϕ, ϕ) + (Rϕ, ϕ).

Vol. 4, 2003

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

427

2 We have |(Rϕ, ϕ)| ≤ O(α )ϕ2 . Also, (U ϕ, ϕ) = wϕ ¯ , where w is a convex combination of the eigenvalues of U . Because these all lie in the arc [0, − arg a] of the unit circle, we have |w| ¯ ≥ cos((− arg a)/2), with arg w ¯ ∈ [0, arg a]. We choose

θ

=

− π−b if −ω ≤ arg a < 3π −b π 4 b 2 43π if ω > arg a > − − 4 2 4 −b

with some ω > 0. If α is chosen large enough, we ﬁnd X +Y

=

¯ + O(α )) + Y ϕ2 (w

satisﬁes |X + Y | ≥ Kϕ2 , with K = O(1). Combining this with (3.17), we ﬁnd Dϕ

≥ K ϕ

with K = O(1). Hence ker D = {0} and D is invertible, with D−1 ≤ 1/K . The factorization property of D−1 depends on the proper deﬁnition of connected components. Choose a term generated by the h-expansion. We deﬁne two of our expansion cells δ and δ to be connected if • the corresponding variable hδδ has been diﬀerentiated, or • they are neighbors and at least one of them contains a painted cube. With these rules we form maximally connected sets of connected δ’s, called connected components. Note that due to the second rule painted regions are surrounded by corridors, one cell deep, of unpainted δ’s. A typical connected component is illustrated in Figure 1. Lemma 3.5 Suppose there are n connected components containing painted cells, with supports Λ1 , . . . , Λn . Then D(s, h) can be written D

= D|Λ1 ⊕ · · · ⊕ D|Λn ⊕ I|Λc ,

(3.18)

where Λc = Λ \ {∪Λi }, and N |Λi is the operator Λi N Λi acting on 2 (Λi ). Proof. Consider x, y ∈ Λ such that A(x, z; h)σ(z)M(z)B(z, y; h) = 0 (cf. (3.13)). Because M(z) is nonzero only on painted ∆’s, we must have z ∈ ∆z , a painted box. By our connection rules, both x and y must be in ∆’s that are connected to ∆z , and hence x and y are in the same connected component, which necessarily contains at least one painted box. The form (3.18) follows. The same argument shows that no Vn can connect two diﬀerent connected components. Hence the chain of operators D−1 V1 · · · Vk D−1 vanishes for any term which contains more than one connected component.

428

Steven E. Golowich

Ann. Henri Poincar´ e

δ1 δ2

δ3

δ4

Figure 1: A typical connected component, consisting of the three painted cells δ2 , δ3 , δ4 , the corridors around them, and the unpainted cell δ1 . A dashed line indicates that the corresponding hδδ has been diﬀerentiated. All boxes within the thick lines are part of the connected component. In this example α = 2.

Vol. 4, 2003

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

429

4 Convergence of the expansion We prove convergence of the expansion and establish Theorem 2.1. We ﬁrst note that the leading behavior of Gλ,Λ is given by the term with no derivatives (|S| = 0); the rest of the expansion will be a small remainder, so we write Gλ,Λ (x, a; 1) =

CΛ (0, x) + Grem (x, a)

and proceed to estimate Grem . Recall that only one connected component contributes to Grem . We begin by pulling out a common factor of exp(−νa |x|), with ν < 1 − η chosen small enough below. We compensate for this small factor by introducing other large factors, which will be cancelled by small factors already present in the expansion. The large factors we introduce are

• a factor of eν a (d(∆j ,∆j )+d(∆j ,∆j )) per triplet, which will later be cancelled by part of the exponential decay in rj−1 , and √

• a factor of eν dα a per ∆ in the support of the connected component, which will later be cancelled by a small factor per ∆ arising from the coupling constants g˜. By our connection rules any two ∆’s in the support of a connected component are connected by a chain of the two types of links, so these two types of factors are suﬃcient to compensate for the decay we have pulled out. We now take absolute values inside most of the integrals, and are left with 1 dz dz ds∆ |Grem (x, a)| ≤ e−ν a |x| Λ

×

1

dhδδ

(δ,δ )∈P

0

×

dµScos−1 (2θ)I (˜ σ)

S:|S|≥1 P ∆∈S

2(|S|+|P |)

k

×

dzj

m=1

|S| cos−1/2 (2θ)

sets of triplets tn (∆n ,∆n ,∆ n)

∆j

gk

k=kmin

0

σ−pairings (α,β)

 

 δ(zα − zβ )

α,β

k √ |Λ0 | eν a (d(∆j ,∆j )+d(∆j ,∆j )) eν dα a j=1

1 dγ1 · · · dγk ν a |x−z | 2 e × B(x, z ) 2πi |γj |=rj j γj −1  γj Vj  A(z , y)eν a |z −y| , × D + j

z ,z

430

Steven E. Golowich

Ann. Henri Poincar´ e

which we estimate by ≤ e−ν a |x|

|Grem (x, a)|

  

dz dz

Λ

−1  ν a |x−z | × supe B(x, z ) D + γj Vj  ×

j

  sup  z ,z

S:|S|≥1 P ∆∈S

2(|S|+|P |)

×

g

k=kmin

×





σ−pairings (α,β)

×

k

k

sets of triplets tn (∆n ,∆n ,∆ n)

z ,z

1

A(z , y)eν a |z

ds∆

0

  −y| 

1

dhδδ

(δ,δ )∈P

σ) dµScos−1 (2θ)I (˜

0 k m=1

dzj

∆j



δ(zα − zβ )

α,β

rj−1 eν a (d(∆j ,∆j )+d(∆j ,∆j ))



Ke

|Λ0 |  √ ν dα a 

j=1

.

(4.1)

σ }, {zj }, The unlabeled supremum is over S, P , {s∆ }, {hδδ }, k, sets of k triplets, {˜ choices of σ ˜ pairings, and {γj }. We have written Λ0 for the support of a term, i.e., the union of boxes in triplets. The ﬁrst term in braces in (4.1) can be bounded by B(x, ·)eν a |x−·| A(·, y)eν a |·−y| (D +

j

γj Vj )−1 ≤

K 1 + |a|

where we have used Lemma A.3 to bound the 2 norms and (D + γj Vj )−1 ≤ (1 + D−1 γj Vj )−1 D−1 j

j

along with applications of Lemmas 3.3 and 3.4 to bound the operator norm. At ˜ (zj ), so this point there is no longer any dependence on any of the s∆ , hδδ , or σ we can do all of those integrals, which result in a factor K per ∆ ∈ Λ0 . Next do the z integrals. Recall that when we removed the σ ˜ ’s, all of the zj ’s were paired up by δ functions: either two σ ˜ ’s contracted with each other, which resulted in a δ function equating the corresponding zj ’s, or a σ ˜ contracted with a D−1 , which created a new V with corresponding z equated to the original z with a δ function. We integrate over all of the z’s pair by pair. In either type of pair, each zj is uniquely associated with a Vj , and each Vj gave rise to a factor of grj−1 ,

Vol. 4, 2003

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

431

so for each triplet there is a factor of g/(1 + |a|). Each pair of z integrals gives αd , so the total factor per triplet is Kg/(1 + |a|). Now bound the number of possibilities for σ ˜ pairing. The choice of typed triplets determines which σ ˜ ’s are available for pairing, so an upper bound on the number of possibilities is the number of pairings if all σ ˜ ’s were allowed to pair up. ˜ (zj ) must have zi and zj in the same ∆. Before σ ˜ In order to pair, σ ˜ (zi ) and σ removal, there are at most 1 + n(∆)σ ˜ ’s per ∆ that are restricted to ∆, so the number of σ ˜ pairings is bounded by ∆ (1 + n(∆))!!. We choose to rewrite the sum as one over supports Λ0 ⊂ Λ, and then sum over all diagrams with support Λ0 . We use the decay from the rj−1 to cancel the exponentially growing terms we introduced earlier, and are left with k K Kg |Grem | ≤ e−ν a |x| sup 1 + |a| 1 + |a| z ,z

×

sets of k triplets tj (∆j ,∆j ,∆ j)

Λ0 ⊃{z ,z } k consistent with Λ0 √ ν dα a

Ke

(1 + n(∆))!! n(∆)2(n(∆)+1)

∆

×

k

e−(η−ν) a (d(∆j ,∆j )+d(∆j ,∆j )) .

j=1

Here the sum over sets of triplets includes all sets with support Λ0 that could have been produced by the expansion. We do not need sums over S and P any more since a set of typed triplets uniquely determines these sets. We bound the n(∆) dependence with a piece of the exponential decay by means of Lemma 4.1 Given a set of triplets, k −ν a d(∆,∆ ) n(∆)! e ≤ K. ∆∈Λ0

pairs (∆,∆ )

∆

where the product over pairs includes all pairs (∆, ∆ ) that arose from the action of a derivative d/dhδ(∆)δ(∆ ) . Proof. This is the standard exponential pinning argument. First, note that the pairs in the product are exactly those in the deﬁnition of n(∆). Next, split each exponential decay factor into a product of two, and assign each half to one of the boxes in the pair. Then each ∆ is connected to exactly n(∆) other boxes ∆ by an exponential decay factor. Choose a ∆. All n(∆) of the associated ∆ ’s must be distinct. So half of them must be further away from ∆ than K(d)n(∆)1/d . The net factor from the exponential decays of the distant ∆ ’s is bounded above by e−K a n(∆)

1+ 1 d

< K (n(∆)!)−k

for any k. Since this is true for each ∆, we are done.

432

Steven E. Golowich

Ann. Henri Poincar´ e

We next replace the sum over sets of triplets with a sum over trees, which is more convenient to estimate. Each node of our trees will be mapped to some ∆ ⊂ Λ. It will be helpful to distinguish the structure of connections in the abstract tree from those of its image in Λ; the latter need not be simply connected. We ﬁrst observe that any set of triplets can be given such a tree structure. Think of each triplet as a subtree with three nodes. Assemble these subtrees into a single tree by introducing new connections: two nodes may be connected if they are mapped to the same ∆, or to neighboring ∆’s. The resulting tree (which is not necessarily unique) has the property that every node is a member of exactly one triplet. In order to recover the entire structure of a set of triplets from a tree, we must assign each node a label according to whether it is a center, left, or right box in the triplet, and whether or not it was diﬀerentiated (the upper or lower possibilities in (3.9)). Also, to each center node there must be indications for which two of the nodes connected to it are the corresponding side nodes. Now we sum over trees in such a way that all allowed sets of typed triplets with support Λ0 in the expansion appear as terms in our sum. For an upper bound we drop the restriction that Λ0 z , and consider summing over rooted trees, with the root being a node with coordination number one connected to a node which maps to the box containing z . We label our nodes starting from 0 (for the root), and let di denote the coordination number of the ith node (so d0 = 1). The estimates at this point do not depend on the types of the triplets, so we introduce a factor of three per node to select for left, right, or center, and a factor of two per node select whether ornot it was diﬀerentiated. We also insert a factor to of 1 + d2i per node; the d2i is to choose between the various possibilities for forming triplets with this node as the center box, and the 1 corresponds to the possibility of this node not being a center box. Since we have seen that the trees corresponding to terms in the expansion satisfy the condition that each node is in exactly one triplet, this factor is suﬃcient. This fact is also the source of the crucial small factor guaranteeing convergence; each triplet in the expansion is associated with a factor Kg/(1 + |a|), so we can insert a factor of (Kg/(1 + |a|))1/3 for each node in our tree. The bound becomes m/3 ∞ √ Kg 1 K |Grem (x, a)| ≤ e−ν a |x| (Keν dα a )|Λ0 | 1 + |a| m! 1 + |a| Λ0 z m=|Λ0 | m di 1 × ··· e− 2 (ν−η) a d(∆,∆ ) 1+ , 2 i=1 d1 ≥1 dm ≥1 trees positions of links ) *+ , T (∆,∆ )∈T nodes i (di −1)=m−1

where the sum on positions of nodes is restricted so at least one node is mapped to each ∆ ⊂ Λ0 . We now observe (Keν

√ dα a |Λ0 |

)

≤ (Ke3ν

√ dα a m/3

)

.

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433

We then remove the sum over Λ0 by allowing the positions of nodes to vary over all of Λ, and summing over all m. Note that the smallest nonvanishing diagram contributing to Grem has m = 6 (and therefore is O(g 2 ) = O(λ)), since a term with only one triplet would correspond to |S| = 1, and the removal of the sole σ ˜ would create another triplet. We now estimate this sum by using the exponential decay factors to do the sums over positions of nodes, getting a factor K per node. By Cayley’s formula and we can then do the sums over the sum on trees yields (m − 1)!/ i (di − 1)!, √ 3ν dα a /(1 + |a|) ≤ O(1), then for any the di . If we choose ν small enough that e λ < λ0 small enough we obtain for our ﬁnal bound |Grem (x, a)|

≤

K g 2 −ν a |x| e . 1 + |a|

(4.2)

Also, by Lemmas A.2 and A.3, we have that CΛ (0, x) → C∞ (0, x) uniformly in a ∈ D as Λ ↑ Zd . We have proven the convergence of the expansion in ﬁnite volume, for the killed process, with estimates uniform in the volume. By our comments after (2.4), (4.2) allows us to take the inﬁnite volume limit, which yields Gλ (x, a), the Green’s function on Zd , and proves Theorem 2.1.

A

Appendix: Covariance operators on 2 (Λ)

The ﬁnite-volume free process is generated by the lattice Laplacian with Dirichlet boundary conditions ∆Λ = Λ∆Λ. We have taken Λ to be a hypercube with sides of length L centered at the origin, with L odd. We deﬁne the boundary ∂Λ to be all z ∈ Zd , not in Λ, that are neighbors of points in Λ. We require estimates involving the kernels of the operators CΛγ ≡ (−∆Λ + a)−γ , with γ = 1, ±1/2 and a ∈ D. We study these operators by explicit diagonalization of −∆Λ + a. Details on some of these considerations may be found in [14]. We adopt the Fourier convention f (x) = ddp fˆ(p)eipx ; fˆ(p) = (2π)−d f (x)e−ipx . [−π,π]d

x∈Zd

To begin, we deﬁne

e(1) pα (xα )

ep (x)

 1/2   2 cos(pα xα ) if pα = L+1 = 1/2   2 sin(pα xα ) if pα = L+1 =

d α=1

e(1) pα (xα ).

(2n+1)π L+1 , 2nπ L+1 ,

n∈Z

n∈Z

434

Steven E. Golowich

Lemma A.1 The set

ep (x) : p ∈

π Zd , 0 < pα < π L+1

Ann. Henri Poincar´ e

diagonalizes −∆Λ , with eigenvalues λ(p)

= 4

d

sin2 (pα /2).

(A.1)

α=1

Proof. By construction we have ep (x) = 0 whenever x ∈ ∂Λ, so Λ∆Λep (x) = d ∗ Λ∆ep (x). The result follows by explicit computation, using −∆ = α=1 ∂α ∂α , where ∂ is the lattice gradient. From this we ﬁnd (λ(p) + a)−γ ep (x)ep (y). (A.2) CΛγ (x, y) = π Zd L+1 0
p∈

It will be convenient to have another representation for CΛγ (x, y), in terms of 2 d C∞ (x, y) = C∞ (x − y) = (−∆ + a)−1 x,y , an operator on (Z ). We use the method of images, with the set of image points consisting of translates by n ∈ 2(L + 1)Zd 2d −1 of the set of 2d points {y (j) }j=0 with coordinates L+1 L+1 yα(j) = ±yα ∓ + 2 2 (j)

where sign ± for yα is chosen based on the value of the αth bit in the binary representation of j (+ for 0). With this deﬁnition y (0) = y. Let j be the number of 1’s in the binary representation of j. Lemma A.2

 2d −1   γ (−1) j C∞ (x − yj + 2(L + 1)n) if x, y ∈ Λ ∪ ∂Λ CΛγ (x, y) = (A.3) d j=0  n∈Z  0 otherwise.

γ Proof. C∞ (x − y) are the Fourier coeﬃcients ddp (2π)−d (λ(p) + a)−γ eip(x−y) .

(A.4)

[−π,π]d

which we insert into the RHS of (A.3), and call the result S(x, y). For x, y ∈ Λ we obtain S(x, y)

=

n∈Zd

d

−d

d p (2π) [−π,π]d

−γ

(λ(p) + a)

d −1 2

(−1) j eip(x−y

j=0

(j)

+2(L+1)n)

Vol. 4, 2003

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

lim

M→∞

   ddp  eip2(L+1)n  

=

435

[−π,π]d

n∈Zd |nα |≤M



× (2π)−d (λ(p) + a)−γ



d −1 2

(j) (−1) j eip(x−y ) 

j=0

The integrand is periodic in any coordinate direction with period 2π. We introduce a partition of unity in p-space given by 1 = ∈Zd χ(p + 2π), with χ ∈ C0∞ (Rd ). Then    ip2(L+1)n  ddp  e S(x, y) = lim (A.5)  M→∞

Rd

n∈Zd |nα |≤M



× (2π)−d χ(p)(λ(p) + a)−γ

d −1 2



(−1) j eip(x−y )  . (j)

j=0

The ﬁrst bracketed term in (A.5) converges in the sense of distributions to (2π)d δ(2(L + 1)p − 2πm), m∈Zd

while the second bracketed term is a C0∞ test function. We do the integral over p-space, and use periodicity again to ﬁnd

−d

S(x, y) = (2(L + 1))

−γ

(λ(p) + a)

π Zd L+1 |pα |≤π

d −1 2

(j) (−1) j eip(x−y ) .

j=0

p∈

Next we use the fact that λ(p) is even with respect to reﬂections about any of the coordinate planes to ﬁnd S(x, y) = (2(L + 1))−d (λ(p) + a)−γ π Zd L+1 0
p∈

×

1

(−1)j1 +···+jd

d

eipα (xα −yα

(j)

)

+ e−ipα (xα −yα

(j)

)

,

α=1

j1 ,...,jd =0

where now the limits on the sum over pα do not include 0 or π, since the contributions from these terms vanished due to our choice of image points. A bit of algebra shows S(x, y) = (λ(p) + a)−γ ep (x)ep (y). π Zd L+1 0
p∈

Comparing with (A.2), we are done.

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Lemma A.3 There exist constants K3 , K4 = O(1), independent of a and Λ, such that for any a ∈ D, |a| > K/L2 and γ < d/2, |CΛγ (x, y)|

≤ K3 (1 + |a|)−γ e− a |x−y| ,

where a = K4 ln(1 + |a|1/2 ). Proof. We prove the result in inﬁnite volume; the ﬁnite volume result follows γ by an application of Lemma A.2. We have a representation of C∞ (x, y) as Fourier coeﬃcients in (A.4). The integrand is analytic as a function of k1 except for branch points at the zeros of a + λ(k). These are at  1/2  d a . + sin2 (kα /2) k1∗ = 2 sin−1 ±i 4 α=2 The location of the closest zero to the real axis is easily seen to satisfy | k1∗ | ≥

K ln(1 + |a|1/2 ).

The integrand is periodic in k1 for all k, so we can shift the contour up or down, depending on the sign of (x1 − y1 ), by half the distance to the nearest zero, say. We ﬁnd γ |C∞ (x − y)| ≤ e− a |x1 −y1 | ddk (a + λ(k + iae1 ))−γ . [−π,π]d

where we have chosen a = K ln(1 + |a|1/2 ) to be half the distance to the zero. As long as γ < d/2, the integral is bounded by O(1) when |a| < O(1) and by O(|a|−γ ) when |a| 1. By symmetry in coordinate directions and with a redeﬁnition of the constant K in the deﬁnition of a , we obtain the stated result.

Acknowledgments The author gratefully acknowledges many useful conversations with John Z. Imbrie, Paulo da Veiga, and Jacques Magnen.

References [1] L.V. Ahlfors, Complex analysis, McGraw-Hill, 1979. [2] T. Balaban, A low temperature expansion for classical N -vector models I: A renormalization group ﬂow., Comm. Math. Phys. (1994). [3]

, A low temperature expansion for classical N -vector models II: Renormalization group equations, Commun. Math. Phys. 182, 675–721 (1996).

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, A low temperature expansion for classical N -vector models III: A complete inductive description, ﬂuctuation integrals, Commun. Math. Phys. 196, 485–521 (1998).

[5] D. Brydges and T. Spencer, Self-avoiding walk in 5 or more dimensions, Commun. Math. Phys. 97, 125–148 (1985). [6] David Brydges, Steven N. Evans, and John Z. Imbrie, Self-avoiding walk on a hierarchical lattice in four dimensions, Ann. Prob. 20, 82–124 (1992). [7] David Brydges and John Z. Imbrie, End-to-end distance from Green’s function for a hierarchical self-avoiding walk in four dimensions, Commun. Math. Phys. (2003, to appear). [8]

, Green’s function for a hierarchical self-avoiding walk in four dimensions, Commun. Math. Phys. (2003, to appear).

[9] P.G. de Gennes, Exponents for the excluded volume problem as derived by the Wilson method, Phys. Lett. A 38, 339–340 (1972). [10] S.F. Edwards, The statistical mechanics of polymers with excluded volume, Proc. Phys. Soc. 85, 613–624 (1965). [11] J. Feldman, J. Magnen, V. Rivasseau, and R. S´en´eor, Construction and Borel summability of infrared Φ44 by a phase space expansion, Commun. Math. Phys. 109, 437–480 (1987). [12] P. Flory, Principles of polymer chemistry, Ithaca Press, Cornell, 1949. [13] K. Gaw¸edzki and A. Kupiainen, Massless lattice φ44 theory: Rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys. 99, 197–252 (1985). [14] J. Glimm and A. Jaﬀe, Quantum physics: A functional integral point of view, Springer-Verlag, 1987. [15] Steven E. Golowich and John Z. Imbrie, A new approach to the long-time behavior of self-avoiding random walks, Ann. Phys. 217, 142–169 (1992). [16]

, The broken supersymmetry phase of a self-avoiding random walk, Commun. Math. Phys. 168, 265–319 (1995).

[17] T. Hara and G. Slade, The lace expansion for self-avoiding walk in ﬁve or more dimensions, Rev. Math. Phys. 4, 235–327 (1992). [18]

, Self-avoiding walk in ﬁve or more dimensions. I. The critical behavior, Commun. Math. Phys. 147, 101–136 (1992).

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[19] T. Hara and H. Tasaki, A rigorous control of logarithmic corrections in fourdimensional φ4 spin systems, J. Stat. Phys. 47, 57–121 (1987). [20] T. Hattori and T. Tsuda, Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski gasket, preprint, mp arc:02-225 (2002). [21] D. Iagnolnitzer and J. Magnen, Polymers in a weak random potential in dimension four: Rigorous renormalization group analysis, Commun. Math. Phys. 162, 85–121 (1994). [22] John Z. Imbrie, End-to-end distance for a four-dimensional self-avoiding walk, CRM Proceedings and Lecture Notes, vol. 7, American Mathematical Society, 1994, pp. 191–196. [23] K.M. Khanin, J.L. Lebowitz, A.E. Mazel, and Ya.G. Sinai, Self-avoiding walks in ﬁve or more dimensions: polymer expansion approach, Russian Math. Surveys 50, 403–434 (1995). [24] R. van der Hofstad, F. den Hollander, and G. Slade, A new inductive approach to the lace expansion for self-avoiding random walks, Probab. Th. Rel. Fields 111, 253–286 (1998). [25] R. van der Hofstad and Gordon Slade, A generalized inductive approach to the lace expansion, Preprint, arXiv:math.PR/0012026 (2000). Steven E. Golowich Bell Laboratories Lucent Technologies 700 Mountain Ave. Murray Hill, NJ 07974, USA email: [email protected] Communicated by Vincent Rivasseau submitted 3/12/02, accepted 24/02/03

Ann. Henri Poincar´e 4 (2003) 439 – 486 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030439-48 DOI 10.1007/s00023-003-0136-6

Annales Henri Poincar´ e

One-particle (improper) States in Nelson’s Massless Model Alessandro Pizzo Abstract. In the one-particle sector of Nelson’s massless model, the one-particle states are constructed for an arbitrarily small infrared cutoﬀ in the interaction term of the Hamiltonian of the system. The performed method is a constructive one which exploits only regular perturbation theory, by a suitable iteration scheme. The disappearance of one-particle states is showed in the limit of no infrared regularization. Constructive features, as regularity in some parameters, are also inquired.

Introduction In this paper we treat some spectral problems in a model describing quantum mechanical matter locally interacting with the quantized relativistic ﬁeld of scalar massless bosons. Such a model was rigorously studied, in the case of massive bosons, by Nelson [Ne], who removed the ultraviolet cutoﬀ in the interaction. Nowadays it is widely considered a toy model for analyzing infrared aspects of radiation theory. The underlying conjecture is that the model retains some of the infrared features of Q.E.D., in spite of the various approximations made: the charge is not described by a ﬁeld (no pair production), an ultraviolet cutoﬀ is (generally) imposed on the interaction, the “photons” are scalar and the “electron” is a spin less non-relativistic particle. In a rigorous analysis of radiation theory, the zero photon mass implies nontrivial mathematical problems at the level of spectrum properties, which are not avoidable for a satisfactory explanation of radiative phenomena at low energies. In this respect many papers have been recently devoted to a rigorous analysis of binding and resonances ([B.F.S], [Ge], [G.L.L], [L.M.S], [Ar]), scattering of photons and relaxation to the ground state for isolated atoms ([Sp], [D.Ge], [Ge], [F.G.S]). In this paper we are concerned with the translation invariant Nelson’s massless model restricted to only one non-relativistic particle interacting with the boson (scalar) ﬁeld: the so-called “one-particle sector”. The aim is to clarify, in an interacting and physically non-trivial model, the phenomenon of the disappearance of a properly deﬁned mass-shell for the electron in Q.E.D. [Sc], [Bu], when no infrared regularization is performed. Such an analysis is also a prerequisite for a rigorous treatment of the counterpart of “Compton scattering” in the given scalar model. It is worth-while to stress that, though important indications come from solvable infrared models, like non-relativistic Q.E.D. in dipole approximation (also called Pauli-Fierz model [P.F], [Bl]) or a simpliﬁed version of Nelson’s model it-

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self (see [Fr.1] page 27), the answers one gets for the mass-shell problem are not satisfactory, because the adopted approximations are too strong. For this reason the analysis of the full interacting Nelson’s model turns out to be a relevant nonperturbative step to analyze some (limited) aspects of the mass-shell problem in Q.E.D. Mathematically, the mass-shell problem is formalized as the absence of the “one-particle subspace” in the Hilbert space of the system; the one-particle subspace is that one generated by vectors on which the Hamiltonian H acts as a function of the total momentum P of the system (the considered model is covariant under translation). The states in the one-particle space (if it exists with some properties) describe a free particle alone in the world, with an expected non-relativistic dispersion for small momenta. For Nelson’s massless model, this subspace exists in the Hilbert space of the system as long as an infrared cutoﬀ on the interaction is imposed, as proved in [Fr.1] and in the present paper. Our main concerns are therefore: to study the fate of one-particle states when the infrared cutoﬀ is removed, and to get a control on the involved vectors for an arbitrarily small infrared cutoﬀ. More technically, such a study concerns the limiting behavior of the ground states of the Hamiltonians HP , at a ﬁxed total momentum P and acting on a copy of the boson Fock space, when no infrared cutoﬀ is introduced in the interaction term. For this purpose we use an iterative procedure diﬀerent from the operatorial renormalization group developed for analogous problems in [B.F.S] and [Ch]. The method we use provides the construction of the ground state for the Hamiltonians HP,σ , obtained from HP by an (arbitrarily small) infrared cutoﬀ σ in the interaction. The method is based on the analytic perturbation [R.S], it works for small values of the coupling constant and in a neighborhood of P = 0, corresponding to a ratio |P| /m strictly less than 1, where m is the Hamiltonian parameter corresponding to the non-relativistic particle. It exploits the “smallness” of the variation of the interaction term when we slightly modify the infrared (energy) scale. By the same method we can prove the strong convergence of the ground state for some w , acting on the Fock space and obtained from HP,σ through a Hamiltonians HP,σ P-dependent coherent transformation of the boson variables, already known from [Fr.1]. This result easily implies the weak limit to the zero vector of the ground eigenvector of the original Hamiltonian HP,σ , for σ → 0. It also prevents any massshell construction by glueing states corresponding to diﬀerent values of P; in fact, because of the inequivalence, in the limit σ → 0, of the coherent transformations for diﬀerent P, it turns out to be physically meaningless (see [Fr.1], page 53). The fundamental results were already discovered by Fr¨ ohlich [Fr.1], [Fr.2], through a non-perturbative method inspired by Glimm and Jaﬀe [G.J], which is not constructive. By the iterated analytic perturbation, the one-particle states are studied here in terms of the ground eigenvectors of related transformed Hamiltoniw ). The new vectors are more regular in their dependence on parameters ans (HP,σ like the infrared cutoﬀ and the total momentum. Such a characterization of oneparticle states, for arbitrarily small infrared cutoﬀs, is a key ingredient for a rather

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complete description of the asymptotic dynamics, both for the massless ﬁeld and for the charged particle [Pi]. In conclusion, the constructive method provides a more explicit control and then more insight into the physical content of the limit states. Indeed, in this framework, regular perturbation theory becomes eﬀective, even though the limiting situation is that of a non-isolated eigenvalue and therefore seemingly not accessible through such a method. Simple questions can be answered, at least partially, by ordinary perturbation techniques; for instance: how two ground states at diﬀerent cutoﬀ or at diﬀerent P are related, in which sense and to what extent the expected regularity, of certain physical quantities, is conserved under the removal of the infrared cutoﬀ. Deﬁnition of the model The physical system consists of a non-relativistic spin less quantum particle of mass m, linearly coupled to a quantized relativistic scalar boson ﬁeld, which is massless and real. The non-relativistic particle is described by position and momentum variables with usual canonical commutation rules (c.c.r.) [xl , pj ] = iδl,j

( = 1) .

The (scalar) boson ﬁeld, which we will call photon ﬁeld, at time t = 0 is d3 k † 1 A (0, y) = √ 3 · a (k) e−ik·y + a (k) eik·y , 2 |k| 2π (c = = 1), where a† (k) , a (k) are standard creation and annihilation operatorvalued tempered distributions obeying the c.c.r. a (k) , a† (q) = δ 3 (k − q) [a (k) , a (q)] = a† (k) , a† (q) = 0 . The spatial translations are implemented by the total momentum P := p + k a† (k) a (k) d3 k . The dynamics of the system is generated by the covariant (under translation, [H, P] = 0) Hamiltonian κ d3 k p2 ph H := +g a (k) eik·x + a† (k) e−ik·x √ 1 + H 2m 0 2 |k| 2 where κ is an ultraviolet cutoﬀ (the integration bounds throughout the paper are referred to the radial part of k, g is the coupling constant and H ph is the free Hamiltonian of the photon ﬁeld H ph := |k| a† (k) a (k) d3 k).

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The Hilbert space of the system is H= L2 R3 , d3 x ⊗ F where F is the Fock space to the creation and annihilation operator-valued distributions † with respect a (k) , a (k) : an element ψ of H is a sequence {ψ n }n∈N of functions on R3(n+1) with ψ < ∞, where 2

ψ =

∞

ψ n (x, k1 , . . . , kn )ψ n (x, k1 , . . . , kn ) d3 k1 . . . d3 kn d3 x

n=0

and each ψ n (x, k1 , . . . , kn ) is symmetric in k1 , . . . , kn . The n = 0 component corresponds to the tensor product of the vacuum subspace of F with the nonrelativistic particle space L2 R3 . Standard results about H and P : i) The operators

P= p+

k a† (k) a (k) d3 k

are essentially self-adjoint (e.s.a.) in D :=

h ⊗ ψn

n∈N

which is the set of the ﬁnite linear combinations of vectors of wave functions h (x) ψ n (k1 , . . . , kn ), where h (x) ∈ S R3 (the space of Schwartz test functions), 3n d n is symmetric in its variables. Since p ≡ −i dx and ψ (k† 1 , . . . , kn ) 3∈ Ss R 3 n ka (k) a (k) d k are e.s.a. in S R and n∈N ψ respectively, the result easily follows for the P operators. The spectrum of each component of P is the real axis, the spectral measure is absolutely continuous with respect to the Lebesgue measure. ii) The interaction term in the Hamiltonian is inﬁnitesimal small with respect to H0 :=

p2 + H ph . 2m

Therefore H is bounded from below, it is e.s.a. in D and its self-adjointness domain (s.a.d.), D (H), coincides with D (H0 ) (s.a.d. of H0 ). iii) The groups eia·P and eiτ H (τ, ai ∈ R) commute. iv) The joint spectral decomposition of the Hilbert space with respect to the P ⊕ operators is H= HP d3 P , where HP is a copy of F . Indeed to the improper eigenvectors of the P operators, ΦnP , where 3

ΦnP (x, k1 , . . . , kn ) := (2π)− 2 ei(P−k1 −···−kn )·x ϕnP (k1 , . . . , kn ) ϕnP (k1 , . . . , kn ) ∈ Ss R3n ,

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we can relate a natural scalar product: m m 3 3 , Φ ) = δ ϕn (Φn n,m P P P (k1 , . . . , kn )ϕP (k1 , . . . , kn ) d k1 . . . d kn . The vector space n∈N ΦnP is deﬁned as the closure of the ﬁnite linear combinations of the wave functions ΦnP (x, k1 , . . . , kn ) in the norm which arises from the scalar product above. Starting from this space, we uniquely deﬁne the linear application IP :

ΦnP → F b

n∈N

by the prescription: IP (ΦnP (x, k1 , . . . , kn )) 1 = √ b† (k1 ) . . . b† (kn ) ϕnP (k1 , . . . , kn ) d3 k1 . . . d3 kn ψ0 , n! where b (k) , b† (k), which formally correspond to a (k) eik·x , a† (k) e−ik·x , are annihilation and creation operator-valued tempered distributions in the Fock space Fb ∼ = F , and ψ0 is the related vacuum. The given norm for ΦnP is equal to IP (ΦnP )F ( ·F is the Fock norm). The application IP is onto and unitary. v) Since [H, P] = 0, we have that H = HP d3 P , where HP : HP → HP is e.s.a. in Db := n∈N ΦnP ; in terms of the variables P, b (k) , b† (k), the operator HP is written as follows: ph 2 κ d3 k P −P +g b (k) + b† (k) HP = + H ph 2m 2 |k| 0 being H ph ≡ |k| b† (k) b (k) d3 k and Pph ≡ k b† (k) b (k) d3 k when applied on the ﬁber spaces HP . Survey of results Our ﬁrst concern (Section 1) is to single out a sequence of infrared cutoﬀs, {σj }, 8 j σj = κ 2 , j ∈ N (natural numbers), 0 < < 15 , and to construct, for g uniform σ in j, the ground eigenvectors ψPj of the Hamiltonians HP,σj acting on HP ∼ = Fb and with the infrared cutoﬀ σj in the interaction term, namely HP,σj

ph 2 κ d3 k P −P +g b (k) + b† (k) = + H ph 2m 2 |k| σj

m where P belongs to Σ ≡ P : |P| ≤ 20 . The constraint on Σ reﬂects the mixed character of the model, which forces to restrict the physical region of the total momentum to the set {P : |P| < m}; the adopted more restrictive constraint, m , is only due to technical reasons. The law in the infrared sequence, P : |P| ≤ 20

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{σj }, derives from the requirement to have, step by step, a relative interaction σ (∆HP )σjj+1 (deﬁned below) “of the same order” as the gap at least. The restrictive constraint on plays a role in the proof of Theorem 2.3 in Section 2; in Section 1 the κ . natural constraint, 0 < < 1, works. Our proofs require in general a small ratio m Concerning terminology, we use the term ground eigenvector to be precise about σ normalization (the vectors ψPj , obtained in the iteration, are not normalized) and because a phase ﬁxing is implicit in the used procedure and for the desired results. We will generally use the term ground state to indicate only the corresponding ray; by the term “unique ground eigenvector” we mean that the corresponding eigenvalue is non-degenerate. σ In constructing the ground eigenvectors ψPj , the underlying idea is to break σ σ the interaction and to construct the vector ψPj+1 in terms of ψPj by iteration of the analytic perturbation [R.S]. The “small” and analytic perturbation for the Hamiltonian HP,σj is represented by the diﬀerence of the interaction terms σj d3 k σj (∆HP )σj+1 := HP,σj+1 − HP,σj = g b (k) + b† (k) 2 |k| σj+1 at subsequent infrared cutoﬀs and at ﬁxed coupling constant g. In developing this technique, the tensorial structure of the Fock space is crucial: it means that if the Hilbert complex space h is given as a direct sum h1 ⊕ h2 , then the bosonic Hilbert space F over h, F (h), is isomorphic to F1 ⊗ F2 , where F1 is the Fock space over h1 and F2 is the Fock space over h2 . The technique essentially relies on the comparison between the resolvents of the Hamiltonians HP,σj and HP,σj+1 ; it recursively uses the Kato-Rellich theorem on the analytic perturbation of isolated eigenvalues (of self-adjoint operators) to σ σ relate the corresponding ground eigenvectors ψPj and ψPj+1 . At each step two pieces of information are required: 1) a lower bound for the gap of the Hamiltonian HP,σj restricted to the subspace Fσ+j+1 := F (h) , h := L2 R3 \ Bσj+1 , d3 k , Bσj+1 := {k : |k| < σj+1 } ; σ

2) an estimate of the diﬀerence, (∆HP )σjj+1 := HP,σj+1 − HP,σj , between two subsequent infrared cutoﬀ Hamiltonians; we need that it is small with respect to HP,σj |Fσ+ in a generalized sense, in order to expand the spectral proj+1

jection of HP,σj+1 |Fσ+

j+1

, on the ground eigenvalue, in a perturbative series

in terms of the resolvent of HP,σj |Fσ+

j+1

σ

and of the diﬀerence (∆HP )σjj+1 .

The requirement 1) is provided by Lemma 1.1, where we study the operator HP,σj applied to the subspace Fσ+j+1 . The result is that, under the constructive hypotheses σ σ for {σj } and Σ, if ψPj is the unique ground eigenvector of HP,σj |Fσ+ of energy EPj with gap bigger than

σj 2

, then HP,σj |Fσ+

j+1

j

σ

has unique ground eigenvector ψPj ⊗ ψ0

(ψ0 vacuum state) with a gap larger than 35 σj+1 .

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The meaning of this result is the following. When the operator HP,σj , initially considered on Fσ+j , is applied on the larger space Fσ+j+1 keeping the interaction ﬁxed above the cutoﬀ σj , new further terms appear; however, the contribution of the new terms is, roughly speaking, positive. More precisely, it turns out that: • the ground state is essentially the same as for HP,σj |Fσ+ ; j • the new gap is of order σj+1 . The requirement 2) is provided by Lemma 1.3, in which, given the estimate for the gap provided in Lemma 1.1 and for properly chosen values of Ej+1 ∈ C (complex numbers), we can establish that the resolvent 1 HP,σj+1 |Fσ+

j+1

has a series expansion in terms of

1 HP,σj |F +

−Ej+1 σ

σj+1

−Ej+1

and the diﬀerence (∆HP )σjj+1 ,

at small, but uniform in j, coupling constant g. Except for the relevant estimate 12

2

|f (k)| 3 3

f (k) b (k) d kψ ≤ d k

|k|

ph 12

H

ψ

(where the expression on the right side is supposed to be well deﬁned) and for an ordinary factorization (see also [B.F.S]) in the series expansion of the resolvent, the result is only due to a crucial consideration based on the joint spectral decomposition of commuting observables. Then we obtain the main result of the section which consists in the deﬁnition of σ σ σ σ ψPj+1 by the pertubation of ψPj (ψPj identiﬁed with ψPj ⊗ ψ0 , vector in Fσ+j+1 ) σj under some hypotheses on ψP ; namely the result is: Theorem 1.4 Under the constructive hypotheses and for g suﬃciently small, if σ σ ψPj is the unique ground eigenvector of HP,σj |Fσ+ with gap larger than 2j , σ

HP,σj+1 |Fσ+

j+1

corresponding gap is bigger than ﬁned σ

j

σ

has a unique ground eigenvector ψPj+1 of energy EPj+1 and the

σ

σj+1 2 ;

ψPj+1 := Pσj+1 ψPj = −

1 2πi

σ where Ej+1 ∈ C and Ej+1 − EPj =

σ

the (unnormalized) vector ψPj+1 is so de

1 σ dEj+1 ψPj HP,σj+1 − Ej+1

11 20 σj+1 . σ

σ

j+1 j According to this theorem, it turns out

ψσPj is given by ψP plus a ﬁnite

σj+1

that

≥ c ψP where 0 < c < 1, provided g is g-dependent remainder so that ψP σ suﬃciently small. Because of this result and the spectral features of ψPj+1 (gap, non-degeneracy), the same operation can be repeated for the next infrared cutoﬀ. σ σ0 Then we can construct the sequence ψPj , by iteration, starting from ψP ≡ ψ0 .

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σj In Section 2 we deal with the problem the convergence of ψP and we σjof areforced to discuss a related sequence φP . Mathematically, the convergence σ of ψPj involves a problem of perturbation of eigenvalues at the threshold of the continuous spectrum, more speciﬁcally the perturbation of the ground energy of the Hamiltonian (P ∈ Σ) 0 HP

ph 2 P −P + H ph. := 2m

If the exponent of |k| in the interaction term of the Hamiltonian HP κ d3 k b (k) + b† (k) g 2 |k| 0 were larger than − 21 , the norm estimates σ for the resolvents would be suﬃcient not only to construct the sequence ψPj but also to gain the convergence. The local interaction (up to the ultraviolet cutoﬀ κ) of the relativistic ﬁeld yields the exponent − 21 . It is a limiting case for the existence of the ground state in the following sense: • the ground state exists in the Fock space F b (∼ =HP ) for arbitrarily small infrared cutoﬀ in the interaction; • when the infrared cutoﬀis removed, it requires non-Fock coherent representations of the variables b (k) , b† (k) , which are also inequivalent at diﬀerent P. Therefore the strategy is to properly transform the Hamiltonians HP,σj and to study the sequence of ground eigenvectors of the so obtained Hamiltonians acting on F b . The known coherent transformation (in this respect see [Fr.1]) is re-obtained thanks to a heuristic proof based on a virial type argument1 . Namely, starting from the assumption of a ground state “coherent in the infrared region”, such an argument works out the representation given by the following intertwiner   κ † 3 b (k) − b (k) d k  , W (∇E (P)) := exp −g 2 |k| 0 |k| 1 − k · ∇E (P) where ∇E (P) is the gradient of the ground energy (as a function of the total momentum P) which is well deﬁned as proved in [Fr.2], at least except a set of measure zero. Taking care of the above expression we turn to consider the transformed Hamiltonians w HP,σ := Wσj (∇E σj (P)) HP,σj Wσ†j (∇E σj (P)) j 1 I am indebted to G. Morchio for having suggested to me this eﬀective argument and for many discussions and advice.

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where  Wσj (∇E σj (P)) := exp −g

κ

σj

 b (k) − b† (k) d3 k  · ∇E σj (P) 2 |k| |k| 1 − k

σ

and EPj is the ground energy of HP,σj . w Then we realize that HP,σ can be put in the following “canonical” form j 1 2m

2 σj σj − σ 2 · φP , ΠP,σj φP

φ j 1

ΠP,σj

∞

+ 0

where • ΠP,σj := Pph − g

P

(|k| − k · ∇E σj (P)) b† (k) b (k) d3 k + cPj

κ σj

σ

σ

√

k(b(k)+b† (k)) 3 σj 2|k| 2 (1−k·∇E (P))

d3 k

w (note however that the • φPj (to be speciﬁed) is ground eigenvector of HP,σ j w “canonical” form of the Hamiltonian HP,σj involves only the ray of the correσ sponding ground eigenvector, the same ray of Wσj (∇E σj (P)) ψPj therefore) σj • cP is an additive constant.

An iteration procedure as in Section 1 can be now carried out for the Hamiltonians w HP,σ , acting on F b , to construct again a sequence of related ground eigenvectors j σj φP . We use an analogous chain of projectors, by exploiting the spectral information known for the Hamiltonians HP,σj . We ﬁnd that thanks to the property σ φPj

⊥

σ ΓiP,σj φPj

σ ΠiP,σj φPj

:=

1

− σj 2 ·

φ P

σ σ φPj , ΠiP,σj φPj

σ φPj

i = 1, 2, 3

the norm of the remainder σ

σ

φPj+1 − φPj σ

is order the ratio κj raised to some positive power, in contrast to the sequence of σj ψP . The ﬁnal result is the content of Theorem 2.3 and Corollary 2.4, in which σ we prove the strong convergence of the sequence φPj to a vector φP , in F b . The key result is just the inequality proved, by induction, in: κ suﬃciently small, the inequality Theorem 2.3 For g and m M 1 σj σj 2 i i φ Γ g · ΓP,σj φP , < j P,σj P w 4 HP,σ − Ej+1 j

(i = 1, 2, 3)

holds uniformly in j and in P ∈ Σ, being M a suﬃciently small constant.

448

A. Pizzo

Ann. Henri Poincar´e

σ As straightforward of φPj , we obtain that σj consequence of the strong convergence in F b , and it converges to a vector ψP the sequence ψP goesweakly to zero in the representation of b (k) , b† (k) given by the coherent transfor non-Fock † mation W (∇E (P)). Since the representations of b (k) , b (k) , associated to the intertwiners W (∇E (P)), are not equivalent for diﬀerent P, the construction of a state “ ψP d3 P ” is physically meaningless (it requires the superselection of the total momentum) [Fr.1]. w , where In Section 3 we deﬁne a normalized ground eigenvector φσP of HP,σ σ σ lies in the continuum. The vector φP is strongly convergent for σ → 0, and it is proved to carry a (strong) H¨ older property with respect to P (in the considered neighborhood of P = 0), uniformly in σ, though, in general, a more regular behavior is expected [Ch]. This is the price that we have to pay, in terms of approximation, by using regular perturbation theory. k and g suﬃciently small, Theorem 3.4 Under the constructive hypotheses, for m σ σ older in |∆P| with coeﬃcient the norm diﬀerence between φP and φP+∆P is H¨ 1 16 − δ, δ > 0 and arbitrarily small. The multiplicative constant, Cδ , is uniform in 0 ≤ σ < κ, in P, P + ∆P ∈ Σ and ∆P ∈ I, I a suﬃciently small ﬁxed ball around ∆P = 0.

The regularity property in P, resulting from analytic perturbation theory, seems to be essential in the construction of the asymptotic states in the scattering theory [Pi].

σ 1 Construction of the sequence ψPj In the present section we only construct the sequence of ground eigenvectors of the Hamiltonians HP,σj . In order to do it, we introduce some preliminary lemmas (1.1, 1.2, 1.3) which are necessaryto perform the projection in Theorem 1.4. Finally, in σ Corollary 1.5, the sequence ψPj , j ∈ N is constructed by iteration. The constructive hypotheses are:

1 8 j I) the considered infrared cutoﬀ are σj = κ 2 where 0 < <m 5 , j ∈ N ; II) the momenta P are restricted to the set Σ ≡ P :|P| ≤ 20 ;

III) the ratio

κ m

fulﬁlls the inequality:

κ m

1

2πg 2 + 35 2

≤

1 200 .

We synthesize the content of the three lemmas: • in Lemma 1.1, starting from HP,σj |Fσ+ , we study the operator HP,σj applied j

to the subspace Fσ+j+1 and we show how to recover the new eigenvector and the new gap for the same operator (HP,σj ) on the larger space Fσ+j+1 , which however does not contain new interacting bosons compared with Fσ+j ; • in Lemma 1.2 the ground energy is checked to be not decreasing in the infrared cutoﬀ: σ σ EPj ≥ EPj+1 ;

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

449

• in Lemma 1.3 the meaning of the “smallness” of (∆HP )σσjj+1 = HP,σj+1 − HP,σj with respect to HP,σj |Fσ+

j+1

is explained.

Remark. The value of g (g > 0) will be constrained several times during the procedure; at each time we call g the maximum value such that the constraint under examination as well as the previous constraints are satisﬁed. In Sections 2, 3 κ to prove the convergence of the transformed we assumea suﬃciently small ratio m σj sequence φP and the regularity properties. The proofs of Section 1 do not require the restrictive constraint on but only 0 < < 1 and moreover can be k if one assumes some results by Fr¨ ohlich extended to arbitrary (ﬁnite) values of m [Fr.2] concerning the ground energy. σ

Lemma 1.1 If ψPj is the unique ground eigenvector of HP,σj |Fσ+ with correspondσ

σ

σ

j

j (deﬁned below), ing gap larger than 2j , then ψPj ⊗ ψ0 , ψ0 vacuum state in Fσj+1 σ is the unique ground eigenvector of HP,σj |Fσ+ with the same eigenvalue EPj (of

σ

ψPj ) and its gap is larger than

j+1

3 5 σj+1 . σ

σ

j j Proof 2 . Let us decompose Fσ+j+1 as Fσ+j ⊗ Fσj+1 , where Fσj+1 is the tensorial subproduct deﬁned as follows j j j ≡ F (h) , h := L2 Bσσj+1 , d3 k , Bσσj+1 := {k : σj+1 < |k| < σj } . Fσσj+1

σ

σ

Clearly the vector ψPj ⊗ ψ0 is an eigenvector of HP,σj |Fσ+ , with eigenvalue EPj , j+1 and σ σ HP,σj : Fσ+j+1 ψPj ⊗ ψ0 → Fσ+j+1 ψPj ⊗ ψ0 σ σ where ψPj ⊗ ψ0 denotes the subspace generated by ψPj ⊗ ψ0 . For this reason the gap we want to estimate can be analyzed starting from inf HP,σj |Fσ+ {ψσj ⊗ψ0 } spec

P

j+1

σ EPj .

if the above quantity is larger than inf HP,σj |Fσ+ spec

j+1

In this case the gap corresponds to σj σj . −E P {ψ ⊗ψ0 } P

Since it is useful in Lemma 1.3, we prove a stronger result: 1 3 σ inf HP,σj |Fσ+ {ψσj ⊗ψ0 } − H ph |σσjj+1 −EPj ≥ σj+1 spec P j+1 5 5 2 I am indebted to J. Fr¨ ohlich for having suggested to me a shorter proof and for a helpful discussion of the lemma.

450

A. Pizzo

having deﬁned H ph |σσjj+1 :=

σj

†

3

|k| b (k) b (k) d k

and N

σj+1

|σσjj+1 :=

σj

Ann. Henri Poincar´e

b† (k) b (k) d3 k.

σj+1

For this purpose note that HP,σj , n (k) = 0 for |k| < σj , where n (k) := b† (k) b (k), in distributional sense; it implies that our search of the inﬁmum can be restricted to the analysis of the expectation value of HP,σj |Fσ+

σ

j+1

{ψPj ⊗ψ0 }

1 − H ph |σσjj+1 5

σ

on vectors like ϕ ⊗ η, ϕ ⊗ η⊥ψPj ⊗ ψ0 and ϕ = η = 1, where ϕ ∈ Fσ+j is in the σj σ is in the domain of H ph |σjj+1 and it is eigenvector of domain of HP,σj , η ∈ Fσj+1 σj N |σj+1 . It is suﬃcient to distinguish two diﬀerent energy regimes, corresponding to vectors σ m η with spectral support, in H ph |σjj+1 , below and above the value 20 respectively; let us deﬁne q := η ,H ph |σσjj+1 η then in the ﬁrst case q ≤

m 20 ,

in the second one q >

m 20 .

m 1) q ≤ 20 For the set {P : |P | ≤ |P| + q , P ∈ Σ} the condition on q implies |∇E σj (P )| < 1 5 , by steps as in Lemma A2, Appendix. Moreover we can estimate 1 σ ϕ ⊗ η , HP,σj − H ph |σσjj+1 −EPj ϕ ⊗ η 5

from below in terms of 1 min σj , 2

q:

inf

m 20 ≥|q|≥σj+1

σj EP−q

−

σ EPj

4 + |q| 5

due to the following facts: σ • the gap of HP,σj |Fσ+ is bigger than 2j , by hypothesis; j • the inequality which holds for η ⊥ ψ0 1 σ ϕ ⊗ η, HP,σj − H ph |σσjj+1 −EPj ϕ ⊗ η 5 2 Pph − P + q ≥ m inf ϕ, q: 20 ≥|q|≥σj+1 2m κ b(k) + b† (k) 3 4 σ j d k + H ph |+∞ +g ϕ σj + |q| − EP 5 2|k| σj 4 σj σ − EPj + |q| . ≥ m inf EP−q 5 q: 20 ≥|q|≥σj+1

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

451

From the constraint on the gradient and being 12 σj ≥ 35 σj+1 , we can conclude that

min

2) q >

1 σj , 2

q:

σ

inf

m 20 ≥|q|≥σj+1

σ

j − EPj + EP−q

3 4 |q| ≥ σj+1 . 5 5

m 20

Let us start observing that HP,σj |Fσ+

j+1

−H ph |σσjj+1 +2πg 2 κ ≥ 0 ,

to provide the bound 1 σ ϕ ⊗ η , HP,σj − H ph |σσjj+1 −EPj ϕ ⊗ η 5

Now, from the constraints on the ratio σ σ0 guarantees EP ≥ EPj , we have

κ m,

4 σ q − 2πg 2 κ − EPj 5 m σ − 2πg 2 κ − EPj . ≥ 25

≥

on Σ, and from next Lemma 1.2 which

m m m m 3 3 σ σ0 − 2πg 2 κ − EPj ≥ − 2πg 2 κ − EP − 2πg 2 κ − ≥ ≥ σ1 ≥ σj+1 . 25 25 25 2 · 202 5 5 where the inequality m 3 m − 2πg 2 κ − ≥ σ1 2 25 2 · 20 5 derives from the assumption 3 1 m . 2πg 2 κ + κ 2 ≤ 5 200 Conclusion inf

spec

HP,σj

1 σ |Fσ+ {ψσj ⊗ψ0 } − H ph |σσjj+1 −EPj P j+1 5 σ

≥

3 σj+1 . 5

σ

Lemma 1.2 The following relation between EPj and EPj+1 (ground energy of HP,σj+1 |Fσ+ ) holds: j+1

EPj ≥ EPj+1 ≥ EPj − 10πg 2 σj . σ

σ

σ

Proof. Considering that HP,σj+1 |Fσ+

j+1

= HP,σj |Fσ+

j+1

+IFσ+ ⊗ g j

d3 k b (k) + b† (k) √ 1 , σj+1 2 |k| 2 σj

452

A. Pizzo

Ann. Henri Poincar´e

(IFσ+ is the identity operator in the space Fσ+j ) the expectation value of HP,σj+1 j

σ

on ψPj ⊗ ψ0

σj σ ψP ⊗ ψ0 , HP,σj+1 ψPj ⊗ ψ0

σj

ψ ⊗ ψ0 2 P

σ

coincides with EPj . σ EPj+1 is the inﬁmum of the expectation value of HP,σj+1 on the vectors, in Fσ+j+1 , σ σ belonging to the operator domain, by deﬁnition. Therefore EPj+1 ≤ EPj and, in σ σ general, EP ≤ EP for σ ≥ σ . Moreover collecting the following results • as proved in the previous lemma 1 σ inf HP,σj |Fσ+ − H ph |σσjj+1 ≥ EPj j+1 spec 5 • being 1 ph σj H |σj+1 +g 5 we can estimate σ EPj+1

d3 k 2 b (k) + b† (k) √ 1 + 10πg (σj − σj+1 ) ≥ 0 σj+1 2 |k| 2 σj

= inf

spec

HP,σj |Fσ+

j+1

≥ inf HP,σj |Fσ+

j+1

spec

≥

σ EPj

d3 k b (k) + b† (k) √ +g 1 σj+1 2 |k| 2 1 − H ph |σσjj+1 −10πg 2 σj 5 σj

2

− 10πg σj

Lemma 1.3 For properly small values of g and under the hypotheses of Lemma 1.1, σj 3k σ b (k) + b† (k) √d2|k| is small with respect to HP,σj |Fσ+ in (∆HP )σjj+1 = g σj+1 j+1 the following sense: For Ej+1 ∈ C such that 11 σj Ej+1 − inf HP,σj | + σj+1 , Fσj+1 = Ej+1 − EP = spec 20 it turns out that

1 1 σj

HP,σ − Ej+1 − (∆HP )σj+1 HP,σ − Ej+1 j j

n

Fσ+j+1

≤

20 (Cg )n σj+1

1 where Cg , 0 < Cg < 12 , is a constant independent of j. It implies the validity of the series expansion:

1 | + HP,σj + (∆HP )σσjj+1 − Ej+1 Fσj+1 +∞

1 1 σ = − (∆HP )σjj+1 HP,σj − Ej+1 n=0 HP,σj − Ej+1

n

|Fσ+

j+1

.

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

453

Proof. The aim of this lemma is to arrive at a norm controlled series expansion of the resolvent, to be exploited in Theorem 1.4. Let us analyze the nth term of the series +∞ n

1 1 σ |Fσ+ , − (∆HP )σjj+1 j+1 HP,σj − Ej+1 n=0 HP,σj − Ej+1 namely  n (−1) 

1 HP,σj − Ej+1

1 

2

 ··· 

1

1 HP,σj − Ej+1

where



2



1

1

σj

(∆HP )σj+1 

HP,σj − Ej+1



2

 ··· 

1

1 HP,σj − Ej+1



2

1 2

1 HP,σj − Ej+1

is deﬁned starting from the spectral representation of HP,σj , by using the convention to take the branch of the square root with smaller argument in (−π, π]. Each term in the series expansion is under control if we are able to estimate the norm:

1 1

2 2 1 1

σj (∆HP )σj+1 .

HP,σj − Ej+1 HP,σj − Ej+1 + Fσ j+1

This norm is less or equal to

1

2 1

2g

HP,σj − Ej+1

+ Fσ j+1

σj 1 b (k) 3

· √ 1 d k

σj+1 2 |k| 2 HP,σj − Ej+1

if the quantities written above exist. Moreover the following estimate holds:

σj 1 d3 k

b (k)

σj+1 H 2 |k| P,σj − Ej+1

1 2

≤

1 2

+ Fσ j+1

√ 1 10π · σj2 .

(1)

Fσ+j+1

% Indeed, starting from vectors belonging to Db Fσ+j+1 (Db := n∈N ΦnP , see point iv) in Deﬁnition of the model ), we get the bound:

1 2

σj 2 1 d3 k

b (k)

σj+1

+ 2 |k| HP,σj − Ej+1 Fσj+1

& 1 '† 1

2 2

ph σ 1 1 j

≤ 2πσj ·

H |σj+1

HP,σj − Ej+1 HP,σj − Ej+1

σ in this respect note that HP,σj , H ph |σjj+1 = 0.

+ Fσ j+1

454

A. Pizzo

Ann. Henri Poincar´e

The operator norm of & H ph |σσjj+1

1 HP,σj − Ej+1

1 2

'†

1 HP,σj − Ej+1

1 2

|Fσ+

j+1

σ σ can be studied on ψPj ⊗ ψ0 and on Fσ+j+1 ψPj ⊗ ψ0 separately. The operator σ σ vanishes on ψPj ⊗ ψ0 (put H ph |σjj+1 on the right). The discussion is then restricted σ j to the subspace Fσ+j+1 ψP ⊗ ψ0 . As already seen in Lemma 1.1, we have 1 3 σ inf HP,σj |Fσ+ {ψσj ⊗ψ0 } − H ph |σσjj+1 ≥ EPj + σj+1 spec P j+1 5 5 from which inf HP,σj |Fσ+

σ

ψ j ⊗ψ0 } j+1 { P

spec

1 − H ph |σσjj+1 −ReEj+1 5

≥

11 3 σj+1 − σj+1 > 0 . 5 20 σ

Going to the joint spectral representation of the operators HP,σj and H ph |σjj+1 , we easily obtain

& 1 '† 1

2 2

ph σ 1 1

H |σj

≤5. (2)

j+1 H − E H P,σ j+1 P,σ j j − Ej+1

+ Fσ j+1

Conclusion For g suﬃciently small but uniform in j, the thesis is proved because the norm

1

2 1

HP,σj − Ej+1 + Fσ j+1

is of order

12

1 σj+1

.

Now, on the basis of the previous results and starting from the relation between the resolvents of the Hamiltonians HP,σj+1 and HP,σj applied on the subσ space Fσ+j+1 , we can construct ψPj+1 and establish that the norm diﬀerence between σj+1 σ the ground eigenvectors, ψP and ψPj , is bounded by a constant strictly less than σ σ 1. Concerning notations, starting from now, we identify ψPj and ψPj ⊗ψ0 as vectors in Fσ+j+1 . Theorem 1.4 Under the constructive hypotheses and for g suﬃciently small, if σ σ ψPj is the unique ground eigenvector of HP,σj |Fσ+ with gap larger than 2j , HP,σj+1 |Fσ+

j+1

σ

j

σ

has a unique ground eigenvector ψPj+1 of energy EPj+1 and the

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

455 σ

σ

j+1 corresponding gap is larger than j+1 is so de2 ; the (unnormalized) vector ψP ﬁned 1 1 σj+1 σ ψP := − dEj+1 ψPj (3) 2πi HP,σj+1 − Ej+1 σ where Ej+1 ∈ C and Ej+1 − EPj = 11 20 σj+1 .

Proof. Continuity argument. We consider the series expansion of the resolvent (inside the integral (3)) which is provided by Lemma 1.3. We distinguish the coupling constant g in HP,σj |Fσ+ j+1

σ

from that one in (∆HP )σjj+1 , and we denote the latter by g . Kato-Rellich theorem ensures that (3) is well deﬁned for suﬃciently small g , since the gap of HP,σj |Fσ+ j+1 σj is bigger or equal to 35 σj+1 (Lemma 1.1) and ∆HP g σj+1 is a small Kato perturbation with respect to HP,σj |Fσ+ . Now look at the ﬁgure, j+1

Im(E(j+1))

sigma j+1 EP

sigma j E P

Re(E(j+1))

3/5 sigma j+1 11/20 sigmaj+1

as g increases, the deﬁnition (3) is surely consistent as long as the perturbed σ eigenvalue EPj+1 g remains inside the circle of integration and the remaining σj spectrum of HP,σj+1 g |Fσ+ := HP,σj |Fσ+ + ∆HP g σ is localized outj+1 j+1 j+1 side. According to the estimates given in Lemma 1.3, we can conclude that: i) the integral 1 − 2πi

1 σj ψP

+ j+1 (g ) − Ej+1 F

HP,σ

exists for 0 ≤ g ≤ g ;

σj+1

dEj+1

456

A. Pizzo

Ann. Henri Poincar´e

σ

ii) the vector ψPj+1 is not zero since σ

ψPj+1 σ = ψPj

+∞

1

− 2πi n=1

1 1 σ −(∆HP )σjj+1 HP,σj − Ej+1 HP,σj − Ej+1

and the norm of the remainder,

σ σ ψPj+1 −ψPj ,

is less than

n σ

dEj+1 ψPj

11Cg

σj 1−Cg · ψP ,

(4) therefore

σj+1 1 − 12Cg σj

ψ

≥ · ψP > 0 ; P 1 − Cg iii) the ground state of HP,σj+1 |Fσ+ iv) since

σ EPj+1

≤

σ EPj

j+1

is unique by continuity;

(Lemma 1.2), the new gap is larger than σ Corollary 1.5 The sequence ψPj , j ∈ N is well deﬁned.

σj+1 2 .

Proof. Thanks to the results of Lemma 1.1, 1.2, 1.3 and Theorem 1.4, it is possible to iterate the procedure at ﬁxed g, starting from the vacuum state (of F b ) ψ0 at the level j = 0 . The iteration is consistent and does not stop since the vector obtained at the step j + 1 has norm larger than a ﬁxed fraction of the norm of the 1 vector at the j step. At each step the infrared cutoﬀ is reduced by a factor 2 .

2 Convergence of the ground eigenvectors of the transformed Hamiltonians As we know from [Fr.1], the Hamiltonians HP have a ground state for representations of b (k) , b† (k) which are coherent non-Fock due to their infrared σ and behavior (k → 0), so that the sequence ψPj cannot converge in F b to a nonzero vector. The correct coherent factor can be re-obtained by a heuristic argument which singles out the expected infrared limit of the coherent factor, explicitly in the case P = 0 and implicitly in the case P = 0, P ∈ Σ. Such an information is then used in a rigorous proof which is based on the iterative procedure of construction of the ground state. In particular, for each infrared cutoﬀ σj , we consider a Hamilw tonian HP,σ unitarily equivalent to HP,σj , according to a coherent transformation j that depends both on P and on σj . The sequence of related ground eigenvectors, σj φP , that we construct σj by iteration, turns out to be useful to characterize the original sequence, ψP , in two respects: σ • the strong convergence of φPj in F b and the fact that the σj -dependent coherent transformation is not unitary in the j → ∞ easily imply the σlimit weak limit to zero, in F b , for the sequence ψPj ; σ • by the regularity property in P of the vectors φPj , uniformly in j, and the explicit knowledge of the coherent transformation, we have a better control σ on the P-dependence of the vectors ψPj and of related quantities.

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

457

Derivation of the coherent factor Let us assume that ψP is an eigenvector of HP and that it is “coherent in the infrared region”, namely b (k) ψP ≈ fP (k) ψP

f or k → 0

where the meaning of the limit is given only “a posteriori”. Then the coherent function fP (k) has to fulﬁll the following relation: (ψP , [HP , b (k)] ψP ) = 0 f or k → 0 . The expected behavior for fP (k) turns out to be g fP (k) ≈k→0 − · 2 |k| |k| +

1 |k|2 2m

−

P·k m

+

,

k·(ψP ,Pph ψP ) m·ψP 2

the conjectured coherent factor is therefore labelled by ( ph ) ( ph ) ψP , Pph ψP P ψP := . P − P ψP , 2 ψP The heuristic argument indicates that, if the ground state is “coherent in the infrared region”, it does not belong to the Fock space. Starting from this result, we essentially perform the proper coherent transformation on the variables b (k) , b† (k) of the Hamiltonian HP , and we search for a ground state of the so transformed Hamiltonian in F b . Actually, we accomplish this reducing the infrared cutoﬀ step by step.

2.1

Transformed Hamiltonians

Let us consider the coherent transformation b (k) −→ b (k) −

√ 3 2 |k| 2

g · 1−k

P− Pph

obtained by the unitary operator  ( ph ) κ σ P− P ψ j  P Wσj = exp  −g  m σj

ψ

σj P

f or k :

σj ≤ |k| ≤ κ

m



b (k) − b† (k)

· |k| 1 − k

P− Pph

σj ψ P

d3 k   2 |k| 

m

which becomes an intertwiner between inequivalent representations (of {b(k), b† (k)}) in the limit j → ∞.

(5)

458

A. Pizzo

Ann. Henri Poincar´e

σ

From the perturbation of the isolated eigenvalue EPj of HP,σj |Fσ+ (see [Fr.1]), j ) ( one can easily check that P − Pph ψσj corresponds to m∇E σj (P), where P |∇E σj (P)| < 1 for P ∈ Σ as proved in Lemma A2 in Appendix. The gradient ∇E σj (P) fulﬁlls the equation: ( ph ) P ψσj = P − m∇E σj (P) P κ σ 1 k σ (6) 3 = σ 2 φPj , ΠP,σj φPj + g 2 2 d k

φ j σ 3 j σj 2 |k| P αP k σ := 1 − k · ∇E σj (P), φσj is a ground eigenvector, not speciﬁed yet, where αPj k P of the transformed Hamiltonian Wσj (∇E σj (P)) HP,σj Wσ†j (∇E σj (P)) and

ΠP,σj := P

ph

κ

−g σj

k b (k) + b† (k) 3 d k. √ 3 σ 2 |k| 2 αPj k

We want now to provide a more explicit expression for the transformed Hamiltonian w := Wσj (∇E σj (P)) HP,σj Wσ†j (∇E σj (P)) HP,σ j which takes into account the relation (6).( ) Let us rewrite HP,σj , P = m∇E σj (P) + Pph ψσj , as P

HP,σj 2

=

=

P − 2m

( ) m∇E σj (P) + Pph ψσj · Pph P

m ( ph ) P ψσj · Pph

2

+

Pph +g 2m

κ

d3 k + H ph b(k) + b† (k) 2|k| σj

∞ 2 P2 Pph P + − + (|k| − k · ∇E σj (P))b† (k)b(k)d3 k 2m 2m m κ κ g g + (|k| − k · ∇E σj (P)) b† (k) + √ b(k) + √ d3 k 3 3 σ σ σj 2|k| 2 αPj k 2|k| 2 αPj k σj κ 1 d3 k. + (|k| − k · ∇E σj (P))b† (k)b(k)d3 k − g 2 2 σj σj 2|k| α 0 P k

The arrangement above aims to show the origin of some terms in the transformed Hamiltonian; in particular, for our purposes, it is important to isolate ) 2 ( Pph − Pph ψσj P Wσ†j (∇E σj (P)) Wσj (∇E σj (P)) 2m and to exploit a related structural property later.

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

459

By performing the coherent transformation (5) on HP,σj , we formally obtain:  2 κ κ † k b(k) + b (k) 3 1  ph k 3  d k + g2 P − g 2 d k  √ 3 σj 2m σ 3 2 j σj σ j 2|k| 2|k| αP k αP k   ( ph ) κ κ P ψ † k b(k) + b (k) 3 k P,σj  3  d k + g2 · Pph − g − 2 d k  √ 3 σ j m σj σj 2|k|3 ασj k 2|k| 2 αP k P κ ∞ 1 P2 d3 k . − g2 (|k| − k · ∇E σj (P))b† (k)b(k)d3 k + + 2 σ 2m 0 σj 2|k| α j k P By substitution it corresponds to σj ∞ σ 2 φP , ΠP,σj φPj 1 σ ΠP,σj − + (|k| − k · ∇E σj (P)) b† (k) b (k) d3 k + cPj

σj 2 2m

φ 0 P (7) κ σ 2 P2 1 1 3 where cPj = 2m − 2m [P − m∇E σj (P)] − g 2 σj k. d σ ) 2|k|2 αPj (k The expression above, not only formally but also from an operatorial point of w view, corresponds to HP,σ and its selfadjointness domain (s.a.d.) coincides with j b D HP,σj , s.a.d. of HP,σj (see an analogous proof in [Ne]). Definitions i) To streamline formulas, in next steps we will use the deﬁnition σ σ φPj ,ΠiP,σj φPj (i = 1, 2, 3 , j ∈ N ). ΓiP,σj := ΠiP,σj −

σj 2

φ P σ ii) In proving the convergence of the sequence φPj (to be constructed) we w , which are introduced take advantage of intermediate Hamiltonians, H P,σj because of the fact that, at ﬁxed P (P = 0), the P-function in the coherent factor changes step by step: w P,σ H := Wσj+1 (∇E σj (P))HP,σj+1 Wσ†j+1 (∇E σj (P)) j+1

= Wσj+1 (∇E σj (P))Wσ†j+1 (∇E σj+1 (P)) w × HP,σ Wσj+1 (∇E σj+1 (P))Wσ†j+1 (∇E σj (P)) j+1 σj σj k b(k) + b† (k) 3 1 2 = ΓP,σj − g d k + g √ 3 σ 2m σj+1 σj+1 2|k| 2 α j k

+ 0

P

∞

(|k| − k · ∇E σj (P))b† (k)b(k)d3 k + cPj+1 . σ

3

2|k|

k

2 3

2 d k σ αPj k

460

A. Pizzo

Here σ

cPj+1 =

1 P2 − [P − m∇E σj (P)]2 − g 2 2m 2m

Ann. Henri Poincar´e

κ

σj+1

1 2 |k|

2

σ αPj

d3 k . k

They are essentially derived by the same steps used for (7). iii) Analogously we deﬁne: i Π P,σj+1 := = Wσj+1 (∇E σj (P)) Wσ†j+1 (∇E σj+1 (P)) × ΠiP,σj+1 Wσj+1 (∇E σj+1 (P)) Wσ†j+1 (∇E σj (P)) κ k i b (k) + b† (k) 3 phi −g =P d k √ 3 σ σj+1 2 |k| 2 αPj k 2 2 σj+1 σj k α 2 κ − α P P k g 3 + ki [ 2 2 ] d k 2 σj+1 σj σj+1 3 αP k |k| αP k σ σj+1 i φPj+1 , Π P,σj+1 φP i i ΓP,σj+1 := ΠP,σj+1 − ,

σj+1 2

φP σ w where φPj+1 is ground eigenvector of H P,σj+1 and is properly deﬁned by the iterative procedure explained in the next paragraph. Note however that, w as for HP,σ , the expression is completely deﬁned in terms of the ray of j+1 σ σj Wσj+1 (∇E (P)) ψPj+1 .

Remarks 1) Note that the two transformations w = Wσj (∇E σj (P)) HP,σj Wσ†j (∇E σj (P)) HP,σj → HP,σ j w P,σ HP,σj+1 → H = Wσj+1 (∇E σj (P)) HP,σj+1 Wσ†j+1 (∇E σj (P)) j+1

are diﬀerent in the infrared cutoﬀ but not in the P-function inside the coherent factor. w w are s.a. on the same domain and and H 2) The Hamiltonians HP,σj , HP,σ P,σj j the formal derivations provided so far are well deﬁned from an operatorial point of view.

2.2

Convergent sequence

σ In order to arrive at a strongly convergent sequence, φPj , of ground eigenvectors w of HP,σ , we start from the vector φσP0 ≡ ψ0 , ψ0 vacuum state in F b . From the j

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results of the previous chapter and by unitarity, the following properties hold for each j ∈ N (these properties are exploited in Lemma A1, Appendix): σ w | + has ground eigenvalue EPj with the corresponding gap larger than i) HP,σ j Fσ ii)

σj 2 ; w HP,σ j

j

|Fσ+

σ

j+1

has ground eigenvalue EPj with the corresponding gap larger

than 35 σj+1 .

w w and H Comparing the resolvents of the Hamiltonians HP,σ P,σj+1 , we can construct j σ σ j+1 j φP in terms of a given φP by projection, thanks to the estimates contained in Lemma A1, in Appendix, which is the analogue of Lemma 1.3 for the Hamiltonians HP,σj . Therefore we obtain n

∞ 1 1 1 σj+1 σj σ w σj φPj dEj+1 −(∆HP )σj+1 w φP := φP − w 2πi n=1 HP,σ − E H − E j+1 j+1 P,σj j (8) where σ σj+1 w σj w w )σj+1 := H . + cPj − HP,σ . (∆HP P P,σj+1 − c j

Then we deﬁne φPj+1 := Wσj+1 (∇E σj+1 (P)) Wσ†j+1 (∇E σj (P)) φPj+1 . σ

σ

σ According to this recipe, the construction of φPj is carried out starting σ from φσP0 ≡ ψ0 (the corresponding sequence of rays is already known from ψPj and the coherent transformation). Outline of the proof of the convergence. In studying the (strong) convergence of σ the vectors φPj for j → ∞, we have to compare the following vectors one after the other: σ σ σ σ σ φPj → φPj+1 → φPj+1 → φPj+2 → φPj+2 σ σ (in the special case P = 0, there is a simpliﬁcation because φPj ≡ φPj , being σ ∇E (0) = 0 by symmetry). First note that one needs a more reﬁned estimate σ σ of the diﬀerence between the generic vectors φPj and φPj+1 , being the estimates provided in Lemma A1 only suﬃcient to construct the sequence. σAt this point a crucial diﬀerence emerges with respect the previous sequence, ψPj . The result of Lemma A1 can improved, diﬀerently from the analogous one (Lemma 1.3) be σ for the sequence ψPj .

To this purpose break the interaction w j w (∆HP )σj+1 := H . P P,σj+1 − c σ

in

σj+1

σ

w + cPj − HP,σ j

/ 0mix / 0quad. w σj w σj (∆HP )σj+1 + (∆HP )σj+1

462

A. Pizzo

Ann. Henri Poincar´e

where 0mix / w σj (∆HP )σj+1 :=  σj k b(k) + b† (k) 3 g 2 σj  g = ΓP,σj · − d k+ 2m σj+1 √2|k| 32 ασj k 2m σj+1 P

 3

2|k|

k σ αPj

3  2 d k  + h.c. k

(h.c. means hermitian conjugate) and consider again the expression (8). Because of the mixed terms, the estimate provided in Lemma A1 for

12 12

1 1 w σj

− (∆HP )σj+1 w

H w − Ej+1 H − E j+1

P,σj P,σj

Fσ+j+1

σ σ does not imply that the norm φPj+1 − φPj is inﬁnitesimal for j → ∞. However

σ σ we are able to give a more reﬁned estimate of the norm φPj+1 − φPj by a careful analysis of the ﬁrst factor in each term of the sum in (8), precisely

1 w HP,σ − Ej+1 j

12

w σj − (∆HP )σj+1

1 w HP,σ − Ej+1 j

σ

φPj .

Note, indeed, that if an inequality like the following were true

12 12

j+1 / 0 mix

8 1 1 σj w σj

≤ (∆H ) φ P σj+1 P

H w − Ej+1 w HP,σ − Ej+1 4

P,σj j

(9)

j+1 σ

σ we could estimate φPj+1 − φPj less or equal to 8 . It is suﬃcient because: • from Lemma A1, the norm of the contribution due to the quadratic terms can be bounded by

j+1 8

40

(for a proper g)

12 12

j+1 / 0quad

8 1 1 σ j w σj

φP ≤ (∆HP )σj+1 ; w

H w − Ej+1 HP,σ − Ej+1 40

P,σj j • for each term of the sum, the norms of the other factors of the product are 1 according to Lemma A1. smaller than 1, in particular less than 12

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Therefore from (9) we would have

n ∞

1 1

σ j w σj − (∆H ) φ

P P σj+1 w w

HP,σ − Ej+1 HP,σ − Ej+1 j j n=1

12 12 ∞ n−1

j+1 1 1 11 1

8 ≤ · · · · σ w

40 12 EPj − Ej+1

HP,σj − Ej+1

+ n=1 Fσj+1 j+1 1 · 8 . ≤ σ EPj − Ej+1

j+1

σ σ Finally, as it is shown in Corollary 2.4, an estimate like φPj+1 − φPj ≤ 8 σ implies the convergence of the sequence φPj . The conclusion of the previous reasoning is that, turning to a strong estimate for the ﬁrst factor in all the terms of the series expansion in (8), we are able to prove the convergence of the sequence if the inequality (9) holds. 0mix / w σj )σj+1 , after few steps and for g Taking care of the expression of (∆HP suﬃciently small, one can check that the inequality (9) is implied by the following M 1 σj σj 2 i i g ΓP,σj φP , φ (i = 1, 2, 3) (10) Γ < j P,σ w j P 4 HP,σ − Ej+1 j σ (Ej+1 is s.t. Ej+1 − EPj = is uniform in j.

11 20 σj+1 )

where M is a suﬃciently small constant, that

We arrive at this conclusion through two technical lemmas: Lemma 2.1 and Lemma 2.2. w , we In Lemma 2.1, starting from the spectral representation of the operator HP,σ j provide a form bound of the type 1 1 ϕ ϕ, w ϕ ≤ a · ϕ, w HP,σj − Ej+1 HP,σj − Ej+1

where a > 0 is uniform in j, for ϕ belonging to the subspace of Fσ+j orthogonal σ w . This argument can be applied to the to the ground eigenvector, φPj , of HP,σ j σ j i vectors ΓP,σj φP (i = 1, 2, 3) because the property σ ΓiP,σj φPj

=

holds by deﬁnition.

σ ΠiP,σj φPj

1

− σj 2 ·

φ P

σ σ φPj , ΠiP,σj φPj

σ φPj

σ

⊥ φPj

464

A. Pizzo

Ann. Henri Poincar´e

In Lemma 2.2 we deal with the relevant term in (9); by calling a term “relevant” we mean that the other ones have a better infrared behavior or can be reduced to the relevant term plus smaller terms. The relevant term, for j ≥ 1, turns out to be

12 12

σj i †

1 g k b (k) 1 σj 3 i

· kΓ φ d P,σj P . w

H w − Ej+1 m σj+1 √2|k| 32 ασj k HP,σ − Ej+1

P,σj j P

In the estimate of the relevant term, we essentially exploit the pull-through formula (see [B.F.S]) combined with the result of Lemma 2.1, in order to pull the operator

σj

σj+1

k i b† (k) d3 k √ 3 σj 2 2 |k| αP k

κ through the resolvent from the right side. In Lemma 2.2 we assume the ratio m suﬃciently small. Being standard computations, the lemma is proved in Appendix (Lemma A3). The ﬁnal step is Theorem 2.3, in which, by induction, we provide the estimate (10) and then the inequality (9).

Lemma 2.1 The following inequalities hold: σj σj σj ki b† (k) 1 3 i I) d k ΓP,σj φP , H w −Ej+1 σj+1 σj+1 √2|k| 32 ασj k P,σj P ( ) ≤

√

122 σσj j+1

√

ki b† (k) 3

σj

σ

d3 k ΓiP,σ φPj , j

2|k| 2 αP (k) σ σ II) ΓiP,σj φPj, H w 1−Ej+1 ΓiP,σj φPj P,σj 1 where Q () ≡

1+

√ 11 √ 10−11

2

σ ki b† (k) d3 k ΓiP,σj φPj √ 3 σ ) 2|k| 2 αPj (k

σj σ ki b† (k) 1 d3 k ΓiP,σ φPj w 3 σj HP,σ −Ej+1 σj+1 √ j ) 2|k| 2 αP (k j

σ σ ≤ Q () · ΓiP,σj φPj, H w 1−Ej+1ΓiP,σj φPj P,σj

.

Proof. Let us deﬁne the wave functions, ζI (z) , ζII (z), of

σj

σj+1

√

k i b† (k) 2 |k|

3 2

σ αPj

d3 k ΓiP,σj φσPj k

and

σ

ΓiP,σj φPj

w respectively, in the spectral variable, z, of the operator HP,σ − ReEj+1 (we do j not explicit the other degrees of freedom).

Note that: w • the operator HP,σ − ReEj+1 , applied to the vector j

σj

σj+1

k i b† (k) d3 k ΓiP,σj φσPj , √ 3 σ j 2 |k| 2 αP k

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

1 takes spectral values larger or equal to 20 σj+1 = 35 σj+1 − of Lemma 1.1 and the property σj k i b† (k) d3 k ΓiP,σj φσPj ⊥ φσPj ; √ 3 σj 2 σj+1 2 |k| αP k

465 11 20 σj+1

because

w • the operator HP,σ − ReEj+1 takes spectral values larger or equal to j √ σ 10−11 1 11 σj = 2 σj − 20 σj+1 when it is applied to the vector ΓiP,σj φPj ; it 20 w follows from the gap estimate (concerning HP,σ | + ) contained in Theorem j Fσj 1.4 and the orthogonality property.

Let us write the scalar products I) and II), in the statement of the lemma, by w using the spectral representation of the operator HP,σ − ReEj+1 with the given j spectral measure dµ (z). We do not make the integration over the remaining degrees of freedom explicit. In the chosen spectral representation and because of the range of the variable z, the following inequalities are clear: |ζI,II (z)|2 dµ (z) z − iIm (Ej+1 ) 2 |ζI,II (z)| z ≥ 2 ·2 dµ (z) 2 2 z 2 + [Im (Ej+1 )] z 2 + [Im (Ej+1 )] 2 zmin |ζI,II (z)| 2 ≥ 2 · dµ (z) σ 2 2 z 2 + [Im (Ej+1 )]2 zmin + Ej+1 − EPj It follows that: • in the case I), being zmin ≥

1 20 σj+1

and then

zmin 1 2 2 ≥ √122 , σ j 2 zmin + Ej+1 − EP we have

2 √ |ζI (z)| 2 dµ (z) dµ (z) ≤ 122 · z − iIm (E ) 2 j+1 z 2 + [Im (Ej+1 )] 2

|ζI (z)|

• in the case II), being zmin ≥

√ 10−11 20

· σj and then

zmin 1 2 ≥ 1 2 , √ σ 2 2 11 √ zmin + Ej+1 − EPj 1 + 10−11

466

A. Pizzo

Ann. Henri Poincar´e

we have

2 |ζII (z)| 2 dµ (z) dµ (z) ≤ Q () · z − iIm (E ) 2 j+1 z 2 + [Im (Ej+1 )] 2

|ζII (z)|

The two inequalities correspond to the ones in the statement of the lemma. Lemma 2.2 For a suﬃciently small ratio potheses, the following inequality holds:

κ m,

in addition to the constructive hy-

2

12 12

σj i †

1 k b (k) 1 σj 3 i

k Γ φ d √ P,σj 3 P w

H w − Ej+1 σ HP,σ − Ej+1 σj+1

P,σj 2 |k| 2 αPj k j √ i 1 1 σj σj σj i Γ Γ ≤ 2Q () · 122 · Zσj+1 · σj φ , φ · P,σ P,σ P P j w j HP,σ − Ej+1 EP − Ej+1 j being j Zσσj+1 =

    

i

σj

σj+1

k 3

2 |k|

i2

σ αPj

  

3 2 d k .   k

Proof. See Lemma A3 in Appendix. Theorem 2.3 For g and

κ m

suﬃciently small, the inequality (10)

M 1 i σj σj i g · ΓP,σj φP , φ Γ < j P,σj P w 4 HP,σ − Ej+1 j 2

(i = 1, 2, 3)

holds uniformly in j and in P ∈ Σ, being M a suﬃciently small constant. Proof. We recall that, due to the result of Lemma 2.2 and the preliminary discussion about the convergence, the inequality above, for M (and g) suﬃciently small, implies the estimate (9) in Outline of the proof of the convergence, namely:

12 1

j+1 / 0 2 mix

8 1 1 σj w σj

≤ (∆H ) φ σ P σj+1 P

H w − Ej+1 4 EPj − Ej+1

P,σj

j+1

σ σ and then the bound φPj+1 − φPj ≤ 8 as previously discussed. In order to prove the inequality (10) in the statement of the present theorem, ﬁrst we relate the expression corresponding to the level j to the one corresponding to j − 1.

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1) We start applying the unitary operator Wσj (∇E σj−1 (P)) Wσ†j (∇E σj (P)) to both the factors of the scalar product to obtain the analogous quantity with the corresponding “ˆ- operators”:

iP,σ φσj, Γ j P

1 w − Ej+1 H P,σj

iP,σ φσj Γ j P

.

(11)

Note now that the circle of integration Cj+1 , related to the integration variable j+1 σ 2 , where is a ﬁxed parameter Ej+1 , has a radius, r = Ej+1 − EPj = 11 20 κ σ in the construction. Therefore, thanks to the result of Lemma 1.2, EPj−1 is inside Cj+1 for g suﬃciently small (in particular of order ). In this respect see the ﬁgure below. ImE(j+1)

11/20 sigma j sigma E j P

sigmaj−1 E P

11/20 sigma j+1

ReE(j+1)

By the same procedure used in Lemma A1 (in Appendix), taking into account Lemma 1.2 and for g suﬃciently small, it is possible to perform a series expansion to eventually arrive at the following bound for the absolute value of the expression (11): 2·

i φσj, Γ P,σj P

1 i σ j φ . Γ w HP,σ − Ej+1 P,σj P j−1 σ

2) Adding and subtracting ΓiP,σj−1 φPj−1 on the left and on the right of the scalar product above, we bound the new terms that we get, using elementary properties

468

A. Pizzo

Ann. Henri Poincar´e

of the scalar product: 1 σ σ j j 2 i i P,σ φ , 2g · Γ φ Γ j P w HP,σ − Ej+1 P,σj P j−1 1 i σj σj−1 σj σj−1 2 i i i ≤ 4g · ΓP,σj φP − ΓP,σj−1 φP , w φ − ΓP,σj−1 φP Γ HP,σj−1 − Ej+1 P,σj P 1 i σj−1 σj−1 i ΓP,σj−1 φP , w φ . Γ HP,σj−1 − Ej+1 P,σj−1 P

+ 4g 2 ·

(12) (13)

At this point, we want to reduce the quantity (13) to the expression (10) at the 1 level j − 1, times a constant less than − 4 , and to estimate the remainder (12) by j a quantity of order − 4 . It requires some technical manipulation and an inductive argument that eventually leads to the thesis. The proof consists in some preliminary results before the induction, and it is so organized: i) Treatment of the remainder (12); ii) Treatment of the expression (13);

σj σj iii) Estimate of φ − φ ; P

P

iv) Inductive proof. Treatment of the remainder (12) The following inequality holds 1 i σj σj−1 σj σj−1 2 i i i φ − ΓP,σj−1 φP 4g · ΓP,σj φP − ΓP,σj−1 φP , w Γ HP,σj−1 − Ej+1 P,σj P

2

12

1 σ σ j i φ j − Γi Γ ≤ 8g 2 · P,σj P P,σj−1 φP

Hw

P,σj−1 − Ej+1

2

12

1 i σj − φσj−1 φ + 8g 2 · Γ P,σj−1 P P

Hw

P,σj−1 − Ej+1

σ 2  σ

φ j j  2 φPj−1 j P

σj

σj−1 8 − + σ σ

φPj φPj−1  R1 (g)  φP − φP + 8  R2 (g)   ≤ · + · j j j j   4 2 8 4 4 8

where R1 (g) and R2 (g) are independent of j and vanish for g → 0. R1 (g) and R2 (g) are obtained considering the following facts:

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i i) from the relation (6) and the deﬁnition of Π P,σj+1 in paragraph 2.1: k i b (k) + b† (k) 3 d k √ 3 σ σj 2 |k| 2 αPj−1 k σj−1 i σj i σj−1 2 + m ∇ E (P) − ∇ E (P) + g

i i Γ P,σj = ΓP,σj−1 − g

σj−1

σj

3

2 |k|

ki σ αPj−1

3 2 d k k

12

1 iP,σ φσj − ΓiP,σ φσj ii) Γ j P j−1 P

Hw

P,σj−1 − Ej+1

 12 

σj−1 i †

b (k) + b k (k) 1 σ j  3

 d k φP g ≤ √ 3 w σ j−1 σj

HP,σj−1 − Ej+1 k 2 |k| 2 αP

12

1 σj σj−1 σj + |m∇E (P) − m∇E (P)| · φ P

Hw

P,σj−1 − Ej+1

1 σj−1

2 i

2

1 k σj d3 k · + g φ P w

3 σ σj

HP,σj−1 − Ej+1 2 |k| αPj−1 k iii) the following estimates hold with constants C ∇E , C , C , uniform in j, for g suﬃciently small

σ σ

φ j j φPj−1 P

• |∇E σj (P) − ∇E σj−1 (P)| ≤ C ∇E · − σj σj−1

φP φP + 8 (see Lemma A2 )

1

2 j

σj−1 ki (b(k)+b† (k)) 3

1 ≤ C · 4 • σj d k Hw

√ 3 σ ) P,σj−1 −Ej+1

+ 2|k| 2 αPj−1 (k Fσj

1

2

ΓiP,σj−1 ≤ Cj

H w 1 −Ej+1 P,σj−1

+ 4 Fσj

by steps as in Lemma A1 and for the result in Lemma 1.2. Treatment of the expression (13) In the next estimate, once we are given a suﬃciently small g, we proceed as follows. We exploit the crucial property

σ σ φPj−1 , ΓiP,σj−1 φPj−1 = 0

470

A. Pizzo

Ann. Henri Poincar´e

so that we can apply Lemma 2.1 in a slightly modiﬁed version, getting a multiplicative factor Q (). Then we expand 1 w HP,σ − Ej+1 j−1 in terms of

w HP,σ

1 j−1

−Ej

and Ej+1 − Ej , thanks to the spectral consequence of

the orthogonality property; in this way we get a multiplicative factor b ( b 2) uniformly (in j) bounded from above. A further application of Lemma 2.1 provides another factor Q (). Namely: 1 σ σj−1 i j−1 2 i 4g · ΓP,σj−1 φP , φ Γ w HP,σ − Ej P,σj−1 P j−1 1 i σj−1 σj−1 2 i ≤ 4g · Q () · ΓP,σj−1 φP , φ Γ P,σj−1 P w HP,σ − Ej+1 j−1 1 σ σj−1 i ≤ 4g 2 · b · Q2 () · ΓiP,σj−1 φPj−1, Γ φ P,σ w j−1 P HP,σ − Ej j−1 Remark. The expression above corresponds, at the level j − 1, to what we want to estimate (see (10)) times the factor 4 · b · Q2 (). Acting on g, at ﬁxed , we 1 can provide a multiplicative factor, 4 · b · Q2 (), less than − 4 (> 1). This fact is crucial in the inductive proof. It is enough because we only require a divergent (for j → ∞) bound for the expression (10).

σ σ Estimate of φPj − φPj Before the inductive proof, a further preliminary step is in order, concerning with

σ σ an upper bound for the norm φPj − φPj . It is the second part of the step φPj−1 → φPj → φPj σ

σ

σ

σ σ and it can be easily related to the ﬁrst one (φPj−1 → φPj ) through the variation of the energy gradient as it is explained below. Note that σ σ φPj ≡ Wσj (∇E σj (P)) Wσ†j (∇E σj−1 (P)) φPj

by deﬁnition, from which

σj σj σj σj

φP − φP = Wσj (∇E σj (P)) Wσ†j (∇E σj−1 (P)) φP − φP

σ σ = Wσ†j (∇E σj−1 (P)) Wσj (∇E σj (P)) ψPj − ψPj

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An upper estimate of the norm above is therefore given by j σ σ g · Z · ∇EPj − ∇EPj−1 · ln 2 where • Z is a constant dependent on m, κ and uniform in j; j • the logarithmically divergent (for j → ∞) quantity, ln 2 , arises from

κ

b (k) ψP,σj 2 d3 k

12

σj

taking into account that g

b (k) ψP,σj = 2 |k|

1 E σj (P) − |k| − HP−k,σj

ψP,σj

for {k : σj ≤ |k| ≤ κ} (proof as in [Fr.1]), ψP,σj ≤ 1, and from   

 κ

σj

 12

2

1 √  d3 k   3 σj−1 σj 2 2 |k| αP k · αP k

• the inﬁnitesimal (for j → ∞) quantity ∇E σj − ∇E σj−1 P P comes from the diﬀerence between the coherent factors in the Weyl operators. Inductive proof Now, let g be suﬃciently small such that: :∞ k k ∇E • G∞ · 8 ln 2 ≤ 1 := k=1 g · Z · 4C

1

• φσP1 − φσP0 ≤ 8

1 12

• the bound (10) is valid for j = 1, the previously established, uniform in j, constraints hold, in particular we require 1 0 < R1 (g) + R2 (g) =: R (g) ≤ 1 − 4bQ2 () 4 · M 1

where 4b · Q2 () 4 < 1, being b 2 and 0 < <

1 8 5

.

472

A. Pizzo

Ann. Henri Poincar´e

Inductive hypotheses Let us assume that for the chosen value of g the property (10) holds for j − 1(> 1) and that j−1

σj−1

k σ0

φ ≤ − φ 8 + Gj−1 1 P P k=1

where

φσP0

≡ ψ0 (vacuum state in F b ) and Gj−1 = g· 1

:j−1 k=1

k k Z · 4C ∇E · 8 ln 2 .

Thesis As consequences of the inductive hypotheses:

j

σ σ • φPj − φPj−1 ≤ 8

σ

σ

: k 2 ∞ 8 • since φPj−1 ≥ 1 − φPj−1 − φσP0 > 1 − ∞ k=1 − G1 > 3

σj σj−1

φσj − φσj−1

φP

j φ

− P

≤ 2 P σ P

≤ 3 8 j−1

σj φσj−1

φ

φP

P P j σ σ • ∇EPj − ∇EPj−1 ≤ 4C ∇E · 8 (see Lemma A2). Then starting from the equality     1 1 σ σ σ σ j j j j i i ,   i  Γ = Γ Γ φ φ φ ΓiP,σj φP , P,σj P P,σj P P,σj P w w HP,σ − Ej+1 − E H j+1 P,σj j

and collecting the results obtained so far (for the remainder (12) and the expression (13)), we have 1 σj σj 2 i i g ΓP,σj φP , ΓP,σj φP w HP,σj − Ej+1 R (g) 1 i σj−1 σj−1 2 2 i ≤ + 4g b · Q () · ΓP,σj−1 φP , φ Γ j P,σ w j−1 P HP,σ − Ej 4 j−1 ≤

R (g)

j 4

+ 4b · Q2 () ·

M

j−1 4

≤

M j

4

.

At the same time:

σj

σj

σ

σ σ σ

φ − φσ0 ≤

φP − φPj + φPj − φPj−1 + φPj−1 − φσP0 P P ≤ g · Z · 4C

∇E

j−1 j

j k 2 8 · ln + + 8 + Gj−1 1 j 8

k=1

which means

j

σj

k

φ − φσ0 ≤ 8 + Gj1 . P P k=1

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

473

σ Corollary 2.4 Given the result of Theorem 2.3, the sequence φPj ( φσP0 ≡ ψ0 , ψ0 vacuum state in F b ) converges strongly to a non-vanishing vector. σ Proof. By the estimates of Theorem 2.3 we easily conclude that φPj is a Cauchy sequence: l

σ k

φ l − φσj ≤ 8 + Glj+1 P P

∀l, j ∈ N

l ≥j+1.

k=j+1

The limit does not vanish because

σj

φ ≥ 1 − P

1

8 1

1 − 8

+

G∞ 1

≥

2 . 3

3 Regularity w In this section we deﬁne a normalized vector φσP , that is ground state of HP,σ |Fσ+ (σ ≤ κ), for σ in the continuum. Starting from infrared sequences of cutoﬀs σ which ﬁll the continuum, we construct a ground eigenvector for each σ, φP , by the projection method performed in the previous section. Having ﬁxed the ground state w |Fσ+ , we deﬁne the vector φσP , by applying the one-dimensional projection, of HP,σ b corresponding to the ray, on the vacuum state (in F ) ψ0 and normalizing. To do σ

it, in advance we check that φP , ψ0 = 0.

By this procedure we get: • the strong convergence of the so-deﬁned vector, φσP , to a vector φP , for σ → 0; • the H¨ older property with respect to P:

σ

1

φP+∆P − φσP ≤ Cδ |∆P| 16 −δ for P, P + ∆P ∈ Σ and ∆P ∈ I, where I ⊂ Σ is a ﬁxed ball around ∆P = 0, for δ > 0 and where Cδ is uniform in P, ∆P and σ. Preliminary conditions

√ Let us consider the infrared sequences starting from {κ : κ ≥ κ ≥ κ } and κ such suﬃciently small values of g and m √ that it is possible to perform the iterative procedure uniformly in , ≥ ≥ , and in P ∈ Σ, with the properties already shown in the case of the factor . Therefore we can assume the results of Theorem 2.3 and Corollary 2.4 for the factor . We also require that for the chosen value of the coupling constant g: 1 κ φP , ψ0 > 3

√ ∀ , ≥ ≥ , ∀P ∈ Σ.

474

A. Pizzo

Ann. Henri Poincar´e

σ

Deﬁnition of φP

j

Given a σ ranging between σj and σj+1 , j ≥ 2, we can always write it as κ (σ) 2 where σ 2j (σ) := . κ By performing the iteration shown in the previous section, we deﬁne j

κ (σ) 2

σ

φP := φP

. σ Lemma 3.1 φP , ψ0 = 0 ∀σ ≤ κ, ∀P ∈ Σ.

j

κ κ 2

Proof. Knowing that φP − φP ≤ 13 (from Corollary 2.4) and being ψ0 = 1, we have:

σ σ κ (σ) κ (σ) , ψ0 − φP − φP , ψ0 > 0 . φP , ψ0 ≥ φP

Deﬁnition of φσP σ w Since φP is ground eigenvector of HP,σ |Fσ+ with a gap larger than tion, thanks to Lemma 3.1 we can deﬁne the normalized vector ; 1 1 − 2πi w −E dE ψ0 HP,σ σ

φP :=

1 ;

− 2πi H w 1 −E dE ψ0

σ 2

by construc-

P,σ

σ |= (where E ∈ C and s.t. |E − EP

σ 4)

w that is ground state of HP,σ |Fσ+ .

Theorem 3.2 For P ∈ Σ, the limit s − limσ→0 φσP =: φP exists. σ2

σ1

Proof. Again write φP − φP in the following way σ2

σ1

σ2

l

κ (σ2 ) 2

φP − φP = φP − φP

l

κ (σ2 ) 2

+ φP

κ (σ1 )

− φP

m 2

κ (σ1 )

+ φP

m 2

σ1

− φP

where l, m are natural numbers. Now, given an arbitrarily small ρ > 0, there exist natural numbers l, m suﬃciently large and a phase eiη(ρ) for which

l

κ (σ1 ) m2 κ (σ2 ) 2

φP

iη(ρ) φP

− e ≤ρ. l

κ (σ ) m2

κ (σ2 ) 2 1

φ

P

φP

This is essentially due to the convergence established in Corollary 2.4 and because of the fact that the ground state is unique as long as there is a cutoﬀ, by construction.

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

475

Therefore we can estimate

σ2 1 σ1

φP σ2 4 −δ σ1 14 −δ φ

− e−iη(ρ) P ≤ cδ + +ρ

σ2

σ1 κ κ

φP

φP with δ > 0 and arbitrarily small, and cδ a δ-dependent constant. Moreover we have that

   

σ1 σ1 σ2 σ2

φP φ φ φ

·  P , ψ0  − P ·  P , ψ0 

σ1 σ1 σ2 σ2

φP

φP

φP

φP

  

σ2 σ1 σ1

φP

φ φ P

− e−iη(ρ) P ≤ ·  , ψ0  + 1

σ2 σ1 σ

φP φP1

φP

σ

Hence it follows that φσP (≡

σ

φP (φP ,ψ0 )

) converges strongly to a nonzero vector (φσP ,ψ0 ) 1 −δ φP in F b , with an error of order σκ 4 at most. σ φP

Lemma 3.3 The following H¨ older estimate holds: 1

|∇E σ (P) − ∇E σ (P + ∆P)| ≤ C |∆P| 16 where the constant in 0 < σ < κ , in P, P + ∆P ∈ Σ and ∆P ∈ C is uniform 83 1 3 |∆P| 1 where I := ∆P : I, , m 4 |∆P| 4 ≤ κ and CI is a constant m ≤ 3C I

suﬃciently larger than 1. Proof. The idea is to perturb, in P, the gradient 1 1 ph 1 |∆P| 4 P − P |∆P| 4 |∆P| 4 ψP ∇E (P) ≡ ψP , m 1

|∆P| 4

where ψP

is the (normalized) ground eigenvector of H

simply denote as H

1

P,|∆P| 4

3

1

P,m 4 |∆P| 4 +

(analogous simpliﬁed notation for F

1

|∆P| 4

, which we

).

For this purpose we use a series expansion of the resolvent 1 H

1

P+∆P,|∆P| 4

1 |∆P| 4 (where E ∈ C and s.t. E − EP = information:

−E 3

|F +

1 |∆P| 4

1

m 4 |∆P| 4 4

) on the basis of the following

476

A. Pizzo

i) H ii) H

1 P+∆P,|∆P| 4 1 P,|∆P| 4

|F +

−H

1 P,|∆P| 4

ph = − ∆P + m ·P

∆P m

·P+

|∆P|2 2m ;

1

1

|∆P| 4

1 |∆P| 4

Ann. Henri Poincar´e

|∆P| 4

has unique ground state ψP 3 4

of energy EP

and its

1 4

(Theorem 1.4 in the continuum case); gap is bounded from below by m |∆P| 2 iii) the quantity

12

∆P · P − Pph

1 3 1 8

√ m 8 |∆P| ·

1 − E H |∆P| m

4 P,|∆P|

F+

1 |∆P| 4

is uniformly bounded in P, P + ∆P ∈ Σ, therefore we can ﬁnd a constant CI suﬃciently larger than 1 such that 3 m4

1 |∆P| 4

 1

 2 

 ∆P · P − Pph  1 |∆P|

    ·  + 

m |∆P| 2m 1 −E

H

P,|∆P| 4

   H

1

2   

1 −E

+ P,|∆P| 4 F 1

< CI ;

1 |∆P| 4

iv) for ∆P belonging to I (deﬁned in the statement of the lemma)

n 3 ∞

1 1

|∆P| 4 1

|∆P| 4 dE ψP · CI >0.

−

≥1− 3

2πi

1 − E H m4 4 P+∆P,|∆P|

n=1

From the above considerations it follows that: • the vector 1 ; |∆P| 4 1 1 dE ψ − P 2πi H 1 1 −E |∆P| 4 P+∆P,|∆P| 4

ψP+∆P := 1

1 ; |∆P| 4 1

−

dE ψP

2πi H

1 −E P+∆P,|∆P| 4

1 is ground state of H P+∆P,|∆P| 4 • the following estimate holds

1

|∆P| 14 3 |∆P| 4

≤ C |∆P| 8

ψ

P+∆P − ψP

where C is a constant uniform in P, P + ∆P ∈ Σ and ∆P ∈ I. Since: 1

1

1) ∇E |∆P| 4 (P + ∆P) − ∇E |∆P| 4 (P) 1 1 1 1 ph ph |∆P| 4 P + ∆P − P |∆P| 4 |∆P| 4 P − P |∆P| 4 ψP+∆P − ψP ψP , = ψP+∆P , m m 2) H

1 P,|∆P| 4

+ 2πg 2 κ −

(Pph −P)2 2m

≥0

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

477

we can conclude that 1 1 3 ∇E |∆P| 4 (P) − ∇E |∆P| 4 (P + ∆P) ≤ C |∆P| 8

(14)

where C is a constant uniform in P , P + ∆P ∈ Σ and ∆P ∈ I. 1

3

Turning to the expression to be proved, if σ < m 4 |∆P| 4 we take advantage of the result of Lemma A2 (in Appendix) together with Theorem 2.3, which implies that 1 ∇E |∆P| 4 (P) − ∇E σ (P) 1

1

3

is of order |∆P| 16 , for P ∈ Σ, at most. If σ ≥ m 4 |∆P| 4 , an estimate analogous to (14) holds. Under the same hypotheses of Lemma 3.3, we have an analogous result about the regularity of the ground eigenvectors in the variable P. k Theorem 3.4 Under the constructive hypotheses, for m and g suﬃciently small, σ σ the norm diﬀerence between φP and φP+∆P is H¨ older in |∆P| with coeﬃcient 1 16 − δ, δ > 0 and arbitrarily small. The multiplicative constant, Cδ , is uniform in 0 ≤ σ < κ, in P, P + ∆P ∈ Σ and ∆P ∈ I, I a suﬃciently small ﬁxed ball around ∆P = 0.

Proof. Preliminary computation: Hw

1

P+∆P,|∆P| 4 1 |∆P| 4

− Hw

= cP+∆P − cP 

 κ 1  + 2m g 3 1  m 4 |∆P| 4

+

+

1 P,|∆P| 4

2m

1

|∆P| 4

k · ∇EP

1

  · g κ 3 1  m 4 |∆P| 4

2 d3 k + ∆Π

  1 1 |∆P| 4 |∆P| 4  b(k)+b† (k) k k·∇E − k·∇E P P+∆P 



1 1 3 √ |∆P| 4   |∆P| 4 2|k| 2  1−k·∇E 1−k·∇E P+∆P P

1

|∆P| 4

where the additive constant cP ∇EP

b† (k) b (k) d3 k

   1 1 3 √ |∆P| 4   |∆P| 4  2|k| 2  1−k·∇E 1−k·∇E P+∆P P

+h.c. 1 |∆P| 4

|∆P| 4

− k · ∇EP+∆P

  1 1 |∆P| 4 |∆P| 4  k k·∇E − k·∇E b(k)+b† (k) P P+∆P

 Γ

1

P,|∆P| 4

1 |∆P| 4

=

1 |∆P| 4 φP ,Π

1 P,|∆P| 4

 d3 k + ∆Π



1 P,|∆P| 4

   

1

3

is a short notation for ∇E σ (P) |

1 P,|∆P| 4

   

corresponds to the infrared cutoﬀ σ=m 4 |∆P| 4 , 3

1

σ=m 4 |∆P| 4

∆Π



1

P,|∆P| 4

1

|∆P| 4

φP

and

1 |∆P| 4 − φP+∆P , Π

1 P+∆P,|∆P| 4

1

|∆P| 4

φP+∆P

.

478

A. Pizzo

Ann. Henri Poincar´e

Considering that for P, P + ∆P ∈ Σ and ∆P ∈ I • the estimate (14), in Lemma 3.3, holds: 1 1 3 ∇E |∆P| 4 (P) − ∇E |∆P| 4 (P + ∆P) ≤ C |∆P| 8 3

• ∆Πi

1

P,|∆P| 4

is estimated of order |∆P| 8 (see equation (6), paragraph 2.1)

• the operator Hw

1

P+∆P,|∆P| 4

− Hw

1

P,|∆P| 4

3

3

m− 8 |∆P| 8 is relatively form-bounded with respect to H w 1 |∆P| 4

• the gap of EP 3

below by

1

P,|∆P| 4

(as ground energy of H w

1

P,|∆P| 4

, uniformly in |∆P|;

|F +

1

1 |∆P| 4

) is bounded from

m 4 |∆P| 4 2

we conclude that the vector 1 |∆P| 4

φP+∆P

; 1 − 2πi =

1 ;

− 2πi

1 P+∆P,|∆P| 4

φP

−E dE

ψ0

dE ψ

w 0 H 1 −E

4 P+∆P,|∆P|

1 |∆P| 4 (where E ∈ C and s.t. E − EP+∆P = 1 |∆P| 4

1

Hw

1

3

1

m 4 |∆P| 4 4

) can be obtained perturbing

I a suﬃciently small ball around ∆P = 0. for ∆P ∈ I ⊂ I,

From the perturbation we also get the estimate:

1

|∆P| 14 1 |∆P| 4

≤ C |∆P| 16

φ

P+∆P − φP

(15)

where the constant C is uniform in P, P + ∆P ∈ Σ and ∆P ∈ I. 1

3

For σ < m 4 |∆P| 4 , the thesis is proved by the norm inequality below

1 1 1 1

σ

φP+∆P − φσP ≤ φσP+∆P − φ|∆P| 4 + φ|∆P| 4 − φ|∆P| 4 + φ|∆P| 4 − φσP P+∆P P

P+∆P

P

and using Theorem 3.2. 3

1

If σ ≥ m 4 |∆P| 4 , an estimate analogous to (15) holds.

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

479

Appendix Preliminary remark on Lemma A1 Analogously to Lemma 1.3, we want to prove that, for P ∈ Σ and g suﬃciently small, the operator w j w w P,σ (∆HP )σj+1 := H − cPj+1 + cPj − HP,σ j+1 j σ

σ

σ

w is small with respect to HP,σ | + in a generalized sense. We aim at expanding j Fσj+1 the resolvent 1 σ σ |Fσ+j+1 w H − Ej+1 + c j+1 − c j P

P,σj+1

P

σj σ 11σ σ σ (where Ej+1 ∈C s.t. Ej+1 −EPj = 20j+1 , and cPj+1 −cPj =−g 2 σj+1 in terms of

1 w HP,σ −Ej+1 j

|Fσ+

σ

j+1

w j and (∆HP )σj+1 .

1 3 σ ) d k) 2|k|2 αPj (k

Lemma A1 Given the spectral properties pointed out in paragraph 2.1, for suﬃw σj w )σj+1 is small with respect to HP,σ in the ciently small g and for P ∈ Σ, (∆HP j following sense: σ given Ej+1 ∈ C s.t.Ej+1 − E j = 11 σj+1 , P

20

n

1 1

w σj − (∆HP )σj+1 w

w

HP,σj − Ej+1 HP,σj − Ej+1

≤

Fσ+j+1

where 0 < Cg < w H P,σ =

=

j+1

w HP,σ j

1 12 ;

20 (Cg )n σj+1

therefore the series expansion below is well deﬁned:

1 |F + σ σ σj+1 − Ej+1 + cPj+1 − cPj +

w σj ∆HP σj+1

1 | + w HP,σ − Ej+1 Fσj+1 j

1

|F + σ σ σ σj+1 + cPj+1 − Ej+1 − cPj + cPj+1 n +∞

1 1 w σj + w |F + . − (∆HP )σj+1 w σj+1 HP,σ − Ej+1 n=1 HP,σ − Ej+1

−

σ cPj

j

j

Proof. Following the proof of Lemma 1.3, we discuss the norm of

1 w HP,σ − Ej+1 j

12

w σj (∆HP )σj+1

1 w HP,σ − Ej+1 j

12 |Fσ+

j+1

480

A. Pizzo

Ann. Henri Poincar´e

0mix / 0quad. / w σj w σj w σj where (∆HP )σj+1 = (∆HP )σj+1 + (∆HP )σj+1 with /

0quad. w σj (∆HP )σj+1 := 2  σj σj k b(k) + b† (k) 3 1  k 3  = (a1) d k − g2 g σ 2 d k √ 3 σj 2m j 3 σj+1 2 σ 2|k| αP (k) j+1 2|k| αP (k) 0mix / w σj )σj+1 := (∆HP       σj σj  † (k) k b(k) + b k 1   3 2 3  = d k + g d k + h.c. . ΓP,σj ·  −g  √ 3 2 σj σ  2m  j 3 σj+1 σj+1 2|k|   2|k| 2 αP (k) αP (k) (a2)

In order to control the above quantities, we use the following estimate again and again

12

σj

3 √ 1

k 1 d 1 i

k b (k) ≤ 10π · σj2 √ 3 w

max σ j 1−v

σj+1

+ 2 |k| 2 αP (k) HP,σj − Ej+1 Fσj+1

which essentially derives from the estimate (1) of Lemma 1.3, by performing a unitary transformation, and from the result in Lemma A2, point 1, which ensures σ that 0 < v max < 1. So we can provide a bound of order κj for the norm of the “quadratic terms”:

12 12

/ 0 κ quad.

1 σj 1 w σj

(∆H ·C . ) ≤ 1 g, P σj+1

H w − Ej+1 w HP,σ − Ej+1 κ m

+

P,σj j Fσj+1

For the mixed terms (a2) containing the operators ΓP,σj , we exploit the fact that the norm



12 i 12 †

i

Γ Γ

1 1 P,σ

P,σ

 √ j = √ j 

w

H w − Ej+1

m

m HP,σj − Ej+1

P,σj

+ + Fσ j+1

is of order

κ σj+1

12

Fσj+1

, which follows from the form inequality

ΓiP,σj

2

σj w . ≤ 2m HP,σ − c P j

Therefore a uniform bound is worked out for them:

12 12

/ 0mix

1 1 w σj

(∆HP )σj+1 w

H w − Ej+1

HP,σj − Ej+1

P,σj

Fσ+j+1

κ ≤ C2 g, . m

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

481

Conclusion For g suﬃciently small, the j-independent constants C1,2 can be tuned so that the thesis is proved. Lemma A2 Under the constructive hypotheses the following results for ∇E σ (P) hold: 1) |∇E σ (P)| < v max < √180 2) for g suﬃciently small

∀σ ;

|∇E σj+1 (P) − ∇E σj (P)| < C ∇E

 

σj+1 σj

φP j+1 φ

8 

− P ·  σj +

σj+1

φP φP

where C ∇E is uniform in j ∈ N . Proof. 1) Let us start from

|∇E σ (P)| =

< 2 = 12 : σ i phi σ ψ , ( P − P ) ψ P P i 1

≤

σ 2 m ψP

σ 12 σ ψP , ( HP,σ + 2πg 2 κ ) ψP σ ψP

2 · m

According to the constructive hypotheses, Section 1, we have: •

σ σ |(ψP , HP,σ ψP )|

ψPσ 2

≤ |(ψ0 , HP,σ ψ0 )| =

P2 2m

≤

m 2·400

|(ψPσ , 2πg2 κ ψPσ )| m ≤ 2·100 2 ψPσ The upper bound for the absolute value of the gradient, v max , is surely smaller than √180 . •

2) Let us analyze the diﬀerence between the gradients of the ground energy at subsequent infrared cutoﬀs. Starting from the relation (6), paragraph 2.1, σj κ σ φP , ΠP,σj φPj k σj 3 − g2 m∇EP = P −

σj 2 2 d k ,

φ σ 3 j σ j 2 |k| αP k P we obtain σ

σ

m∇EPj − m∇EPj+1 − g 2

κ

k

2 |k|

σ αPj+1

3 2 2 d k + g k

κ

k

3 2 d k σ 2 |k| αPj k σ 1 P,σ σj+1 − 1 · φσj , ΠP,σ , φσj . j+1 , Π (a3) = φ

· φ j+1 P j P P P

σj+1 2

σj 2

φP

φP σj+1

3

σj

3

482

A. Pizzo

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By simple steps we relate the diﬀerence between the two gradients to the norm diﬀerence of the corresponding ground eigenvectors:

· ∇E σj+1 + k · ∇E σj 2 − k · ∇E σj+1 · ∇E σj − k k −k P P P P +g d3 k σ 2 σ 2 j+1 j 3 σj k αP k 2|k| αP    σj σj σj+1 φ φ 1 k σ j+1 3 P,σ  P − P  + g2

· φ = σ ,Π σ 2 d k j+1 P

j+1

σj+1 σj 3 σ j+1 2|k|

φP

φP φP αPj+1 k σ σ 1 1 σj σj

j+1

j+1 +

σj+1 σj · φP , ΠP,σj+1 φP − σj+1 σj · φP , ΠP,σj φP

φP φP

φP φP σ 1 j+1 , ΠP,σ φσj − 1 · φσj , ΠP,σ φσj .

· φ + σ j P j P P P

j+1 σj σj 2

φP φP

φP

σ m∇EPj

σ − m∇EPj+1

2

κ

Considering that iP,σ Π − ΠiP,σj j+1 σj i k b (k) + b† (k) 3 = −g d k √ 3 σ σj+1 2 |k| 2 αPj k i · ∇E σj+1 + k · ∇E σj − k · ∇E σj 2 − k · ∇E σj+1 − k 2 κ k P P P P g d3 k + 2 2 2 σj+1 σj σj+1 3 |k| αP k αP k the equation (a3) can be written in the following way σj+1 σj σj σj+1 2−k·∇E k −k·∇E +k·∇E −k·∇E P P P P σ 2 σ 2 d3 k j+1 j 2|k|3 αP αP (k (k) ) σj+1 σj σj σj+1 σ k −k·∇E +k·∇E −k·∇E 2−k·∇E P P P P j+1 ,φσj · g 2 κ σ 2 σ 2 d3 k −

σj+11

σj

· φ P P σj+1 j+1 j

φP

· φP αP (k 2|k|3 αP (k) ) σ

σ

m∇EPj − m∇EPj+1 + g 2

= +g 2

σ

j+1 φ P

σ

j+1 , ΠP,σj+1

φP

σj σj+1

κ

σj

σ

j+1 φ P

σ

j+1

φP

σk 2 d3 k − j+1 2|k|3 αP (k)

σj

φP

σ

j+1 φ P

σ

j+1

φP

σj

σj

φP

+

−

φP

σj

φP

σ

j

φP σj k b(k)+b† (k) 3 σj σj+1 1

σ

d kφP . ,g σ 3 σj

j+1 σj · φP j+1 √ )

φP

· φP 2|k| 2 α (k

−

σj

φP

, ΠP,σj

P

Then the thesis follows from the expression above and from the following considerations. i) On the left-hand side of the equation there is a quantity whose absolute value is larger than σ σ C ∇EPj − ∇EPj+1 for g suﬃciently small, where C is a positive constant that is uniform in j and converges to m for g → 0. It is due to the result in point 1).

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483

ii) On the right-hand side there is a quantity whose absolute value is bounded by a g-dependent, uniform in j, constant times the factor

 

σj+1 σj

φP

j+1 φ P

 + 8  ; −

σj+1

φσj

φP

P

it is due to the bounds below, where (ﬁrst estimate) we exploit the form inequality σ σ w b† (k) b (k) d3 k ≤ H P,σ |k| αPj k − cPj+1 j+1 and (second estimate) we use the known unitary transformations to switch from operators to the “Π” ones and ﬁnally to Pph : the “Π”

•

σj+1 , g φ P =g

σj

σj+1 √

≤g

σj σj+1

≤g

ki b(k)+b† (k) 3 k φσ j d 3 σj P σj+1 √ ) 2|k| 2 αP (k

σj

σj σj+1

ki

σ 3 j 2|k|2 αP (k ) 2

σj+1 , φ P

(ki )2 σ 3 d3 k j 2|k|4 αP (k ) (ki )2 σ 3 d3 k j 2|k|4 αP (k )

1 2

1 2

|k|

1 2

=

σj+1 , g φ P

σ αPj

σj

σj+1 √

ki b† (k) 3

σj

2|k| 2 αP

1 2 b† (k) φσj k P

σ

) (k

d3 k φPj

d3 k

1 σ σ

b† (k) b (k) d3 k φ j+1 , |k| ασj k σj+1 2 · · φ

φPj P P P 1 σ σ 1 σ

2 σj+1 2 · j+1 , φ · EPj+1 − cP (j + 1) · φ

φPj P P

σj+1 i

i σj+1 σj+1 σj+1 • Π

= ΠiP,σj+1 φP ≤ P ph − P i ψP + b · ψP P,σj+1 φP

σ √ 1 σ σ ≤ 2m · 2πg 2 κ + EPj+1 2 · ψPj+1 + b · ψPj+1 where b is a constant uniform in j and P ∈ Σ .

κ , in addition to the constructive hyLemma A3 For a suﬃciently small ratio m potheses, the following inequality holds:

2 12 12

σj i †

1 k b (k) 1 σj 3 i

d k ΓP,σj φP √ 3 w

H w − Ej+1

σj H − E 2 j+1 σj+1

P,σj P,σ 2 |k| αP k j √ 1 1 σ σ j · ΓiP,σ φ j, ≤ 2Q () · 122 · Zσσj+1 · σj ΓiP,σj φPj j P w HP,σj − Ej+1 EP − Ej+1

being

σj Zσj+1

=

: σj i σj+1

2

ki d3 k σ )2 2|k|3 αPj (k

.

Proof. Let us start from

2 1

2 σj

σj ki b† (k) 3 k Γi φ = d

H w 1−E 3 σj √ P,σj P σ j+1 j+1

P,σj 2|k| 2 αP (k) σ σj σ ki b† (k) j d3 k ΓiP,σ φPj , H w 1−E = 3 σj σj+1 √ j j+1 σj+1 ) P,σj 2|k| 2 αP (k

√

ki b† (k) 3

σj

2|k| 2 αP

σ

) (k

d3 k ΓiP,σ φPj j

.

484

A. Pizzo

Ann. Henri Poincar´e

Now, from Lemma 2.1 

 σ σj σj   σj ki b† (k) j ki b† (k) 3 i 3 i 1 d k Γ φ , d k Γ φ  σj+1 √ P,σj P H w P,σj P  3 σj 3 σj −Ej+1 σj+1 √ P,σj k k 2|k| 2 α 2|k| 2 α P P   σ √ σj σj σj   ki b† (k) ki b† (k) j 3 i 3 i 1 d k ΓP,σ φP , d k ΓP,σ φP  . ≤ 122 ·  σj+1 w σ 3 3 j+1 H −E σ σ √ √ j j j+1 j j P,σj 2|k| 2 α 2|k| 2 α k k P P

Starting from the expression w = HP,σ j

1 2 Γ + 2m P,σj

σ b† (k) b (k) d3 k + cσj |k| αPj k P

the following identity holds in distributional sense, for {k : |k| ≤ σj }:

1 w HP,σ − Ej+1 j 

 = b† (k) 

1 2m

b† (k)

ΓP,σj + k

 2

+

1 σ |q| αPj

( q) b†

(q) b (q) d3 q

+

σ |k| αPj

 . σj k + cP − Ej+1

σ κ Moreover, for m suﬃciently small, due to the prescription for EPj − Ej+1 , due to the constraints on |∇E σj (P)|, being σj+1 ≤ |k| ≤ σj and taking into account σ σ E j − cPj , the following bound holds:

P

1

2

1 1

k · ΓP,σj +

H w +|k|ασj (k m )−Ej+1

P,σj P

k2 2m

1 σj w )−Ej+1 HP,σ +|k|αP (k j

1 2

≤

1 2

.

(a4)

+ Fσ j

Therefore the series expansion 



 w HP,σ

j

+∞



2   − 1 k · ΓP,σ + k j σj m 2m + |k|α k − Ej+1 n=0

1

P

n 1

 σ w − Ej+1 HP,σ + |k|αPj k j

is well deﬁned in Fσ+j . σ Then we can write (note that b (k)φPj = 0 for |k| ≤ σj ): σj σj σ σ ki b† (k) ki b† (k) d3 kΓiP,σj φPj , H w 1−Ej+1 d3 kΓiP,σj φPj σj+1 √2|k| 32 ασj k σj+1 √2|k| 32 ασj k P,σ j P ( ) P ( ) 2 σj :∞ ki 3 = σj+1 M (k) d k σ n n=0 )2 2|k|3 αPj (k where Mn (k) corresponds to n 2 2k · Γ k P,σ + σ σ j k k ΓiP,σj φPj , RH (Ej+1 ) − (Ej+1 ) ΓiP,σj φPj RH w w P,σj P,σj 2m 1 k with RH (E ) := σ w j+1 w )−Ej+1 . P,σj HP,σ +|k|αPj (k j Exploiting the Schwartz inequality, we have that n 2 2k · Γ k P,σ + i σj σ j j k k i (E ) − (E ) Γ φ R w w ΓP,σj φP , RH j+1 j+1 H P,σ P j P,σj P,σj 2m

Vol. 4, 2003

One-particle (improper) States in Nelson’s Massless Model

485

can be bounded by the product of the following quantities: • •



2

1 †

2 σ

 j 1 i  Γ φ

σj P,σ P w j

HP,σ +|k|αP (k)−Ej+1 j

1 1

n 2 2

1 1 k2 1

k · ΓP,σj + 2m . σj

H w +|k|ασj (k

m w )−Ej+1 )−Ej+1 HP,σ +|k|αP (k

+ P P,σj j F σj

Hence, due to the estimate (a4) and Lemma 2.1, the absolute value of the scalar product   

σ j σj+1

ki b† (k)

d 3 √ σj 2 |k| 2 αP k

 3

σj i k ΓP,σ φP j

, 

1 w HP,σ − Ej+1 j

 

σ j σj+1

ki b† (k) √

3

σj

2 |k| 2 αP

 d k

3

σj  i k ΓP,σ φP  j

is bounded by



2 1

  : 2

1 n σj 1 i 3k

Γ φ d · ∞ σ 2 j n=0 2 P,σj P ) Hw   +|k|α −E k k ( ( ) j+1

  P,σj P

2

1

2 2

σ ki 1 3 ≤ 2 σσj · ΓiP,σ φPj σj σj 2

d k

3 w j j+1 2|k| α k

HP,σj +|k|αP (k)−Ej+1 P ( ) 2 σj σj ki 1 3 k · Γi i . ≤ 2 · Q () · σσj Γ d φ , φ σj w 2 P,σ P,σ P P H −E j j j+1 2|k|3 α j+1 k P,σj P ( )   

2 σj ki σj+1 2|k|3 ασj P

Collecting all the estimates the thesis follows.

Acknowledgments. The content of this paper is the ﬁrst part of my P.H.D. dissertation in S.I.S.S.A, Trieste. I would like to thank G. Morchio and F. Strocchi for having suggested to me the problem and for discussions and advice. It is a pleasure to thank V. Bach, G. Dell’Antonio and J. Fr¨ ohlich for their interest in my work and for many enjoyable and interesting discussions. I have to very warmly thank J. Fr¨ ohlich for having explained to me some points of his old papers [Fr.1, Fr.2], for his helpful suggestions, criticism and advice. I am also grateful to C. Gerard, G.M. Graf and M. Griesemer for fruitful discussions.

References [Ar]

A. Arai, mp arc 00–478 (2000).

[B.F.S] V. Bach, J. Fr¨ ohlich and I.M. Sigal, Adv. Math. 137(2), 299–395 (1998) and 137, 205–298 (1998). [Bl]

Ph. Blanchard, Comm. Math. Phys. 15, 156 (1969).

[Bu]

D. Buchholz, Phys. Lett. B174, 331 (1986).

[Ch]

T. Chen, preprint mp arc 01-310 (2001).

[D.Ge]

J. Derezinski and C. Gerard, Rev. Math. Phys 11, 383–450 (1999).

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[Fr.1]

J. Fr¨ ohlich, Ann. Inst. H. Poincar´e, Sect. A, XIX (1), 1–103 (1973).

[Fr.2]

J. Fr¨ ohlich, Fort. der Phys. 22 , 158–198 (1974).

[F.G.S] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Los Alamos Preprint Archive, math-ph/0009033 (2000). [F.G.S] J. Fr¨ ohlich, M. Griesemer and B. Schlein, math-ph/0103048 (2001). [F.P]

M. Fierz and W. Pauli, Nuovo Cimento 15, 167 (1938).

[Ge]

C. Gerard, preprint mp arc 99–158 (1999).

[Ge]

C. Gerard, mp arc 01–103 (2001).

[G.L.L] M. Griesemer, E.H. Lieb and M. Loss, preprint, Los Alamos Preprint Archive, math-ph/0007014 (2000). [G.J]

J. Glimm and A. Jaﬀe, Ann. Math. 91, 362 (1970).

[L.M.S] J. Lorinczi, A. Minlos and H. Spohn. Los Alamos Preprint Archive, mathph/0011043 (2000). [Ne]

E. Nelson, J. Math. Phys. 5, 1190–1197 (1964).

[Pi]

A. Pizzo (in preparation).

[R.S]

M. Reed, B. Simon, Methods of modern mathematical physics, Volumes 4, Academic Press, 1975.

[Sc]

B. Schoer, Fortschr. Physik 11, 1–31 (1963).

[Sp]

H. Spohn, J. Math. Phys. 38(5), 2281–2296 (1997).

Alessandro Pizzo FB Mathematik Johannes Gutenberg Universit¨ at D-55099 Mainz, Germany email: [email protected] Communicated by Gian Michele Graf submitted 29/04/02, accepted 11/10/02

Ann. Henri Poincar´e 4 (2003) 487 – 512 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030487-26 DOI 10.1007/s00023-003-0137-5

Annales Henri Poincar´ e

The Dirac Equation in Geometric Quantization Andrej B´ ona∗

Abstract. The coadjoint orbit of the restricted Poincar´e group corresponding to a mass m and spin 1/2 is described. The orbit is quantized using the geometric quantization. To include the discrete symmetries, one has to induce the irreducible representation of the restricted Poincar´e group obtained by the quantization procedure to the full Poincar´e group. The new representation is reducible and the reduction to an irreducible representation corresponds to the Dirac equation.

1 Introduction The Dirac equation is one of the most important equations in relativistic quantum physics as it describes particles of spin 1/2, and both electron and protons have spin 1/2. When P.A.M. Dirac introduced this equation [5] in 1928, while his reasoning was original and insightful, his derivation of the equation did not shed much light on the meaning of the equation. The purpose of this paper is to interpret the Dirac equation from geometric-quantization point of view and thus provide more understanding of the topic.

1.1

Group theory

In quantum mechanics, the states of a system are represented by vectors (or rays) of a Hilbert space and observables correspond to self-adjoint operators. In the case of a free (isolated) relativistic system, one requires the theory to be invariant under the Poincar´e group. More precisely, there is a one-to-one correspondence between states corresponding to any two diﬀerent observers observing the same system and the transition probabilities for the two observers are equal. These conditions imply an existence of a unitary (or antiunitary) representation of the Poincar´e group in the Hilbert space. If one assumes from the very beginning the invariance of the theory under the Poincar´e group (or some other group), one can look at possible unitary representations of the group and decide if a given representation is physically meaningful or not. Moreover, if one is interested in description of elementary quantum systems, one restricts the representations to only the irreducible ones [8]. This way, using only the representation theory, one can obtain physically meaningful results. Perhaps the most important one stems from the time translations given by the group ∗ The

´ author would like to thank J¸edrzej Sniatycki for his help and support.

488

Andrej B´ ona

Ann. Henri Poincar´e

representation. This one parameter subgroup of unitary operators corresponds to the time evolution of the system, or, in other words, represents the equation of motion for the free system. It is possible to use the pure group theory to ﬁnd the Dirac equation or other relativistic wave equations. This has been done, for example, in [3]. The knowledge of the representations of the group, however, does not constitute a full physical theory. A model based solely on a representation of the group does not contain any information on the observable quantities of the system. To obtain this information, one needs to consult the corresponding classical system. To relate the classical system with a quantum one, one can quantize the corresponding classical system. It was noted by Souriau [13] and Kostant [7], that coadjoint orbits have a symplectic structure and thus represent a classical mechanical system. The underlying classical system in the case of a free relativistic spinning particle is a coadjoint orbit of the Poincar´e group. To quantize this system, the method of geometric quantization is used.

1.2

Geometric quantization

The canonical quantization is perhaps the most known method of relating classical and physics. It associates to a classical observable f (x, p) an operator quantum ∂ on the Hilbert space of square integrable functions ψ (x). This method f x, −i ∂x has many drawbacks, among which the most obvious one is the problem of ordering of xs and qs. The geometric quantization generalizes the canonical quantization to a coordinate free theory and all the structures have a nice geometrical meaning. Historically, the geometric quantization started by publications of Souriau [13] and Kostant [7] and later that of Blattner [4]. A prequantization of the Poincar´e group was done already in 1969 [10], but in this paper the construction of the prequantization is done in a diﬀerent way.

1.3

Summary of the results

One can represent the trivial line bundle over the coadjoint orbit O of the Poincar´e group, corresponding to mass m and spin s, by the space of sections λ of a trivial line bundle over the space M = (x, p, z) ∈ R1,3 × R1,3 × C2 which are restricted by the conditions ∇Xp2 λ = 0 ∇Xzpz¯ λ = 0 and Pp2 −m2 λ = 0 Pzp¯z −s/2π λ = 0,

Vol. 4, 2003

The Dirac Equation in Geometric Quantization

489

where Pf is a prequantum operator corresponding to function f . Here the vector ﬁelds Xp2 and Xzp¯z are the Hamiltonian vector ﬁelds, corresponding to functions p2 and zp¯ z , given by the symplectic form ω, which, restricted to the orbit O, is the Kostant-Soriau form. The operators P are the prequantized operators in the trivial bundle over M . The quantization in this representation is given by introducing the polariza¯A ∂z¯A . The quantized space is thus represented tion spanned by the vectors ∂x and w by sections of the trivial line bundle over M with restrictions ∇∂x λ = 0 ∇w¯ A ∂z¯A λ = 0 and ∇Xzpz¯ λ = 0 Pp2 −m2 λ = 0 Pzp¯z −s/2π λ = 0. Here, formally, the prequantized operators are the same as quantized operators, since the ﬂows of the Hamiltonian vector ﬁelds Xp2 −m2 and Xzp¯z−s/2π preserve the ¯A ∂z¯A . Note also that the condition polarization spanned by the vectors ∂x and w on the sections from the prequantization ∇Xp2 λ = 0 has vanished, since this condition is now included in the condition of covariant constancy along the polarization. This is implied by the fact that the vector ﬁeld Xp2 = −2p∂x belongs to the polarization. This way the space of quantized sections Hp is of the following form λ | ∇∂x λ = 0, ∇w¯ A ∂z¯A λ = 0, ∇Xzpz¯ λ = 0, Hp = . Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0 The action of the one parameter subgroup of the restricted Poincar´e group, generated by an element ξ = (A, a) of the Lie algebra, on the elements of Hp is given by operators Uξt acting as Uξt λ = (ψ ◦ φt ) λ0 , where λ = ψλ0 and φt is the ﬂow of the vector ﬁeld generated by ξ = (A, a). The action of the discrete symmetries is not, however, well deﬁned on this space. The discrete symmetries act on the spinorial part as antilinear operators, and thus they do not preserve the polarization. To deﬁne the action of the discrete symmetries, one can induce the representation of the Poincar´e group. To do this, one can take the representation of the restricted Poincar´e group which lives in the

490

Andrej B´ ona

Ann. Henri Poincar´e

space Hp . One then takes the image of Hp under the discrete symmetries in H, space of sections of the line bundle over M . The product of these four spaces forms the new representation space, in which the discrete symmetries act as they act in the prequantized bundle: Tψ = ψ ◦ T P ψ = ψ ◦ P. One can further combine the elements of the four spaces into one vector-valued function. This way one can write A z ξA (p) ψ = e2πip,x z¯A η A (p) and t U(A,a)

e

2πip,x

T At eA t ξA 2πip,a e p (p) = e ¯ A −At e η z¯A η A (T p) 2πip,x T ψ (x, p, z) = e z A ξA (T p) z¯A η A (P p) P ψ (x, p, z) = e2πip,x . z A ξA (P p) ξA ηA

This representation is not irreducible; there is the operator D that commutes with all elements of the representation of the Poincar´e group. The operator D acts on the sections represented as λ = ψλ0 as follows: A A

z ξA (p) z pAA η A (p) 2πip,x 2πip,x Dψ (x, p, z) = D e . =e z¯A η A (p) ξA (p) pAA z¯A To get an irreducible representation, one has to investigate the eigenspaces of the operator D. The operator D has eigenvalues ±m. Thus, the wave functions of the irreducible representation must satisfy the equation Dψ = ±mψ. This is the Dirac equation.

1.4

The organization of the paper

This paper is divided into two main parts. In the ﬁrst part, the Poincar´e group is discussed. This part is mainly a technical elaboration of the construction of the coadjoint orbits and the related symplectic structure. The construction of the orbit is done with help of two-spinors. This construction has been motivated by [14].

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The Dirac Equation in Geometric Quantization

491

The second part is devoted to the geometric quantization of the Poincar´e group. It is in this part of the paper that the most of the new and interesting work is presented. At the beginning of the paper, a short summary of the geometric quantization is presented.

2 Short introduction to geometric quantization Let M be a 2n dimensional symplectic manifold with a symplectic structure given by a 2-form ω. One considers a complex line bundle (L, π, M ) over the manifold M together with a connection α ∈ Λ (L) such that dα = π ∗ ω. The existence of such a line bundle is guaranteed provided that so-called integrality condition is satisﬁed:

ω ∈ Z, (2.1) where the integration is over each closed oriented two-surface in M . One can construct a Hilbert space H of square integrable sections λ of L, where the scalar product is given by

(λ1 (m) , λ2 (m))m ω n . (λ1 , λ2 ) = M

Here the symbol (., .)m represents the product on the ﬁbre π −1 (m) ∼ = C. For each diﬀerentiable function f on M , there is a Hamiltonian vector ﬁeld Xf , such that Xf ω = −df . It is possible to lift the Lie algebra of the Hamiltonian vector ﬁelds [Xf , Xg ] = X{f,g} , here {., .} is the Poisson bracket, to the line bundle L to obtain a faithful representation of the Poisson algebra {f, g}1 . To do that, one deﬁnes to each f a self-adjoint operator Pf acting on H as Pf = ∇Xf − 2πif, where ∇ is the covariant derivative given by the connection α. The representation Pf of the Poincar´e algebra is called prequantization. The Hilbert space H is still too big to satisfy the Heisenberg principle. To deal with this diﬃculty, one introduces a polarization on the manifold M and restricts the space of sections to sections that are covariantly constant along this polarization. This, however, destroys the Hilbert space, since these sections are not square integrable with respect to the previously introduced scalar product. One can still introduce a scalar product with respect to which the covariantly constant sections along the polarization can be square integrable. This is done with help of half-forms. The topic is too extensive to be covered in a brief overview, hence the reader is referred to literature [4]. The object similar to a half-form is a half-density, which is much easier to deﬁne and suﬃces for the purpose of this overview. 1 The Hamiltonian vector fields are not a faithful representation of the Poisson algebra because all constant functions map to the zero vector field.

492

Andrej B´ ona

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A d-density is an element of the associated ﬁbre bundle to the frame bundle F N of N with the ﬁbre R and action of Gl (n, R) on the ﬁbre being d

gx = |det g| x, where x ∈ R and g ∈ Gl (n, R). A d-density µ can be represented by a complexvalued function µ# on the frame bundle F N , such that for g ∈ Gl (n, R) and each w ∈ FN, d µ# (wg) = |det g| µ (w) . The pair of two half-densities µ1 and µ2 can deﬁne a one-density (µ1 , µ2 ) as #

# (µ1 , µ2 ) (w) = µ# 1 (w)µ2 (w) .

A one-density can be integrated over N . One can deﬁne for a polarization F two other distributions, namely, D = F ∩ F¯ ∩T M and E = F + F¯ ∩T M . For so-called strongly admissible polarization F , D and E are involutive distributions, the spaces M/D and M/E are manifolds and the projections πE : M → M/E, πD : M → M/D and πED : M/D → M/E are submersions. If F is strongly admissible, then for each integral manifold P −1 of E, Q = πED (P ) is a symplectic manifold with a polarization FQ such that ¯ FQ ∩ FQ = 0 and FQ = T πD F |P with the symplectic form ωQ such that ω|P = ∗ πD ωQ . This allows to deﬁne a scalar product for covariantly constant sections along the polarization F in an extended space with elements λ ⊗ µ

n−d ∗ (λ1 ⊗ µ1 , λ2 ⊗ µ2 ) = πD (λ1 , λ2 ) ωQ (µ1 , µ2 ) , M/E

Q

where µi are densities on M/E and d is dimension of D. This scalar product can be used to deﬁne the Hilbert space H. It is possible to deﬁne quantized operators on H corresponding to a function whose Hamiltonian vector ﬁeld preserves the polarization. The details can be found in [12].

3 The Poincar´e group and algebra The Poincar´e group is a group of linear isomorphisms of the Minkowski space that preserve the distance. More precisely, one can express the elements of the Poincar´e group as follows. An element g of the Poincar´e group G can be represented by a pair g = (Λ, l), where Λ is a matrix and l is a vector from the Minkowski space. The action ρ of the group on the vector x from the Minkowski space is (ρ(g)x)α = Λαβ xβ + lα .

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The Einstein summation convention is used here, as well as in the rest of this document. The multiplication and the inverse element are given for g = (Λ, l) and h = (∆, d) by gh = (Λ∆, Λd + l), g −1 = (Λ−1 , −Λ−1 l). Since the scalar product on the Minkowski space R1,3 is given by the tensor gαβ , that has signature +, −, −, −, and Λ preserves the scalar product, the matrix Λ must satisfy the following equation: Λαβ Λγβ = g αγ . In this expression, the lowering of the indices with help of gαβ is used, e.g., xα = gαβ xβ . An element ξ of the Poincar´e algebra g can be represented by the pair ξ = (Aαβ , aα ), where the tensor Aαγ = Aαβ g βγ must be antisymmetric, Aαγ = −Aγα , 1,3 and a is a vector from the Minkowski space R . In this case, the exponential tA map exp : g −→ G is given by exp tξ = e , ta . Here, the expression etA is to be understood as the standard exponentiation of the matrix Aαβ . An element α of the dual space g∗ to the space g can be expressed as the pair α = (Mαβ , pα ) that acts on the element ξ = (A, a) ∈ g as α(ξ) = M, A − p, a = 1/2T r M A − pa = −1/2Mαβ Aαβ − pα aα , where Mαβ gβγ = Mαγ = −Mγα = −Mγβ gβα with p ∈ R1,3 . The adjoint action Ad(Λ,l) (A, a) =

d (Λ, l) exp {t (A, a)} (Λ, l)−1 = (ΛAΛ−1 , −ΛAΛ−1 l + Λa), dt t=0

together with the evaluation α(ξ), gives the coadjoint action: Ad∗(Λ,l) (M, p), (A, a) = (M, p), Ad(Λ,l)−1 (A, a) = (M, p), (Λ−1 AΛ, Λ−1 Al + Λ−1 a) = ΛM Λ−1 , A − Λp, Al − Λp, a . αβ

If one deﬁnes the wedge product of x and y as (x ∧ y) the following identity holds:

(3.1) (3.2)

= xα y β − y α xβ , then

1 α x, (p ∧ q)y = xα (p ∧ q)αβ y β = x (p ∧ q)αβ y β − y β (p ∧ q)βα xα 2 1 αβ = − p ∧ q, x ∧ y . (p ∧ q)αβ (x ∧ y) = 2

(3.3)

Using this formula, one can write the expression (3.1) as ΛM Λ−1 , A − Λp, Al − Λp, a = ΛM Λ−1 , A + A, Λp ∧ l − Λp, a = ΛM Λ−1 + Λp ∧ l, A − Λp, a = ΛM Λ−1 + Λp ∧ l, Λp , (A, a) ,

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where one uses the fact that any antisymmetric tensor, and thus also A, can be expressed as a sum of tensors in a form of p ∧ q. This way one can conclude that Ad∗(Λ,l) (M, p) = (ΛM Λ−1 + Λp ∧ l, Λp). To express the elements of the dual space to the Poincar´e algebra in more convenient way, one uses a notion of the dual tensor ∗ A to the tensor A given by the expression: ∗ Aαµ = 12 εαµνλ Aνλ . The symbol ε denotes the Kronecker totally antisymmetric tensor. Then, one can use the following theorem: Theorem 1 Any antisymmetric tensor M αβ = −M βα can be written in a form M = (x ∧ p +∗ (p ∧ u)) for a given p, where xα =

1 p2

αβ M pβ + tpα , uα = ∗

1 p2

(∗ M )αβ pβ + t pα

for

some real t and t . Then automatically M p, p = ( M ) p, p = 0. Proof. The proof can be simply obtained using formulas in [2]. Based on the previous theorem, one can reparametrize the dual of the Poincar´e algebra as follows (M, p) → (xα (t) =

(∗ M )δε pε + t pδ M αβ pβ + tpα , p , u (t ) = ), γ δ p2 p2

for some real t, t , where all points on the lines x (t) and u (t) describe one point on the orbit. The coadjoint action in this representation is given by the formula Ad∗(Λ,l) (x, p, u) = (Λx + l, Λp, Λu) since, recalling (3.1), Ad∗(Λ,l) (x ∧ p +∗ (p ∧ u), p) = (Λ(x ∧ p)Λ−1 + Λ∗ (p ∧ u)Λ−1 − l ∧ Λp, Λp) = (Λx ∧ Λp + Λp ∧ l +∗ (Λp ∧ Λu), Λp) ←→ (Λx + l, Λp, Λu) using Λ(x ∧ p)Λ−1 = Λx ∧ Λp. The coadjoint orbit Oα = {β = Ad∗Λ,l α|α ∈ g∗ , (Λ, l) ∈ G} of Poincar´e group G in the dual g∗ of its algebra passing through the point α = (M, p) is then O(M,p) = {(Λx + l, Λp, Λu)| x =

1 1 (M p + tp) , u = 2 ((∗ M ) p + t p)}/ ∼, (3.4) 2 p p

where the equivalence relation ∼ is given by x ∼ x + tp, u ∼ u + t p. The coadjoint action preserves the equivalence classes. The group action preserves p2 as well as p ∧ u, p ∧ u , since p ∧ u, p ∧ u = 2 −p2 u2 + (pu)2 = −p2 (u + tp) + (p·(u + tp))2 . Any orbit is completely determined

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by these two numbers, since one can choose a representant u from the equivalence class u + tp such that pu = 0 and thus p ∧ u, p ∧ u with p2 determine u2 . Since in this paper only the orbits with positive p2 are treated, the numbers p2 and p ∧ u, p ∧ u are going to be called m2 and (s/π)2 respectively, for m and s real. One would like to express the Kostant-Souriau form on the coadjoint orbits. The coordinates (x, p, u) are not the most convenient ones. Better coordinates are expressed with help of two-spinors. Recall that the space S C2 of spinors z A is endowed with a symplectic form given by tensor 0 1 ,

AB = −1 0 where the indices take values 0 and 1. With the help of one can deﬁne the dual space S ∗ to S: S ∗ zA = z B BA , z B ∈ S. There are other complex spaces associated to S: complex conjugate space S¯ and dual conjugate space S¯∗ . To distinguish between elements of these spaces we must introduce another type of indices (not only upper and lower), namely primed indices taking value as well 0 or 1. ¯ z¯A ∈ S¯∗ . z A ∈ S, zA ∈ S ∗ , z¯A ∈ S, The symplectic forms in S ∗ , S¯ and S¯∗ are AB , A B and A B , which are equal to 0 1 .

A B = AB = A B = −1 0

We can lower and lift indices with help of as follows (since εT ε = id):

zB = z A AB , wA = AB wB , uB = uA A B , v A = A B vB . One can deﬁne isomorphism of the Minkowski space R1,3 and the subspace of S ⊗ S¯ represented by Hermitian matrices. This can be done by taking a distinguished basis of the Hermitian matrices to be the σαAA Pauli matrices: 1 1 0 1 0 1 AA AA = √ =√ σ0 , σ1 2 0 1 2 1 0 1 1 1 0 0 i AA AA = √ = √ σ2 , σ3 . 2 −i 0 2 0 −1 and identify it the standard basis of the Minkowski space R1,3 . This way the with 0 1 2 3 1,3 element x = x , x , x , x ∈ R is identiﬁed with the element

xAA = xα σαAA . One can read more on spinors in [14] or [9].

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Since p ∧ u is antisymmetric, one can write it with help of spinors (see [2], [14]) as:

(A

B ) AB

(p ∧ u)αβ σαAA σβBB = pAA uBB − uAA pBB =z

(A

w

B) A B

+ z¯

w ¯

(3.5)

for some spinors z, w ∈ C2 , satisfying z (A wB) =

1 AB C (p ∧ u) C . 2

It is important to note, that the spinors z, w are given by p ∧ u up to the equivalence class (z, w) ∼ (λz, λ−1 w), for any nonzero λ ∈ C. One can check, using simple algebra, that 1 (A B) A B z w + z¯(A w ¯B ) AB z(A wB) A B + z¯(A w ¯B ) AB 2 2 s 2 = 4 zA w A = π

p ∧ u, p ∧ u = −

This way, one can set zA wA = ±s/2π using the orbit invariant s. The meaning of the plus-minus sign will be discussed shortly. It is also possible to show that one can express wA as wA = pAA z¯A . This, however, is not preserved within an equivalence class (z, w) ∼ (λz, λ−1 w). This relation restricts the equivalence (z, w) ∼ (λz, λ−1 w) to (z, w) ∼ (eiφ z, e−iφ w) for some real φ. Using the expression wA = pAA z¯A , one can see that for p = (±m, 0, 0, 0) one has to choose the sing in zA wA = ±s/2π to be the same as the sign of p0 . To conclude, one can write the orbit given by p2 = m2 and 2 p ∧ u, p ∧ u = (s/π) as

O = {(xα , pβ , z A ) | p2 = m2 , zA pAA z¯A = ±

s }/ ∼ 2π

where the equivalence relation ∼ is given as α β A α x , p , z ∼ x + tpα , pβ , eiφ z A and the plus-minus sign corresponds to diﬀerent connected components of the orbit. One has to still determine the coadjoint action of the Poincar´e group in these new coordinates.

3.1

Action of the Lorentz group on spinors

To determine the action of the Lorentz group on spinors, one has to investigate how the action changes the spinorial version pAA = pα σαAA of a vector pα from the

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Minkowski space. The determinant of pAA is unchanged under the action of the ¯ A pBB of pBB Lorentz group on pα , since det pAA = p2 . The multiplication LAB L B ¯ A = LA clearly preserves the determinant by matrices LAB ∈ Sl (2, C) and L B B ¯ A pBB is Hermitian. This way ±L correspond to some Lorentz of pAA and LAB L B transformation and one gets a two-to-one correspondence between Sl (2, C) and the Lorentz group. ¯ A Since the Lorentz group acts on the spinor version of p as p˜AA = LAB L B pBB and it can be shown that pAA = ±2π/s wA w ¯A + p2 /2 z A z¯A , one infers that the action of the Lorentz group on spinors is z A −→ LAB z B

¯ AB z¯B . z¯A −→ L The uncertainty which element of SL (2, C) to choose is absorbed in the equivalence class z ∼ eit z. From these relations one can get the transformation rules for the dual spinors: z˜A = LAB z B z˜A = z˜B BA = LBC z C BA = − AB LBC CD zD . Here is the symplectic form on the space of two spinors S and is used to lift the indices. Thus a spinor zA transforms as: B zA −→ − AD LDC CB zB = zB L−1 A .

3.2

Tangent vectors

The tangent vectors generated by the inﬁnitesimal action of ξ = (A, a) ∈ g on the space R1,3 × R1,3 × S are (3.6) Xξ (α) = Aαβ xβ + aα ∂xα + Aγδ pδ ∂pγ + AAB z B ∂zA + A¯AB z¯B ∂z¯A

β where α = (x, p, z), AAB = Aαβ σαAA σBA e and the action of the restricted Poincar´ group is given by ρ(Λ, l)α = (Λx + l, Λp, ±Lz)

with ±L being elements of Sl(2, C) corresponding to Λ. Similarly, for the space R1,3 × R1,3 × S¯∗ the tangent vectors corresponding to the action T −1 ¯ ρ(Λ, l)α = (Λx + l, Λp, ± L z¯) are given by Xξ (α) = Aαβ xβ + aα ∂xα + Aγδ pδ ∂pγ + A¯AB z¯B ∂z¯A + AA B zB ∂zA .

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Discrete symmetries

To determine the action of the full Poincar´e group on spinors, one must ﬁnd the parity P and time reversal T transformations. From the action of the parity transformation on the Minkowski space P : (u0 , u1 , u2 , u3 ) → (u0 , −u1 , −u2 , −u3 ) one can ﬁnd the following coordinate expression: AA

(P (u))

= u A A .

Similarly for the time reversal T : (u0 , u1 , u2 , u3 ) → − (u0 , −u1 , −u2 , −u3 ) the following expression holds:

(T (u))AA = −uA A . This suggests P : (x, p, z A ) → (P x, P p, ηP z¯A ) T : (x, p, z A ) → (T x, T p, ηT z¯A ) here ηs are some phase factors. To see that P and T really correspond to antilinear transformations on spinors, one can write the elements of the Lie algebra in a form   x2 x3 0 x1  x1 0 y3 y2   Aαβ =   x2 −y3 0 y1  x3 −y2 −y1 0 − − where (→ x ) and (→ y ) are some real vectors. From the relation

β AAB = Aαβ σαAA σBA

one obtains AAB

=

1 2 x1

+

1 2 x3 −iy3 y2 + i 12 x2

− y1

1 2 x1

− y2 − i 12 x2 + y1 . − 12 x3 + iy3

Moreover, the adjoint action GAG−1 of g on A ∈ g is given in the matrix form for the time reversal   −1 0 0 0  0 1 0 0   T =  0 0 1 0  0 0 0 1

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

α T AT −1 β

0  −x1 =  −x2 −x3

−x1 0 −y3 −y2

−x2 y3 0 −y1

499

α −x3 y2   y1  0 β

or T Xa T −1 = −Xa T Ya T −1 = Ya where Xa is the matrix corresponding to xa = 1 and all other elements zero. Ya denotes similar matrix with ya = 1 and all other elements zero. Since for the spinor equivalents Xa , Y a of Xa , Ya one gets the same relations T Xa T −1 = −Xa T Ya T −1 = Ya , R

and moreover Xa ∼ iYa , the time reversal acts on spinors as an antilinear operator. Since P T = −id, the parity is antilinear as well.

3.4

The orbit

To conclude the above discussion, the orbit corresponding to m and s can be represented as ±ms O = {(xα , pβ , z A ) | p2 = m2 , zA pAA z¯A = √ }/ ∼ 2

where the equivalence relation ∼ is z A ∼ eit z A , x ∼ x + t p

(3.7)

for some real t and t . The coadjoint action of the Poincar´e group on the orbit is given by the formulas: ρ(Λ, l) (x, p, z) = (Λx + l, Λp, ±Lz) ρ (P ) (x, p, z A ) = (P x, P p, ηP z¯A ) ρ (T ) (x, p, z A ) = (T x, T p, ηT z¯A ). We see that the factorization by the equivalence classes reduces the sign ambiguity in ±L corresponding to Λ as well as the phase factors ηP and ηT .

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Coordinates

One can choose local coordinate systems on the orbit O to be φ1 (x, p, z) = (x1 , x2 , x3 , p1 , p2 , p3 ,

z0 ) z1

on the open set U1 = {z1 = 0}, and φ0 (x, p, z) = (x1 , x2 , x3 , p1 , p2 , p3 ,

z1 ) z0

on the open set U0 = {z0 = 0}. It is convenient to introduce new coordinates on each of these coverings, namely c0 = zz10 on U0 and c1 = zz01 on U1 . On the overlap of these two sets the transition function φ0 φ−1 1 is 1 −1 φ0 φ1 (x1 , x2 , x3 , p1 , p2 , p3 , c1 ) = x1 , x2 , x3 , p1 , p2 , p3 , , c1 Similarly φ1 φ−1 0

1 (x1 , x2 , x3 , p1 , p2 , p3 , c0 ) = x1 , x2 , x3 , p1 , p2 , p3 , . c0

The last component transforms according to the transition function for a sphere in stereographic coordinates. Thus one can conclude that O = T P × S2, where T P is the tangent bundle of two-sheeted hyperboloid given by p2 = m2 , since one can choose a representant of the equivalence class x ∼ x + tp to be such x that px = 0.

3.6

Symplectic structure

One wants to ﬁnd the symplectic structure on the orbit O. To do so, the following theorem proved in [14] can be applied. Theorem 2 Let (O, ω) be a coadjoint orbit of G and W a manifold where G acts as g ξ → Xξ with a surjection π : W → O and θ is a one form on W such • π −1 (α) is connected for all α ∈ O • π∗ Xξ = Xξ • for all x ∈ W the following holds Xξ θ (x) = α (ξ) where α = π (x) ∈ O ⊂ g∗ then θ is invariant under action of G and dθ = π ∗ ω.

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Since one can considerthe surjection π : W → O to be the equivalence class 3.7 identiﬁcation in W = (x, p, z) ∈ R1,3 × R1,3 × C2 , one can use the above theorem provided that there is a 1-form θ on W such that θα (Xξ ) = α(ξ). Since α(ξ) = p ∧ x +∗ (p ∧ u), A − p, a 1 = − p, Ax + a − i(z (A wB) A B − z¯(A w ¯B ) AB )AABA B 2 i = −pα dxα (Xξ ) − (z (A wB) AABAA − z¯(A w ¯B ) AAAA B ) 2 = −pα dxα (Xξ ) + i(z A AAB wB − z¯A A¯A B w ¯B )

zB (Xξ )w ¯B ), = −pα dxα (Xξ ) + i(dzB (Xξ )wB − d¯ using the expression (3.6) for Xξ and the formula (see [14]) ∗ (p ∧ u) = i z (A wB) A B − z¯(A w ¯B ) AB . Then

θ = −ipAA (z A d¯ z A − z¯A dz A ) − pα dxα

(3.8)

deﬁnes a symplectic potential on O. The symplectic form on the orbit is ω = dθ. Note, that by this deﬁnition θ is well deﬁned as an element of T ∗ O, since there is a diﬀerence of the complex conjugate elements multiplied by i, or as an element of the complexiﬁed space TC∗ O.

3.7

Hamiltonian vector fields

The Hamiltonian vector ﬁeld is deﬁned as Xf ω = −df One can easily verify that the Hamiltonian ﬁeld for the function p2 is Xp2 = −2p∂x .

z ) one gets the Hamiltonian vecSimilarly for the function z A pAA z¯A (denoted zp¯ tor ﬁeld i Xzp¯z = − (z A ∂zA − z¯A ∂z¯A ) 2 θ (i.e., £ because the ﬂow of this ﬁeld is (e−it/2 z A , eit/2 z¯A ) and it preserves Xzpz¯ θ =

0). Thus Xzp¯z ω = £Xzpz¯ θ − d(Xzp¯z θ) = −d(Xzp¯z θ) = −d z A pAA z¯A . Again, even though the vector Xzp¯z formally belongs to the complexiﬁed tangent space, it is an element of the real tangent space, since it is a diﬀerence of the complex conjugate vectors multiplied by i. Note also, that these vector ﬁelds are well deﬁned as both elements of T O and T W .

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4 Quantization 4.1

Prequantization

The purpose of prequantization is to construct a faithful representation of the algebra of the smooth functions on a symplectic manifold (M, ω) with the bracket operation being the Poisson bracket given by the symplectic structure on M . To do this, one can consider a line bundle over M . To be able to diﬀerentiate the sections of this bundle, one must introduce a connection. If the curvature of the connection is equal to the symplectic form, the line bundle together with the connection is called the prequantum bundle. There is a necessary condition on the curvature to construct a line bundle with this connection. This condition is generally referred to as the integrality condition (see e.g., [7]): Condition 1 The integral over any closed oriented 2-surface in M of the curvature form is a natural number. In the case of the symplectic form given by the potential (3.8), the integrality condition implies that 2s ∈ N. From now on, only the trivial line bundle is considered. To deﬁne a scalar product on the space of the sections λ, one can set

(λ1 , λ2 )m ω n . (λ1 , λ2 ) = O

Here (λ1 , λ2 )m denotes the scalar product of λ1 (m) and λ2 (m) in the ﬁbre π −1 (m), where m ∈ O . This integral thus deﬁnes a scalar product and the space H of the square integrable sections. The trivializing section λ0 is deﬁned so that ∇λ0 = 2πiθ ⊗ λ0 . Note that here λ0 is deﬁned up to a constant multiplier. One can compute Hermitian structure (., .) as ¯ (λ0 , λ0 ) = 0 . d (λ0 , λ0 ) = (∇λ0 , λ0 ) + (λ0 , ∇λ0 ) = 2πi(θ − θ) Thus one can set (λ0 , λ0 ) = 1. The prequantized operator Pξ corresponding to the generator ξ ∈ g acts on a section λ = ψλ0 as Pξ (ψλ0 ) = ∇Xξ − 2πiJξ (ψλ0 ) = ∇Xξ − 2πiθ (Xξ ) (ψλ0 ) = (Xξ ψ + 2πiθ (Xξ ) ψ − 2πiθ (Xξ ) ψ) λ0 = (Xξ ψ) λ0 , where Jξ is the momentum map2 evaluated at ξ and satisﬁes Jξ = θ (Xξ ). 2 More

on momentum maps can be found in [1]

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The prequantized operator corresponding to the time reversal T (or any other discrete symmetry) can be computed as follows. An action of T on the trivializing section λ0 can be expressed as T λ0 = f λ0 ◦ T where f is to be determined. This way one can write T ∗ ∇λ0 = ∇T λ0 = (df + 2πif θ) ⊗ λ0 ◦ T or equivalently T ∗ ∇λ0 = T ∗ (2πiθ ⊗ λ0 ) = 2πif T ∗θ ⊗ λ0 ◦ T. Comparing the right sides of the last expressions and since T ∗ θ = θ, one infers that df = 0, or f = const. The action of T on an arbitrary section is ˜ λ0 ◦ T T (ψλ0 ) = ψf where ψ˜ corresponds to the transformed ψ. Using the fact that T commutes with the connection, one can that ˜ λ0 ◦ T ) = f dψ˜ ⊗ λ0 ◦ T + 2πif ψθ ˜ ⊗ λ0 ◦ T . T (∇ψλ0 ) = ∇(ψf The above can be expressed also as ˜ ∗ θ ⊗ f λ0 ◦ T. T (∇ψλ0 ) = T (dψ ⊗ λ0 + 2πiψθ ⊗ λ0 ) = (T ∗ dψ) ⊗ f λ0 ◦ T + 2πiψT ˜ or From this follows that T ∗ dψ ≡ dT ∗ ψ = dψ, T ψ = ψ˜ = T ∗ ψ + const. = ψ ◦ T + const. If one requires that T preserve the norm, one gets Tψ = ψ ◦ T and f = 1. By similar reasoning, one can write that P ψ = ψ ◦ P.

4.2

Polarization

To quantize the prequantum line bundle, one has to introduce a polarization and then reduce the prequantum bundle to the sections that are constant along the introduced polarization. The spinorial part of the orbit is diﬀeomorphic to a sphere. On a sphere, however, all the polarizations have nonzero real and imaginary components. The parity P is an antilinear map, and thus does not preserve these

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polarizations. There is no invariant polarization under the full Poincar´e group on the orbit O. There is no problem in studying only the irreducible representations of the restricted Poincar´e group, i.e., the connected component of the Poincar´e group, since the polarization is invariant on the connected component O0 of the orbit O, since O0 is the orbit of the restricted Poincar´e group. One can choose to work with the component corresponding to p0 > 0. To recall, this orbit is given by s O0 = (x, p, z) | p2 = m2 , zp¯ / x ∼ x + tp, z ∼ eit z . z= 2π The condition p0 > 0 is implicitly included in the constraint zp¯ z = s/2π. The line bundle over this orbit can be constructed as a line bundle over one connected component of the constraint manifold s C = x, p, z A | p2 = m2 , zp¯ z= 2π with the following restriction on the sections ∇Xp2 λ = 0

(4.1)

∇Xzpz¯ λ = 0 where λ is a section over C and Xf is a restriction of the Hamiltonian vector ﬁeld corresponding to the function f in M to C. The ﬂow of X p2 is (x, p, z) → (x − 2tp, p, z) and the ﬂow of Xzp¯z is (x, p, z) → x, p, exp − 2i t z . These ﬂows are considered as ﬂows of restrictions of the Hamiltonian vector ﬁelds on W to C. Here one replaces the factorization of the constraint space C by the equivalence relations with the requirement of the sections to be constant along the ﬂows given by the factorizing equivalence relations. One can make this construction, since the ﬂow of Xp2 is not closed and the parallel transport along the ﬁeld Xzp¯z changes the phase by factor exp 2π θ , what, by the quantization condition, is equal to exp (4πs) = 1. The equivalence (4.1) can be written as p∂x ψ = 2πip2 ψ = 2πim2 ψ (z∂z − z¯∂z¯) ψ = 4πzp¯ zψ = ±2sψ, where λ = ψλ0 . The problem with this construction is that C together with ω extended to it is a presymplectic manifold and thus one cannot use the quantization theory on this space. There is, however, a way around this problem by considering commutation of reduction and quantization (see [6] and the references therein) and taking the space W . One can construct a line bundle over this space and restrict the sections of this line bundle to sections satisfying ∇Xp2 λ = 0 ∇Xzpz¯ λ = 0

(4.2)

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as before, plus condition that restricts the support of the sections to the constraint manifold C Pp2 −m2 λ = 0 Pzp¯z −s/2π λ = 0. Here P s are the prequantized operators corresponding to the indicated functions. To see how these operators restrict the support of the sections, one expresses the formulas explicitly as Pp2 −m2 λ = ∇Xp2 λ − 2πi p2 − m2 λ = 0 s λ=0 z− Pzp¯z −s/2π λ = ∇Xzpz¯ λ − 2πi zp¯ 2π which with (4.2) yields that the support of a section is given by equations p2 = m2 and zp¯ z = s/2π. The distribution given by {∂c¯}C = {w∂ ¯ z¯}C is well deﬁned (∂c¯ is a well-deﬁned complex polarization on a sphere) and is invariant under restricted Poincar´e group since it is a restriction of {∂z¯}C to C , [∂z¯, Xξ ] ∈ ∂z¯ and C is an invariant manifold. This way one can deﬁne a polarization on the orbit of the restricted Poincar´e group corresponding to m and s and denoted by O0 as ¯ z¯}C FO0 = {u∂x , w∂ where u is such, that u∂x ∈ T O0 . It is important that FO0 is restriction of F = {∂x , ∂z¯}C ⊂ T R1,3 × S = W to the orbit O0 . This way, one can represent the line bundle over the orbit O0 as the line bundle over T R1,3 × S such that λ| ∇Xp2 λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0 and deﬁne the polarization to be F = {∂x , ∂z¯}C . 4.2.1 Quantization One takes the prequantum space H = λ |∇Xp2 λ = 0 , ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0 and restricts it to the sections that are covariantly constant along the polarization. The restriction of H to the sections covariantly constant along the polarization

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F = {∂x , ∂z¯}C will be denoted Hp . Since the polarization F contains vector Xp2 , one can write the quantized space as Hp = λ |∇∂x λ = 0, ∇∂z¯ λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0 . For the polarization F = {∂x , ∂z¯}C one writes the quantization restriction on sections as 0 = ∇F λ = (2πiθ(F ) + F )ψ ⊗ λ0 or ∂z¯ψ = 0 ∂x ψ = 2πipψ After setting ∇Xzpz¯ λ = 0, one obtains (z∂z − z¯∂z¯) ψ = 4πzp¯ zψ . √ This can be written with help of the constraint zp¯ z − ms/ 2 λ = 0 on the support of λ and the requirement of covariant constancy along the polarization as z∂z ψ = 2sψ . From the integrality condition 2.1 follows that 2s ∈ N. Thus, after taking into account that the sections must be constant also along ∂x , one can express the elements of Hp as ψ = e2πip,x z A1 z A2 . . . z A2s ξA1 A2 ...A2s (p). One wants to determine the action of the quantized operator Uξt corresponding to the one parameter subgroups generated by ξ ∈ g.3 −1 Uξt λ = φ# ◦ λ ◦ φt t where φt is the ﬂow of Xξ and φ# t is the ﬂow of the lifted vector ﬁeld Zξ = d 2πiJm (ξ) λ (m)t=0 of Xξ to the line bundle. For a general section ψλ0 one gets dt e −1 Uξt λ = φ# ◦ λ ◦ φt = φ# t −t ((ψ ◦ φt ) λ0 ◦ φt ) = (ψ ◦ φt ) φ# −t ◦ λ0 ◦ φt . The ﬂow of Zξ can be split into the horizontal (parallel) and vertical parts. The parallel section ψλ0 along a path φt is given by4 ψ (c (t)) = ψ0 e 3 More

details can be found in [12]. = ∇X λ = (Xψ + 2πiθ (X) ψ) λ0 Xψ = −2πiθ (X) ψ 40

ψ=e

−2πi φ θ(X)dt t ψ0

−2πi

φt

θ

.

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The vertical part of φ# t can be computed as z → e

2πi

and hence Uξt λ = (ψ ◦ φt ) e

2πi

φt

φt

Jξ dt

(Jξ −θ(Xξ ))dt

λ0

where one integrates along the ﬂow φt of the vector ﬁeld Xξ . This formula can be written also for the operator corresponding to a function f as Uft λ = (ψ ◦ φt ) e

2πi

φt

(f −θ(Xf ))dt

λ0

(4.3)

where φt is in this case the ﬂow of the vector ﬁeld Xf . Since, from the construction of θ, one has that θ (Xξ ) (α) = α (ξ) and Jξ (α) = α (ξ), one obtains Uξt λ = (ψ ◦ φt ) λ0 . Note that this cannot be generalized to the case of an arbitrary function f , since, in general, f = θ (Xf ). Note also that considering only sections covariantly constant along the polarization F , one looses the scalar product. One can ﬁx this problem by using so-called half-densities; here they are, however, not treated and the construction of the scalar product is omitted. 4.2.2 Free Dirac equation In the following only the case when s = 1/2 is treated. The action of the discrete symmetries does not preserve the polarization (it is impossible to ﬁnd a polarization that would be preserved by this action since T and P act as antilinear operators on the spinorial part of the orbit and the only polarization on a sphere is given by ∂z or ∂z¯). To allow the discrete symmetries, one can extend the Hilbert space in such a way, that the discrete symmetries preserve the polarization. This can be done by inducing the representation of the restricted Poincar´e group obtained in the previous section to the representation of the full Poincar´e group. There is a well-deﬁned action of the discrete symmetries on the sections of the representation space Hp which can be considered as a subspace of H. Thus, one can construct three other spaces corresponding to all discrete symmetries HpP = P (Hp ) HpT = T (Hp ) HpT P = T P (Hp ) . The explicit realization of the spaces corresponding to the transformed space Hp = λ |∇∂x λ = 0, ∇∂z¯ λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0

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by the discrete symmetries is as follows: HpT P = λ |∇∂x λ = 0, ∇∂z¯ λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z +s/2π λ = 0 HpT = λ |∇∂x λ = 0, ∇∂z λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z +s/2π λ = 0 HpP = λ |∇∂x λ = 0, ∇∂z λ = 0, ∇Xzpz¯ λ = 0, Pp2 −m2 λ = 0, Pzp¯z −s/2π λ = 0 . To see why this is so, one can recall the action of the discrete symmetries P : (x, p, z A ) → (P x, P p, ηP z¯A ) T : (x, p, z A ) → (T x, T p, ηT z¯A ) and thus the transformations P and T must map the polarization to its conjugate and the transformations T and T P transform the connected component of p2 = m2 with p0 > 0 to the connected component with p0 < 0. The typical elements of these spaces are ψ T P = e2πip,x z A ξA (p) with the support on p2 = m2 , p0 < 0, zp¯ z = s/2π

ψ T = e2πip,x z¯A η A (p) z = −s/2π and with the support on p2 = m2 , p0 < 0, zp¯

ψ P = e2πip,x z¯A η A (p) z = −s/2π. This way, one can represent with the support on p2 = m2 , p0 > 0, zp¯ all the wave functions as one multi-valued function form the space Hp = Hp ⊕ HpT ⊕ HpP ⊕ HpT P : A z ξA (p) 2πip,x ψ=e z¯A η A (p) with the support on the whole p2 = m2 and zp¯ z = s/2π for the ﬁrst component t and zp¯ z = −s/2π for the second component. The action of the operators U(A,a) generated by the inﬁnitesimal generators (A, a) is given on each component separately. To determine U one follows the same procedure as in the case of the space HP to ﬁnd the action on the ﬁrst component as t U(A,a) ψ + = e2πie

At

e zξ eAt p .

p,eAt x+at At

This operator acts similarly on the other component as well. After recalling the action of the Lorentz group on dual conjugate spinors one can summarize:

AT t At ξ e ξ A A t U(A,a) e p . e2πip,x (p) = e2πip,a ¯ A −At ηA e η

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This representation is however not irreducible. Since this construction should correspond to a single spinning particle, one would like to obtain an irreducible representation, as discussed in [8]. There is an operator which commutes with all the t and all the discrete symmetries. This operator can be written as U(A,a) Dψ (x, p, z) = e

2πip,x

z A pAA η A (p) ξA (p) pAA z¯A

,

or in the matrix form acting only on the spinor part of the function, given by ξA vector : ηA 0 pAA T A A D= . p 0 To show that this operator commutes with both U and the discrete symmetries, one has to determine explicitly the action of the discrete symmetries: z¯A η A (T p) T ψ (x, p, z) = e2πip,x z A ξA (T p) since T acts as a pull-back and maps z A to z¯A . Similarly z¯A η A (P p) 2πip,x P ψ (x, p, z) = e . z A ξA (P p) One can easily show that z¯A η A (T p) 2πip,x DT ψ (x, p, z) = D e z A ξA (T p) AA (T p) ξ (T p) z ¯ A A = e2πip,x z A (T p)AA η A (T p) A z pAA η A (p) 2πip,x T Dψ (x, p, z) = T e ξA (p) pAA z¯A AA ξA (T p) z¯A (T p) 2πip,x =e . z A (T p)AA η A (T p) Thus DT = T D. Similarly DP = P D. To show that D commutes with all U ’, it is enough to prove that D preserves the ﬂow of Xξ . Since D acts as a pull-back in the spinorial part of the coordinates D S z A ←→ z A pAA ∈ S¯∗

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one has to check the action of D only on the spinor part of the wave function A A D eA B t z B = eA B t z B eAt p AA A C ¯C = eA B t z B pCC e−A A t e−A A t ¯C

= z C pCC e−A A t ¯C

= (Dz)C e−A A t = φtξ (Dz) . Similarly, the dual-conjugate part of the ﬂow is preserved under the action of the operator D. To construct an irreducible representation of the Poincar´e group, one must restrict the investigation to a eigenspaces of the operator D. Only these subspaces have a chance to form an irreducible representation. The operator D has an eigenvalue ±m, since 0 pAA T A A det − λI = λ4 − p4 = λ4 − m4 . 0 p The eigenvector equation Dψ = ±mψ can be rewritten as √ 2pα

0 σ αAA

σ αAA 0

ξA ηA

ξA = ±m A η

.

To write the above in the matrix multiplication notation, one can transpose the appropriate block as A A √ 0 σ αAA η η 2pα = ±m . σ ¯ αA A 0 ξA ξA Denoting γα =

√ 2

0

σ ¯ αA A

σ αAA , 0

one can write the equation Dψ = ±mψ as (pα γ α ∓ m) ψ = 0. One can show, that the matrices γ α satisfy the relation γ α γ β + γ β γ α = 2g αβ ,

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writing

0

The Dirac Equation in Geometric Quantization

σ αAA 0

0

σ αAA + σ ¯ αA A 0 σ ¯ βA A 0 αβ 1 0 = 2g 0 1 σ ¯ αA A σ βAA + σ ¯ βA A σ αAA

σ βAA 0

σ ¯ αA A σ ¯ βA A α σ AA σ ¯ βA A + σ βAA σ ¯ αA A = 0

511

0

σ βAA 0

0

using the formula σ αAA σ ¯ βA A + σ βAA σ ¯ αA A = 2g αβ . The above relations are the anticommutation relations for the Dirac matrices. This way one obtains the Weyl representation of the Dirac spinors [11].

References [1] R. Abraham and J.E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publishing Company, second revised, enlarged and reset edition (1978). [2] H. Bacry, Lectures on Group Theory and Particle Theory, Gordon and Breach Science Publishers (1977). [3] A.O. Barut and R. R¸aczka, Theory of Group Representations and Applications, PWN – Polish Scientiﬁc Publishers, Warszawa, second revised edition (1980). [4] R.J. Blattner, Quantization and representation theory, In Proc. Sympos. Pure Math., volume 26, pages 145–165. Amer. Math. Soc., R.I. (1974). [5] P. Dirac, The quantum theory of the electron, Proc. Roy. Soc. 117, 610–624 (1928). ´ [6] C. Duval, J. Elhadad, M. J. Gotay, J. Sniatycki, and G. M. Tuynman, Quantization and bosonic BRST theory, Annals of Physics 206, (1), 1–26 (1991). [7] B. Kostant, Quantization and uintary representations, Lecture notes in mathematics 170 (1970). [8] G.W. Mackey, Induced Representaitons of Groups and Quantum Mechanics, Bejamin (1968). [9] R. Penrose and W. Rindler, Spinors and Space-Time, volume 1. Cambridge University Press (1987). [10] P. Renouard, Vari´et´es Symplectiques et Quantification, PhD thesis, Universit´e de Paris (1969). [11] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row, Peterson and Co. (1961).

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´ [12] J. Sniatycki, Geometric Quantization and Quantum Mechanics, SpringerVerlag (1980). [13] J.-M. Souriau, Structure des Syst`emes Dynamiques, Dunod, Paris (1970). [14] N.M.J. Woodhouse, Geometric Quantization, Clarendon Press, Oxford, second edition (1992).

Andrej B´ ona Department of Earth Sciences Memorial University St. John’s, NL A1C 5S7, Canada email: [email protected] Communicated by Rafael D. Benguria submitted 02/01/02, accepted 14/02/02

Ann. Henri Poincar´e 4 (2003) 513 – 552 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030513-40 DOI 10.1007/s00023-003-0138-4

Annales Henri Poincar´ e

A Landau-Zener Formula for Non-Degenerated Involutive Codimension 3 Crossings Clotilde Fermanian Kammerer and Patrick G´erard Abstract. In this paper, we study 2 × 2 systems of semi-classical pseudo-diﬀerential evolution equations which display eigenvalue crossing. We assume that the classical trajectories which hit the crossing set form the union of two involutive manifolds, with some non-degeneracy hypothesis, which covers the case of the Schr¨ odinger equation with a matrix potential displaying a generic codimension 3 crossing. We derive an explicit formula for the energy transfer induced by the crossing, using two-scale Wigner measures. The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau-Zener formula. R´esum´e. Cet article est consacr´e ` a l’´ etude d’un syst` eme de deux ´equations d’´evolution pseudo-diﬀ´erentielles semi-classiques pr´esentant un croisement de modes. Nous supposons que l’ensemble des trajectoires classiques qui rencontrent le croisement est l’union de deux sous-vari´et´ es involutives v´eriﬁant une hypoth`ese de non-d´eg´ en´ erescence; cette situation contient le cas de l’´equation de Schr¨ odinger avec un potentiel matriciel pr´esentant un croisement g´en´ erique de codimension 3. Nous obtenons une formule explicite d´ecrivant le transfert d’´energie au-dessus du croisement, ` a l’aide des mesures de Wigner ` a deux ´ echelles. La d´emonstration repose sur un th´eor` eme de forme normale qui permet de se ramener ` a une formule de Landau-Zener ` a valeurs op´erateurs.

1 Introduction The problems induced by eigenvalue crossings occur in diﬀerent areas of Mathematical Physics. A large class of examples arises in the semiclassical approximation of equations of Quantum Mechanics: in the Born-Oppenheimer approximation, where these crossings have been classiﬁed by G. Hagedorn (see [22]); or in the study of the electronic motion in a crystal (see [8], [18], [33] and [4]). Similar diﬃculties also arise in high frequency wave propagation in periodic media ([3], [20]). In this paper, we focus on crossings for 2 × 2 hermitian matrices. However it is likely that our methods can be applied to the other cases of crossings classiﬁed in [22]. We consider the evolution equations oph (H0 )ψ h = o(h),

(1)

where h is a small parameter and oph (p) denotes the semi-classical pseudodiﬀerential operator of Weyl symbol p, i.e., the operator of kernel y + y dξ , hξ eiξ·(y−y ) p , (y, y ) ∈ Rd+1 , K(y, y ) = d+1 d+1 2 (2π) R

514

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and where H0 is the Hamiltonian p1 (t, x, ξ) H0 (t, x, τ, ξ) = τ + k(t, x, ξ) + p2 (t, x, ξ) − ip3 (t, x, ξ)

Ann. Henri Poincar´e

p2 (t, x, ξ) + ip3 (t, x, ξ) −p1 (t, x, ξ)

,

where k ∈ C ∞ (R2d+1 , R), p ∈ C ∞ (R2d+1 , R3 ). The eigenvalues of H0 , λ± = τ + k ± | p |, are distinct outside {p = 0}. In this situation, it is natural to expect that the structure of the set {p = 0} will play a major role. We shall say that the crossing is of codimension l if the rank of the diﬀerential dp is l on the set {p = 0}. Of typical interest are the following cases : l = 3 (generic crossings for 2 × 2 hermitian matrices) and p3 = 0, l = 2 (generic crossings for 2 × 2 real symmetric matrices). In this context, previous works have been devoted to the semiclassical evolution through crossings in particular examples. The earlier works are those of Landau and Zener (see [30] and [37]), then come the articles of G. Hagedorn and A. Joye (see [21], [22], [28], [23] and [24]) for linear functions p and special initial data. In the case of stationary systems in one space dimension, a complete analysis of the solutions has been performed in [11]. In view of these results, some energy transfer between the two modes is expected to happen above the crossing. For codimension two crossings in two space dimensions, this transfer was calculated by the authors in [15], for any bounded data in L2 but for the special Hamiltonian p1 = ξ1 , p2 = ξ2 , p3 = 0, k = V (x). This latter result has been recently generalized to codimension 2 symmetric generic crossings in [16] by means of some normal form method. Similar normal forms have been obtained recently independently by Y. Colin de Verdi`ere in [9] (see the remark at the end of Section 1.4.). It should also be mentioned that formal normal forms have already been derived by P.P. Braam and J.J. Duistermaat for symmetric crossings in [5] (see also the references therein). In this paper, under some additional assumptions which will be speciﬁed below, we shall deal with the case of codimension 3 crossings, namely we suppose that dp is of rank 3 on {p = 0}. (H1) As in [15], [16], the energy transfer will be described in terms of semi-classical measures, of which we now brieﬂy recall the deﬁnition. Given a uniformly bounded family (ψ h ) in L2 (Rd+1 , C2 ), one can associate some positive Radon measures, called semi-classical measures or Wigner measures of this family and satisfying 0 and a 2 × 2 matrix-valued positive Radon such that for all a ∈ C0∞ R2d+2 , M2 (R) ,

There exists a sequence hk ∗

measure µ on T R

d+1

−→

k→+∞

ophk (a)ψ hk | ψ hk −→ tr (a , µ) . k→+∞

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Moreover, if (ψ h ) satisﬁes an assumption of h-oscillation as in [19], we have hk 2 tr(µ)(t, x, dτ, dξ), w − lim| ψ (t, x) | dt dx = Rd+1 τ,ξ

k→+∞

which explains its interest for studying the evolution of the energy density. Assume now that ψ h satisﬁes equation (1). Since outside {p = 0} the eigenvalues of H0 are of constant multiplicity 1, the evolution of µ outside {p = 0} is well known (see [20]). Let us denote by Π± the spectral projectors associated with the eigenvalues λ± , then µ decomposes as µ = µ+ Π+ + µ− Π− , where µ± are scalar Radon measures satisfying the localization property Supp(µ± ) ⊂ {λ± = 0}. Moreover µ± propagates along the Hamiltonian curves associated with both eigenvalues λ± , {λ± , µ± } = 0, outside S := {p = 0, τ + k = 0}, where {f, g} denotes the Poisson bracket {f, g} := ∇(τ,ξ) f · ∇(t,x) g − ∇(t,x) f · ∇(τ,ξ) g := Hf (g) = σ(Hf , Hg ), and where Hf is the Hamiltonian vector ﬁeld associated with f . The problems arise as classical trajectories reach S. This is precisely the problem treated in this article. In the rest of the introduction, we discuss the existence of these curves and their geometrical features, which allow to precise the assumptions. Then, we state our main result, a Landau-Zener formula for some two-scale Wigner measures, the deﬁnition of which we recall below. Finally, we shortly discuss the proof of this theorem.

1.1

The geometry of the crossing, assumptions

Of course, the ﬁrst important result consists of the existence of Hamiltonian curves ∗ d+1 of λ± which go through S, i.e., of functions ρ± , satisfying the s valued in T R diﬀerential system ± ± (2) ρ˙ ± s = Hλ± (ρs ), (ρs )|s=0 = ρ0 ∈ S. More precisely, we focus on Hamiltonian curves which are transverse to S, in the sense that lim± dp(ρ± s ) = 0. s→0

If we introduce the following three-vectors, E = {τ + k, p}, B = ({p3 , p2 }, {p1 , p3 }, {p2, p1 }) ,

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it is not diﬃcult to see that such curves may exist only if one of the following two conditions holds E · B = 0 and | E |2 >| B |2 , E · B = 0.

(H2) (H2 )

Proposition 1 If ρ0 is such that (H2) or (H2 ) holds in ρ0 , then there exist two ± ∓ curves ρ± s solution to (2). Moreover (ρs )s≤0 can be smoothly continued by (ρs )s≥0 . We shall say that the crossing is non degenerated if there exists some neighborhood V of ρ0 in S such that ((H1) and (H2)) or ((H1) and (H2 )) hold in V. We ±,out the union deﬁne J ±,in as the union of the curves (ρ± s )s<0 which reach V, and J ± of the curves (ρs )s>0 issued from V. Because of Proposition 1, J = J +,in ∪ J −,out on one hand and J = J −,in ∪ J +,out on the other hand are smooth submanifolds of T ∗ Rd . Proposition 2 If (H1) and (H2) hold on Ω, then J and J are codimension 3 involutive submanifolds of T ∗ Rd+1 t,x for the symplectic form σ = dτ ∧ dt + dξ ∧ dx. At this stage, we take up with a speciﬁc feature of codimension 3 crossings. Actually, for codimension 2 crossings (p3 = 0) with some opened condition ensuring the existence of the Hamiltonian curves, the manifolds J and J are always involutive. It is not the case in codimension 3 since if (H1) and (H2 ) hold, S is symplectic and thus J and J cannot be involutive. In this article, we focus on the case of codimension 3 non-degenerated crossings of involutive type, namely, we suppose that (H1) and (H2) hold in a neighborhood of some ρ0 in S. Observe that Hagedorn’s codimension 3 crossing of type B (see page 19 in [22]) corresponds to the case d = 3, k(t, x, ξ) =

1 | ξ |2 , p(t, x, ξ) = p(x), 2

for which (H2) holds for ξ = 0. Therefore signiﬁcant situations are covered by our result. Besides, for codimension 2 generic crossings, J and J are involutive submanifolds and the proof below applies, with slight modiﬁcations, to the results announced in [16]. On the other hand, the symplectic case (H1)–(H2 ) covers, for example, situation where k = V (t, x) and p = ξ − A(t, x), with E · B = 0, as in [14] where the case of constant electromagnetic vector ﬁelds is studied. In this setting, E and B respectively are the electric and the magnetic ﬁeld, which explains the choice of the notations. It happens that a second level of observation is required to describe the repartition of the incident energy on each of the outgoing trajectories. In order to x2 ) for which √ we explain that fact, consider the simple case k = 0 and √ p = (t, x1 , √ have B = 0 and |E| = 1 = 0. The scaling s = t/ h, z1 = x1 / h, z2 = x2 / h

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eliminates the parameter h of the equations and leads to a scattering problem for the system of ordinary diﬀerential equations s z1 + iz2 i∂s u = u, (3) z1 − iz2 −s where u(s, z1 , z2 , x3 ) = ψ h (t, x). The scattering matrix of (3) can be exactly computed in terms of z1 and z2 (see [30], [37], [28] or [15]). It is easy to see that, in this context, J ∪ J = {x1 = x2 = 0} ∩ Σ with Σ = {τ 2 = t2 + x21 + x22 }. h Thus √ the transfer of energy depends on the way the family (ψ √ ) concentrates at the scale h on J and J . The role of this characteristic scale h already appeared in the coherent state approach of G. Hagedorn ([22]) and for avoided crossings ([21], [23], [24] and [11]). In the semi-classical approach which is ours, its role has been emphasized for special cases: in [15] for d = 2, k = V (x), p = ξ and in [14] for d = 3, k = E · x, p = ξ − 12 B × x where E and B are constant vectors of R3 with E · B = 0. Because of the geometric feature of J and J , in the involutive case, the concentration of (ψ h ) on them can be described by the two-scale Wigner measures introduced by L. Miller in [32] and developed by the authors in [15]. In the symplectic case, operator-valued two-scale semi-classical measures as in [14] are expected to be signiﬁcant.

1.2

Two-scale Wigner measures for involutive submanifolds

Let us consider for a while a more general setting and let us recall some results of [15]. Let I be a codimension m involutive submanifold of the cotangent space T ∗ RD . We suppose that I is given by some system of equations f = 0 where f ∈ C ∞ (R2D , Rm ), Rank(df ) = m on f = 0. Let us denote by Rm the closed ball obtained by adding a sphere at inﬁnity to Rm . We consider the set A of symbols a = a(z, ζ, η) ∈ C ∞ (RD × RD × Rm ) which are uniformly compactly supported in the variables (z, ζ) with respect to η and m which can be extended as a function of C ∞ (RD × RD × R ) by a(z, ζ, ω ∞) = lim a(z, ζ, R ω), ∀ω ∈ Sm−1 . R→+∞

With any a ∈ A, we associate the two-scaled pseudo-diﬀerential operator, f (z, ζ) opIh (a) =: oph a(z, ζ, √ ) . h By the Calderon-Vaillancourt theorem, the family of operators opIh (a) is a bounded family of bounded operators in L2 (Rd ). If (φh ) is a bounded family in L2 (Rd , CN ),

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√ N ∈ N, its concentration on I at the scale h is characterized two-microlocally by a positive Radon measure on N (I), the compact normal bundle to I, which describes the evolution as h goes to 0 of Kh (a) := opIh (a)φh | φh . Let us explain what is the bundle N (I). We associate with I its tangent bundle T I. By taking the quotient of the tangent space T (Rd) |ρ above some point ρ of I by T I |ρ , we obtain the ﬁber above ρ of N (I), the normal bundle to I. Then N (I)|ρ is the closed m-dimensional ball obtained by adding a sphere at inﬁnity to N (I) |ρ . The choice of the equation f induces local coordinates on N (I)|ρ given by the continuation χ of the isomorphism χ, χ : [δρ] ∈ N (I)|ρ → η = df (ρ)δρ ∈ Rm . If ν is a measure on N (I), we denote by νf the measure on Rm which is the image of ν by χ. Let us come back to the limit of Kh (a). There exists a sequence hk such that for all a ∈ A, Khk (a) −→ tr k→+∞

R

−→ 0, a positive Radon measure ν on N (I)

k→+∞

m

a dνf

f (x, ξ) ∞ dµ , a x, ξ, | f (x, ξ) | f =0

+ tr

where µ is a semi-classical measure of (φh ). We point out that ν determines µ above I by µ1I = ν(x, ξ, dη). I

Our aim in the following is to calculate two-scale Wigner measures associated with a family (ψ h ) solution to (1), and with the involutive submanifolds J and J .

1.3

The Landau-Zener formula

We denote by ν (resp. ν ) the measures associated with (ψ h ) and J (resp. J ). We denote by Σ+ , Σ− , J˙±,in and J˙±,out the sets Σ± = {λ± = 0}, J˙±,in = J ±,in \S, J˙±,out = J ±,out \S. Let us denote by N Σ± (J˙±,in ) and N Σ± (J˙±,out ) the sub-bundles of N (J˙±,in ) and N (J˙±,out ) obtained respectively by adding a sphere at inﬁnity to the ﬁbers of T Σ± /T (J˙ ±,in ) and of T Σ± /T (J˙ ±,out ). Because of the localization properties of semi-classical measures, there exist scalar positive Radon measures ν ±,in and ν ±,out supported in N Σ± (J˙±,in ) and N Σ± (J˙±,out ) respectively such that ν = ν +,in Π+ + ν −,out Π− , ν = ν −,in Π− + ν +,out Π+ .

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Moreover, the measures ν ±,in and ν ±,out are invariant by the linearized Hamiltonian ﬂow of λ± transversally to J ±,in and J ±,out respectively, in Σ± . These propagation properties have been precisely studied in local coordinates in [15] and the reader can refer to this article for more details. Because the Hamiltonian ﬂows are transverse to S, these measures have traces in the sense of distributions on S, which we denote by νS±,in and νS±,out . These four traces can be identiﬁed to measures on one set in which we can study the existing link between (νS+,out , νS−,out ) and (νS+,in , νS−,in ). Let us describe more precisely this identiﬁcation. First, we determine the limits of the ﬁbers of N Σ± (J˙±,in ) and N Σ± (J˙±,out ) above some point ρ which tends to S. Denote by H and H the limits of the two Hamiltonian vector ﬁelds along J and J above S. We have − lim Hλ+ (ρ+ s ) = lim+ Hλ− (ρs ) = H,

s→0−

s→0

− lim+ Hλ+ (ρ+ s ) = lim− Hλ− (ρs ) = H .

s→0

s→0

Let H ⊥ and (H )⊥ be the orthogonal subspaces of H and H respectively for the symplectic form on T (T ∗ Rd+1 ). Then νS+,in and νS−,out are measures on the compactiﬁcation of H ⊥ /T J, and similarly, νS−,in and νS+,out live on the compactiﬁcation of (H )⊥ /T J . Note that T J = T S ⊕ RH and T J = T S ⊕ RH , therefore if we set F = T J + T J = T S ⊕ RH ⊕ RH , both planes H ⊥ /T J and (H )⊥ /T J can be identiﬁed to T (T ∗Rd+1 )/F . Since F ⊥ = {W =

(E ∧ B) · X Hτ +k + X · Hp , X · E = 0, X ∈ R3 }, | E |2

the sets F ⊥ and the normal plane to E for the Euclidean metric in R3 are isomorphic in a natural way. Once given an orthonormal basis (X1 , X2 ) of this plane, we obtain a basis (W1 , W2 ) of F ⊥ ; the map φ : T (T ∗(Rd+1 )) → R2 [δρ] → (σ(W1 , δρ), σ(W2 , δρ)) then deﬁnes a system of coordinates on T (T ∗(Rd+1 ))/F . We choose some orthonormal basis (X1 , X2 ) of the normal plane to E for the Euclidean structure. More precisely, if B = 0, we choose X1 =

E∧B E , X2 = ∧ X1 . | E || B | |E|

Having in mind this identiﬁcation, the connection between νS±,in and νS±,out is described by the following theorem.

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Theorem 1 If νS+,in and νS−,in are mutually singular, then +,out +,in νS νS 1−T T = , T 1−T νS−,out νS−,in with

T = exp −π

| E |2 (| E |2 − | B |2 )

3/2

(4)

| B |2 2 σ(W1 , δρ) + 1 − σ(W2 , δρ) . | E |2 2

Remark. One can give a slightly more elementary statement of the previous theorem, which does not involve the symplectic form σ. The map Y : δρ → dp δρ ∧ E + d(τ + k)δρ B induces an isomorphism between T (T ∗Rd+1 )/F and the plane normal to E for the Euclidean structure of R3 . In these coordinates,

2 π E∧B 2 T (Y ) = exp − |Y | − Y · . |E|2 (|E|2 − |B|2 )3/2 Remark. If B = 0, which is the case for Hagedorn’s crossings, we have π π | Y |2 σ(W1 , δρ)2 + σ(W2 , δρ)2 T = exp − . = exp − |E| | E |3 Observe that in this case, T does not depend on the choice of the orthonormal basis (X1 , X2 ). Remark. At this stage, it should be observed that the evolution structure of the Hamiltonian H0 was not really used in the above analysis. In fact, one can check easily that our method would apply more generally to the Hamiltonian p1 (y, η) p2 (y, η) + ip3 (y, η) , H0 (y, η) = p0 (y, η) + −p1 (y, η) p2 (y, η) − ip3 (y, η) with the assumption that (dpj )0≤j≤3 are independent on {p0 = p1 = p2 = p3 = 0}. One can deﬁne similarly E = {p0 , p} and B = ({p3 , p2 }, {p1 , p3 }, {p2 , p1 }) and argue as above.

1.4

A reduction theorem

The crucial point of the proof consists of a reduction to a model problem by means of a canonical transform in all variables (t, x, τ, ξ). If κ is a canonical transform of T ∗ Rd+1 , there exist semi-classical Fourier integral operators U = Uh which are associated with κ in the following sense: ∀f ∈ L2 (Rd+1 ), ∀a ∈ C0∞ (R2d+2 ), U ∗ oph (a)U f = oph (a ◦ κ)f + O(h2 ) ||f ||L2 , in L2 (Rd+1 ).

(5)

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The reader can refer to [35] for a complete study of semi-classical Fourier Integral Operators, or to [15] where a self-contained approach is included, and where this claim is proved. Theorem 2 Consider ρ0 ∈ S such that (H1) and (H2) hold in a neighborhood V of ρ0 in S. There exist a local canonical transform κ from a neighborhood of ρ0 into some neighborhood Ω of 0, κ : (t, x, τ, ξ) → (s, z, σ, ζ), κ(ρ0 ) = 0, a Fourier integral operator U associated with κ and a smooth invertible matrix A such that v h = U oph (A)ψ h satisﬁes for all φ ∈ C0∞ (Ω), −σ + s oph (φ)oph γ 1 ζ1 + γ 2 ζ2

γ1 ζ1 + γ2 ζ2 −σ − s

v h = O(h) in

L2 (Rd+1 ),

(6)

where γj = γj (s,z,σ,ζ), j ∈ {1, 2} are complex-valued functions with Im(γ1 γ 2 ) = 0. Moreover, if I = {ζ1 = ζ2 = 0}, J ∪ J = Σ ∩ I = {σ 2 = s2 } ∩ I, ±,in

J = {σ ∓ s = 0} ∩ I ∩ {s > 0}, J ±,out = {σ ± s = 0} ∩ I ∩ {s < 0}.

(7) (8) (9)

Remarks 1) Because of the invariance of two-scale Wigner measures through canonical transforms which is proved in Lemma 2 in [15], it is equivalent to study two-scale Wigner measures for (v h ) and J and J , or for (ψ h ) and the same sets. 2) The set I is an involutive submanifold of T ∗ Rd+1 . Moreover, the intersection in (7) is transverse outside S and (I ∩Σ)\S is involutive. Thus, if νI is the twoscale measure associated with the concentration of (ψ h ) on I, we can identify measures ν (resp. ν ) with νI above N (I) |J (resp. N (I) |J ). Actually, if N Σ (J) is the bundle above Σ obtained by adding a sphere at inﬁnity to the ﬁbers of T (Σ)/T (J), the canonical isomorphism from T (Σ)/T (J) onto T (T ∗ Rd+1 ) |J /T (I) |J extends to some isomorphism θΣ,I : N Σ (J) → N (I) |J , which can be used in order to identify ν and νI 1N(I)|J . The reader can refer to Lemma 4 in [15] for a proof of this fact. Notice that N (I)|S is exactly the bundle T (T ∗Rd+1 )|S /F introduced in Section 1.3. for stating the Landau-Zener formula (4). This makes eﬀective the identiﬁcation used in the statement

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of Theorem 1, since, in formula (4), νS±,in , νS±,out are exactly the traces on N (I)|S of tr(νI ) above J ±,in , J ±,out . Thus, in our proof below, it will be suﬃcient to calculate the two-scale Wigner measure νI of (ψ h ) for I. 3) After this work was written, Y. Colin de Verdi`ere proved a more reﬁned normal form result for codimension 3 crossings (see [9] and [10]). In view of the invariance of two-scale Wigner measures through canonical transforms, it is reasonable to expect that his result implies ours. After some analysis of the geometry of the crossing performed in Section 2, Section 3 is devoted to the proof of Theorem 2. Then, the last part of the proof follows the strategy developed in [15]. In Section 4, we study directly on system (6) the reﬂection of the measure ν1|η|=+∞ . In Section 5, we obtain a normal form which holds in {| η |< +∞} in the same spirit as in [1], [36] and [29]. At this stage of the proof, we are left with the system √ h s hG h √ ∂s u = (10) uh , h G∗ −s i where G is a two-scale pseudo-diﬀerential operator, G = opIh (φ(η) ((γ1 )S η1 + (γ2 )S η2 )) , where φ ∈ C0∞ (R2 ). Section 6 is devoted to the proof of the Landau-Zener formula for this system by a strategy inspired by [14]. Notations. For v = (v1 , v2 , v3 ) ∈ R3 , we denote by M (v) the matrix v1 v2 + iv3 M (v) = , v2 − iv3 −v1 so that the Hamiltonian we study is H0 (τ, x, ξ) = τ + k + M (p) . If v = v2 + iv3 , we shall write M (v) = M (v1 , v ).

2 The geometry of generic codimension 3 crossings In this section we prove Propositions 1 and 2. We consider ρ0 ∈ S and we suppose that assumptions ((H1) and (H2)) or ((H1) and (H2 )) hold in a neighborhood of ρ0 in S. Proof of Proposition 1. Assume there exist Hamiltonian curves of λ± , ± ± ± ρ± s = (t0 + s, xs , τs , ξs ),

with

lim±

s→0

d ± p(ρs ) = 0. ds

Then there exist r, r > 0 and ω, ω ∈ S2 such that 1 1 ± ∓ p(t0 + s, x± p(t0 + s, x∓ s , ξs ) −→∓ rω, s , ξs ) −→∓ r ω . s s s→0 s→0

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In view of system (2), this implies by an elementary computation that r, r , ω, ω must satisfy rω = E + B ∧ ω, r ω = E − B ∧ ω . (11) If (H2) or (H2 ) holds, (11) has a unique solution (r = r , ω, ω ) ∈ R∗+ × S2 × S2 , which, in the case of (H2), reads B∧E r B∧E r E + , ω = E − , r = | E |2 − | B |2 . (12) ω= | E |2 | E |2 | E |2 | E |2 Moreover, if E · B = 0 and | B |≥| E |, (11) has no solution. Suppose that (H2) or (H2 ) hold. We concentrate on ﬁnding the curve ρ˜s =

˜s , τ˜s , ξ˜s deﬁned by t˜s , x ρ˜s = ρ+ s if s < 0,

ρ˜s = ρ− s if s > 0.

− The curves (ρ+ s )s>0 and (ρs )s<0 would be deﬁned similarly. Let us denote by g the function 1 g = p(t0 + s, x ˜s , ξ˜s ) s p g = sgn(s) |g| . The system (2) becomes so that |p|

d g x ˜s = ∇ξ k(t0 + s, x , ˜s , ξ˜s ) + t ∇ξ p ds |g| d ˜ g , ξs = −∇x k(t0 + s, x ˜s , ξ˜s ) − t ∇x p ds |g| d g (sg) = E(t0 + s, x . ˜s , ξ˜s ) + B(t0 + s, x ˜s , ξ˜s ) ∧ ds |g| Plugging ω in the latter equation and writing g(s) = rω + y(s), we obtain s

d y + (1 − Q0 )y = F0 (s, x ˜s , ξ˜s ) + G(y)yF1 (s, x˜s , ξ˜s ) + H(y)y · y, ds

where Q0 is the matrix deﬁned by Q0 y = 1r B(t0 , x0 , ξ0 ) ∧ (y − (ω · y)ω), Fj , G and H are smooth functions such that Fj (0, x0 , ξ0 ) = 0 for j ∈ {0, 1}. Therefore (y, x ˜s , ξ˜s ) satisﬁes some system of the form s

d y + (1 − Q0 )y ds d x ˜s ds d ˜ ξs ds

=

A(s, x ˜s , ξ˜s , y(s)),

(13)

=

B1 (s, x ˜s , ξ˜s , y(s)),

(14)

=

B2 (s, x ˜s , ξ˜s , y(s)).

(15)

−Q0 ln|s| The eigenvalues of Q0 are 0 and ± i E·B r 2 . Therefore, the function s → se is absolutely continuous and (13) reads d −Q0 ln|s| se y = e−Q0 ln|s| A(s, x˜s , ξ˜s , y(s)). ds

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Integrating between 0 and s, we obtain s e−Q0 ln|σ| A(σ, x˜σ , ξ˜σ , y(σ))dσ , s e−Q0 ln|s| y(s) = 0

i.e., y(s) =

0

1

e−Q0 ln θ A(sθ, x˜sθ , ξ˜sθ , y(sθ))dθ.

The proof is completed by applying a ﬁxed point argument to the map

1 F: x ˜s , ξ˜s , y(s) → e−Q0 ln θ A(sθ, x˜sθ , ξ˜sθ , y(sθ))dθ, 0 s s ˜ B1 (σ, x˜σ , ξσ , y(σ))dσ , ξ0 + B2 (σ, x˜σ , ξ˜σ , y(σ))dσ , x0 + 0

0

on the set Bδ,s0 deﬁned for δ, s0 > 0 by

Bδ,s0 = Sup | x˜s − x0 | + | ξ˜s − ξ0 | + | y(s) | < δ, |s|<s0

y(0) = 0, x ˜0 = x0 , ξ˜0 = ξ0 .

Remark 1. For further reference, we give the explicit expression of vector ﬁelds H and H in terms of ω and ω , which can be obtained from an easy calculation,

H

:=

H

:=

3

− lim Hλ+ (ρ+ s ) = lim Hλ− (ρs ) = Hτ +k − Σ ωj Hpj ,

s→0−

s→0+

j=1 3

− lim Hλ+ (ρ+ s ) = lim Hλ− (ρs ) = Hτ +k + Σ ωj Hpj .

s→0+

s→0−

j=1

Proof of Proposition 2. Let us focus on J. The case of J is similar. By deﬁnition, J is a codimension 3 submanifold of T ∗ Rd+1 . Moreover, since Hλ± are Hamiltonian vector ﬁelds, J is an involutive submanifold if and only if for any ρ ∈ S, T J|ρ is an involutive subspace. Since T J|ρ = T S|ρ ⊕ RH, ⊥ ∩ H ⊥ . Consider the four-dimensional vector space we have (T J|ρ )⊥ = T S|ρ ⊥ . V = T S|ρ

We have H ∈ V and the vector ﬁelds Hτ +k , Hp1 , Hp2 and Hp3 generate V . Writing the matrix of the restriction to V of the symplectic form σ in this basis, we obtain that dim Ker(σ|V ) = 2 if E · B = 0, E = 0, Ker(σ|V ) = 0 if E · B = 0.

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In particular, in the second case, V is symplectic and the intersection of (T J)⊥ = V ∩ H ⊥ with T J = V ⊥ ⊕ RH is reduced to RH. Thus J cannot be involutive. We focus now on the ﬁrst case. Because the Hamiltonian ﬂows are transverse to S, H ∈ / Ker(σ|V ) and we obtain, by means of dimension arguments, H ⊥ ∩ V = Ker(σ|V ) ⊕ R H ⊂ V ⊥ ⊕ R H. Hence (T J|ρ )⊥ ⊂ T S|ρ ⊕ R H = T J and J is involutive as soon as (H2) holds. In the following, it will be convenient to have equations of J and J . This is the purpose of our next result. Proposition 3 There exist smooth functions u, u ∈ C ∞ (R2d+2 , S2 ) such that u|S = ω, u|S = −ω .

(16)

and such that J and J have equations of the form J = {p = (τ + k)u}, J = {p = (τ + k)u }. Moreover, if L = {(p − (τ + k)u)) ∧ (u − u ) = 0} , then Σ ∩ L = J ∪ J . Proof. Since dp is of rank 3 on {p = 0}, there exists a function y = y(t, x, τ, ξ) ∈ R2d−2 such that (t, x, τ, ξ) → (τ + k, p, y) is a local diﬀeomorphism near some point ρ0 of S. We claim that the map J → R2d−1 ρ → (τ + k, y) is also a local diﬀeomorphism. Indeed, let δρ ∈ T J be a null vector of the diﬀerential of this mapping at ρ0 ∈ S. We have δρ = δs + λH, with δs ∈ Tρ0 S, λ ∈ R, and 0 = d(τ + k)δρ = λ d(τ + k)H = λ σ(H, Hτ +k ) = λ ω · E, in view of Remark 2. Since ω · E = r > 0, we conclude λ = 0, and therefore dy δs = 0, which implies δs = 0 by deﬁnition of y. Therefore, we can ﬁnd equations of J of the form p = θ(τ +k, y). Since | p |=| τ +k | on J, there exists some smooth vector u = u(t, x, τ, ξ) in S2 such that J = {p(t, x, ξ) = (τ + k(t, x, ξ)) u(t, x, τ, ξ)}. Similarly, we can write, J = {p(t, x, ξ) = (τ + k(t, x, ξ)) u (t, x, ξ)}.

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Let us compute u and u above S. We have + + + rω = lim− dp(ρ+ s )Hλ+ (ρs ) = lim− d(τ + k)(ρs )Hλ+ (ρs ) u s→0 s→0 p = − lim− {τ + k , p} · (ρ+ s )u = (E · ω)u = ru, |p| s→0 + + + rω = lim dp(ρ+ s )Hλ+ (ρs )u = lim+ d(τ + k)(ρs )Hλ+ (ρs ) u + s→0 s→0 p = − lim {τ + k , p} · (ρ+ s )u = −(E · ω )u = −ru . |p| s→0+ Whence (16). To achieve the proof, let us discuss the links between L and J ∪ J . It is clear that J ∪ J ⊂ L. Moreover, if ρ = (t, x, τ, ξ) ∈ L ∩ Σ with τ + k = 0, then ρ ∈ S ⊂ J ∩ J . Consider now ρ = (t, x, τ, ξ) ∈ L ∩ Σ with τ + k = 0. Then p = (τ + k)θ, θ ∈ S2 with (θ − u) × (u − u ) = 0. Let l ∈ R be such that θ = u + l(u − u ), we have 0 =| θ |2 −1 = 2l(1 − u · u )(1 + l). Since u · u = 1, either l = 0 or l = −1, which yields θ = u or θ = u and completes the proof of Proposition 3. Remark. The introduction of the manifold L is a ﬁrst step in the direction of the involutive manifold I, which will be crucial for our analysis. The next section will explain how to modify L in order to get I.

3 Reduction to a model system In this section, we focus on the proof of Theorem 2. In a ﬁrst time, we prove some algebraic lemma which uses speciﬁc properties of H0 and turns H0 into a Hamiltonian in which some equations of L appear. Then, we deﬁne the canonical transform κ. Finally, the end of the section is devoted to the proof of Theorem 2.

3.1

An algebraic lemma

Let us ﬁrst introduce some notations. We denote by e1 the vector deﬁned by e1 = e1 (t, x, τ, ξ) := We have (e1 )|S = −

u − u . | u − u |

E . |E|

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We choose some smooth functions (e2 , e3 ) so that (e1 , e2 , e3 ) is a direct orthonormal basis of R3 with moreover (e2 )|S =

E∧B if B = 0. | E || B |

(18)

Consider now the complex-valued function θ = θ(t, x, τ, ξ) deﬁned by θ = u · e2 + iu · e3 . Observe that near ρ0 , | θ |< 1. Actually, using (12) and (16), we obtain 2 θ|S ∈ R and θ|S =

| B |2 < 1. | E |2

(19)

Lemma 1 There exists some smooth invertible matrix A1 = A1 (t, x, τ, ξ) such that H0 = A∗1 H1 A1 with H1 = τ + k + α · g +

√p·e1

1−|θ|2

g

g

− √p·e1

where g = g1 + ig2 is a complex equation of L satisfying 2 dg1 |S = |E| r 2 d [(p − (τ + k)u) · e2 ] , dg2 |S = |E| r d [(p − (τ + k)u) · e3 ] . The complex-valued vector α = α(t, x, τ, ξ) = (α1 , α2 ) satisﬁes α1 |S = −θ, α2 |S = 0. Moreover, the matrix A1 satisﬁes  1  + ∗ 2    A1 Π A1 = (1 − |θ| ) 0     0  − ∗ 2   A1 Π A1 = (1 − |θ| ) 0 0   A1 Π+ A∗1 = (1 − |θ|2 )     0   1    A1 Π− A∗1 = (1 − |θ|2 ) 0

.

1−|θ|2

0 0 0 1 0 1 0 0

(20)

(21)

on J +,in , on J −,in , on J +,out ,

(22)

on J −,out .

Proof of Lemma 1. We use the strategy developed in [14]. The ﬁrst step consists in taking advantage of the quaternion structure of matrix M (v) and the second one, in multiplying both sides of the Hamiltonian by some appropriate matrix.

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The matrix M (v) is a generic point of the space H0 of self-adjoint matrices with 0 trace, H0 = {M (v), v ∈ R3 }. It is well known that the unitary group SU (2) acts by conjugation on H0 by SU (2) × H0 → H0 (U, M (v)) → U M (v)U −1 so that if we denote by φU (v) the vector of R3 such that ∀U ∈ SU (2), U M (v)U −1 = M (φU (v)), the map φ : U → φU , is a two-sheet covering of the orthogonal group SO(3): the group SU (2) is the universal covering of SO(3). Therefore, we have the following lemma. Lemma 2 Consider t → P (t) some smooth map from a simply connected neighborhood V ⊂ Rk of t = 0 into SO(3), then there exists some smooth map t → Q(t) from V into SU (2) such that ∀v ∈ R3 , Q(t)M (v)Q(t)−1 = M (P (t)v) . Lemma 2 yields the existence of a smooth unitary matrix Q = Q(t, x, τ, ξ) such that the Hamiltonian H0 (t, x, τ, ξ) := QH0 Q∗ satisﬁes H0 (t, x, τ, ξ) = τ + k +

p · e1 p · e2 − ip · e3

p · e2 + ip · e3 −p · e1

.

We aim now to introduce in the Hamiltonian the equations of L f

= =

(p − (τ + k)u) · e2 + i (p − (τ + k)u) · e3 p · e2 − (τ + k)θ1 + i (p · e3 − (τ + k)θ2 ) .

We set S(θ) = so that we have

H0 = (τ + k)S(θ) +

1 θ θ¯ 1

p · e1 f

, f −p · e1

.

The matrix S(θ) is a smooth invertible (since | θ |< 1) positive self-adjoint matrix. Let us denote by N the set of such matrices 1 z N = S(z) = , z∈C . z 1

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Lemma 3 Let S = S(z) ∈ N , then for all v = (v1 , v2 , v3 ) = (v1 , v ) ∈ R3 , S(z)M (v1 , v )S(z) = zv + zv + M (1 − |z|2 )v1 , v + z 2 v , The proof is straightforward. We use matrix S(θ)−1 to get rid of the matrix term (τ + k)S(θ) in the Hamiltonian H0 . Notice that S(θ)−1 =

−1 S(−θ) , S(θ) = a S(−bθ), 2 1− | θ |

with a=

Deﬁne A1 = H1 =

1 (1− | θ |2 )−1/2 1+ | θ | + 1− | θ | , 2 1 b= . 1 + 1− | θ |2

S(θ)Q and H1 = (A∗1 )−1 H0 A−1 1 . Lemma 3 yields

S(θ)−1 H0 S(θ)−1 = τ + k − a2 b(θf + f θ) + a2 M (1 − b2 | θ |2 )p · e1 , f + b2 θ2 f .

We check that a2 (1 − b2 | θ |2 ) = √ Since θ|S ∈ R and a2 (1 + b2 |θ|2 ) =

1 . We set 1−|θ|2 1 1−|θ|2 , we have

g = g1 + ig2 = a2 (f + b2 θ2 f ).

1 df1 |S , 1 − θ2 |S 1 = √ df2 |S , 1 − θ2 |S

dg1 |S

= a2 (1 + b2 θ2 )|S df1 |S =

dg2 |S

= a2 (1 − b2 θ2 )|S df2 |S

whence (20). Moreover, if α1 g1 + α2 g2 = −a2 b(θf + θf ) = −2a2 bRe(θf ), then we have (α1 dg1 + α2 dg2 )|S = −2a2 b θdf |S . Therefore, we obtain α2 = 0 and α1 = −2(1− | θ |2 )a2 bθ above S, hence (21). We prove now formula (22). Observe that 1 1 A1 Π± A∗1 = S(θ)QM (p)Q∗ S(θ) . S(θ) ± 2 |p| Using that

QM (v)Q∗ = M (v · e1 , v · e2 , v · e3 ),

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that u · e1 = −u · e1 = − 1− | θ |2 (since u · e1 < 0) and that τ + k = ∓ | p | on J ±,in ∪ J ±,out , we obtain 1 A1 Π+ A−1∗ S(θ) − S(θ)M (− 1− | θ |2 , θ) S(θ) on J +,in , = 1 2 1 − −1∗ S(θ) − S(θ)M (− 1− | θ |2 , θ) S(θ) on J −,out , A1 Π A1 = 2 1 + −1∗ S(θ) + S(θ)M (− 1− | θ |2 , θ) S(θ) on J +,out , A1 Π A1 = 2 1 − −1∗ S(θ) + S(θ)M (− 1− | θ |2 , θ) S(θ) on J −,in . A1 Π A1 = 2

1 Since S(θ) = 2 1+ | θ | + 1− | θ | S(bθ), by Lemma 3 and (19), we get (22) through straightforward computations.

3.2

The canonical transform

Proposition 4 There exist some function λ, λ > 0 near ρ0 , and some local canonical transform κ, κ(ρ0 ) = 0, (t, x, τ, ξ) → (s, z, σ, ζ), such that σ = −λ (τ + k + α1 g1 + α2 g2 ) , p · e1 s=λ , 1− | θ |2 λg(t, x, τ, ξ) = γˇ1 (s, z, σ, ζ)ζ1 + γˇ 2 (s, z, σ, ζ)ζ2 + (σ 2 − s2 )β, where β, γˇ 1 and γˇ2 are complex-valued functions with Im(ˇ γ1 γˇ2 ) = 0. Moreover, λ2|S =| E |−2 | E |2 − | B |2 . Proof. We set σ ˇ = − (τ + k + α1 g1 + α2 g2 ) , p · e1 sˇ = . 1− | θ |2 A simple calculation yields 1 | E |2 (E + B ∧ u) · e = > 0, {ˇ σ , sˇ}|S = − 1 (1− | θ |2 )3/2 | E |2 − | B |2 |S

(23)

where we used (16), (17) and (19). Moreover σ ˇ and sˇ vanish in ρ0 , therefore we can use Lemma 21.3.4 in [26] and we obtain the existence of some function λ, λ > 0 near ρ0 such that {λˇ σ , λˇ s} = 1.

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In view of (23), we ﬁnd the value of λ above S as claimed in the proposition. Then, by the Darboux theorem, we ﬁnd z˜ and ζ˜ such that ˜ κ ˜ : (t, x, τ, ξ) → (s, z˜, σ ζ) is a canonical transform with s = λˇ s and σ = λˇ σ. ˜ the equations of L such that f˜ = λg in the new We denote by f˜ = f˜(s, z˜, σ, ζ) variables, and we focus on the equations of J and J in these new variables. Remark that, if ρ ∈ L, λ2 | τ + k |2 = σ 2 , λ2 | p |2 = s2 (1− | θ |2 ) + σ 2 | θ |2 . Therefore L ∩ {s2 = σ 2 } = L ∩ {|p|2 = (τ + k)2 } = J ∪ J . We identify each part of J ∪ J by looking at the sign of σ which is −sgn(τ + k), i.e., ± on J ±,in ∪ J ±,out , and at the sign of s which is sgn(p · e1 ), i.e., −sgn(τ + k) on J and +sgn(τ + k) on J (since u · e1 < 0 and u · e1 > 0). For example, J = {σ = s} ∩ {f˜ = 0} and J +,in ⊂ {s > 0, σ > 0}. We focus now on ﬁnding convenient spatial coordinates. In this purpose, we use the geometric properties of J and J . Actually, these involutive submanifolds are given by J = {σ − s = 0, f˜ = 0}, J = {σ + s = 0, f˜ = 0}. Therefore, we have {σ + s, f˜} = 0 on σ + s = 0, f˜ = 0, {σ − s, f˜} = 0 on σ − s = 0, f˜ = 0. Lemma 4 There exist three complex-valued functions β, γ˜1 and γ˜2 on Ω such that ˜ = Re(f˜)(0, 0, z˜, ζ)˜ ˜ γ1 + Im(f˜)(0, 0, z˜, ζ)˜ ˜ γ2 + (σ 2 − s2 )β. f˜(s, σ, z˜, ζ) Proof of Lemma 4. Let us use for a while the coordinates (y, η) where √ √ 2 η = σ − s, 2 y = σ + s. In these coordinates, we have (24) ∂y f˜ = 0 on η = 0, f˜ = 0 and ∂η f˜ = 0 on y = 0 f˜ = 0.

We consider the vector-valued function F = Re(f˜), Im(f˜) . We claim that y, η, Re(f˜) and Im(f˜) are local coordinates. Actually, it is enough to check that s, σ, Re(f˜) and Im(f˜) are local coordinates and, for this, we study above S, the diﬀerentials of τ + k, p · e1 , (p − (τ + k)u) · e2 and (p − (τ + k)u) · e3 . The fact that dp is of rank 3 on S, yields that the system of these four diﬀerentials is of rank 4,

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˜ and some which proves the claim. Therefore, there exist some matrices B and B ˜ vectors A and A such that ∂y F ∂η F

= B(y, η)F + ηA, ˜ η)F + y A. ˜ = B(y,

(25) (26)

˜ dependance of F and look for some function F = We can now forget the (˜ z , ζ) F (y, η) satisfying (25) and (26). Equation (25) yields that F is of the form y R(y, y , η)A(y , η)dy , F (y, η) = R(y, 0, η)F (0, η) + η 0

where

∂y R(y, y , η) = B(y, η)R(y, y , η), R(y, y, η) = Id.

Hence, there exists some vector-valued function C such that F (y, η) = R(y, 0, η)F (0, η) + yη C(y, η). By using (26) for y = 0, we obtain the existence of some invertible matrix S satisfying F (0, η) = S(η)F (0, 0). Therefore, F (y, η) = R(y, 0, η)S(η, 0, y)F (0, 0) + yη C(y, η).

Let us deﬁne now the coordinates z, ζ. For simplicity, we denote by φ1 and φ2 the functions ˜ = Re(f˜)(0, 0, z˜, ζ), ˜ φ2 (˜ ˜ = Im(f˜)(0, 0, z˜, ζ). ˜ z , ζ) z , ζ) φ1 (˜ Then, a system of equations of J is σ − s = φ1 = φ2 = 0. Since J is an involutive submanifold of T ∗ Rd+1 , we have {φ1 , φ2 } = 0 on φ1 = φ2 = 0. Therefore, we can ﬁnd symplectic coordinates z, ζ such that φ1 and φ2 are linear functions of ζ1 and ζ2 . We denote by M (z, ζ) the invertible matrix such that φ = (φ1 , φ2 ) = M (z, ζ)(ζ1 , ζ2 ). We deﬁne γˇ = (ˇ γ1 , γˇ2 ) by γˇ = t M γ˜ . Then, using γ˜1 |s=σ=0 = 1 and γ˜2 |s=σ=0 = i, we obtain γ1 γ˜2 )|S = det(M )|S = 0, Im(ˇ γ1 γˇ2 )|S = det(M )|S Im(˜ which completes the proof of Proposition 4.

In the following, we keep the notation f˜ = f˜(s, z, σ, ζ), despite the change of variables, so that we have f˜(s, z, σ, ζ) = λg(t, x, τ, ξ) = γˇ1 ζ1 + γˇ2 ζ2 + (σ 2 − s2 )β.

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Proof of Theorem 2

Set

A = λ−1/2 A1 .

(27)

In the following calculations, we shall denote by O(h) either a family (Rh ) of operators or a family (rh ) of functions such that, for every φ ∈ C0∞ , 1 || oph (φ)Rh ||L(L2 ) + | oph (φ)rh |L2 is bounded. h We have

oph (λH1 ) = oph (A∗−1 )oph (H0 )oph (A−1 ) + O(h).

Consider now some Fourier integral operator U associated with κ and v h = U oph (A)ψ h . Then, oph

−σ + s f˜

f˜ vh −σ − s

=

U oph (λH1 )v h

=

U oph (A∗−1 )oph (H0 )ψ h + O(h)

=

O(h),

where we have used (5) and (1). In the coordinates y, η introduced in the proof of Lemma 4, we are left with the system √ − 2η γˇ1 ζ1 + γˇ√ 2 ζ2 + yηβ oph (28) v h = O(h). γˇ1 ζ1 + γˇ2 ζ2 + yηβ − 2y We set fˇ = γˇ1 ζ1 + γˇ2 ζ2 and √ 0 fˇ η 0 0 D= 2 , B= , F = ˇ β 0 y f 0

β 0

.

With these notations, system (28) writes oph (D − DBD)v h = oph (F )v h + O(h). By symbolic calculus, we infer oph (D)v h − oph (D)oph (B)oph (D)v h = oph (F )v h + O(h). Plugging this latter equation into itself, we obtain oph (D) − [oph (D)oph (B)]2 oph (D)v h − oph (D)oph (B)oph (F )v h = oph (F )v h + O(h).

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Since BF =

β fˇ 0 0 β fˇ

Ann. Henri Poincar´e

is diagonal,

oph (D)oph (B)oph (F ) = oph (B)oph (F )oph (D) + O(h). 2

Therefore, if we set P = 1 − oph (B)oph (F ) − (oph (D)oph (B)) , i.e., P = oph 1 − (DB)2 − BF + O(h), we get P oph (D)v h = oph (F ) + O(h). Near ρ0 , the matrix 1 − (DB)2 − BF is invertible. Therefore, P has a parametrix P of the form P = oph (1 − (DB)2 − BF )−1 + O(h).

−1 , Observe that, if d = 1 − yη|β|2 − β fˇ d 0 (1 − (DB)2 − BF )−1 = . (29) 0 d By using this parametrix, we obtain oph (D)v h = P oph (F )v h + O(h). Because of (29),

0 dfˇ + O(h). dfˇ 0 This completes the proof of Theorem 2 with γ1 = d γˇ1 and γ2 = d γˇ2 . P oph (F ) = oph

(30)

4 Analysis at infinity In the following, we shall work with the family (v h ) satisfying system (6). By standard properties of Wigner measures and because of (22), the two-scale Wigner measure ν˜ associated with the concentration of (v h ) on the involutive submanifold I satisﬁes ˜ + + ν˜− Π ˜ −, ν˜ = ν˜+ Π ˜ ± are the spectral projectors with ν˜± supported on J ±,in ∪ J ±,out and where Π ±,in associated with system (6). We denote by ν˜ (resp. ν˜±,out ) the traces of ν˜± on + − s = 0 (resp. s = 0 ) and we focus on the reﬂection of the part supported in {| η |= +∞} of measures ν˜± , i.e., ν˜±,out (η) = ν˜±,in (η) if | η |= +∞.

(31)

We begin by proving some hyperbolic energy estimate which is crucial in our analysis. In the following, we set ˜ ζ ), ζ˜ = (ζ1 , ζ2 ), ζ = (ζ, so that I = {ζ˜ = 0}.

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A hyperbolic energy estimate

2 Proposition 5 Let φ ∈ C0∞ (R = 1,there exists h0 and δ0 > 0 such that for

), φ(0) ˜2 |ζ| is bounded in L∞ Rs , L2 (Rdz ) . all δ < δ0 , the family oph φ(s, δ ) v h h
Arguing as in [15], the following corollary is straightforward. Corollary 1 If bh ∈ C0∞ (R2d+2 ) is supported near s = 0 and ζ˜ = 0, there exists some constant C such that +∞ | oph (bh )v h | v h |≤ C Sup Sup | ∂zβ ∂σγ bh (s, z, σ, ζ) | ds. −∞

γ+|β|≤N z,σ,ζ

˜2 Proof of Proposition 5. Consider wδh = oph φ(s, |ζ|δ ) v h . Then, we have oph

−σ + s γ · ζ˜

γ · ζ˜ −s

wδh = hrδh ,

with | rδh |L2 (Rd+1 ) ≤ C(δ). Recall that, since we work locally, γ1 and γ2 can be supposed to be compactly supported. We have, d | wδh (s) |2L2z ≤ C|wδh (s)|L2 |rδh (s)|L2 ds +

2 0 γ · ζ˜ h h Im oph (s) | w (s) . w δ δ γ · ζ˜ 0 ih L2

(32)

z

For j ∈ {1, 2}, consider gj = gj (s, t, z, ζ) such that γj (s, z, σ, ζ) =

+∞

−∞

eitσ gj (s, t, z, ζ)dt,

and let us denote by G = G(s, t, z, ζ) the matrix G=

0 g(s, t) · ζ˜ . g(s, −t) · ζ˜ 0

Then we have 0 γ · ζ˜ h h A(s) := oph wδ (s) | wδ (s) γ · ζ˜ 0 +∞ t h h = dt. oph G(s + h , t) wδ (s + ht) | wδ (s) 2 −∞ L2 z

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In the following, for simplicity, we drop the index L2z . We have Im(A)(s) =

1 (A(s) − A(s)) 2i 1 +∞ t h h = oph G(s + h , t) wδ (s + ht) | wδ (s) dt 2i −∞ 2 +∞ 1 t − wδh (s) | oph G(s + h , t) wδh (s + ht) dt. 2i −∞ 2

We remark that G(s, t)∗ = G(s, −t), therefore turning t into −t in the last integral, we obtain 1 +∞ t h h Im(A)(s) = oph G(s + h , t) wδ (s + ht) | wδ (s) dt 2i −∞ 2 1 +∞ t − oph G(s − h , t) wδh (s) | wδh (s − ht) dt. 2i −∞ 2 s0 Let us now estimate −∞ Im(A)(s)ds for s0 ∈ R. By the Fubini theorem, we have s0 Im(A)(s)ds −∞

=

1 2i

+∞

s0

dt −∞

ds −∞

t oph G(s + h , t) wδh (s + ht) | wδh (s) 2 t − oph G(s − h , t) wδh (s) | wδh (s − ht) . 2

We perform the change of variables s = s − ht in the last integral, whence s0 Im(A)(s)ds −∞

t h h dt ds oph G(s + h , t) wδ (s + ht) | wδ (s) 2 −∞ −∞ +∞ s0 −ht 1 t h h − dt ds oph G(s + h , t) wδ (s + ht) | wδ (s ) . 2i −∞ 2 −∞

1 = 2i

+∞

s0

Finally, s0 Im(A)(s)ds −∞

=

1 2i

+∞

s0

dt −∞

ds s0 −ht

t oph G(s + h , t) wδh (s + ht) | wδh (s) . 2

Ats0 the term of this calculation, we have got a power of h in the hestimate of −∞ Im(A)(s)ds. Going back to (32), and using that on Supp(wδ ), we have

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| γ · ζ˜ |2 ≤ Cδ, we obtain √ Sup| wδh (s) |2L2z ≤ C Sup| wδh (s) |L2z +[O( δ) + hC(δ)] Sup| wδh (s) |2L2z , s

s

s

which is enough to conclude the proof.

4.2

Reflection at infinity

In this section, we prove relations (31). We follow the proof of [15] with some modiﬁcations induced by two facts. First, we have to take into account some O(h) term in (6). Moreover, the involutive submanifold I is of codimension 2 (1 in [15]) and thus ν˜± 1|η|=+∞ is not utterly determined by its mass ν˜± (dη)1|η|=+∞ . In the following we focus on the + mode, the proof is similar for the − mode. Let us introduce ﬁrst some notations. For q = q(s, z, σ, ζ, η) ∈ C ∞ , we set ζ˜ qh (s, z, σ, ζ) = q(s, z, σ, ζ, √ ), h √

˜ qh (s, z, σ, ζ) = q(s, z, hσ, hζ, hζ), so that we have for q ∈ A, opIh (q) = oph (qh ) = op1 (qh ). We denote by A = A(s, z, σ, ζ) the matrix s γ · ζ˜ A= , γ · ζ˜ −s and we write

h ∂s v h = oph (A)v h + hf h , i with | oph (φ)f h |L2 = O(1) for all φ ∈ C0∞ (Ω). We set   1 1 ˜ ± = 1 ± Π A . 2 2 2 ˜ s + |γ · ζ| We consider a0 = a0 (z, σ, ζ, ω) ∈ C0∞ (R2d+1 × S2 ), ρ ∈ C0∞ (R), ρ = 1 in a neighborhood of 0, δ < δ0 (where δ0 is given by Proposition 5), R > 0 and ε > 0 and we set a(s, σ, z, ζ, η) = a0 (z, σ, ζ,

s | ζ˜ |2 |η| η )ρ( )ρ( )(1 − ρ)( ). |η| ε δ R

η The function a is a symbol of A. Actually, a is smooth despite the term in |η| ˜ + is smooth, because η = 0 on Supp(a). For the same reason, the function a Π

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˜ + ) even ˜ + ) is well deﬁned. We will denote it by opI (aΠ thus the operator oph (a Π h + + ˜ ∈ ˜ ∈ A.). This diﬃculty though aΠ / A (note that, on the other hand, ∂s aΠ compels us to use Weyl-H¨ormander’s symbolic calculus for some convenient metric which we shall now introduce. ˜ we have the estimate ˜ + is an homogeneous function of (s, γ · ζ), Since Π β α ˜+ | ∂σ,z,ζ ∂ ˜ (ah (s, z, σ, ζ)Π ) |≤ Cα,β (ε, δ) s,ζ

1 √ R h

|α| .

√ (Note that we implicitly assume R h ≤ 1.) This estimate allows to use WeylH¨ ormander’s symbolic calculus (see Sections 18.4, 18.5 and 18.6 in [26]). The ˜ + belongs to the class S(1, gh ) where gh is the metric symbol a Π gh = dz 2 + Since

gh σ gh

≤

√ 2 h R

ds2 hdζ˜2 + h2 (dσ 2 + d(ζ )2 ) + . 2 R h R2

, the gain of this symbolic calculus is

√ h R .

The relation (31) is equivalent to the fact that for all a0 as before, s lim ∂s ν˜+ 1|η|=+∞ , aρ( ) = 0. ε

ε→0

Since

s s ˜+ h ∂s ν˜+ 1|η|=+∞ , a0 ρ( ) = − lim lim lim opIh (∂s aρ( )Π )v | v h , R→+∞ δ→0 h→0 ε ε it is enough to study the limit of

s ˜+ h Lε,R,δ,h (a0 ) = − opIh (∂s aρ( )Π )v | v h . ε Then, Lε,R,δ,h is the sum of three terms Lε,R,δ,h = L1 + L2 + L3 , with L1 L2 L3

i I ˜+ [oph (aΠ ) , oph (A)]v h | v h ,

h ˜ + )v h | v h , = opIh (a∂s Π

˜ + )f h | v h − i opIh (aΠ ˜ + )v h | f h . = i opIh (aΠ =

Let us study now Lj for j ∈ {1, . . . , 3}.

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• We begin with L3 . By the Weyl-H¨ ormander symbolic calculus, if χ ∈ C0∞ (Ω) with χ = 1 near the support of a, we have  √  N h ˜ + ) = opIh (aχΠ ˜ + ) = opIh (aΠ ˜ + )oph (χ) + op1 S opIh (aΠ , gh  , R for all N ∈ N. On the other hand, we can apply Corollary 1 to the symbol ˜ + . Using the explicit expressions of a and of Π ˜ + , we get b h = a Π  √  N +∞ h s 3  . | ρ( ) | ds + O | L |≤ C ε R −∞ • Let us consider now L1 . Observe that A h ∈ S(λ h , gh ), therefore by symbolic calculus, we obtain

1 ˜ + , A} − {A, a Π ˜ + } v h | v h + O( 1 ). L1 = oph {a h Π h 2 R2

We set J=

1 0 0 −1

, K =

0 1 1 0

, L =

0 i −i 0

,

so that A = sJ + α · ζ˜ K + β · ζ˜ L . Then, using that ˜ + − λ, (Π ˜ + )2 = Π, ˜ A = 2λΠ ˜ + } − {Π ˜ + , A} = 0, and thus we have {A, Π ˜ + , A} − {A, a Π ˜ +} = Π ˜ + {a , A} − {A, a }Π ˜ + = b , {a h Π h h h h with b

=

˜ +J + J Π ˜ +) ∂σ a(Π

+

˜ +) ˜ +K + K Π {a, α}s,σ,z,ζ · ζ˜ − α · ∂z˜a +t ∇z˜α η · ∇η a (Π

+

˜ + ). ˜ + L + L Π {a, β}s,σ,z,ζ · ζ˜ − β · ∂z˜a +t ∇z˜β η · ∇η a (Π

As for L3 , we apply Corollary 1. Therefore, as h goes to 0, √ +∞ δ s 1 s ρ ( ) ds. ρ( ) + +C | L1 |≤ O 2 R ε ε ε −∞ Hence limsup

limsup limsup | L1 |= 0.

(ε,δ)→(0,0) R→+∞

h→0

(33)

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• Finally, let us deal with the remainder term L2 . Using again (33), we get A + + ˜ ∂s Π = A, 2 ∂s Π . 2λ √ In the metric gh , λ h ≥ C hR on Supp(a h ) −2 1 ˜+ ∈ S √ aλ A h ∂s Π , gh . hRλ h We obtain

˜ + ) ∈ 1 op (A)opI aλ−2 A∂s Π ˜ + − 1 opI aλ−2 A∂s Π ˜ + op (A) opIh (a∂s Π h h h h 2 2 1 +op1 S , g . h R2 Therefore, we can use again the equation, which yields 1 h I −2 ˜ + vh | vh . L2 = O + O(h) − oph ∂s aλ A∂s Π 2 R 2i

β ˜ + |≤ 2 C 2 3/2 , as h goes to 0, Since | ∂σ,z ∂s aλ−2 A∂s Π (s +hR ) h +∞ 1 1 ds 2 = o(1) + O L =O + o(1) + Ch . 2 2 2 3/2 R R2 −∞ (s + hR ) Hence,

limsup limsup L2 = 0. R→+∞

h→0

This completes the proof of (31).

5 Analysis at finite distance In this section, our aim is to reduce to a simpler system by getting rid of the (s, σ) dependence of function γ for {| η |< +∞}. We introduce the following notations, −σ + s γ · ζ˜ −σ + s γ 0 · ζ˜ H2 = = , H , 3 γ · ζ˜ −σ − s γ 0 · ζ˜ −σ − s −σ + s γ 00 · ζ˜ H4 = , γ 00 · ζ˜ −σ − s with γ 0 (s, z, ζ) := γ(s, 0, z, ζ) and γ 00 := γ(0, 0, z, ζ). Proposition 6 For every ball B of R2η , there exist matrices R = R(s, z, σ, ζ, η) and ˜ = R(s, ˜ z, σ, ζ, η) such that for all a ∈ C0∞ (R2d+2 × Bη ), R s,σ,z,ζ ! √ √ I ˜ = O(h). (34) oph (a) opIh (1 + hR)oph (H2 ) − oph (H4 )opIh (1 + hR) 2 L(L )

We prove this proposition in the following two elementary steps.

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The normal form in σ

Let us denote by J the matrix J=

1 0

0 −1

.

Lemma 5 For every ball B of R2η , there exists a matrix C1 = C1 (s, z, σ, ζ, η) such that for all a ∈ C0∞ (R2d+2 s,σ,z,ζ × Bη ) ! √ √ I oph (a) opIh (J(1 + hC1 )J)oph (H2 ) − oph (H3 )opIh (1 + hC1 ) 2 = O(h). L(L )

(35) Proof. We set

K=

0 1 −1 0

, L=

0 i i 0

.

Equation to (35) is equivalent √ I I oph (a) oph (1 + hC1 )oph (JH2 ) −oph (JH3 )opIh (1 +

! √ hC1 )

L(L2 )

√ = O(h h),

(36)

with, for η ∈ B, JH2 JH3

= −σJ + s + α · ζ˜ K + β · ζ˜ L + hR, = −σJ + s + α0 · ζ˜ K + β 0 · ζ˜ L + hR,

√ where γ = α + iβ and γ 0 = α0 + iβ 0 . We use symbolic calculus to expand in h the left-hand side. We notice that since a is compactly supported in the variable η, √ √ opIh (a)oph (ζj b) = h opIh (aηj b) + O( h), whence

I oph (a)oph (ζj b)

L(L2 )

√ = O( h),

for j ∈ {1, 2} and b ∈ C0∞ (R2d+2 s,σ,z,ζ ). We obtain the equation σ[J, C1 ] = (−α · η + α0 · η )K + (−β · η + β 0 · η )L. a 1 c1 Setting C1 = , this system reduces to b1 d1 2σb1 2σc1 which can be solved easily.

= −γ · η + γ 0 · η, = −γ · η + γ 0 · η, η ∈ B,

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The normal form in s

Lemma 6 For every ball B of R2η , there exists a matrix D1 = D1 (s, z, ζ, η), such that for all a ∈ C0∞ (R2d+2 s,σ,z,ζ × Bη ), ! √ √ I oph (a) opIh (1 + hD1 )oph (H3 ) − oph (H4 )opIh (1 + hD1 )

L(L2 )

= O(h).

Proof. We argue as in Lemma 5. We set 0 1 0 i K = , L = , 1 0 −i 0 so that H3 H4

= −σ + sJ + α0 · ζ˜ K + β 0 · ζ˜ L , = −σ + sJ + α00 · ζ˜ K + β 00 · ζ˜ L .

We obtain the following equation for η ∈ B s[D1 , J] = (α00 · η − α0 · η) K + (β 00 · η − β 0 · η) L . We use the same notation as before D1 =

a1 b1

c1 d1

and we solve similarly 2s b1 = −γ 0 · η + γ 00 · η, 2s c1 = γ 0 · η − γ 00 · η, η ∈ B.

6 Landau-Zener formula Let us summarize the results of the previous section. We may localize η in some η ball B of R2 applying a cutoﬀ function, φ( R )γ(z, ζ) for R > 0 and φ ∈ C0∞ (R). √ h ˜ h , and we have Then, applying Proposition 6, we set u ˇ = opIh (1 + hR)v ∀a ∈ C0∞ (R2d+2 × B), opIh (a)oph (H4 )ˇ uh = O(h), (37) −σ + s φ(η/R)γ 00 · ζ˜ , γ 00 (z, ζ) = γ(0, 0, z, ζ). Notice that with H4 = −σ − s φ(η/R)γ 00 · ζ˜ (37) also holds with γ 00 (z, ζ) = γ|S , however this latter improvement does not provide major simpliﬁcation in the arguments below. From now on, we shall drop the superscript 00 for simplicity.

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Obviously, the two-scale measures of (ˇ uh ) and (v h ) are the same in B. We are left with a system of the form η s φ( R )γ · ζ˜ h h I ∂s uˇ = oph uˇh + hf h (s, z), η i )γ · ζ˜ −s φ( R where the family (f h ) is uniformly bounded in L2s,z . More precisely, we aim to describe the traces on s = 0− in terms of the traces on s = 0+ of the two-scale semiclassical measures ν˜ of (ˇ uh ) for I = {ζ1 = ζ2 = 0}. We claim that (f h ) does not contribute to this description. Indeed, denote by Sh (s, s ) the evolution operator associated with the free system (10) and let (uh ) be a solution of this system deﬁned more precisely by, η ˜ s opIh (φ( R ) γ · ζ) h ∂s uh = ˇh|s=0 . (10) uh , uh|s=0 = u η ˜ opIh (φ( R i )γ · ζ) −s

Then we have h

h

u ˇ (s) = u (s) + i 0

s

Sh (0, t)f h (t)dt.

Since the family (f h ) is uniformly bounded in L2s,z , we obtain that, uˇh (s) = uh (s) + O( | s |) in L2 (Rdz ). uh ) on s = 0± Therefore, the traces of the two-scale Wigner measure of (uh ) and (ˇ are the same. In the following, we focus on system (10), which allows to calculate the transfer coeﬃcient in the variables (s, z, σ, ζ) and then, in variables (t, x, τ, ξ).

6.1

An operator-valued Landau-Zener formula

Proposition 7 Let H be a Hilbert space, and G = Gh be a bounded family of operators on H. Let uh ∈ C (Rs , H × H) be a solution of the system √ h hG s h √ (38) uh ∂s u = h G∗ −s i with Sup| uh |L∞ (Rs ,H×H) < +∞. h

There exist families of vectors in H, (αhj ), (ωjh ), j ∈ {1, 2}, such that, as h goes to 0, for any χ ∈ C0∞ (R), χ(GG∗ )αh1 , χ(G∗ G)αh2 , χ(GG∗ )ω1h and χ(G∗ G)ω2h are bounded and for s < 0, ∗

χ(GG

)uh1 (s, z)

=

χ(G∗ G)uh2 (s, z) =

GG∗ s i 2 χ(GG )e √ αh1 + o(1), h −i G∗2 G s2 s ∗ −i 2h √ χ(G G)e αh2 + o(1), h ∗

2

s i 2h

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for s > 0, i GG2 ∗ s χ(GG∗ )uh1 (s, z) = χ(GG∗ )e √ ω1h + o(1), h −i G∗2 G s2 s ∗ h ∗ −i 2h χ(G G)u2 (s, z) = χ(G G)e ω2h + o(1), √h h h ω1 α1 h Moreover = S with ω2h αh2 a(GG∗ ) −b(GG∗ )G h S = , b(G∗ G)G∗ a(G∗ G) s2 i 2h

with a(λ) = e−π 2 , b(λ) = λ

π

i 2ie √4 λ π

(39)

2 2 2−iλ/2 e−π 4 Γ(1 + i λ2 ) sh( πλ 2 ), a(λ) + λ|b(λ)| = 1. λ

Proof. We shall solve explicitly the system (38) by means of the following integral operator √ 2 2 1 i e 2h (y +s −2 2ys) f (s)ds. Kf (y) = √ π 2 √ Notice that K sends S (Rs , H × H) onto S (Ry , H × H) and that K −1 = π 2K ∗ . We set √ U h (y) = K[uh ( h·)] so that the system becomes

√ 2 √i

∂y U1h = G U2h , 2 y U2h = G∗ U1h .

(40)

In particular i GG∗ U1h , 2 which has the following general solution, in view of the spectral theorem for GG∗ , y ∂y U1h =

U1h (y) =| y |iGG

∗

/2

Ah + | y |iGG

∗

/2

sgn(y)B h ,

(41)

where Ah , B h ∈ H. Since U1h is bounded in S (Ry , H), we infer that, for every χ ∈ C0∞ (R), χ(GG∗ )Ah , χ(GG∗ )B h are bounded in H. Indeed, given ψ ∈ C0∞ (R) " such that ψ(λ) > 0 for all λ ∈ R, we observe that +∞ GG∗ iGG∗ /2 dy " =ψ − ψ(Ln y)y . y 2 0 Hence, by integration of (41) against y ∈ R → 1y>0 ψ(Ln y)y −1 and y ∈ R → 1y<0 ψ(Ln (−y))y −1 ,

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" GG∗ )(Ah ± B h ) is bounded in H. Then, one only has to we conclude that ψ(− 2 observe that χ = χ ˜ψ" with χ ˜ = χψ"−1 ∈ C0∞ (R). We now seek for a similar expression of U2h (y). Let us introduce the two continuous functions λ ∈ R → Pλ ∈ S (Ry ) and λ ∈ R → Qλ ∈ S (Ry ) deﬁned by the formulas: ∀λ ∈ R, ∀φ ∈ S(R), +1 iλ−1 |y| [φ(y) − φ(0)]dy + Pλ , φ = Qλ , φ

−1 +∞

= 0

|y|>1

| y |iλ−1 φ(y)dy,

| y |iλ−1 [φ(y) − φ(−y)]dy.

It is easy to check the following identities, ∂ | y |iλ sgn(y) = iλPλ + 2δ(y), ∂y ∂ | y |iλ = iλQλ . yQλ =| y |iλ , ∂y

yPλ =| y |iλ sgn(y),

Plugging the expression (41) of U1h into the second equation of system (40), we get G∗ U2h (y) = √ QGG∗/2 Ah + PGG∗ /2 B h + δ(y)C h , 2

(42)

for some C h ∈ H. Comparing with the ﬁrst equation of (40), we obtain √ 2 2 h B = GC h . (43) i Coming back to (42), and using the identity G∗ f (GG∗ ) = f (G∗ G)G∗ , we conclude U2h

= =

1 √ QG∗ G/2 G∗ Ah + 2 1 √ QG∗ G/2 G∗ Ah + 2

i PG∗ G/2 G∗ GC h + δ(y)C h 4 ∗ 1 ∂ | y |iG G/2 sgn(y) C h . 2 ∂y

(44)

At this stage, we observe that, for every χ ∈ C0∞ (R), χ(G∗ G)C h is bounded in H. Indeed, if χ = 0 near 0, (43) implies ∗ χ(G∗ G)C h = G∗ χ(GG ˜ )B h √ χ(λ) where 2iλ ˜ = χ(λ), hence χ(G∗ G)C h is bounded. On the other hand, if χε is 2 supported in [−ε, ε]

| G∗ Gχε (G∗ G)C h |H ≤ ε | χε (G∗ G)C h |H ,

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hence, integrating (44) against φ ∈ S(Ry ) with φ(0) = 1, we conclude, for ε small enough, Sup| χε (G∗ G)C h |H < +∞. h

It remains to convert the expressions (41) and (42) into information on (uh1 ) and (uh2 ). We use the following lemma, which is essentially Lemma 11 in [15]. Lemma 7 As s tends to ±∞, we have, locally uniformly in λ ∈ R+ ,

√ 2 K −1 | y |iλ/2 = eis /2 | s |iλ/2 2iλ/4 2π e−iπ/4 + o(1), √ 2 i i π −iπ/4 iλ/4 √ sgn(y) | y |iλ/2 = ± eis /2 | s |iλ/2 K −1 e 2 + o(1), 2 2 2 √ 1 πλ λ −1 −is2 /2 −iλ/2 2i 2 −iλ/4 √ Qλ = ± e 2 + o(1), |s| Γ(1 + i ) sh K λ 2 4 2 2 1 ∂ πλ λ K −1 | y |iλ/2 sgn(y) + o(1). = e−is /2 | s |−iλ/2 2−iλ/4 Γ(1 + i ) ch 2 ∂y 2 4 In view of this lemma and of the estimates for Ah and C h , we obtain, for every χ ∈ C0∞ (R), if h → 0+ and ±s > 0,

iGG∗ /2 ∗ h ∗ is2 /2h s ∗ h ∗ h p(GG )A ± r(GG )GC + o(1), χ(GG )u1 (s) = χ(GG ) e √h χ(G∗ G)uh2 (s) ∗

= χ(G G) e

−is2 /2h

∗ s −iG G/2 ∗ ∗ h ∗ h √ ± q(G G)G A + t(G G)C + o(1), h

where p(λ)

=

r(λ)

=

q(λ)

=

t(λ)

=

√ 2iλ/4 2π e−iπ/4 , √ i π −iπ/4 iλ/4 e 2 , 2√ 2i 2 −iλ/4 πλ λ 2 Γ(1 + i ) sh , λ 2 4 λ 2−iλ/4 Γ(1 + i ) ch(πλ/4). 2

It remains to notice that p(GG∗ ) p(GG∗ ) r(GG∗ )G h =S q(G∗ G)G∗ −q(G∗ G)G∗ t(G∗ G) where S h is given by (39).

−r(GG∗ )G t(G∗ G)

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Two-scale analysis of the model system (10)

In order to calculate the two-scale Wigner measure of family (uh ), we set

η γ(z, ζ) · η , G = opIh φ R and we use the following two results. Lemma 8 Consider χ ∈ C0∞ (R), then we have 2 η opIh χ φ( )γ(z, ζ) · η = χ(GG∗ ) + o(1) = χ(G∗ G) + o(1). R Proof of Lemma 8. We prove this lemma thanks to the Helﬀer-Sj¨ostrand formula (see [25] and [12]): consider χ ∈ C0∞ (R) and a positive integer N0 , to be ﬁxed later; there exists a continuation of χ, χ ˜ ∈ C0∞ (C), such that we have ∀w ∈ C, | ∂ χ(w) ˜ |≤ C | Im(w) |N0 . Then, ∀λ ∈ R, χ(λ) = −

1 π

C

(45)

∂ χ(w) ˜ dL(w), λ−w

where dL(w) denotes the Lebesgue measure on C. Consider now some self-adjoint two-scale pseudo-diﬀerential operator Q, √ Q = opIh (bh ) = oph (q0 ) + O( h) in L(L2 ), with q0 ∈ C0∞ . By the spectral theorem, we have 1 χ(Q) = − ∂ χ(w)(Q ˜ − w)−1 dL(w). π C

(46)

In order to estimate the resolvent of Q, we use the following formula √ opIh (a)opIh (b) = opIh (ab) + hR1 , with ||R1 ||L(L2 ) ≤ C

Sup |α|+|β|+|γ|≤M

| ∂zα ∂ζβ ∂ηγ a |

Sup |α|+|β|+|γ|≤M

| ∂zα ∂ζβ ∂ηγ b | .

Since q0 is real-valued, we infer, for w ∈ C \ R, √ (w − Q)−1 = opIh (w − q0 )−1 + hR, with ||R||L(L2 ) ≤ C | Im(w) |−N1 , N1 ∈ N. Choosing N0 ≥ N1 in (45), (46) yields √ χ(Q) = opIh (χ(q0 )) + O( h) in L(L2 ). Applying this result to GG∗ and G∗ G, we obtain Lemma 8.

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Lemma 9 For every χ ∈ C0∞ (R2d+4 )), we have in L(L2 ), GG∗ GG∗ s −i 2 s i 2 I √ oph (χ) √ h h ∗ G G G∗ G s −i 2 s i 2 I = √ oph (χ) √ + o(1) = opIh (χ) + o(1). h h Proof. Consider χ ∈ C0∞ (R2d+2 ). For t ∈ R, ∗ ∗ ∗ d −iGG∗ t I e oph (χ)eiGG t = ie−iGG t opIh (χ) , GG∗ eiGG t dt √ −iGG∗ t ∗ = he Rh eiGG t where Rh is a uniformly bounded operator in√L2 (Rd ), independent of t. Therefore, ∗ ∗ ˜ h (t), with for t ∈ R, e−iGG t opIh (χ)eiGG t = opIh (χ) + h R ˜ ≤C|t|. Rh (t) L(L2 )

Setting t = 12 ln √sh , we obtain the result. Observe that ˜ + +,out = Π ˜ − −,in = Π |J |J

1 0 0 0

˜ − −,out = Π ˜ + +,in = , Π |J |J

0 0 0 1

.

Therefore, by Lemma 9 and Lemma 8, ν˜+,out is the two-scale Wigner measure of the family (αh2 ), ν˜−,out of (αh1 ), ν˜+,in of (ω1h ) and ν˜−,in of (ω2h ). Using the scattering operator S h and again Lemma 8, we infer that if measures ν˜+,in and ν˜−,in are mutually singular, then for | η |< +∞, +,out +,in ν˜ 1 − T˜ T˜ ν˜ = , (47) T˜ 1 − T˜ ν˜−,out ν˜−,in with T˜ = exp − | γ1 η1 + γ2 η2 |2 .

6.3

Conclusion: Proof of Theorem 1

Because of the invariance through canonical transforms of two-scale Wigner measures, ∀a ∈ A, a , ν˜ = a ◦ N (κ) , AνI A∗ , where A is the matrix of Theorem 2 deﬁned in (27), and N (κ)(ρ, η) = (κ(ρ), η). Using the decomposition of ν and of ν˜ on modes ± and the formulas (27) and (22), we have a , ν˜±,out = a ◦ N (κ) , λ−1 (1 − θ2 )νS±,out , a , ν˜±,in = a ◦ N (κ) , λ−1 (1 − θ2 )νS±,in .

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Therefore, in view of (47), we are able to link νS±,out and νS±,in . It remains to specify the coordinates on N (I) generated by the coordinates (s, z, σ, ζ) and to compare them with those in which we wrote Theorem 1. Consider some point ρ of S, its coordinates (s, z, σ, ζ) and (t, x, τ, ξ) in both systems and δρ ∈ T (T ∗Rd )|ρ . Then, by the deﬁnition of γ1 , γ2 (see Proposition 4) and by (20), we have γ.dζ˜|S = λdg|S = λ

|E|2 |E| d [(p − (τ + k)u) · e2 ] . d [(p − (τ + k)u) · e2 ] + i 2 r r

Observe that d [p − (τ + k)u] δρ = σ(Hτ +k , δρ)ω − σ(Hp , δρ), with (ω · ej )|S = − E∧B |E|2 · ej because of (12). Therefore, d [p − (τ + k)u] δρ.(e2 + ie3 ) E∧B = − σ(Hτ +k , δρ) + σ(H , δρ) · (e2 + ie3 ) p | E |2 = − (σ(W1 , δρ) + iσ(W2 , δρ)) , for the choice of the basis (W1 , W2 ) associated with the orthonormal basis of the plane normal to E, (X1 , X2 ) = (e2 , e3 ). In view of the value of λ above S (see Proposition 4), we conclude | E |2 r2 2 2 , δρ) + σ(W , δρ) | γ · η |2 = σ(W , 1 2 r3 | E |2 ˜ with η = dζ[δρ]. Coming back to (47), we obtain r2 | E |2 2 2 σ(W2 , δρ) T = exp −π 3 σ(W1 , δρ) + , r | E |2 which completes the proof of Theorem 1.

Acknowledgements. Part of this work was realized during the stay of the two authors at the Schr¨ odinger Institute in Vienna, which we thank warmly for its hospitality.

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[30] L. Landau, Collected papers of L. Landau, Pergamon Press, 1965. [31] P-L. Lions, T. Paul, Sur les mesures de Wigner, Revista Matem´ atica Iberoamericana, 9, pp. 553–618 (1993). [32] L. Miller, Propagation d’onde semi-classiques `a travers une interface et mesures 2-microlocales, Th`ese de l’Ecole Polytechnique, (1996). [33] P.A. Markowich, N.J. Mauser, F. Poupaud, A Wigner function approach to semi-classical limits: electrons in a periodic potential, J. Math. Phys., 35, pp. 1066–1094 (1994). [34] F. Poupaud, C. Ringhofer, Semi-classical limits in a crystal with exterior potentials and eﬀective mass theorems, Comm. Part. Diﬀ. Eq., 21, N◦ 11–12, pp. 1897–1918 (1996). [35] D. Robert, Autour de l’approximation semi-classique, Birkh¨ auser, 1983. [36] M. Rouleux, Tunelling eﬀects for h pseudo-diﬀerential operators, Feshbach resonances, and the Born-Oppenheimer approximation, in Evolution equations, Feshbach resonances, Hodge theory, p. 131–242, Math. Top., 16, WileyVCH, Berlin (1999). [37] C. Zener, Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond. 137, pp. 696–702 (1932). C. Fermanian Kammerer Universit´e de Cergy-Pontoise Math´ematiques 2, avenue Adolphe Chauvin BP 222, Pontoise F-95302 Cergy-Pontoise cedex France email: [email protected] P. G´erard Universit´e Paris XI Math´ematiques Bˆat. 425 F-91405 Orsay France email: [email protected] Communicated by Bernard Helﬀer submitted 16/09/02, accepted 11/03/03

Ann. Henri Poincar´e 4 (2003) 553 – 600 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030553-48 DOI 10.1007/s00023-003-0139-3

Annales Henri Poincar´ e

Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators Near a Threshold Xue Ping Wang∗

Abstract. In this paper, we analyze the properties of threshold energy eigenvalue and threshold energy resonance of N -body Schr¨ odinger operators at the bottom of essential spectrum, and give the resolvent expansions in weighted Sobolev spaces.

1 Introduction The low-energy spectral analysis of two-body Schr¨odinger operators has been extensively studied. See [5, 11, 12, 16, 17] and references therein. These results are useful in two-body threshold scattering ([3, 5, 6, 17]). Threshold spectral properties of two-body Subhamiltonians play an important role in Faddeev’s three-body theory ([7]). Their detailed analysis leads to interesting applications to three-body Schr¨ odinger operators. See [21, 26] for the Eﬁmov eﬀect, and [10] for the structure of three-body scattering matrix. Spectral structure of two-body Schr¨odinger operators with decreasing potentials is simple: 0 is the only threshold. For N -body Schr¨ odinger operators, thresholds are the union of eigenvalues of all Subhamiltonians and, generically, are inﬁnite. Although remarkable achievements have been obtained in N -body scattering theory during the past two decades, little is known about the properties of N -body Schr¨ odinger operators in neighborhood of thresholds. We are only able to quote [13] on coupling constant limit of eigenvalues absorbed into a simple two-cluster threshold for C0∞ interactions, and [18] on the absence of embedded eigenvalues in a small interval on the right of thresholds. To our knowledge, there does not yet exist a result concerning threshold energy resonance in N -body problems. However, thresholds are the exceptional points where interesting physical phenomena appear (see [25]). This work is an initial step in spectral analysis of N -body Schr¨ odinger operators near thresholds. Let P denote the N -body Schr¨ odinger operator with the mass-center removed from the total energy operator N 1 − ∆xj + 2m j j=1

Vij (xi − xj ),

xj ∈ R3 ,

(1.1)

1≤i<j≤N

∗ The author thanks the Erwin Schr¨ odinger Institute and the organizers of the special semester “Scattering Theory”, where the results of this work were reported. Research partially supported by a grant from the Chinese Academy of Sciences.

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where xj and mj denote the position and mass of the j-th particle, Vij is assumed to be real and relatively compact with respect to −∆ in L2 (R3 ) and satisﬁes the decay (1.2) |Vij (y)| ≤ Cij |y|−ρ for y ∈ R3 , |y| > R for some R > 0. In this paper we assume at least ρ > 2 and ρ varies according to the results to obtain. P is regarded as a self-adjoint operator in L2 (X), where X is the 3(N − 1)-dimensional real vectorial space: N X = {(x1 , . . . , xN ) ∈ R3N ; j=1 mj xj = 0}. Let A denote the set of all cluster decompositions of the N -particle system. For a ∈ A, let #a denote the number of clusters in a and P a the Subhamiltonian associated with a. For i, j ∈ {1, . . . , N }, we write (ij) ⊂ a if i and j belong to a same cluster in a. If a is a k-cluster decomposition a = (a1 , . . . , ak ), we denote Xa = {x ∈ X; l∈aj ml xl = 0, j = 1, . . . , k}. and Xa = {x ∈ X; xi = xj if (ij) ⊂ am for some m ∈ {1, . . . , k}}. Xa and Xa give an orthogonal decomposition for X relative to the quadratic x ∈ X. For x ∈ X, we have the corresponding form q(x) = ( j 2mj |xj |2 ), orthogonal decomposition: x = xa + xa with xa ∈ Xa and xa ∈ Xa . The N -body Schr¨ odinger operator P introduced above can be written in the form P = P0 + V (x) where P0 is the Laplace-Beltrami operator on the Euclidean space (X, q) and V (x) =

Va (xa )

a∈A

with Va (xa ) = Vij (xi − xj ) if a is an (N − 1)-cluster decomposition and (ij) ⊂ a, 0 otherwise. Let −∆a ( −∆a , resp.) denote the restriction of P0 on Xa ( on Xa , resp.). For a ∈ A, denote P a = −∆a +

b⊆a

Vb (xb ),

Pa = P a − ∆a ,

Ia (x) =

Vb (xb ).

b⊆a

P a is the Subhamiltonian associated with the cluster decomposition a and Ia is the sum of all inter-cluster interactions. Let T = ∪a∈A,#a≥2 σp (P a ) be the set of thresholds of P . The well-known HVZ theorem gives the bottom of the essential spectrum E0 ≡ inf σess (P ) of P by the formula E0 = mina∈A,#a≥2 inf σ(P a ) = mina∈A,#a=2 inf σ(P a ). Under rather general conditions, the positive eigenvalues of P are absent and its negative eigenvalues can only accumulate at the thresholds ([8], [15], [19]). Less is known about the spectral properties of P near a threshold (see, however, [13, 18] ). The purpose of this paper is to study the spectral properties of P near its ﬁrst threshold E0 and to establish the asymptotic expansions of the resolvent there. We refer to [3], [5], [11], [12], [16], [17] for related results on two-body Schr¨ odinger

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operators and their applications. The following assumption is made throughout this work: there exists a unique decomposition, a0 ∈ A, such that E0 = inf σ(P a0 ).

(1.3)

This assumption is used in [13] on the absorbtion of eigenvalues into thresholds. It should be noted that [13] is a work in L2 -setting and the notion of threshold energy resonance does not appear there. For N ≥ 3, by the HVZ Theorem, the assumption (1.3) implies that #a0 = 2, E0 is in the discrete spectrum of P a0 and there is 0 > 0 such that inf σ(P a ) ≥ E0 + 0 , for a ∈ A, #a ≥ 2, a = a0 . Let H m,s , m, s ∈ R, be the weighted Sobolev space on X equipped with the norm ||u||m,s = ( |(1 + |xa0 |2 )s/2 (1 − x )m/2 u|2 dx)1/2 . X

Sometimes, we denote also by the symbol H m,s the weighted Sobolev space on Xa0 with x replaced by a0 . Let ·, · ( (· · · , ·)a , ·, · a , resp.) denote the scalar product on L2 (X) ( on L2 (Xa ), L2 (Xa ), resp.). In this paper, we identify H 1,−s , s ≥ 0, to the dual space of H −1,s with the usual L2 product as the dual product. Let L(m, s; m , s ) = L(H m,s , H m ,s ). It will be convenient to regard P as an operator in L(1, 0; −1, 0). Definition. ([10],[17]) We say that E0 is a resonance of P if the equation P u = E0 u has a solution u ∈ H 1,−s (X)\L2 (X) for any s > 1/2. Following [17, 11], we call E0 a regular point of P if E0 is neither eigenvalue, nor resonance of P ; an exceptional point of the ﬁrst kind (respectively, the second kind; the third kind) of P if it is a resonance but not an eigenvalue (respectively, an eigenvalue but not a resonance; both an eigenvalue and a resonance) of P . Let R(z) = (P − z)−1 , z ∈ σ(P ). To prove the existence of the asymptotic expansions of R(z) for z near E0 , we need a detailed analysis of the spectral properties of P at E0 . The following result proved in Sections 3 and 4 plays an important role. Theorem 1.1 Assume (1.2) with ρ > 2 and (1.3). Let ϕ0 ∈ D(P a0 ) be an eigenfunction of P a0 P a0 ϕ0 = E0 ϕ0 , ϕ0 = 1. (1.4) If E0 is an eigenvalue of P in L2 , its multiplicity is ﬁnite. If u ∈ H 1,−s , ∀s > 1/2 and P u = E0 u, then, u is a resonant state of P if and only if Ia0 (x)u(x)ϕ0 (xa0 ) dx = 0. (1.5) X

Here ϕ0 is a normalized eigenfunction of P a0 associated with E0 . The resonance of P at E0 , if it does exist, is simple, which means that the eigenspace of P associated with E0 is of codimension 1 in the kernel of P − E0 in H 1,−s , ∀s > 1/2.

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Let δ > 0 be small. Denote Uδ (E0 ) = {z ∈ C; z > 0, |z − E0 | < δ} and Uδ = Uδ (0). The latter will be used for the eﬀective Hamiltonian obtained after the ﬁrst reduction through a Grushin problem. The main result of this work is the following Theorem 1.2 Let λ = z − E0 and (λ)1/2 be the branch such that (λ)1/2 > 0 for

λ > 0. (a) Assume that E0 is a regular point of P . Then, for k ∈ N∗ , ρ > max {2, 2k + 1} and (2k − 1)/2 < s < ρ − (2k − 1)/2 for k ≥ 2 and 1 < s < ρ − 1 for k = 1, one has R(z) =

k−1

(0)

λj/2 Bj

+ O(|λ|(k−1)/2+ ).

(1.6)

j=0

(b) Suppose that E0 is an exceptional point of the ﬁrst kind of P . Assume ρ > max{2, 2k − 1}, k ∈ N with k ≥ 1. Let s > 1 if k = 1 and (2k − 1)/2 < s < ρ − (2k − 1)/2 if k ≥ 2. We have R(z) =

k−2

(1)

λj/2 Bj

+ O(|λ|(k−2)/2+ ).

(1.7)

j=−1

Here

(1)

B−1 = i ·, u u,

(1.8)

and u is a resonant state of P at E0 normalized by 1 √ Ia (x)u(x)ϕ0 (xa0 ) dx = 1. 2 π X 0

(1.9)

(c) Suppose that E0 is an exceptional point of the second kind (l = 2) or the third kind (l = 3) of P . Assume ρ > 2k − 1, k ∈ N with k ≥ 2. Let (2k − 1)/2 < s < ρ − (2k − 1)/2. One has R(z) =

k−4

(l)

λj/2 Bj + O(|λ|(k−4)/2+ ),

l = 2, 3.

(1.10)

j=−2

Here

(2)

(3)

B−2 = B−2 = −ΠE0 , and ΠE0 is the orthogonal projection onto the eigenspace of P associated with E0 . The above expansions are valid in L(−1, s; 1, −s) for λ ∈ Uδ with δ > 0 small enough and ∈]0, 1/2] veriﬁes < (2s − 2k + 1)/4, if k ≥ 2; and 0 < < (s − 1), if k = 1.

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Remarks (a) Under the assumption (1.3), the methods of this paper can be used to establish the asymptotics of the resolvent near the thresholds E ∈ σd (P a0 ) with E < mina∈A,#a≥2,a=a0 inf σ(P a ). The projection Π used in Section 2 should then be replaced by the spectral projection associated to all eigenvalues less than or equal to E. For an N -body system with identical particles, if the degeneracy of E0 only arises from the internal symmetries, this degeneracy can be removed by the method of [23] and the analysis of this paper can be carried over to this case. (b) In the case where min

a∈A,#a=2

inf σ(P a ) =

min

a∈A,#a≥3

inf σ(P a ),

the spectral properties of an N -body Schr¨ odinger operator with rapidly decreasing pair potentials are sometimes similar to two-body Schr¨ odinger operators with slowly decreasing potentials (ρ < 2). In fact, the Eﬁmov eﬀect in three-body problems shows that in this case, even for compactly supported pair potentials, the discrete spectrum can be inﬁnite. In these cases, the results of Theorem 1.2 cannot hold. (c) It is an interesting open question to establish the resolvent expansions near E0 when it is attained by several two-cluster Subhamiltonians. Even though our results look similar to those known for two-body Schr¨ odinger operators ([5], [11], [12], [16], [17]), the proof is technically involved. It is only after a tedious calculation that we arrive at the simple leading term in the case l = 1, 2, 3. (l) In addition, the expression of the coeﬃcients Bj depends not only on l which classiﬁes the spectral nature of P at E0 , but also on the fact whether or not E0 belongs to the discrete spectrum of an auxiliary operator P deﬁned in Section 2. (0) For example, in the case E0 ∈ σd (P ), B0 is given by (0)

B0 = R (E0 ) + + (E0 )(1 + G0 V0 )−1 G0 − (E0 ) (0)

(see Proposition 3.3) and if E0 ∈ σd (P ), the expression of B0 is very complicated and is given by (4.75) in Theorem 4.11 (a). An immediate consequence of Theorem 1.2 is large time expansions of scattering solutions to time-dependent N -body Schr¨ odinger equation with energy supported near E0 . These expansions are similar to three-dimensional two-body Schr¨ odinger operators, instead of 3(N − 1)-dimensional two-body ones. This is to compare with the phenomenon of eigenvalue absorbtion into two-cluster threshold. The resolvent estimates up to the threshold imply the absence of embedded eigenvalues of P just above E0 . See also [18]. Theorem 1.1 and 1.2 play an important role in the proof of the existence of the N -body Eﬁmov eﬀect for N ≥ 4 in [25]. See also [21] and [26] for the role of

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two-body resolvent asymptotics in the proof of three-body Eﬁmov eﬀect. We also expect that these results will be useful in N -body threshold scattering (see [3], [5], [11], [17] and [24] for two-body problems, and [10], [22] for results on many-body scattering matrices). Let us now explain the ideas used in the proof of Theorem 1.2. Under the assumption (1.3), by solving a Grushin problem associated to P − z, we obtain a representation for the resolvent R(z) in the form R(z) = E(z) − E+ (z)E+− (z)−1 E− (z) for z near E0 , where E(z), E± (z) and E+− (z) are all holomorphic near E0 . E+− (λ + E0 ) is of the form −∆ + V (λ) C λ − (−∆ + V (λ)) or λ − , C∗ 0 depending on whether E0 is an eigenvalue of P or not. Here ∆ is the 3-dimensional Laplacian, C : Cm → H 1,s , ∀s > 0, and V (λ) is a family of non-local operators admitting a convergent expansion of the form V (λ) =

∞

λj Vj ,

|λ| < δ.

j=0

In the case E0 ∈ P , −∆ + V (λ) is an operator pencil. In order to obtain the asymptotics of the inverse of the operator −∆ + V (λ) − λ = −∆ + V0 + λ(V1 − 1) +

∞

λj Vj ,

j=2

it is important that V1 − 1 be invertible on the generalized kernel of −∆ + V0 . Fortunately, we can show by explicit calculation that V1 ≤ 0. The case E0 ∈ σd (P ) is even more complicated (see (a) of Theorem 4.11 for the simplest case where E0 is neither eigenvalue nor resonance of P ). In this case, C may not be injective. This implies that the resolvent of the unperturbed operator already has a pole at zero. In the both cases, to prove the existence of the asymptotic expansion of E+− (z)−1 , we solve another appropriate Grushin problem in weighted Sobolev spaces to obtain a representation of the operators (1 + (−∆ − λ)−1 V (λ))−1 and

1+

−∆ C C∗ 0

−1 V (λ) −λ 0

0 0

−1 .

The singularities of E+− (z)−1 near z = E0 can then be calculated from those of the inverse of a holomorphic matrix-valued function E+− (λ) for λ near 0. To prove the

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existence of the asymptotic expansions of E+− (λ)−1 near λ = 0, we need detailed information on eigenfunctions and resonant states of the operators −∆ + V0

and

−∆ + V0 C∗

C 0

at 0 and to relate them to those of P at E0 . Notice that in the presence of threshold eigenvalue, the operator V1 intervenes in the leading term. Our approach allows to optimize the decay assumption on ρ. For example, to obtain the leading term, we just need ρ > 2 in the case of an exceptional point of the ﬁrst kind, and ρ > 3 in the case of an exceptional point of the second or the third kind. The plan of this paper is as follows. In Section 2, we introduce an auxiliary operator P and study its spectral properties near E0 . In Section 3, under the assumption that E0 is not an eigenvalue of the operator P , we study two Grushin problems to reduce the study of R(z) to the inverse of a matrix-valued holomorphic function and establish the asymptotic expansion of R(z) according to the spectral properties of E0 with respect to P . In Section 4, the case where E0 is an eigenvalue of P is studied. The zero energy eigenvalue and resonance of a matrix-valued operator are analyzed and the asymptotics of R(z) are proved. Theorems 1.1 and 1.2 are proved in Section 3 and Section 4, respectively, according to the cases whether or not E0 belongs to σd (P ).

2 Preliminaries Throughout this paper, conditions (1.2) and (1.3) are assumed. From now on, we denote a = a0 . Let ϕ0 be a normalized eigenfunction of P a associated with E0 . Let Π be the projection in L2 (X) deﬁned by Πf = ϕ0 ⊗ (f, ϕ0 )a , f ∈ L2 (X). Put Π = 1 − Π and P = Π P Π . Lemma 2.1 There exists 0 > 0 such that the essential spectrum of P = Π P Π satisﬁes σess (P ) ⊂ [E0 + 0 , ∞[. Proof. Let {Jb ; b ∈ A, #b = 2} be a partition of unity such that Jb (x)2 = 1 on X, Jb (x) = 1 for x outside a compact set and |xb | ≤ c0 |x| for some c0 > 0 and |∂ α Jb (x)| ≤ Cα x −|α| on X, for all α ∈ N3(N −1) . Write P = (−b + P b ) + Ib = Pb + Ib .

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From the IMS-localization formula (cf. Lemma 3.2 in [6]), we have P

Π (

=

Jb P Jb −

#b=2

=

Π(

≡

|∇Jb |2 )Π

#b=2

#b=2

Π(

Jb (Pb + Ib )Jb −

|∇Jb |2 )Π

#b=2

Jb Pb Jb )Π + K.

#b=2

Due to the assumption (1.2), Π |∇Jb |Π and Π Ib Jb2 Π are relatively compact perturbations of Π P0 Π and of P , and so is K. For b = a, by (1.3), inf σ(Pb ) = inf σ(P b ) ≥ E0 + 0 . Therefore, Π Jb Pb Jb Π ≥ (E0 + 0 ) Π Jb2 Π . (2.1) #b=2,b=a

#b=2,b=a

For b = a, we decompose Π Ja Pa Ja Π as Π Ja Pa Ja Π = Ja Π Pa Π Ja + K1 where

K1 = [Π , Ja ]Pa Ja Π + Ja Π Pa [Π , Ja ].

Note that [Π , Ja ] = [Π, 1 − Ja ]. Let A = (1 − Ja )Π. AA∗ is of the integral kernel (1−Ja (x))ϕ0 (xa )ϕ0 (y a )(1−Ja (y)). Since ϕ0 is exponentially decreasing on Xa and 1 − Ja (x) has support in {|xa | > δ|x|}, δ > 0, AA∗ is a Hilbert-Schmidt operator. In particular, A is compact, and so is the commutator [Π , Ja ]. This proves that K1 is relatively compact with respect to P . Since E0 is the lowest eigenvalue of P a , there exists 1 > 0 such that Π Pa Π ≥ (E0 + 1 )Π . We obtain Π Ja Pa Ja Π

≥ =

where

(E0 + 1 )Ja Π Ja + K1

(E0 + 1 )Π

Ja2 Π

(2.2)

+ K2 ,

K2 = (E0 + 1 )(Ja Π Ja − Π Ja2 Π ) + K1

is still a relatively compact perturbation of P by the arguments used above. Let 0 = min{0 , 1 }. One deduces from (2.1) and (2.2) that P = Π Jb Pb Jb Π + K ≥ (E0 + 0 ) + K3 #b=2

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where K3 = K2 + K is relatively compact. By Weyl’s Theorem on essential spectrum, we obtain σess (P ) ⊂ [E0 + 0 , ∞[. In the following, in order to give the asymptotics of the resolvent R(z) = (P − z)−1 for z ∈ Uδ (E0 ), we shall distinguish the cases E0 ∈ σd (P ) and E0 ∈ σd (P ). Let R0 (z) = (−∆a − z)−1 , z > 0. Since dim Xa = 3, R0 (z) can be continuously extended to C+ as operator in L(−1, s; 1, −s), s, s > 1/2, s + s > 2 and has an asymptotic expansion near 0 of the form R0 (λ) =

k

λj/2 Gj + O(λk/2 ), in L(−1, r ; 1, −r), r , r > k + 1/2,

(2.3)

j=0

where

G0 ∈ L(−1, s ; 1, −s), s, s > 1/2, s + s > 2, Gj ∈ L(−1, s ; 1, −s),

s, s > j + 1/2, for j ≥ 1.

The Gj ’s are integral operators whose kernel can be computed explicitly. Remark that if a = (a0,1 , a0,2 ) and x denotes the relative position of the two mass centers, then 1 1 ∆a = ( + )∆x , 2Ma0,1 2Ma0,2 Ma0,j = k∈a0,j mk . It is more convenient to use the orthonormal coordinates x0 on Xa so that the kernel of Gj is just given by Kj (x0 , y0 ) =

ij |x0 − y0 |j−1 , 4πj!

j ≥ 0.

(2.4)

The following lemma is useful to optimize the remainder estimates. Lemma 2.2 For s > 1/2, 0 < r < 2s − 1, 0 < r ≤ 1, we have || x −s R0 (λ + i0) x −s || ≤ Cs,r |λ|−1+r , λ ∈]0, 1].

(2.5)

More generally, let s > k + 1/2 for k ∈ N∗ and s > 1 for k = 0. One has, || x −s (R0 (λ + i0) −

k

λj/2 Gj ) x −s || ≤ Cs,s ,r |λ|k/2+ , λ ∈]0, 1],

(2.6)

j=0

for 0 < < min{1/2, (2s − 2k − 1)/4}. can be taken to be 1/2 if (2s − 2k − 1)/4 > 1/2. Here x ∈ R3 and · denotes the norm on L(L2 (R3 )). Lemma 2.2 can be easily deduced by dilation and complex interpolation. In fact, since R0 (1 + i0) ∈ L(−1, s; 1, −s), s > 1/2, we deduce by a dilation || λ1/2 x −s R0 (λ + i0) λ1/2 x −s || ≤ Cs |λ|−1 .

(2.7)

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For s > 1/2, || x −s R0 (λ + i0) x −s || ≤ Cs |λ|−1 , 0 < λ ≤ 1. On the other hand, we have, for s > 1, || x −s R0 (λ + i0) x −s || ≤ Cs ,

(2.8)

for 0 < λ ≤ 1. (2.5) follows from a complex interpolation. Using (2.3), (2.6) can be proved similarly.

3

The case E0 ∈ σd (P )

In this section, we suppose that E0 ∈ σd (P ) and δ > 0 is chosen small so that σ(P ) ∩ {|z − E0 | < δ} = ∅. In this case, we can reduce the N -body problem to a generalized two-body operator with non-linear spectral parameter.

3.1

A reduction

The method used is reduction through appropriate Grushin problems. This method is a reﬁned version of Feshbach’s method used, for example, in Jensen-Kato [11] and consists in reducing the N -body problem to a two-body type operator with a non-linear spectral parameter. Grushin’s method already reveals powerful in the study of resonances ([9]). We begin with the study of the Grushin problem P(z) =

P −z

ϕ0 ⊗ ·

(·, ϕ0 )a

0

: D(P ) × L2 (Xa ) → L2 (X) × L2 (Xa ),

(3.1)

where D(P ) is the domain of the operator P . In order to obtain a good representation of the resolvent, we want to prove that P(z) is invertible in a small neighborhood of E0 . To do this, let P = Π P Π ,

and R (z) = (Π P Π − z)−1 Π ,

where Π = ϕ0 ⊗ (·, ϕ0 )a : L2 (X) → L2 (X) and Π = 1 − Π. Let Q(z) be deﬁned by Q(z) =

R (z)

ϕ0 ⊗ ·

(·, ϕ0 )a

z − (P (ϕ0 ⊗ ·), ϕ0 )a

.

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Then, in L2 (X) × L2 (Xa ), (P − z)R (z) + Π P (ϕ0 ⊗ ·) − ΠP Π(ϕ0 ⊗ ·) P(z)Q(z) = 0 1 ΠIa R (z) Π Ia (ϕ0 ⊗ ·) = 1+ . 0 0

Let B=

ΠIa R (z) Π Ia (ϕ0 ⊗ ·) 0

0

.

It is easy to check that B 3 = 0. Therefore, 1 + B is invertible. Consequently, P(z) is invertible on the right. Similarly, one can show that P(z) is invertible on the left. Therefore, P(z) is invertible and one has P(z)−1 = Q(z)(1 − B + B 2 ). Writing the inverse of P(z) in the form (z) + (z) P(z)−1 = , − (z) +− (z)

(3.2)

(3.3)

we can calculate that (z) = + (z) = − (z) = +− (z) =

R (z), −R (z)Ia (ϕ0 ⊗ ·) + (ϕ0 ⊗ ·), (·, ϕ0 )a − (Ia R (z)·, ϕ0 )a , (z − E0 ) − (−xa + I˜a (xa )) + (Ia R (z)Ia (ϕ0 ⊗ ·), ϕ0 )a ,

where I˜a (xa ) = (Ia ϕ0 , ϕ0 )a . Making use of the relations P(z)P(z)−1 = 1 and P(z)−1 P(z) = 1, we obtain the following expression of the resolvent R(z) (see [9]) R(z) = (z) − + (z)+− (z)−1 − (z).

(3.4)

The formula (3.4) is valid for any z with z > 0 without the assumption that E0 ∈ σ(P ). Under the assumption E0 ∈ σ(P ), R (z) is holomorphic in a small neighbourhood of E0 , so are + (z), +− (z) and − (z). The asymptotic behavior of R(z) near E0 is then determined by +− (z)−1 . To study the resonance of P at E0 , we remark that by the method of commutators, the relation (3.4) initially established in L2 remains valid in any weighted-L2 space.

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Set λ = z − E0 and write +− (z) as +− (z) = where

λ − (−xa + V (λ)),

V (λ) = I˜a (xa ) − (Ia R (λ + E0 )Ia (ϕ0 ⊗ ·), ϕ0 )a

(3.5)

is holomorphic near λ = 0. V (λ) has a convergent expansion near λ = 0 V (λ) =

∞

λj Vj

j=0

with V0 Vj

= =

I˜a (xa ) − (Ia R (E0 )Ia (ϕ0 ⊗ ·), ϕ0 )a −(Ia (R (E0 ))j+1 Ia (ϕ0 ⊗ ·), ϕ0 )a , j ≥ 1.

Note that for j ≥ 1, Vj = (Ia (R (E0 ))j+1 Ia (ϕ0 ⊗ ·), ϕ0 )a ∈ L(0, s; 0, s + 2ρ), ∀s ∈ R. In this work, it is important to note that V1 = −(Ia (R (E0 ))2 Ia (ϕ0 ⊗ ·), ϕ0 )a is a negative operator on L2 (Xa ).

(3.6)

Making use of the relation +− (z)−1 = −(1 + R0 (λ)V (λ))−1 R0 (λ), we are reduced to establishing the asymptotic expansion of (1 + R0 (λ)V (λ))−1 in suitable spaces. Proposition 3.1 For k ∈ N, ρ > max{2, 2k+1} and (2k+1)/2 < s < ρ−(2k+1)/2, one has for W (λ) = 1 + R0 (λ)V (λ) W (λ) =

k

λj/2 Wj + O(|λ|k/2+ ),

(3.7)

j=0

in L(1, −s; 1, −s), for λ ∈ Uδ and ∈]0, 1/2] veriﬁes < (2s − 2k − 1)/4 if k ≥ 1; < (s − 1) if k = 0. Here W0 = 1 + G0 V0 , W1 = G1 V0 , W2 = G2 V0 + G0 V1 , W3 = G3 V0 + G1 V1 . Proposition 3.1 follows from the results of Section 2 and the expansion of V (λ) near 0. From (3.4), we can deduce the following asymptotic expansion of the resolventR(z) in the case where E0 is a regular point.

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Proposition 3.2 Assume that E0 is a regular point for P . Then, for k ∈ N, ρ > max{2, 2k + 1}, s > 1 if k = 0 and (2k + 1)/2 < s < ρ − (2k + 1)/2 if k ≥ 1, one has k (0) R(z) = λj/2 Bj + O(|λ|k/2+ ), (3.8) j=0

in L(−1, s; 1, −s), for λ = z − E0 ∈ Uδ and > 0 is the same as in Proposition 3.1. Here (0)

B0

=

R (E0 ) + + (E0 )C0 G0 − (E0 ),

(3.9)

(0) B1

=

+ (E0 )(C0 G1 + C1 G0 )− (E0 ),

(3.10)

with C0 = (1 + G0 V0 )−1 and C1 = −C0 G1 V0 C0 . Proof. By the assumption and Proposition 3.1, 0 is a regular point for +− (E0 ) = −∆ + V0 . In this case, C0 ∈ L(1, −s; 1, −s) for 1/2 < s < ρ − 1/2 and one has (1 + R0 (λ)V (λ))−1 = C0 + λ1/2 C1 + · · · where C0 , C1 are as above. Consequently, one has in suitable spaces −−1 +− (λ)

=

(1 + R0 (λ)V (λ))−1 R0 (λ)

=

C0 G0 + λ1/2 (C0 G1 + C1 G0 ) + · · ·

Proposition 3.2 follows from (3.4).

3.2

Spectral properties of P at the threshold

Definition. We say that 0 is a resonance of +− (E0 ) if the equation +− (E0 )v = 0 has a solution v ∈ H 1,−s (Xa ) for any s > 1/2 and v ∈ L2 (Xa ). As for P , we call 0 a regular point or an exceptional point of the ﬁrst, the second or the third kind of +− (E0 ) according to the spectral properties of 0 with respect to +− (E0 ). It is well known that E0 is an eigenvalue of P if and only if 0 is an eigenvalue of +− (E0 ) ([9]). As to resonance, we have the similar result. Proposition 3.3 E0 is a resonance of P if and only if 0 is a resonance of +− (E0 ). Proof. Using (3.3) for P(E0 )−1 , we obtain the following equations (P − E0 )+ (E0 ) + ϕ0 ⊗ +− (E0 ) = 0, − (E0 )(P − E0 ) + +− (E0 )(·, ϕ0 )a = 0. These relations remain valid in L2,s , s ∈ R.

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Ann. Henri Poincar´e

Assume that zero is a resonance of +− (E0 ). Let v be a resonant state v ∈ H 1,−s (Xa ), ∀s > 1/2, and +− (E0 )v = 0. Then,

u ≡ + (E0 )v = −R (E0 )(Ia (ϕ0 ⊗ v)) + ϕ0 ⊗ v

is in H 1,−s (X), ∀s > 1/2, and satisﬁes (P −E0 )u = 0. We claim that u ∈ L2 (X). In fact, ϕ0 ⊗ v ∈ L2,−s \ L2 and since Ia (ϕ0 ⊗ v) ∈ L2,ρ−s , so does R (E0 )(Ia (ϕ0 ⊗ v)). So, u ∈ L2,−s \ L2 . Conversely, if E0 is a resonance of P and u an associated resonant state, set v = (u, ϕ0 )a . v ∈ H 1,−s , ∀s > 1/2, and +− (E0 )v = 0. v ∈ L2 , because by the relation (z)(P − z) + + (z)(·, ϕ0 )a = I, u = + (E0 )v. If v ∈ L2 , u would also be in L2 , which contradicts the fact that u is a resonant state, which shows that 0 is a resonance of +− (E0 ). The proof of Proposition 3.3 shows that the dimensions of resonant spaces are the same. Theorem 1.1 in the case E0 ∈ σd (P ) can be derived from Proposition 3.3. Proof of Theorem 1.1 in the case E0 ∈ σd (P ). Remark that +− (E0 ) = −(−∆a + V0 ) is a two-body Schr¨ odinger-type operator. Although V0 is non-local, the analysis of [11] is still valid in this case. In particular, one can show that the kernel of +− (E0 ) in H 1,−s , ∀s > 1/2, is of ﬁnite dimension and that if v is a resonant state of +− (E0 ), then V0 v, 1 a = 0. As a consequence, the zero energy resonance of +− (E0 ) is at most simple, so is the resonance of P at E0 , by Proposition 3.3. It follows that if E0 is an eigenvalue of P , its multiplicity is ﬁnite. By the proof of Proposition 3.3, u is resonant state of P if and only if it can be written in the form u = + (E0 )v, where v is a resonant state of +− (E0 ). Since u ∈ H 1,−s , ∀s > 1/2, ρ > 2 and ϕ0 is rapidly decreasing on Xa , Ia uϕ0 ∈ L1 (X). This gives Ia uϕ0 dx = Ia (+ (E0 )v)ϕ0 dx = [(Ia − Ia R (E0 )Ia )(ϕ0 ⊗ v)]ϕ0 dx = V0 v, 1 a = 0,

3.3

Asymptotic expansions of (1 + R0 (λ)V (λ))−1

If E0 is not an eigenvalue of P , the asymptotic expansion of (1 + R0 (λ)V (λ))−1 can be obtained by the method of perturbation. If E0 is an eigenvalue of P , or equivalently, 0 is an eigenvalue of +− (E0 ) = −∆ + V0 , due to the presence of the term λV1 in V (λ), the method of perturbation does not work. In the rest of this

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section, we present a self-contained proof in this general case, using once again the Grushin’s method. Recall that 1 G0 =∈ L(−1, s ; 1, −s), s + s > 2, s, s > , 2 and (1 + G0 V0 ) ∈ L(1, −s; 1, −s) for 1/2 < s < ρ − 1/2. Let K = Ker (1 + G0 V0 ) ⊂ H 1,−s , K ∗ = Ker (1 + G0 V0 )∗ ⊂ H −1,s . As in [11], one can show that K is independent of s > 1/2 and is spanned by the resonant states and the eigenfunctions of −∆ + V0 . One can easily check that V0 is injective from K into K ∗ and V0∗ = V0 is injective from K ∗ into K. Consequently, V0 is bijective from K onto K ∗ . This shows that dim K = dim K ∗ and the form φ → φ, −V0 φ a is non-degenerate on K. Using the positivity of −∆, one can prove that this form is positive deﬁnite. Let µ = dim K. We can choose a basis {φ1 , . . . , φµ } of K such that φi , −V0 φj a = δij .

(3.11)

Without loss, we can assume that φj , j ≥ 2, are eigenfunctions. Denote φ∗j = −V0 φj , j = 1, . . . , µ,

(3.12)

which form a basis of K ∗ . In order to get the expansion of (1 + R0 (λ)V (λ))−1 , we consider another Grushin problem related to W (λ) T A(λ) = : H 1,−s × Cµ → H 1,−s × Cµ , S 0 where s > 1/2, operators T and S are deﬁned by:     c1 c1 µ     cj φj ,  ...  ∈ Cµ , T  ...  = j=1 cµ cµ and

 f, −V0 φ1 a   .. Sf =  , . f, −V0 φµ a 

f ∈ H 1,−s .

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Ann. Henri Poincar´e

Deﬁne Q : H 1,−s → H 1,−s by Qf =

µ

f, φ∗j a φj .

j=1

Then,

T S = Q on H 1,−s and ST = Iµ on Cµ .

Lemma 3.4 One has the decomposition H 1,−s = K ⊕ Range (1 + G0 V0 ).

(3.13)

Q is the projection from H 1,−s , s > 1/2, onto K with Range Q = Range (1 + G0 V0 ). Proof. For u ∈ K ∩ Range (1 + G0 V0 ), we have, for some v, (1 + G0 V0 )u = 0 and u = (1 + G0 V0 )v. Thus, u = −G0 V0 u and −V0 u, u a = −V0 u, v + G0 V0 v a = − V0 (1 + G0 V0 )u, v a = 0. Since −V0 deﬁnes a positive deﬁnite form on K, this implies u = 0. This proves that K ∩ Range (1 + G0 V0 ) = {0}. Since G0 V0 is compact on H 1,−s , Range (1 + G0 V0 ) is closed and is equal to ( Ker (1 + G0 V0 )∗ )⊥ . For any u ∈ H 1,−s , we write u = Qu + (u − Qu) and we can check that u − Qu ∈ ( Ker (1 + G0 V0 )∗ )⊥ . This proves H 1,−s = K ⊕ Range (1 + G0 V0 ). It is easy to verify that Q is the projection onto K corresponding to this decomposition of H 1,−s . Note that Q = 1−Q is a projection from H 1,−s onto F = Range (1+G0 V0 ). Proposition 3.5 Q (1+G0 V0 )Q is bijective on F . For λ ∈ Uδ , δ > 0 small enough, Q W (λ)Q is invertible on F . Let D(λ) = (Q W (λ)Q )−1 Q . Then if ρ > 2, one has

D(λ) = D0 + O(|λ| ),

(3.14)

in L(1, −s; 1, −s), s > 1/2. Here 0 < < min{1/2, (2s − 1)/4}. More generally, if k ≥ 1 and ρ > 2k + 1, one has D(λ) =

k

λj/2 Dj + O(|λ|k/2+ ),

(3.15)

j=0

in L(1, −s; 1, −s), (2k + 1)/2 < s < ρ − (2k + 1)/2. Here > 0 is the same as in Lemma 2.2 and D0 = (Q (1 + G0 V0 )Q )−1 Q , D1 = −D0 W1 D0 .

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Proof. For u ∈ F , such that Q (1 + G0 V0 )Q u = 0, we have Q u = u and 0 = Q (1 + G0 V0 )Q u = (1 + G0 V0 )u − Q(1 + G0 V0 )u = (1 + G0 V0 )u. This means u ∈ K and u = 0, by Lemma 3.4. This proves that Q (1 + G0 V0 )Q is injective on F . Since Q G0 V0 Q is compact on F , the Fredholm’s alternative theorem shows that Q (1 + G0 V0 )Q is also surjective on F . Consequently, D0 = (Q (1 + G0 V0 )Q )−1 Q ,

(3.16)

belongs to L(1, −s; 1, −s), s > 1/2. The second part of the proposition follows from Lemma 2.2 and Proposition 3.1 . Using the operator D(λ), we can construct an approximate inverse of A(λ). Set

B(λ) =

D(λ) S

T −SW (λ)T

: H 1,−s × Cµ → H 1,−s × Cµ , s > 1/2.

Proposition 3.6 For λ ∈ Uδ , the operator A(λ) is invertible from H 1,−s × Cµ to H 1,−s × Cµ . The inverse A(λ)−1 can be written in the form E(λ) E+ (λ) −1 A(λ) = , E− (λ) E+− (λ) where E(λ) = D(λ), E− (λ) = S − SX, with

E+ (λ) = T − D(λ)Y,

(3.17)

E+− (λ) = −SW (λ)T + SXY,

(3.18)

X = QW (λ)Q D(λ), Y = Q W (λ)T.

Proof. By direct calculation, we have, in L(H 1,−s × Cµ , H 1,−s × Cµ ), W (λ)D(λ) + T S W (λ)T − T SW (λ)T A(λ)B(λ) = . SD(λ) ST

(3.19)

Since ST = 1 and T S = Q, we can simplify the above expression to obtain X Y A(λ)B(λ) = 1 + . (3.20) 0 0 Here X and Y are deﬁned as above. We can check that X 2 = 0 and Y 2 = 0. So the left-hand side of the last equation is invertible. This proves A(λ) has a right inverse. Similarly, we can show that it has also a left inverse. A(λ) is thus invertible and −1 X Y −1 . (3.21) A(λ) = B(λ) 1 + 0 0 An easy calculation gives the desired result.

570

Xue Ping Wang

Ann. Henri Poincar´e

As before, we obtain from Proposition 3.6 a representation formula for the inverse of W (λ): W (λ)−1 = E(λ) − E+ (λ)E+− (λ)−1 E− (λ).

(3.22)

Remark also that if ρ > 2k + 1, E± (λ) have an asymptotic expansion of the form E± (λ) =

k

λj/2 E±,j + O(|λ|k/2+ ),

(3.23)

j=0

in L(Cµ ; H 1,−s ) or L(H 1,−s ; Cµ ), s > k + 1/2, where E−,0 = S,

E−,1 = −SW1 D0 ,

E+,0 = T,

E+,1 = −D0 W1 T.

and −1

The asymptotics of W (λ) near 0 are determined by that of E+− (λ)−1 near 0, which is just a µ × µ-matrix-valued function. Lemma 3.7 Assume ρ > 2k − 1 and k ≥ 2. One has: E+− (λ) =

k

λj/2 E+−,j + O(|λ|k/2+ ),

(3.24)

j=1

for 0 < < min{1/2, (ρ − 2k + 1)/4}. Here E+−,1 E+−,2

= =

−SW1 T, −SW2 T + SW1 D0 W1 T,

(3.25) (3.26)

E+−,3

=

−SW3 T + S(W1 D1 W1 + W2 D0 W1 + W1 D0 W2 )T

(3.27)

and > 0 is the same as in Proposition 3.5. In addition, if 0 is an exceptional point of the ﬁrst kind (in this case µ = 1), E+−,1 = c0 :=

i | V0 φ1 , 1 a |2 = 0. 4π

(3.28)

If 0 is an exceptional point of the second kind, then, E+−,1 = 0,

E+−,2 = ( (1 − V1 )φi , φj a )1≤i,j≤µ .

If 0 is both an eigenvalue and a resonance, E+−,1 form:  c0 0 · · ·  0 0 ···  E+−,1 =  . . . ..  .. .. 0

0

···

(3.29)

and E+−,2 can be written in the 0 0 .. . 0

   , 

(3.30)

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   E+−,2 =  

v1 v¯2 .. .

v2 (1 − V1 )φ2 , φ2 a .. .

··· ··· .. .

vµ (1 − V1 )φ2 , φµ a .. .

v¯µ

(1 − V1 )φµ , φ2 a

···

(1 − V1 )φµ , φµ a

571

   . 

(3.31)

Here c0 = 0 is the same as in (3.28) and vj ∈ C. Proof. From Proposition 3.6, one has E+− (λ) = −SW (λ)T + SW (λ)D(λ)W (λ)T, where W (λ) = 1 + R0 (λ)V (λ). −SW (λ)T = ( W (λ)φi , V0 φj a )1≤i,j≤µ . Note that in general, φj ∈ H 1,−s for any s > 1/2 and φj ∈ H 1,0 if it is an eigenfunction. For ρ > k + 1, we have an expansion of the form (3.24) for −SW (λ)T and if 0 is not a resonance, we just need ρ > k + 1/2 to obtain the same result. Remark that S(1 + G0 V0 ) = (1 + G0 V0 )T = 0. SW (λ) and W (λ)T have asymptotic expansions beginning with λ1/2 . In order to obtain an expansion of SW (λ)D(λ)W (λ)T up to the oder λk/2 , we just need to expand each factor up to the oder λ(k−1)/2 . By Proposition 3.1 and Proposition 3.5, the condition ρ > 2(k − 1) is suﬃcient. This proves (3.24). To give the leading term, we remark that in the case of an exceptional point of the ﬁrst kind, the 0 resonance is simple and the resonant state φ1 is characterized by the condition V0 φ1 , 1 a = 0. Since G1 is an integral operator with the constant integral kernel −SW1 T = G1 V0 φ1 , V0 φ1 a =

i 4π ,

i | V0 φ1 , 1 a |2 = 0. 4π

In the case of an exceptional point of the second kind, we have φj ∈ L2 for all j and thus, V0 φj , 1 a = 0, j = 1, · · · , µ. The same calculation as above shows E+−,1 = 0. For the same reason, we have also SW1 D0 W1 T = 0. Consequently, E+−,2 = −S(G2 V0 + G0 V1 )T = ( (G2 V0 + G0 V1 )φi , V0 φj a )i,j=1,...,µ Moreover, since φj ∈ L2 and V0 φj , 1 a = 0, j = 1, . . . , µ, we can check that G2 V0 φi , V0 φj a = φi , φj a . In fact, writing G2 = λ−1 (R0 (λ) − G0 − λ1/2 G1 ) + O(|λ| ), one has G2 V0 φi = −R0 (λ)φi + O(|λ| )V0 φi ,

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Xue Ping Wang

Ann. Henri Poincar´e

and G2 V0 φi , V0 φj a = − φi , R0 (λ)V0 φj a + O(|λ| ). R0 (λ)V0 φj = −φj − λR0 (λ)φj . Since φj ∈ L2 , for λ = iγ with γ ∈ R+ , γR0 (−iγ) converges to 0 in the strong operator topology on L2 . Therefore, taking the limit λ = iγ → 0, γ ∈ R+ , we obtain G2 V0 φi , V0 φj a = φi , φj a . It is clear that G0 V1 φi , V0 φj a = − V1 φi , φj a . This proves (3.29). The case of an exceptional point of the third kind can be calculated in the same way. Note in particular that in this case, V0 φ1 , 1 a = 0,

V0 φj , 1 a = 0, for j = 2, . . . , µ,

which implies that W1 φj = 0 for j ≥ 2. This allows to show that the (i, j)-entry, i ≥ 2, j ≥ 2, of E+−,1 is zero, while that of E+−,2 is (1 − V1 )φi , φj a . Remark If 0 is an exceptional point of the ﬁrst kind, under the assumption ρ > 2, one can obtain the leading term in (3.24). In fact, W (λ)T = ((R0 (λ) − G0 )V (λ) + G0 V1 (λ))T , where V1 (λ) = V (λ) − V0 = O(λ x −2ρ ). Applying (2.6) with k = 0 and s = ρ/2 W (λ)T = O(λr ) in L(C; H 1,−ρ/2 ) for any r < (ρ − 1)/4. Similarly, one has SW (λ) = O(λr ) in L(H 1,−ρ/2 ; C). This gives SW (λ)D(λ)W (λ)T = O(λ2r ). Since ρ > 2, we can take r such that 2r > 1/2. We obtain from the proof of Lemma 3.7 that E+− (λ) = λ1/2 E+−,1 + O(|λ|1/2+ ). This will allow us to obtain the leading term of the asymptotics of the resolvent in the case of an exceptional point of the ﬁrst kind under the assumption ρ > 2. Note that in the case of an exceptional point of the second or the third kind, the span of {φ1 , . . . , φµ } or {φ2 , . . . , φµ } is just the 0-eigenspace of −∆ + V0 . Let Π0 be the spectral projection in L2 onto the 0-eigenspace of −∆ + V0 . Proposition 3.8 (a) Suppose that 0 is an exceptional point of the ﬁrst kind. For ρ > 2k − 1, k ≥ 2, (2k − 1)/2 < s < ρ − (2k − 1)/2, we have (1 + R0 (λ)V (λ))−1 =

k−2

(1)

λj/2 Cj

+ O(|λ|(k−2)/2+ )

(3.32)

j=−1

where

(1)

C−1 = −c−1 0 Q.

−1/2+ If ρ > 2, one has (1 + R0 (λ)V (λ))−1 = −λ−1/2 c−1 ). 0 Q + O(λ

(3.33)

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(b) Suppose that 0 is an exceptional point of the second kind. For ρ > 2k − 1, k ∈ N with k ≥ 2, (2k − 1)/2 < s < ρ − (2k − 1)/2, we have k−4

(1 + R0 (λ)V (λ))−1 =

(2)

λj/2 Cj

+ O(|λ|(k−4)/2+ )

(3.34)

j=−2

where (2)

(2)

(2)

(2)

C−2 = (Π0 (1 − V1 )Π0 )−1 Π0 V0 , C−1 = −C−2 G3 V0 C−2 .

(3.35)

(c) Suppose that 0 is an exceptional point of the third kind. For ρ > 2k −1, k ∈ N with k ≥ 2, (2k − 1)/2 < s < ρ − (2k − 1)/2, we have k−4

(1 + R0 (λ)V (λ))−1 =

(3)

λj/2 Cj

+ O(|λ|(k−4)/2+ )

(3.36)

j=−2

where (3)

C−2

(3) C−1

=

(Π0 (1 − V1 )Π0 )−1 Π0 V0 ,

=

(3) (3) −C−2 G3 V0 C−2

+

c−1 0 (1

(3.37) +

A)T1 T1∗ (1

∗

+ A )V0

(3.38)

(3)

with A = C−2 W2 . Here λ ∈ Uδ with δ > 0 small, ∈]0, 1/2] veriﬁes < (2s − 2k + 1)/4 and the expansions are valid in L(1, −s; 1, −s). Proof. (a) If 0 is an exceptional point of the ﬁrst kind, then, Lemma 3.7 shows that for ρ > 2k − 1, k ≥ 2, E+− (λ) = c0 λ1/2 (1 +

k−1

(k−1)/2+ λj/2 c−1 )). 0 E+−,j+1 + O(|λ|

j=1

Since c0 = 0, E+− (λ) is invertible for λ suﬃciently small and E+− (λ)−1 has an −1/2 expansion up to the order λ(k−2)/2 with the leading term c−1 . To obtain 0 λ (3.32), we apply (3.22) and Proposition 3.6 to the order k − 2 and (3.23) to the order k − 1. The case ρ > 2 follows from the remark after Lemma 3.7. (b) In this case, for ρ > 2k − 1, k ≥ 2, one has E+− (λ) = λE+−,2 +

k

λj/2 E+−,j + O(|λ|k/2+ ).

j=3

Let T ∗ : L2 → Cµ be the adjoint of T . By Lemma 3.7, E+−,2 = T ∗ (1 − V1 )T . Since V1 ≤ 0, T ∗ (1 − V1 )T is invertible. So, E+− (λ) is invertible for λ small and we have an asymptotic expansion for E+− (λ)−1 to the order λ(k−4)/2 with the leading

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term λ−1 (T ∗ (1 − V1 )T )−1 . Then, (3.34) follows from (3.22) and Proposition 3.6 to the order k − 4 and (3.23) to the order k − 2. Since S = −T ∗ V0 , we have (2)

C−2 = −T (T ∗(1 − V1 )T )−1 S = T (T ∗(1 − V1 )T )−1 T ∗ V0 . (2)

To express C−2 in terms of the spectral projection Π0 , we remark that Π0 = T (T T ∗)−1 T ∗ ,

Π0 T = T,

T ∗ Π0 = T ∗ .

It is then easy to check that T (T ∗ (1 − V1 )T )−1 T ∗ = (Π0 (1 − V1 )Π0 )−1 Π0 and (2) C−2 = (Π0 (1 − V1 )Π0 )−1 Π0 V0 . (2)

For C−1 , we note that E+−,3 is reduced to −SW3 T and we have E+− (λ)−1 = λ−1 (T ∗ (1 − V1 )T )−1 + λ−1/2 (T ∗ (1 − V1 )T )−1 SW3 T (T ∗ (1 − V1 )T )−1 + · · · Hence,

(2)

C−1 = −(T ∗ (1 − V1 )T )−1 SW3 T (T ∗ (1 − V1 )T )−1 S (2)

(2)

(2)

(2)

= −C−2 W3 C−2 = −C−2 G3 V0 C−2 . (c) In the case where 0 is both an eigenvalue and a resonance, both E+−,1 and E+−,2 are non zero. To prove the existence of an asymptotic expansion, we need to study λ1/2 E+−,1 + λE+−,2 . Note that ρ > 2k − 1, k ≥ 2, one has E+− (λ) = λ1/2 E+−,1 + λE+−,2 +

k

λj/2 E+−,j + O(|λ|k/2+ ).

j=3

λ1/2 E+−,1 + λE+−,2 can be written in the form 1/2 λ c0 + λv1 1/2 λ E+−,1 + λE+−,2 = λv

λv ∗ λM

,

where v ∈ Cµ−1 and M = ( (1−V1 )φi , φj a )2≤i,j≤µ . Since V1 ≤ 0, M is invertible. We can compute −1/2 −1 1/2 −1 λ c0 v1 v ∗ M −1 λ c0 0 1/2 (λ E+−,1 + λE+−,2 ) =1+ 0 λ−1 M −1 λ1/2 c−1 0 0 v The right-hand side of the above equation is invertible for λ small with the inverse given by b(λ) −b(λ)v ∗ M −1 , ∗ −1 −λ1/2 c−1 1 + c−1 0 b(λ)v 0 b(λ)v v M

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T −1 −1 where b(λ) = (1+λ1/2 c−1 v )) . This shows that λ1/2 E+−,1 +λE+−,2 0 (v1 −v M is invertible for λ small and its inverse is given by

Ψ(λ) ≡ (λ1/2 E+−,1 + λE+−,2 )−1 1 0 0 −1 −1/2 −1 = λ c0 b(λ) +λ −M −1 v 0 M −1

−v ∗ M −1 −1 ∗ M v v M −1

.

Therefore, E+− (λ) is invertible for λ ∈ Uδ and k λj/2 E+−,j + O(|λ|k/2+ ))Ψ(λ))−1 E+− (λ)−1 = Ψ(λ)(1 + ( j=3 k = Ψ(λ)(1 − ( λj/2 E+−,j + O(|λ|k/2+ )Ψ(λ)) + · · · ) j=3

0 0 1 −v ∗ M −1 −1/2 −1 =λ c0 +λ −M −1 v M −1 v v ∗ M −1 0 M −1 0 0 0 0 −1/2 −λ E+−,3 + O(1) . 0 M −1 0 M −1 −1

The existence of the expansion (3.36) is deduced as in the ﬁrst two cases. Remark (3) (3) that C−2 and C−1 are determined by (3)

(3)

−T E+− (λ)−1 S = λ−1 C−2 + λ−1/2 C−1 + · · · To compute them, write T = (T1 , T ),

S=

S1 S

,

where T1 and S1 are just the ﬁrst components in T and S. As in (b), we have, T M −1 T ∗ = (Π0 (1 − V1 )Π0 )−1 Π0 . (3)

(2)

(3)

So C−2 = (Π0 (1−V1 )Π0 )−1 Π0 V0 . Compared with C−1 , C−1 has only an additional term from the zero energy resonance 1 −v ∗ M −1 −c−1 T S 0 −M −1 v M −1 v v ∗ M −1 ∗ ∗ −1 ∗ T − T M −1 v T1∗ + T M −1 v v ∗ M −1 T ∗ )V0 . = c−1 0 (T1 T1 − T1 v M (3)

Note that v = S W2 T1 = −T ∗ V0 W2 T1 and T M −1 T ∗ V0 = C−2 . The above equation is equal to (3)∗

(3)

(3)

(3)∗

∗ ∗ ∗ ∗ ∗ ∗ = c−1 0 (T1 T1 + T1 T1 W2 C−2 + C−2 W2 T1 T1 + C−2 W2 T1 T1 W2 C−2 )V0 . (3)

This proves the formula for C−1 .

Note that if V1 = 0, one can choose suitably the resonant state φ1 such that v = 0.

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Ann. Henri Poincar´e

Asymptotic expansions of R(z) when E0 ∈ σd (P )

Now we are ready to deduce the asymptotics of the resolvent R(z) near E0 . This is based on the following representation and Proposition 3.8. R(z) = =

(z) − + (z)+− (z)−1 − (z)

(3.39) −1

(z) + + (z)(1 + R0 (λ)V (λ))

R0 (λ)− (z),

with λ = z − E0 . Theorem 3.9 (a) Assume ρ > max{2, 2k − 1} for some k ∈ N with k ≥ 1. Suppose that E0 is an exceptional point of the ﬁrst kind of P . We have R(z) =

k−2

(1)

λj/2 Bj

+ O(|λ|(k−2)/2+ )

(3.40)

j=−1

where

(1)

B−1 = i ·, u u, with u an E0 resonant state of P satisfying 1 √ Ia (x)u(x)ϕ0 (xa ) dx = 1. 2 π X

(3.41)

(3.42)

(b) Assume ρ > 2k − 1 for some k ∈ N with k ≥ 2. Suppose that E0 is an exceptional point of the second kind (l = 2) or the third kind ( l = 3) of P . One has k−4 (l) λj/2 Bj + O(|λ|(k−4)/2+ ) (3.43) R(z) = j=−2

where (l)

B−2 (l) B−1

= −ΠE0 , =

(2) + (E0 )(C−1 G0

(3.44) +

(2) C−2 G1 )− (E0 ).

(3.45)

Here ΠE0 denotes the orthogonal spectral projection of P onto its E0 -eigenspace. The expansions are valid in L(−1, s; 1, −s) for (2k − 1)/2 < s < ρ − (2k − 1)/2 if k ≥ 2, for 1 < s < ρ − 1 if k = 1, λ ∈ Uδ with δ > 0 small enough and ∈]0, 1/2] veriﬁes < (2s − 2k + 1)/4. Proof. Since (z), + (z) and − (z) are holomorphic near E0 , making use of (3.39), the existence of the expansions (3.40) follows from Lemma 2.2 and Proposition 3.8. We need only to give the formula for the leading terms.

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577

(a) By Proposition 3.8 (a), we have (1)

B−1

=

−c−1 0 + (E0 )QG0 − (E0 )

=

−c−1 0 G0 − (E0 )·, −V0 φ1 a + (E0 )φ1 .

The proof of Proposition 3.3 shows that u0 = + (E0 )φ1 is a resonant state of P . Since − (E0 )∗ = + (E0 ) and G0 V0 φ1 = −φ1 . G0 − (E0 )·, −V0 φ1 a = ·, + (E0 )φ1 . (1)

So B−1 = −c−1 0 ·, u0 u0 . Let u=

√ 2 π + (E0 )φ1 . V0 φ1 , 1 a (1)

Then u is a resonant state of P and B−1 = i ·, u u. To verify the condition (3.42) on u, put √ 2 π φ= φ1 . V0 φ1 , 1 a Then, u = + (E0 )φ and

1 √ V0 φ, 1 a = 1. 2 π As in the proof of Corollary 3.4, we have 1 1 √ Ia (x)u(x)ϕ0 (xa ) dx = √ V0 φ, 1 a = 1. 2 π X 2 π

(b) Let l = 2. In this case, (2)

B−2

=

+ (E0 )(Π0 (1 − V1 )Π0 )−1 Π0 V0 G0 − (E0 ).

=

−+ (E0 )(Π0 (1 − V1 )Π0 )−1 Π0 − (E0 ).

Denote vj = + (E0 )φj . Then {vj , j = 1, . . . , µ} is a basis of the E0 -eigenspace of P . Let U = + (E0 )T . Then, T ∗ − (E0 ) = U ∗ . We can calculate U ∗U

=

T ∗ (1 − (R (E0 )Ia (ϕ0 ⊗ ·), ϕ0 )a

=

−(Ia R (E0 )(ϕ0 ⊗ ·), ϕ0 )a + (Ia R (E0 )2 Ia (ϕ0 ⊗ ·), ϕ0 )a )T T ∗ (1 − V1 )T,

because (R (E0 )Ia (ϕ0 ⊗·), ϕ0 )a = (Ia R (E0 )(ϕ0 ⊗·), ϕ0 )a = 0. As seen in the proof of Proposition 3.8, (2)

B−2 = −+ (E0 )T (T ∗ (1 − V1 )T )−1 T ∗ − (E0 ) can be simpliﬁed to −U (U ∗ U )−1 U ∗ = −ΠE0 . The case l = 3 can be proved in the same way. The details are omitted.

578

Xue Ping Wang

Ann. Henri Poincar´e

4 The case E0 ∈ σd (P ) In this section, suppose E0 ∈ σd (P ). Let m be its multiplicity and ψ1 , . . . , ψm be the orthonormal eigenfunctions of P : P ψj = E0 ψj ,

j = 1, . . . , m.

Denote Π1 = Π, Π2 u =

m

u ∈ L2 (X).

u, ψj ψj ,

j=1

Recall that Π is deﬁned in Section 3. Let Πj = 1 − Πj , j = 1, 2, and Π = 1 − Π1 − Π2 . Since the ranges of Π1 and Π2 are orthogonal, Π is an orthogonal projection and Π = Π1 Π2 = Π2 Π1 . Let P = Π P Π and R (z) = (P − z)−1 Π . R (z) is well deﬁned in a small neighborhood of E0 and is holomorphic there. Denote Hr,s = H r,s (Xa ) × Cm equipped with the norm (u2r,s + |v|2 )1/2 for (u, v) ∈ H r,s (Xa ) × Cm . Let H = L2 (Xa ) × Cm . Consider the operator 0 P − z E+ (4.1) P(z) = : D(P ) × H → L2 (X) × H, 0 E− 0 where

0 = E+

0 E+,1

0 E+,2

: H → L2 (X)

(4.2)

is deﬁned by 0 f E+,1 0 E+,2 c

f ∈ L2 (Xa ),   c1 m   = cj ψj , c =  ...  ∈ Cm ,

= ϕ0 ⊗ f,

j=1

and 0 = E−

One has

cm

0 E−,1 0 E−,2

0 0 E− = Π1 + Π2 E+

0∗ : L2 (X) → H. = E+

on L2 (X),

0 0 E− E+ = 1

on H.

(4.3)

As in Section 3, we can construct the inverse of P(z) by ﬁrstly considering an approximate one. Let 0 R (z) E+ , (4.4) R0 (z) = 0 0 E− E+−

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579

where 0 0 0 E+− (z) = E− (z − P )E+ .

A direct calculation shows that P(z) is invertible and −1

P(z)

R0 (z) 1 +

=

E(z)

≡

−A AB − B 0

0

E+ (z)

E− (z) E+− (z)

,

(4.5)

where A B

= Π1 Ia R (z),

= Π

(4.6)

0 P E+ .

(4.7)

and E(z) = E− (z) =

R (z), 0 E− (1

0 E+ (z) = (1 − R (z)Π P )E+ ,

− A),

0 0 E+− (z) = E+− (z) + E− (AB − B).

As usual, we derive from the above formulae the representation for the resolvent R(z): R(z) = E(z) − E+ (z)E+− (z)−1 E− (z).

(4.8)

0 E(z), E+ (z), E− (z) and E+− (z) are all holomorphic near E0 . Writing E± in blocs, 0 0 0 E+ = (E+,1 , E+,2 ),

0 E− =

0 E−,1 0 E−,2

,

we have E+− (z) = ≡

0 0 E−,1 (z − P + Π1 Ia R (z)Ia )E+,1 0 0 −E−,2 P E+,1 −C Pe (z) , ∗ −C z − E0

0 0 −E−,1 P E+,2

0 0 E−,2 (z − P )E+,2

where 0 0 Pe (z) = E−,1 (z − P + Ia R (z)Ia )E+,1

= z − E0 − (−∆a + (Ia ϕ0 , ϕ0 )a − (Ia R (z)Ia (ϕ0 ⊗ ·), ϕ0 )a )

(4.9)

580

Xue Ping Wang

Ann. Henri Poincar´e

0 0 0 0 and C = E−,1 Ia E+,2 : Cm → L2 (Xa ) and C ∗ = E−,2 Ia E+,1 : L2 (Xa ) → Cm are given by

Cv

=

m

vj φj ,

φj = (Ia ψj , ϕ0 )a , j = 1, . . . , m,

(4.10)

j=1

 Ia ϕ0 φ, ψ1

  .. =  . . Ia ϕ0 φ, ψm



C ∗φ

(4.11)

Note that ψj , j = 1, . . . , m, are rapidly decreasing on X. C ∗ extends to an operator from H 1,−s to Cm , ∀s. Lemma 4.1 Assume that rank {φ1 , . . . , φm } = m1 < m. Then E0 is an eigenvalue of P with multiplicity at least equal to m − m1 . Proof. Let (c1 , . . . , cm ) ∈ Cm \ {0} be such that

Put ψ =

c1 φ1 + · · · + cm φm = 0.

m

j=1 cj ψj .

Then, (Ia ψ, ϕ0 )a = 0 and P ψ = E0 ψ.

One can check that (P ψ, ϕ0 )a = ((−∆a + P a + Ia )ψ, ϕ0 )a = (−∆a + E0 )((ψ, ϕ0 )a ) = 0 because ψ is in the range of Π1 , which implies (ψ, ϕ0 )a = 0 in L2 (Xa ). The above equation shows that P ψ is in the range of Π1 and that P ψ = Π1 P ψ = P ψ = E0 ψ. ψ is an eigenfunction of P associated with E0 . The claim on the multiplicity is easy to verify. As before, set λ = z − E0 . We write E+− (z) in the form −∆ + V (λ) C E+− (z) = λ − C∗ 0 where

(4.12)

V (λ) = (Ia ϕ0 , ϕ0 )a − (Ia R (λ + E0 )Ia (ϕ0 ⊗ ·), ϕ0 )a

is holomorphic in λ near 0. We shall often expand it in the form V (λ) = V0 + λV1 + λ2 V2 + · · · , λ → 0. In the following, ∆ denotes ∆a . Put −∆ + V (λ) C P(λ) = , C∗ 0 We shall treat P(λ) as a perturbation of P0 .

P0 =

−∆ C C∗ 0

(4.13) .

(4.14)

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

581

Asymptotic expansions of (P0 − λ)−1

Lemma 4.2 Let Q be a self-adjoint operator in L2 (Xa ). Let λ ∈ C, λ > 0, be small enough. Then, Ker (C ∗ (Q − λ)−1 C) = Ker C. Proof. If v ∈ Cm is in Ker C ∗ (Q − λ)−1 C, (Q − λ)−1 Cv, Cv = 0. Note that (Q − λ)−1 > 0 for λ > 0. This gives Cv = 0. So v ∈ Ker C. This shows that Ker C ∗ (Q − λ)−1 C ⊆ Ker C. The inclusion in the other direction is trivial. Let π0 be the orthogonal projection from Cm onto Ker C and π0 = 1 − π0 . is the projection onto Range C ∗ . Lemma 4.2 implies that C ∗ (Q − λ)−1 C is bijective on Range C ∗ and that

π0

(π0 C ∗ (Q − λ)−1 Cπ0 )−1 π0

(4.15)

exists. In the following, we shall use this result with Q = −∆. In this case, Lemma 4.2 remains true for λ = 0. In fact, if C ∗ G0 Cv = 0, v ∈ Cm , then G0 Cv, Cv a = 0 which implies Cv = 0 in H 1,s . So v ∈ Ker C. This shows that Ker C ∗ G0 C ⊂ Ker C. The implication in the other direction is trivial. This proves that Ker C ∗ G0 C = Ker C. To give the asymptotic expansion of R0 (λ) = (P0 − λ)−1 , we resolve the system: u1 u1 v1 −∆ C −λ = . (4.16) u2 u2 v2 C∗ 0 u = (u1 , u2 ), v = (v1 , v2 ) are in H. This is equivalent to u1

=

(−∆ − λ)−1 [−Cu2 + v1 ],

v2

=

C ∗ (−∆ − λ)−1 [−Cu2 + v1 ] − λu2 .

C ∗ (−∆ − λ)−1 C + λ is invertible for λ ∈ Uδ . To see this, we note that C ∗ (−∆ − λ)−1 C + λ can be decomposed as C ∗ (−∆ − λ)−1 C + λ = λπ0 + π0 (C ∗ (−∆ − λ)−1 C + λ)π0 and has an asymptotic expansion to any order given by C ∗ G0 C + λ1/2 C ∗ G1 C + λ(1 + C ∗ G2 C) + · · ·

(4.17)

Since, by Lemma 4.2 with Q = −∆ and λ = 0, π0 C ∗ G0 Cπ0 is invertible on the range of π0 , π0 (C ∗ (−∆ − λ)−1 C + λ)π0 is also invertible there for λ small and we obtain (C ∗ (−∆ − λ)−1 C + λ)−1 = λ−1 π0 + (π0 (C ∗ (−∆ − λ)−1 C + λ)π0 )−1 π0 . (4.18)

582

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Ann. Henri Poincar´e

The solution of (4.16) is given by u1 u2

(−∆ − λ)−1 [−Cu2 + v1 ], (C ∗ (−∆ − λ)−1 C + λ)−1 [C ∗ (−∆ − λ)−1 v1 − v2 ].

= =

(4.19) (4.20)

The expressions for u1 and u2 in terms of v1 and v2 give the following Proposition 4.3 For λ ∈ Uδ , (−∆ − λ)−1 − D(λ)F (λ)D† (λ) R0 (λ) = F (λ)D† (λ)

D(λ)F (λ) −λ−1 π0 − F (λ)

(4.21)

where D(λ)

= (−∆ − λ)−1 C

D† (λ)

= C ∗ (−∆ − λ)−1

F (λ)

= (π0 (C ∗ (−∆ − λ)−1 C + λ)π0 )−1 π0 .

All these operators admit a complete expansion in the powers of λ1/2 in suitable spaces. In particular, F (λ) = F0 + λ1/2 F1 + λF2 · · · : Cm → Cm with

4.2

F0 F1

= (π0 C ∗ G0 Cπ0 )−1 π0 , = −F0 C ∗ G1 CF0 ,

F2

= −F0 (1 + C ∗ G2 C)F0 + (F0 C ∗ G1 C)2 F0 .

Spectral properties of P at the threshold

Let P = P(0) =

−∆ + V0 C∗

C 0

.

0 is called a resonance of P if the equation −∆ + V0 C u1 =0 C∗ 0 u2

(4.22)

has a solution for u = (u1 , u2 ) ∈ H1,−s = H 1,−s (Xa ) × Cm for any s > 1/2 and u1 ∈ L2 . As in Section 3, one can show the following Lemma 4.4 E0 is an eigenvalue (resonance, resp.) of P if and only if 0 is an eigenvalue (resonance, resp.) of P and the dimension of the eigenspaces (the resonant spaces, resp.) is the same.

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

(4.22) is equivalent to

or

(−∆ + V0 )u1 + Cu2 = 0 C ∗ u1 = 0

(1 + G0 V0 )u1 + G0 Cu2 = 0 C ∗ u1 = 0

583

(4.23)

.

(4.24)

This shows that u1 ∈ Ker C ∗ , C ∗ G0 Cu2 = −C ∗ G0 V0 u1 . Write u2 = v1 + v2 with v1 ∈ Ker C and v2 ∈ Range C ∗ . Since C ∗ G0 C is invertible on the range of C ∗ , v2 is uniquely determined by u1 v2 = −(π0 C ∗ G0 Cπ0 )−1 π0 C ∗ G0 V0 u1 = −F0 C ∗ G0 V0 u1 . Therefore, u1 satisﬁes

Set

(−∆ + V0 − CF0 C ∗ G0 V0 )u1 = 0 C ∗ u1 = 0

.

π1 = C(π0 C ∗ G0 Cπ0 )−1 π0 C ∗ G0 .

Then

π12

= π1 . π1 is the projection of H

−1,s

(4.25)

(4.26)

(4.27)

onto Range (C) with the kernel

Ker π1 = Ker (C ∗ G0 ). Let π1 = 1 − π1 . π1 is the projection H −1,s → Ker C ∗ G0 . Put V0# = π1 V0 . Summing up, we have proved the following Proposition 4.5 Let u = (u1 , u2 ) ∈ H1,−s = H 1,−s (Xa ) × Ck . Then, −∆ + V0 C u1 =0 C∗ 0 u2

(4.28)

if and only if u1 satisﬁes

(−∆ + V0# )u1 = 0 C ∗ u1 = 0

(4.29)

and then u2 is given by u2 = v1 + v2 ,

v1 ∈ Ker C, v2 = −π1 V0 u1 .

(4.30)

584

Xue Ping Wang

Ann. Henri Poincar´e

In the following, we shall say that u1 is an eigenvalue or a resonance of −∆ + V0# if there exists u1 ∈ H 1,−s , s suitable, satisfying (4.29). The spectral nature of 0 with respect to −∆ + V0# is classiﬁed as for P . Although V0# is nonlocal and non-selfadjoint, we can still mimic the proof of Lemma 3.3 of [11] for two-body Schr¨ odinger operators to derive the following Corollary 4.6 The kernel of P in H1,−s , ∀s > 1/2, is of ﬁnite dimension. The resonance of P at 0 (hence the resonance of P at E0 ) is at most simple and if u1 ∈ H 1,−s , ∀s > 1/2, is a solution of (4.29), then u1 ∈ L2 if and only if V0# u1 , 1 = 0 and in this case, (u1 , u2 ) with u2 given by (4.30) is an eigenfunction of P. Theorem 1.1 in the case E0 ∈ σd (P ) can be derived from Proposition 4.5 and its Corollary. Proof of Theorem 1.1 in the case E0 ∈ σd (P ). Note that u = (u1 , u2 ) is a solution of (4.28) if and only if w = E+ (E0 )u is in the kernel of P − E0 in H 1,−s and that w ∈ L2 if and only if u ∈ H. Making use of the expression of E+ (E0 ), one has Ia (x)w(x)ϕ0 (xa ) dx = (Ia (E+,1,0 u1 + E+,2,0 u2 ), ϕ0 )a , 1 a X

=

(1 − CF0 C ∗ G0 )V0 u1 , 1 a = V0# u1 , 1 a = 0.

This ﬁnishes the proof of Theorem 1.1.

As in Section 3, the main step to obtain an asymptotic expansion of the resolvent R(z) is to study E+− (z)−1 . As before, we shall apply the method of Grushin to reduce to a holomorphic family of matrices whose behavior near 0 depends on the threshold properties of P . We write V (λ) 0 . (4.31) P(λ) = P0 + V(λ), where V(λ) = 0 0 Then,

−E+− (z)−1 = (1 + R0 (λ)V(λ))−1 R0 (λ).

(4.32)

Using Proposition 4.3, we can compute T(λ) ≡ R0 (λ)V(λ) =

A(λ) B(λ)

0 0

where A(λ) B(λ)

= =

((−∆ − λ)−1 − D(λ)F (λ)D† (λ))V (λ) †

F (λ)D (λ)V (λ).

(4.33) (4.34)

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

An elementary calculation shows that (1 + A(λ))−1 (1 + T(λ))−1 = −B(λ)(1 + A(λ))−1 Set

T0 = T(0) =

(G0 − D(0)F0 D† (0))V0 F0 D† (0)V0

0 1 0 0

585

(4.35) .

T0 is a compact operator on H1,−s , 1/2 < s < ρ − 1/2. The kernel, K, of 1 + T0 in H1,−s is related to that of P. Let (u, v) ∈ Ker (1 + T0 ). Then,

(1 + (G0 − D(0)F0 D† (0))V0 )u F0 D† (0)V0 u + v

= 0 = 0.

Therefore, v is uniquely determined by u v = −F0 D† (0)V0 u.

(4.36)

It is clear that v ∈ range C ∗ . The equation for u shows that C ∗ u = −(C ∗ G0 − C ∗ G0 CF0 C ∗ G0 )V0 u As in the proof of Proposition 4.5, we can show that C ∗ G0 CF0 = C ∗ G0 C(π0 C ∗ G0 Cπ0 )−1 π0 = π0 . This proves C ∗ u = −(C ∗ G0 − π0 C ∗ G0 )V0 u = 0. u satisﬁes the equation (−∆ + V0 − CF0 C ∗ G0 V0 )u = 0. It follows that (u, v) ∈ Ker (1 + T0 ) if and only if (−∆ + V0# )u = 0,

v = −F0 C ∗ G0 V0 u.

(4.37)

This shows that (u, v) is in Ker P with v ∈ Range C ∗ . These arguments can be reversed. The relation between T0 and A(0) gives immediately the following Proposition 4.7 The kernel of 1 + T0 in H1,−s is given by (4.37) and coincides with that of P in H 1,−s × Range C ∗ . u ∈ Ker (1 + A(0)) if and only if (u, v) ∈ Ker (1 + T0 ) with v given by v = −F0 C ∗ G0 V0 u .

586

Xue Ping Wang

Ann. Henri Poincar´e

Note that we can decompose Ker P = Ker (1 + T0 ) ⊕ ({0} × Ker C). The part {0} × Ker C of the kernel of P corresponds to part of the contribution from the zero eigenvalue of P and is already taken into account in R0 (λ). This separation is necessary for our method to work. Proposition 4.8 Let A0 = A(0). Then one has (a) π1∗ = 0, π1∗ = 1 on Ker (1 + A0 ). (b) G0 π1 = π1∗ G0 , G0 π1 = π1∗ G0 and C ∗ G0 π1 = 0. (c) G2 V0# u, V0# v = u, v for u, v ∈ Ker (1 + A0 ) ∩ L2 . Proof. Recall that π1 = 1 − π1 and π1 = C(π0 C ∗ G0 Cπ0 )−1 π0 C ∗ G0 . Since Ker (1 + A0 ) ⊂ Ker C ∗ by Proposition 4.7, (a) is evident. To see (b), we remark G0 π1 = G0 C(π0 C ∗ G0 Cπ0 )−1 π0 C ∗ G0 = π1∗ G0 and

C ∗ G0 π1 = C ∗ G0 C(π0 C ∗ G0 Cπ0 )−1 π0 C ∗ G0 = π0 C ∗ G0 = C ∗ G0 .

Therefore, C ∗ G0 π1 = 0. (c). It can be proved as in Lemma 3.8.

4.3

Asymptotic expansions of (1 + A(λ))−1

The asymptotic expansion of (1 + A(λ))−1 can be obtained as in Section 3.3 by the method of Grushin, but the calculation is more complicated. We begin with the following preparatory result. Lemma 4.9 Let ρ > max{2, 2k + 1}, k ∈ N. A(λ) =

k

λj/2 Aj + O(λk/2+ ), λ → 0,

(4.38)

j=0

in L(1, −s; 1, −s) for 1 < s < ρ − 1 if k = 0 and (2k + 1)/2 < s < ρ − (2k + 1)/2 if k ≥ 1. Here > 0 can be chosen as in Lemma 2.2 and A0 A1 A2 with

= G0 π1 V0 = G1 π1 V0 + G0 M1 V0

= G0 π1 V1 + G0 M2 V0 + G1 M1 V0 + G2 π1 V0 M1 = −CF0 C ∗ G1 π1 ,

M2 = −CF2 C ∗ G0 − CF1 C ∗ G1 − CF0 C ∗ G2 .

(4.39) (4.40) (4.41)

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

587

Proof. We can write A(λ) = R0 (λ)(1 − CF (λ)C ∗ R0 (λ))V (λ) . Notice that C ∗ : H 1,−s → Cm is well deﬁned for any s ∈ R. Therefore, for any k ∈ N and s > (2k + 1)/2, ∗

1 − CF (λ)C R0 (λ) =

k

λj/2 Mj + O(λk/2+ ), in L(−1, s; −1, s),

(4.42)

j=0

where

M0 = 1 − CF0 C ∗ G0 = π1 ,

π1 being the projection onto Ker C ∗ G0 and M1 = −CF1 C ∗ G0 − CF0 C ∗ G1 , M2 = −CF2 C ∗ G0 − CF1 C ∗ G1 − CF0 C ∗ G2 . Note that F1 = −F0 C ∗ G1 CF0 and CF0 C ∗ G0 = π1 . One has M1 = CF0 C ∗ G1 CF0 C ∗ G0 − CF0 C ∗ G1 = CF0 C ∗ G1 (π1 − 1) = −CF0 C ∗ G1 π1 . (4.38) then follows from the asymptotic expansions of R0 (λ) and V (λ) in suitable spaces. Let ν = dim Ker (1 + A(0)) = dim Ker (1 + T0 ) and θ = dim Ker C. Then, ν + θ = dim Ker P. (4.43) By Proposition 4.8, π1∗ = 1 on Ker (1 + A0 ). The map (Ker (1 + A0 ))2 (u, v) → u, −V0 v a = u, −V0# v a ∈ C is a Hermitian form and is positive, since −V0# u, u a = −∆u, u a is positive on Ker (1+A0 ). As in Section 3, we can show that it is non-degenerate. Consequently, we can choose a basis {u1 , . . . , uν } of Ker (1 + A0 ) such that ui , −V0 uj a = δij ,

(4.44)

and if there is the zero energy resonance of −∆ + V0# , we assume without loss that u1 is a resonant state. Proposition 4.10 (a) Suppose that 0 is a regular point for −∆ + V0# (in this case, ν = θ = 0). Then for ρ > max{2, 2k + 1}, k ∈ N, we have (1 + A(λ))−1 =

k j=0

(0)

λj/2 Aj + O(|λ|k/2+ )

(4.45)

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Xue Ping Wang

Ann. Henri Poincar´e

in L(1, −s; 1, −s) for 1 < s < ρ − 1 when k = 0 and (2k + 1)/2 < s < ρ − (2k + 1)/2 when k ≥ 1. Here (0)

A0 = (1 + A0 )−1 .

(4.46)

(b) Suppose that 0 is an exceptional point of the ﬁrst kind of −∆ + V0# (in this case, ν = 1, θ = 0). For ρ > max{2, 2k − 1}, k ≥ 1, we have −1

(1 + A(λ))

=

k−2

(1)

λj/2 Aj + O(|λ|(k−2)/2+ )

(4.47)

j=−1

in L(1, −s; 1, −s) for 1 < s < ρ − 1 when k = 1 and (2k − 1)/2 < s < ρ − (2k − 1)/2 when k ≥ 2, where (1)

A−1 = −c−1 0 ·, −V0 u1 a u1 with c0 =

i | V # u1 , 1 a |2 . 4π 0

(4.48)

(c) Suppose that 0 is an exceptional point of the second (l = 2) or the third (l = 3) kind of −∆ + V0# . For ρ > 2k − 1, k ∈ N with k ≥ 2, (2k − 1)/2 < s < ρ − (2k − 1)/2, we have (1 + A(λ))−1 =

k−4

(l)

λj/2 Aj + O(|λ|(k−4)/2+ )

(4.49)

j=−2

where

(l)

A−2 = (Π0 (1 − V1 + W ∗ W )Π0 )−1 Π0 V0 ,

l = 2, 3.

(4.50)

Here Π0 is the orthogonal projection in L2 onto the 0 eigenspace of −∆ + V0# and W = F0 C ∗ G0 V0 . The above expansions are valid in L(1, −s; 1, −s) for λ ∈ Uδ with δ > 0 small and > 0 is the same as in Proposition 3.9. Proof. The method is the same as in Subsection 3.3. We just sketch the main steps for (b) and (c). In order to get the expansion of (1 + A(λ))−1 , consider 1 + A(λ) T F (λ) = : H 1,−s × Cν → H 1,−s × Cν , S 0 where s > 1/2, operators T and S are deﬁned by     c1 c1 ν     cj uj ,  ...  ∈ Cν , T  ...  = j=1 cν cν

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589

 f, −V0 u1 a   .. 1,−s Sf =  . , f ∈ H . f, −V0 uν a 

and

Let Qf =

ν

j=1 f, −V0 uj a uj .

Then,

T S = Q on H 1,−s and ST = Iν on Cν . Q is the projection from H 1,−s onto the range of T with Ker Q = Ker S. The following relations can be easily veriﬁed: S(1 + A0 ) = 0, SQ = S,

(1 + A0 )T = 0,

(4.51)

QT = T .

(4.52)

Let Q = 1 − Q. Then one can show that D(λ) = (Q (1 + A(λ))Q )−1 Q . is well deﬁned near λ = 0 and D(λ) =

k

λj/2 Dj + O(|λ|k/2+ ),

(4.53)

j=0

in L(1, −s; 1, −s), (2k + 1)/2 < s < ρ − (2k + 1)/2 and D0 = (Q (1 + A0 )Q )−1 Q , D1 = −D0 A1 D0 . In the case Ker (1 + A0 ) = {0}, ν = 0 and Q = 1. In this case, (a) follows from (4.53). In the general case, we construct the inverse of F (λ) and give a representation formula for (1 + A(λ))−1 . Using the operator D(λ), we can ﬁrst construct an approximate inverse G(λ), D(λ) T G(λ) = : H 1,−s × Cν → H 1,−s × Cν , s > 1/2. S −S(1 + A(λ))T of F (λ) to show that F (λ) is invertible and then to calculate explicitly the inverse E(λ) E+ (λ) −1 F (λ) = , E− (λ) E+− (λ) where E(λ) E+ (λ)

= =

D(λ), T − D(λ)Q (1 + A(λ))T ,

E− (λ)

=

S − SQ(1 + A(λ))Q D(λ),

E+− (λ)

=

(4.54) (4.55) (4.56)

S(−1 + (1 + A(λ))Q D(λ)Q )(1 + A(λ))T .

(4.57)

590

Xue Ping Wang

Ann. Henri Poincar´e

As before, we obtain from this inverse a representation formula for the inverse of 1 + A(λ): (1 + A(λ))−1 = E(λ) − E+ (λ)E+− (λ)−1 E− (λ). (4.58) Remark also that if ρ > 2k + 1, E± (λ) have an asymptotics expansion up to an error O(|λ|k/2+ ). k λj/2 E±,j + O(|λ|k/2+ ), (4.59) E± (λ) = j=0

in L(C ; H µ

1,−s

) or L(H

1,−s

; C ), s > k + 1/2, where µ

E−,0 = S,

E−,1 = −SA1 D0 ,

E+,0 = T ,

E+,1 = −D0 A1 T .

and For E+− (λ), we have a similar result of Lemma 3.7. Assume ρ > 2k − 1 and k ≥ 2. One has: k E+− (λ) = λj/2 E+−,j + O(|λ|k/2+ ). (4.60) j=1

Here E+−,1 E+−,2

= −SA1 T , = −SA2 T + SA1 D0 A1 T ,

(4.61) (4.62)

E+−,3

= −SA3 T + S(A1 D1 A1 + A2 D0 A1 + A1 D0 A2 )T

(4.63)

and > 0 is the same as in Proposition 3.5. We should calculate the leading terms of E+− (λ) to show that E+− (λ)−1 admits an asymptotic expansion. We carry out the computation in the most general case, i.e., 0 is an eigenvalue and a resonance of −∆ + V0# . Since A1 = G1 V0# + G0 M1 V0 , E+−,1

= ( (G1 V0# + G0 M1 V0 )ui , V0 uj a )i,j=1,...,ν = ( π1∗ G1 V0# ui , V0 uj a )i,j=1,...,ν

(4.64)

G1 being an integral operator with the constant integral kernel i/(4π), G1 V0# ui , V0# uj a =

i V # ui , 1 a V0# uj , 1 a . 4π 0

By the characterization of resonant states, V0# ui , 1 a = 0 if and only if i = 1. So we obtain   c0 0 · · · 0  0 0 ··· 0    E+−,1 =  . . . , (4.65) . . ...    .. .. 0

0

···

0

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with c0 =

591

i | V # u1 , 1 a |2 = 0 . 4π 0

Write E+−,2 as E+−,2 = ( A2 ui , V0 uj a − A1 D0 A1 ui , V0 uj a )i,j=1,...,ν . Note that

A1 = G1 V0# − G0 CF0 C ∗ G1 V0# = π1∗ G1 V0# .

Making use of the fact V0# uj , 1 a = 0 for j ≥ 2, we can compute A1 D0 A1 ui , V0# uj a = G1 V0# D0 π1∗ G1 V0# ui , V0# uj a = 0, if (i, j) = (1, 1). Set

b11 = G1 V0# D0 π1∗ G1 V0# u1 , V0# u1 a .

Then,

   ( A1 D0 A1 ui , V0# uj a )i,j=1,...,ν =  

b11 0 .. . 0

0 ··· 0 ··· .. . . . . 0 ···

0 0 .. .

   . 

(4.66)

0

The operator A2 is given by A2 = G0 π1 V1 + G2 V0# + G1 M1 V0 + G0 M2 V0 with M1

=

M2

=

−CF0 C ∗ G1 π1 ,

= −CF2 C ∗ G0 − CF1 C ∗ G1 − CF0 C ∗ G2 .

Using the formulae of F1 and F2 , we can calculate G0 M2 V0

= =

−G0 C(F2 C ∗ G0 + F1 C ∗ G1 + F0 C ∗ G2 )V0 (G0 CF02 C ∗ G0 − π1∗ G2 π1 + π1∗ G1 CF0 C ∗ G1 π1 )V0 .

A2 can be written as A2 = G0 π1 V1 + G0 CF02 C ∗ G0 V0 + π1∗ G2 π1 V0 − π1∗ G1 CF0 C ∗ G1 π1 V0 .

(4.67)

The contribution to E+−,2 of each term of A2 can be calculated as follows. Since G0 π1 = π1∗ G0 , G0 π1 V1 ui , V0 uj a

=

π1∗ G0 V1 ui , V0 uj a

= V1 ui , G0 V0# uj a

=

− V1 ui , uj a .

592

Xue Ping Wang

Ann. Henri Poincar´e

Let W = F0 C ∗ G0 V0 . Then G0 CF02 C ∗ G0 V0 ui , V0 uj a = W ∗ W ui , uj a . By Proposition 4.8 (c), one has π1∗ G2 V0# ui , V0 uj a

=

G2 V0# ui , V0# uj a

=

ui , uj a

for i, j = 2, . . . , ν. By the characterization of resonant state, − π1∗ G1 CF0 C ∗ G1 V0# ui , V0 uj a = − G1 CF0 C ∗ G1 V0# ui , V0# uj a = 0, for (i, j) = (1, 1). In the basis (u1 , . . . , uν ), E+−,2 can be written in the form    E+−,2 =   Here aj ∈ C and

a1 a ¯2 .. .

a2 e22 .. .

··· ··· .. .

aν e2ν .. .

a ¯ν

eν2

···

eνν

    . 

eij = (1 − V1 + W ∗ W )ui , uj a .

(4.68)

(4.69)

Here, it is important to notice that V1 ≤ 0 which ensures that the matrix e = (eij )2≤i,j≤ν is invertible. The expressions for E+−,1 and E+−,2 are similar to those obtained in Section 3. Following the discussions in the proof of Proposition 3.9, we can show that λ1/2 E+−,1 + λE+−,2 is invertible for λ near 0 and that E+− (λ)−1 admits an asymptotic expansion near λ = 0. From (4.58), we deduce the existence of the asymptotic expansion of (1 + A(λ))−1 near λ = 0. The leading term in the expansions and the condition on ρ can be checked as in Proposition 3.9. The details are left to the reader.

4.4

Asymptotic expansions of R(z) when E0 ∈ σd (P ).

To establish the expansion of R(z) we use the relation R(z) = E(z) + E+ (z)(1 + R0 (λ)V(λ))−1 R0 (λ)E− (z).

(4.70)

The operator M(λ) ≡ (1 + R0 (λ)V(λ))−1 R0 (λ) can be computed as follows: m11 (λ) m12 (λ) (4.71) M(λ) = m21 (λ) m22 (λ)

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593

where m11 (λ)

= (1 + A(λ))−1 (G0 − G0 CF0 C ∗ G0 + O(λ1/2 ))

m12 (λ)

= (1 + A(λ))−1 (G0 π1 + O(λ1/2 )), = (1 + A(λ))−1 (G0 CF0 + O(λ1/2 )),

m21 (λ) m22 (λ)

= −(F0 C ∗ G0 V0 + O(λ1/2 ))m11 (λ) + F0 C ∗ G0 + O(λ1/2 ), = −(F0 C ∗ G0 V0 + O(λ1/2 ))m12 (λ) − λ−1 π0 − F0 + O(λ1/2 ).

Let k ∈ N. Under suitable assumptions on ρ and s depending on k and on the nature of the threshold, we have mij (λ) =

k

(l)

λl/2 mij + O(λk/2+ ).

l=−2

M(λ) has an asymptotic expansion of the form M(λ) =

k

λj/2 Mj + O(λk/2+ ),

(4.72)

j=−2

where Mj =

(j)

m11 (j) m21

(j)

m12 (j) m22

.

M−2 = M−1 = 0 if E0 is a regular point of P . M−2 = 0 if E0 is an exceptional point of the ﬁrst kind. Note that E(z), E± (z) are holomorphic near z = E0 . Thus we can write E(z) =

R (z) =

∞

λj Ej ,

j=0

E+ (z) =

0 = (1 − R (z)Π P )E+ =

∞

λj E+,j ,

j=0

E− (z) =

0 E− (1 − Π1 Ia R (z)) =

∞

λj E−,j .

j=0

Set E+,j = (E+,1,j , E+,2,j ),

E−,j =

E−,1,j E−,2,j

.

It follows that in suitable spaces R(z) has an asymptotic expansion of the form R(z) = λj/2 (Ej/2 + E+,l1 /2 Ml2 E−,l3 /2 ). (4.73) j≥−2

l1 +l2 +l3 =j

Here Ej/2 , E±,j/2 are set to 0 if j/2 ∈ N. More precisely, one has the following

594

Xue Ping Wang

Ann. Henri Poincar´e

Theorem 4.11 (a) Assume that E0 is neither eigenvalue, nor resonance of P . Then, for k ∈ N, ρ > max{2, 2k + 1} and (2k + 1)/2 < s < ρ − (2k + 1)/2 for k ≥ 1 and 1 < s < ρ − 1 for k = 0, one has R(z) =

k

(0)

λj/2 Bj

+ O(|λ|k/2+ ),

(4.74)

j=0

in L(−1, s; 1, −s). Here (0)

B0

=

∗ R (E0 ) + C1 (G0 π1 E−,1,0 + C2∗ ) − C2 V0 (1 + G0 V0# )−1 C(4.75) 2

+C2 (1 − V0 (1 + G0 V0# )−1 G0 π1 E−,1,0 − K0 with E−,1,0

=

((1 − Ia R (E0 ))·, ϕ0 )a ,

C1 C2

= =

(1 − R (E0 )Ia )(ϕ0 ⊗ ((1 + G0 V0# )−1 ·)), K0 (ϕ0 ⊗ G0 ·) and

K0

=

0 0 (Π2 Ia E+,1 G0 E−,1 Ia Π2 )−1 Π2 .

(b) Suppose that E0 is an exceptional point of the ﬁrst kind of P . Assume ρ > max{2, 2k − 1}, k ∈ N with k ≥ 1. We have R(z) =

k−2

(1)

λj/2 Bj

+ O(|λ|(k−2)/2+ )

(4.76)

j=−1

in L(−1, s; 1, −s) with s > 1 if k = 1 and (2k − 1)/2 < s < ρ − (2k − 1)/2 if k ≥ 2. Here (1) B−1 = i ·, u u, (4.77) and u is an E0 resonant state of P satisfying 1 √ Ia (x)u(x)ϕ0 (xa ) dx = 1. 2 π X (c) Suppose that E0 is an exceptional point of the second or the third kind of P . Assume ρ > 2k − 1, k ∈ N with k ≥ 2. One has R(z) =

k−4

(l)

λj/2 Bj + O(|λ|(k−4)/2+ ),

l = 2, 3,

(4.78)

j=−2

in L(−1, s; 1, −s) for (2k − 1)/2 < s < ρ − (2k − 1)/2, λ ∈ Uδ with δ > 0 small enough and ∈]0, 1/2] verifying < (2s − 2k + 1)/4. Here (2)

(3)

B−2 = B−2 = −ΠE0 , with ΠE0 the orthogonal projection onto the eigenspace of P associated with the eigenvalue E0 .

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

595

Proof. We just give the symbolic calculus of the leading term in each case and leave to the reader the care to check the condition on ρ and s. (a) In the case where E0 is a regular point of P , Ker (1 + A0 ) = {0}, π0 = 0 and M−2 = M−1 = 0. The entries of M0 are given by (0)

m11

(0)

m12

(0)

m21

(0)

m22

=

(1 + G0 V0# )−1 G0 π1 ,

=

(1 + G0 V0# )−1 G0 CF0 ,

=

F0 C ∗ G0 (1 − V0 m11 ),

=

−F0 C ∗ G0 V0 m12 − F0 .

(0)

(0)

Here F0 = (C ∗ G0 C)−1 . Note that 0 , E+,1,0 = (1 − R (E0 )Ia Π1 )E+,1

0 E+,2,0 = E+,2 ,

0 (1 − Π1 Ia R (E0 )), E−,1,0 = E−,1

0 E−,2,0 = E−,2 .

Therefore (0)

B0

=

R (E0 ) + E+,0 M0 E−,0

=

R (E0 ) +

=

R (E0 ) + C1 (G0 π1 E−,1,0 + C2∗ ) + C2 (1 − V0 (1 + G0 V0# )−1 G0 π1 E−,1,0

2

(0)

E+,i,0 mij E−,j,0

i,j=1

−C2 V0 (1 + G0 V0# )−1 C2∗ − E+,2,0 F0 E−,2,0 with C1 C2

= =

E+,1,0 (1 + G0 V0# )−1 , E+,2,0 F0 C ∗ G0 .

0 G0 = ϕ0 ⊗ G0 · : H −1,s (Xa ) → H 1,−s (X). Let K0 = E+,2,0 F0 E−,2,0 and K1 = E+,1 Then, 0 0 G0 E−,1 Ia Π2 )−1 Π2 K0 = (Π2 Ia E+,1

and C2 = K0 K1 . This proves (a). (b) Suppose that E0 is an exceptional point of the ﬁrst kind of P . Then π = 0 (1) (1) and dim Ker (1 + A0 ) = 1. In this case, B−2 = 0 and B−1 is given by (1)

B−1 = E+,0 M−1 E−,0 .

596

Xue Ping Wang

Ann. Henri Poincar´e

(1)

To give the desired formula for B−1 , we write down the entries of M−1 (−1)

m11

(−1)

m12

(−1)

m21

(−1)

m22 (−1)

m11

(1)

=

A−1 G0 π1 ,

=

A−1 G0 CF0 ,

=

−F0 C ∗ G0 V0 m11 ,

=

−F0 C ∗ G0 V0 m12 .

(1)

(−1) (−1)

−1 = −c−1 0 G0 π1 ·, −V0 u1 a u1 = −c0 ·, u1 a u1 ,

where u1 is a resonant state normalized by u1 , −V0 u1 a = 1. (−1)

m12

−1 = −c−1 0 G0 CF0 ·, −V0 u1 a u1 = −c0 ·, v1 a u1 ,

where v1 = −F0 C ∗ G0 V0 u1 . It follows that

(−1)

= −c−1 0 ·, u1 a v1 ,

(−1)

= −c−1 0 ·, v1 a v1 .

m21 and

m22

Note that according to Proposition 4.7, w = (u1 , v1 ) ∈ Ker (1 + T0 ) is a resonant state of P. This proves that M−1 : H−1,s → H1,−s can be written in the form M−1 = −c−1 0 ·, w w. Let u=

√ 2 π V0# u1 , 1 a

E+,0 w =

√ 2 π V0# u1 , 1 a

(E+,1,0 u1 + E+,2,0 v1 ).

Then, u is a resonant state of P and (1)

B−1 = E+,0 M−1 E−,0 = i ·, u u. The normalizing condition on u becomes now √ √ 2 π a Ia (x)u(x)ϕ0 (x ) dx = π1 V0 u1 , 1 a = 2 π. # V0 u1 , 1 a X This proves (b). (c) Suppose that E0 is an exceptional point of the second kind. Then, (2)

B−2 = E+,0 M−2 E−,0 ,

(4.79)

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Asymptotic Behavior of Resolvent for N -Body Schr¨ odinger Operators

597

where the entries of M−2 are given by (−2)

= XV0 G0 π1 = −X,

(−2)

= XV0 G0 CF0 = XW ∗ ,

(−2)

= −F0 C ∗ G0 V0 m11

m11 m12 m21

(−2) m22

(−2)

= −F0 C

∗

(−2) G0 V0 m12

= −W X, − π0 = −W XW ∗ − π0 ,

with X W

= =

(Π0 (1 − V1 + W ∗ W )Π0 )−1 Π0 , F0 C ∗ G0 V0 . (2)

Suppose ﬁrst π0 = 0. Then, B−2 can be written in the form 1 (2) B−2 = −E+,0 · X · (1, −W ∗ )E−,0 . −W In the case π0 = 0, Ker (1 + T0 ) = Ker P and E0,+ : Ker P → Ker (P − E0 ) is bijective. Note also that u ∈ Ker (1 + T0 ) if and only if u ∈ Ker (1 + A0 ) and v = −W u. v This shows that

K ≡ E+,0

1 −W

: Range Π0 → Ker (P − E0 )

is bijective. (1, −W ∗ )E−,0 = K∗ . We can calculate K = E+,1,0 − E+,2,0 W and K∗ K

=

E−,1,0 E+,1,0 − W ∗ E−,2,0 E+,1,0 − E−,1,0 E+,2,0 W + W ∗ E−,2,0 E+,2,0 W

=

E−,1,0 E+,1,0 + W ∗ E−,2,0 E+,2,0 W.

Making use of the formulae for E±,j,0 , j = 1, 2, one derives that E−,1,0 E+,1,0 ∗

W E−,2,0 E+,2,0 W

0 0 = E−,1 (1 + Π1 Ia R (E0 )2 Ia )E+,1 = 1 − V1

= W ∗ W.

(2)

Therefore, −B−2 = KXK∗ satisﬁes (KXK∗ )2 = KX(1 − V1 + W ∗ W )XK∗ = KXK∗ .

598

Xue Ping Wang

Ann. Henri Poincar´e

Therefore, KXK∗ is the projection onto the range of KΠ0 with the kernel equal to Ker Π0 K∗ . KXK∗ is thus the orthogonal projection onto the range of KΠ0 . Since the range of KΠ0 is just the eigenspace of P associated with E0 , we conclude KXK∗ = ΠE0 . This proves (c) for l = 2 in the case π0 = 0. In the case π0 = 0, 1 (2) B−2 = −E+,0 X(1, −W ∗ )E−,0 − E+,2,0 π0 E−,2,0 −W −KXK∗ − E+,2,0 π0 E−,2,0 .

=

The fact that π0 W = 0 implies that K∗ E+,2,0 π0 = −W ∗ 0π0 = 0 and π0 E−,2,0 K = −π0 W = 0. This allows to verify

(2)

(2)

(−B−2 )2 = −B−2 . (2)

This shows that −B−2 is a projection. Note that E+,0 : Ker P = Ker (1 + T0 ) ⊕ ({0} × Range π0 ) → Ker (P − E0 ) (2)

is bijective. One can check that the range of −B−2 is equal to Range KΠ0 ⊕ (2) Range E+,2,0 π0 which is equal to Ker (P − E0 ) and the kernel of −B−2 is equal to ∗ Ker (Π0 K∗ ) ∩ Ker (π0 E0,+ ), which is orthogonal to Range KΠ0 ⊕ Range E+,2,0 π0 . (2)

This proves that −B−2 = ΠE0 for l = 2 in the general case. The case of an exceptional point of the third kind (l = 3) can be treated in the same way as for l = 2, since by Proposition 4.10, the resonance at E0 does not contribute to the leading term.

References [1] S. Agmon, Spectral properties of Schr¨ odinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa, Ser. IV2, 151–218 (1975). [2] S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Math. Notes 29, Princeton Univ. Press. [3] S. Albeverio, D. Boll´e, F. Gesztesy and R. Hoegh-Krohn, Low-energy parameters in nonrelativistic scattering theory, Ann. Physics 148, 308–326 (1983).

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[4] M. Ben-Artzi, Y. Dermenjian and J.-C. Guillot, Analyticity properties and estimates of resolvent kernels near thresholds, Commun. in PDE 25, 1753– 1770 (2000). [5] D. Boll´e, Schr¨odinger operators at threshold, pp. 173–196, in Ideas and Methods in Quantum and Statistical Physics, Cambridge Univ. Press, Cambridge, 1992. [6] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, with Application to Quantum Mechanics and Global Geometry, SpringerVerlag, 1984. [7] L.D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory, Israel Program for Sci. Translations, Jerusalem, 1965. [8] R.G. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N -body Schr¨ odinger operators, Comm. Math. Phys. 87, 429–447 (1982). [9] B. Helﬀer and J. Sj¨ ostrand, R´esonance en limite semiclassique, Bull. Soc. Math. France, M´emoire 24/25 (1986). [10] H. Isozaki, Structure of S-matrices for three-body Schr¨odinger operators, Commun. Math. Phys. 146, 241–258 (1992). [11] A. Jensen and T. Kato, Spectral properties of Schr¨ odinger operators and time decay of wave functions, Duke Math. J. 46, 583–611 (1979). [12] A. Jensen and G. Nenciu, A uniﬁed approach to resolvent expansions at thresholds, Reviews in Math. Phys. 13, 717–754 (2001). [13] M. Klaus, B. Simon, Coupling constant thresholds in nonrelativistic quantum mechanics II. Two cluster thresholds in N -body systems, Commun. Math. Phys. 78, 153–168 (1980). [14] M. Klein, A. Martinez and X.P. Wang, On the Born-Oppenheimer approximation of wave operators in molecular scattering, Commun. Math. Phys. 152, 73–95 (1993). [15] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys. 78, 391–408 (1981). [16] M. Murata, Asymptotic expansions in time for solutions of Schr¨ odinger type equations, J. Funct. Analysis 49, 10–53 (1982). [17] R.G. Newton, Noncentral potentials: the generalized Levinson theorem and the structure of the spectrum, J. Math. Phys. 18, 1582–1588 (1977).

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[18] P. Perry, Exponential bounds and semiﬁniteness of point spectrum for N -body Schr¨ odinger operators, Commun. Math. Phys. 92, 481–483 (1984). [19] P. Perry, I.M. Sigal and B. Simon, Spectral analysis of N -body Schr¨ odinger operators, Ann. Math. 114 , 519–567 (1981). [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, 1978. [21] A.V. Sobolev, The Eﬁmov eﬀect. Discrete spectrum asymptotics, Commun. Math. Phys. 156, 101–126 (1993). [22] A. Vasy, Scattering matrices in many-body scattering, Commun. Math. Phys. 200, 105–124 (1999). [23] S.A. Vugal’ter and G.M. Zhislin, On the asymptotics of the discrete spectrum of a given symmetry of multiparticle Hamiltonians, Tans. Moscow Math. Soc. 54, 165–189 (1993). [24] X.P. Wang, Continuity of time-delay operators and low energy resolvent estimates, Proc. of the Royal Society of Edinburgh 105A, 229–242 (1987). [25] X.P. Wang, On the existence of the N -body Eﬁmov eﬀect, Journal of Functional Analysis, in press. [26] D.R. Yafaev, On the theory of the discrete spectrum of the three-particle Schr¨ odinger operator, Math. USSR-Sb. 23, 535–559 (1974). Xue Ping Wang D´epartement de Math´ematiques Laboratoire Jean Leray UMR 6629 du CNRS Universit´e de Nantes F-44322 Nantes Cedex 3 France email: [email protected] Communicated by Bernard Helﬀer submitted 22/10/02, accepted 27/03/03

Ann. Henri Poincar´e 4 (2003) 601 – 611 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/030601-11 DOI 10.1007/s00023-003-0140-x

Annales Henri Poincar´ e

The Virasoro Algebra and Sectors with Infinite Statistical Dimension Sebastiano Carpi∗

Abstract. We show that the sectors with lowest weight h ≥ 0, h = j 2 , j ∈ 12 Z of the local net of von Neumann algebras on the circle generated by the Virasoro algebra with central charge c = 1 have infinite statistical dimension.

1 Introduction The notion of statistical dimension of superselection sectors, introduced by Doplicher, Haag, and Roberts in [6] is one of the most important concepts emerging in the formulation of Quantum Field Theory in therms of local nets of operator algebras (see [10] for a general reference on this subject). The deep connection with Jones’ theory on index for subfactors [11, 15], established by Longo [16] is a remarkable illustration of the relevance of this notion. For an irreducible representation π of the algebra of observables A satisfying the DHR selection criterion the ﬁniteness of the (statistical) dimension d(π) is equivalent to the existence of a conjugate representation π corresponding to the particle-antiparticle symmetry [6], a condition which is very natural on physical grounds. In fact for local nets over a four-dimensional Minkowski space-time no example of (irreducible) sector with inﬁnite dimension is known and the possibility that in this context the existence of such sectors can be excluded for physically reasonable algebras of observables is still open. The situation is diﬀerent in the case of conformal nets on S 1 , i.e., nets associated to chiral components of 2D conformal ﬁeld theories, where irreducible representations with inﬁnite dimension seem to appear naturally. Examples have been found by Fredenhagen [7] and Rehren has given arguments indicating that for the nets generated by the Virasoro algebra with central charge c ≥ 1 most of the irreducible representations should have inﬁnite dimension [20]. Moreover the analysis of these representations in a model independent framework has been initiated in [1]. In this note we show (Theorem 4.4), in agreement with the arguments in [20], that the representations of the Virasoro algebra with central charge c = 1 and lowest weight h ≥ 0, h = j 2 , j ∈ 12 Z give rise to representations with inﬁnite dimension of the corresponding conformal net AVir . ∗ Supported

in part by the Italian MIUR and GNAMPA-INDAM

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Our strategy of proof diﬀers from the one adopted in [7] where (partial) computation of fusion rules is used to infer inﬁnite dimension. Part of the fusion rules for the Virasoro algebra with c = 1 have been recently computed by Rehren and Tuneke [22] but we shall not use their results. Instead of the fusion structure we use a formula, which appeared in [21], giving the dimension of the restriction of a representation of a net A to a subsystem B ⊂ A (Proposition 3.1 in this note) and well-known results on the representation theory of the Virasoro algebra [12]. As another interesting application of this formula we show, generalizing a result in [24], that for ﬁnite index subsystems of certain rational nets twisted sectors always exist (Proposition 3.3).

2 Conformal nets, their representations and subsystems Let I be the set of nonempty, nondense, open intervals of unit circle S 1 . A conformal net on S 1 is a family A = {A(I)|I ∈ I} of von Neumann algebras, acting on a inﬁnite-dimensional separable Hilbert space HA , satisfying the following properties: (i) Isotony. (ii) Locality.

A(I1 ) ⊂ A(I2 ) for I1 ⊂ I2 , I1 , I2 ∈ I.

(1)

A(I1 ) ⊂ A(I2 ) for I1 ∩ I2 = ∅, I1 , I2 ∈ I.

(2)

(iii) Conformal covariance. There exists a strongly continuous unitary representation U of PSL(2, R) in HA such that U (α)A(I)U (α)−1 = A(αI) for I ∈ I, α ∈ PSL(2, R),

(3)

where PSL(2, R) acts on S 1 by Moebius transformations. (iv) Positivity of the energy. The conformal Hamiltonian L0 , which generates the restriction of U to the one-parameter group of rotations has non-negative spectrum. (v) Existence of the vacuum. There exists a unique (up to a phase) U -invariant unit vector Ω ∈ HA . (vi) Cyclicity of the vacuum. Ω is cyclic for the algebra A(S 1 ) := I∈I A(I) Some consequences of the axioms are [8, 9]: (vii) Reeh-Schlieder property. For every I ∈ I, Ω is cyclic and separating for A(I). (viii) Haag duality. For every I ∈ I A(I) = A(I c ), where I c denotes the interior of S 1 \I. (ix) Factoriality. The algebras A(I) are type III1 factors.

(4)

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A conformal net A is said to be split if given two intervals I1 , I2 ∈ I with the closure of I1 contained in I2 , there exists a type I factor N(I1 , I2 ) such that A(I1 ) ⊂ N(I1 , I2 ) ⊂ A(I2 ).

(5)

Moreover, if for every I, I1 , I2 ∈ I with I1 , I2 obtained by removing a point from I we have (6) A(I1 ) ∨ A(I2 ) = A(I), then A is said to be strongly additive. The split property and strong additivity do not follow from the axioms of conformal nets but they are satisﬁed in many interesting models. A representation of a conformal net A is a family π = {πI | I ∈ I} where πI is a representation of A(I) on a ﬁxed Hilbert space Hπ , such that πJ |A(I) = πI for I ⊂ J.

(7)

Irreducibility, direct sums and unitary equivalence of representations of conformal nets can be deﬁned in a natural way, see [8, 9]. The unitary equivalence class of an irreducible representation π on a separable Hilbert space is called a sector and denoted [π]. The identical representation of A on HA is called the vacuum representation and it is irreducible. The corresponding sector is called the vacuum sector. If Hπ is separable then π is automatically locally normal, namely πI is normal for each I ∈ I and hence πI (A(I)) is a type III1 factor. A representation π is said to be covariant if there is a strongly continuous unitary representation Uπ on Hπ R) of PSL(2, R) such that of the universal covering group PSL(2, AdUπ (α)πI = παI AdU (α),

(8)

R). If a covariant representation π is irreducible where U has been lifted to PSL(2, then there is a unique Uπ satisfying Eq. (8). Hence, in this case, the corresponding generator of rotations Lπ0 is completely determined by π. Given a covariant representation π of A on a separable Hilbert space Hπ one has the (isomorphic) inclusions πI (A(I)) ⊂ πI c (A(I c )) , I ∈ I [8]. Then the Jones (minimal) index [πI c (A(I c )) : πI (A(I))] is independent of I ∈ I and the statistical dimension d(π) of π is deﬁned by 1

d(π) = [πI c (A(I c )) : πI (A(I))] 2 .

(9)

The relation of the above deﬁnition with the one in [6] is given by the indexstatistics theorem [9, 16]. More precisely, in analogy with [6], a deﬁnition of statistical dimension using left inverse and statistics parameter can be given in the context of conformal nets on S 1 [9] (cf. also [8]). It has been shown by Guido and Longo that for a covariant representation π with positive energy the statistical 1 dimension is ﬁnite if and only if [πI c (A(I c )) : πI (A(I))] 2 is ﬁnite [9, Subsection

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2.4] and that in this case these two numbers coincide [9, Corollary 3.7]. Actually, as a consequence of [9, Proposition 2.14] and [1, Corollary 4.4], the positivity of the energy needs not to be assumed a priori. In view of the above considerations it should be clear that the use of formula (9) as a deﬁnition is appropriate both in the ﬁnite and in the inﬁnite case. A conformal subsystem of a conformal net A is a family B = {B(I)| I ∈ I} of nontrivial von Neumann algebras acting on HA such that: B(I) ⊂ A(I) for I ∈ I; U (α)B(I)U (α)−1 = B(αI) for I, ∈ I;

(10) (11)

B(I1 ) ⊂ B(I2 ) for I1 ⊂ I2 , I1 , I2 ∈ I.

(12)

We shall use the notation B ⊂ A for conformal subsystems. Note that B is not in general a conformal net since Ω is not cyclic for the algebra B(S 1 ) := I∈I B(I) unless B = A. However one gets a conformal net B0 by restriction of the algebras B(I), I ∈ I, and of the representation U to the closure HB of B(S 1 )Ω. Since the map b ∈ B(I) → b|HB ∈ B0 (I) is an isomorphism for every I ∈ I, we shall, as usual, use the symbol B instead of B0 , specifying, if necessary, when B acts on HA or on HB . Given a conformal subsystem B ⊂ A the index of the subfactor B(I) ⊂ A(I) does not depend on I and is denoted [A : B].

3 Restricting representations We now consider restriction of representations. Given a subsystem B ⊂ A and a representation π of A one can deﬁne a representation π rest by πIrest = πI |B(I)

I ∈ I.

(13)

Then the following holds [21] (cf. also [23, Section 3]). We include the proof for the convenience of the reader. Proposition 3.1. For every conformal subsystem B ⊂ A and covariant representation π of A on a separable Hilbert space we have d(π rest ) = [A : B]d(π).

(14)

Proof. For I ∈ I we have d(π rest )2 = [πI c (B(I c )) : πI (B(I))]. Consider the inclusions πI (B(I)) ⊂ πI (A(I)) ⊂ πI c (A(I c )) ⊂ πI c (B(I c )) . Then, the multiplicativity of the index [17] implies that d(π rest )2 is equal to [πI c (B(I c )) : πI c (A(I c )) ][πI c (A(I c )) : πI (A(I))][πI (A(I)) : πI (B(I))].

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Since πI is an isomorphism for every I ∈ I we have [πI c (B(I c )) : πI c (A(I c )) ] = [πI c (A(I c )) : πI c (B(I c ))] = [A : B] and similarly [πI (A(I)) : πI (B(I))] = [A : B]. It follows that

d(π rest )2 = [A : B]2 d(π)2 .

Remark 3.2. If N ⊂ M is an inclusion of inﬁnite factors acting on a separable Hilbert space and ρ is a (normal, unital) endomorphism of M one can deﬁne an endomorphism ρrest of N by ρrest := γ ◦ ρ|N ,

(15)

where γ is Longo’s canonical endomorphism [16]. As discussed in [19] the mapping ρ → ρrest (called σ restriction in [2]) corresponds in a natural way to the restriction of representations of a net. In fact a similar argument to the one used in the proof of the previous proposition shows that d(ρrest ) = [M : N ]d(ρ).

(16)

Here the dimension d(ρ) of an endomorphism ρ of a factor M is given by the square root of the index of the subfactor ρ(M ) ⊂ M . Let I1 , I2 ∈ I have disjoint closures, let I3 , I4 ∈ I be the interiors of the connected components of S 1 /(I1 ∪ I2 ) and let A be a conformal net on S 1 . The inclusion (17) A(I1 ) ∨ A(I2 ) ⊂ (A(I3 ) ∨ A(I4 )) is called a 2-interval inclusion. A conformal net A is said to be completely rational if it is split, strongly additive and there is a 2-interval inclusion with ﬁnite index µA (in this case every 2-interval inclusion has the same index [14]). It has been shown in [14] that a completely rational net has ﬁnitely many sectors which are all covariant with ﬁnite dimension. Furthermore the following holds µA = d(πi )2 , (18) i

where for each sector of A a representation πi has been chosen. We now consider a conformal subsystem B of a completely rational net A such that the index [A : B] is ﬁnite. Then B is completely rational [18] and the index µB is given by ([14, Proposition 24.]) µB = [A : B]2 µA .

(19)

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We say that a sector of B is untwisted if it is contained in π rest for some irreducible representation π of A on a separable Hilbert space. If it is not untwisted we say that it is twisted. For every sector of B we choose a corresponding representation σi of B. Let U, (T) be the set of untwisted (twisted) sectors of B. We deﬁne d(σi )2 , (20) µu B = [σi ]∈U t

µ

B

=

d(σi )2 .

(21)

[σi ]∈T

Clearly µB = µu B + µt B . In the case where B ⊂ A is an orbifold inclusion, namely B is the ﬁxed points net for the action of a (non-trivial) ﬁnite group G of internal symmetries of A, it has been shown by Xu [24] that the set of twisted sectors is not empty. Actually Proposition 3.1 implies the existence of such sectors even when there is no underlying group action. Proposition 3.3. Let B be a proper conformal subsystem of a completely rational net A, with finite index [A : B]. Then the set of twisted sectors of B is not empty and in fact µt B ≥ 2. Proof. Let πi , i = 0, 1, . . . , n be inequivalent irreducible representations exausting all sectors of A and let π0 be the vacuum representation. The set U of untwisted sectors of B can be decomposed into disjoint subsets Ui , i = 0, 1, . . . , n in the following way: U0 is the set of sectors of B which are contained in π0 rest and Uk , k = 1, . . . , n is the the set of sectors contained in πk rest but not in πi rest , i = 0, . . . , k − 1. It follows from Proposition 3.1 and Eq. (19) that d(πi rest )2 = [A : B]2 · d(πi )2 i

i

= [A : B]2 µA = µB . Therefore µt B = µB − µu B = d(πi rest )2 − d(σk )2 i

≥

i

≥(

(

[σk ]∈U

d(σk ))2 −

[σk ]∈Ui

[σk ]∈U0

d(σk ))2 −

(

i

d(σk )2 )

[σk ]∈Ui

d(σk )2 ≥ 2,

[σk ]∈U0

where the last inequality follows from the fact that U0 has two or more elements when B = A.

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4 Virasoro algebra and infinite dimension We begin this section with the following easy consequence of Proposition 3.1. Proposition 4.1. Let B be a conformal subsystem of a net A with infinite index [A : B]. Assume that there exists a covariant representation π of A on a separable Hilbert space whose restriction to B is irreducible. Then [π rest ] is a covariant sector of B with infinite statistical dimension. We now come to the sectors of the conformal net AVir generated by the Virasoro algebra with c = 1. We shall use the fact that AVir can be considered has a conformal subsystem of the net A generated by a U(1) current J(z). The net A is deﬁned as follows, see [3, 5] for more details. The Hilbert space HA carries a strongly continuous unitary representation U of PSL(2, R) with positive energy and a unique (up to a phase) U -invariant unit vector Ω. The U(1) current J(z), z ∈ S 1 is deﬁned as operator valued distribution on HA . Namely the operators dz J(z)u(z) u ∈ C ∞ (S 1 ) (22) J(u) = 2πi have a common invariant dense domain D containing Ω which is also U -invariant. For each ψ ∈ D the mapping u → J(u)ψ is linear and continuous from C ∞ (S 1 ) to HA . Moreover the vacuum Ω is cyclic for the polynomial algebra generated by the smeared currents J(u), u ∈ C ∞ (S 1 ). The current J(z) satisﬁes the canonical commutation relations [J(z1 ), J(z2 )] = −δ (z1 − z2 ),

(23)

where the Dirac delta function δ(z1 − z2 ) is deﬁned with respect to the complex dz measure 2πi , the hermiticity condition J(z)∗ = z 2 J(z),

(24)

U (α)J(u)U (α)∗ = J(uα ), u ∈ C ∞ (S 1 ),

(25)

and the covariance property

where uα (z) := u(α−1 z). For every real test function u ∈ C ∞ (S 1 ) the operator J(u) is essentially self-adjoint and the unitaries W (u) := eiJ(u) satisfy the Weyl relations A(u,v) (26) W (u)W (v) = W (u + v)e− 2 , dz where A(u, v) := 2πi u (z)v(z). For every I ∈ I the local von Neumann algebra A(I) is deﬁned by A(I) = {W (u)|u ∈ C ∞ (S 1 ) real, supp u ⊂ I}

(27)

and one can show that the family A(I), I ∈ I is a conformal net on S 1 . Next we deﬁne the conformal subsystem AVir generated by the Virasoro algebra with

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central charge c = 1. First consider the (formal) Fourier expansion of the U(1)current J(z) = Jn z −n−1 , (28) n

where the Fourier modes Jn , n ∈ Z satisfy [Jn , Jm ] = nδn+m,0

(29)

Jn ∗ = J−n .

(30)

One can deﬁne an energy-momentum tensor T (z) by the Sugawara construction T (z) =

1 1 : J(z)2 := (J+ (z)J(z) + J(z)J− (z)), 2 2

where, J+ (z) = J(z) − J− (z) =

∞

J−n z n−1 .

(31)

(32)

n=1

The Fourier modes in the expansion T (z) = Ln z −n−2

(33)

n

satisfy the Virasoro algebra relations [Ln , Lm ] = (n − m)Ln+m +

c n(n2 − 1)δn+m,0 12

(34)

with central charge c = 1, and the hermiticity condition Ln ∗ = L−n .

(35)

According to our previous notations (the closure of) L0 is the positive self-adjoint generator of the restriction of U to the one-parameter subgroup of rotations. For f ∈ C ∞ (S 1 ) the operator dz T (z)f (z) (36) T (f ) = 2πi is well deﬁned on D and is essentially self-adjoint when z −1 f (z) is real. The conformal subsystem AVir ⊂ A is then deﬁned by AVir (I) = {eiT (f ) |f ∈ C ∞ (S 1 ), z −1 f (z) real, supp f ⊂ I} , I ∈ I.

(37)

Representations of the net A have been studied in [3]. For every q ∈ R one can deﬁne a covariant irreducible representation (BMT-automorphism) αq on Hq = HA such that αq I (W (u)) = eq

dz −1 u(z) 2π z

W (u),

(38)

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for I ∈ I, u ∈ C ∞ (S 1 ) with support in I. Such representations have dimension d(αq ) = 1 and correspond to (unitary) positive energy representations of the Lie algebra (29) with lowest weight q [4]. Note that α0 is the vacuum representation of A. Analogously to each representation of the Virasoro algebra (34) with central charge c = 1 and lowest weight h ∈ R+ one can associate a covariant irreducible representation πh of AVir which can be realized has a subrepresentation of αq rest if h = 12 q 2 , see [5]. The characters of the representations αq , q ∈ R, are given by (see, e.g., [13, Section 2.2.] ) αq

∞

1

2

χq (t) = Tr(tL0 ) = t 2 q p(t) t ∈ (0, 1),

(39)

where p(t) = n=1 (1 − tn )−1 . Moreover, for the representations πh , h ∈ R+ and t ∈ (0, 1), by the results in [12] the following hold 1 Z, 2 πh 1 χh (t) := Tr(tL0 ) = th p(t), h = j 2 , j ∈ Z. 2 πh

2

χh (t) := Tr(tL0 ) = tj (1 − t2|j|+1 )p(t), h = j 2 , j ∈

(40) (41)

Lemma 4.2. [A : AVir ] = ∞. Proof. As a consequence of Proposition 3.1 we have [A : AVir ] = d(α0 rest ). More j2 over it follows from the equality χ0 (t) = ∞ j=0 χ (t) that α0 rest = ⊕∞ j=0 πj 2 and this implies inﬁnite index. Lemma 4.3. (cf. [13, Theorem 6.2.]) If h = 12 q 2 , q ∈ /

√1 Z, 2

then π h = αq rest .

Proof. If h = 12 q 2 π h is a subrepresentation of αq rest on a Uαq -invariant subspace Hh ⊂ Hq . Moreover, if q ∈ / √12 Z then χh (t) = χq (t) and hence Hh = Hq . Accordingly we have π h = αq rest . The following theorem is a direct consequence of Proposition 4.1 and the previous two lemmata. Theorem 4.4. If [πh ] belongs to the continuum sectors of AVir , i.e., h ∈ R+ , h = j 2 , j ∈ 12 Z, then it has infinite statistical dimension. Remark 4.5. It has been shown by Rehren [20] that if h = j 2 , j ∈ Z, then d(πh ) = 2|j| + 1 and the same formula is expected to hold for every j ∈ 12 Z. Acknowledgments. The author would like to thank Roberto Longo for some stimulating conversations. He also thanks Roberto Conti and Karl-Henning Rehren for useful comments on the manuscript. This work was essentially completed while the author was at the Dipartimento di Matematica of the Universit` a di Roma Tre thanks to a Post-Doctoral grant of this University.

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References [1] P. Bertozzini, R. Conti, R. Longo, Covariant sectors with inﬁnite dimension and positivity of the energy, Comm. Math. Phys. 193, 471–492 (1998). [2] J. B¨ ockenhauer, D.E. Evans, Modular invariants graphs and α-induction for nets of subfactors I., Comm. Math. Phys. 197, 361–386 (1998). [3] D. Buchholz, G. Mack, I.T. Todorov, The current algebra on the circle as a germ of local ﬁeld theories, Nucl. Phys. B (Proc. Suppl.) 5B, 20–56 (1988). [4] D. Buchholz, G. Mack, I.T. Todorov, Localized automorphisms of the U(1)current, in D. Kastler ed., The algebraic theory of superselection sectors. World Scientiﬁc, Singapore, 1990, pp. 356–378. [5] D. Buchholz, H. Schulz-Mirbach, Haag duality in conformal quantum ﬁeld theory, Rev. Math. Phys. 2, 105–125 (1990). [6] S. Doplicher, R. Haag, J.E. Roberts, Local observable and particle statistics I, Comm. Math. Phys. 23, 199–230 (1971), and II Comm. Math. Phys. 35, 49–85 (1974). [7] K. Fredenhagen, Superselection sectors with inﬁnite statistical dimension, In Subfactors, H. Araki et al. eds., World Scientiﬁc, Singapore 1995, pp. 242–258. [8] F. Gabbiani, J. Fr¨ ohlich, Operator algebras and conformal ﬁeld theory, Comm. Math. Phys. 155, 569–640 (1993). [9] D. Guido, R. Longo, The conformal spin and statistic theorem, Comm. Math. Phys. 181, 11–35 (1996). [10] R. Haag, Local Quantum Physics. 2nd ed. Springer-Verlag, New York Berlin Heidelberg 1996. [11] V. Jones, Index of subfactors, Invent. Math. 72, 1–25 (1983). [12] V.G. Kac, Contravariant form for the inﬁnite-dimensional Lie algebras and superalgebras, in W. Beiglb¨ock et al. eds. Group Theorethical Methods in Physics, Lecture Notes in Phys. 94 Springer-Verlag, New York, 1979, pp. 441–445. [13] V.G. Kac, A.K. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientiﬁc, Singapore, 1987. [14] Y. Kawahigashi, R. Longo, M. M¨ uger, Multi-interval subfactor and modularity of representations in conformal ﬁeld theory, Comm. Math. Phys. 219, 631–669 (2001).

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[15] H. Kosaki, Extension of Jones’ theory on index to arbitrary subfactors, J. Funct. Anal. 66, 123–140 (1986). [16] R. Longo, Index of subfactors and statistics of quantum ﬁelds. I, Comm. Math. Phys. 126, 217–247 (1989), and II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130, 285–309 (1990). [17] R. Longo, Minimal index and braided subfactors, J. Funct. Anal. 109, 98–112 (1992). [18] R. Longo, Conformal subnets and intermediate subfactors, MSRI preprint 2001, math.OA/0102196. [19] R. Longo, K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7, 567–597 (1995). [20] K.-H. Rehren, A new view of the Virasoro algebra, Lett. Math. Phys. 30, 125–130 (1994). [21] K.-H. Rehren, Subfactors and coset models, in Generalized symmetries in physics World Scientiﬁc, River Edge, NJ, 1994, pp. 338–356. [22] K.-H. Rehren, H.R. Tuneke, Fusion rules for the continuum sectors of the Virasoro algebra with c = 1, Lett. Math. Phys. 53, 305–312 (2000). [23] F. Xu, Algebraic coset conformal ﬁeld theories, Comm. Math. Phys. 211, 1–44 (2000). [24] F. Xu, Algebraic orbifold conformal ﬁeld theories, Proc. Natl. Acad. Sci. 97, 14069–14073 (2000). Sebastiano Carpi Dipartimento di Scienze Universit` a “G. d’Annunzio” di Chieti-Pescara Viale Pindaro 42 I-65127 Pescara Italy email: [email protected] Communicated by Klaus Fredenhagen submitted 28/11/02, accepted 10/03/03

Ann. Henri Poincar´e 4 (2003) 613 – 635 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040613-23 DOI 10.1007/s00023-003-0141-9

Annales Henri Poincar´ e

Generalized Free Fields and the AdS-CFT Correspondence Michael D¨ utsch and Karl-Henning Rehren

Abstract. Motivated by structural issues in the AdS-CFT correspondence, the theory of generalized free ﬁelds is reconsidered. A stress-energy tensor for the generalized free ﬁeld is constructed as a limit of Wightman ﬁelds. Although this limit is singular, it fulﬁlls the requirements of a conserved local density for the Poincar´e generators. An explicit “holographic” formula relating the Klein-Gordon ﬁeld on AdS to generalized free ﬁelds on Minkowski space-time is provided, and contrasted with the “algebraic” notion of holography. A simple relation between the singular stress-energy tensor and the canonical AdS stress-energy tensor is exhibited.

1 Introduction According to Maldacena’s conjecture [31], type IIB string theory on 5-dimensional asymptotically Anti-deSitter (AdS) backgrounds with ﬁve compactiﬁed dimensions is equivalent to a maximally supersymmetric Yang-Mills theory in physical Minkowski space-time. In the limit of large number of colors N → ∞ and large ’t Hooft coupling θ = N g 2 → ∞, it is conjectured that the string may be replaced by classical supergravity on AdS. Roughly speaking, 1/N corrections correspond to quantum corrections, and the strong coupling expansion in 1/θ corresponds to the perturbative incorporation of string corrections measured by the string tension α . The “dual” correspondence between ﬁelds on AdS and conformal ﬁelds on Minkowski space-time was made concrete by a proposal in [22, 36]. As a scalar model, the conformal ﬁeld dual to the free Klein-Gordon ﬁeld on AdS has been considered as an exercise in [36]. It is a generalized free ﬁeld [20, 27, 29]. It follows that the perturbative treatment of the AdS-CFT correspondence amounts to an expansion around a generalized free ﬁeld. For this reason, we believe it worthwhile to reconsider the properties of generalized free ﬁelds. Generalized free ﬁelds have been introduced by Greenberg [20] as a new class of models for local quantum ﬁelds, and have been further studied by Licht [29] as candidates for more general asymptotic ﬁelds as required by the LSZ asymptotic condition. They can be characterized in several equivalent ways: the commutator is a numerical distribution; the truncated (connected) n-point functions vanish for n = 2; the correlation functions factorize into 2-point functions; the generating functional for the correlation functions is a Gaussian. But in distinction from a canonical free ﬁeld, a generalized free ﬁeld is not the solution to

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an equation of motion, and its 2-point function is not supported on a mass shell. It rather has the form of a superposition Ω, ϕ(x)ϕ(x )Ω = dρ(m2 ) Wm (x − x ) (1.1) R+

where Wm are the 2-point functions of the Klein-Gordon ﬁelds of mass m in d-dimensional Minkowski space-time. The (positive and polynomially bounded) weight dρ(m2 ) occurring in the superposition is known as the K¨ allen-Lehmann weight [28]. In fact, this is the most general form of the 2-point function of any scalar quantum ﬁeld [28], and a general theorem [21] states that a ﬁeld whose 2-point function is supported within a ﬁnite interval of masses, is necessarily a (generalized) free ﬁeld. There are several ways by which generalized free ﬁelds arise. E.g., from any Wightman ﬁeld φ on a Hilbert space H, one can obtain Wightman ﬁelds φ(N ) , N ∈ N, as the normalized sum of replicas of φ on H⊗N . This suppresses the truncated n-point functions with a factor N (2−n)/2 . In the “central” limit N → ∞, one obtains a generalized free ﬁeld which has the same 2-point function as the original (interacting) ﬁeld. Similarly, in the large N limit of O(N ) or U (N ) symmetric theories, all truncated functions of gauge invariant (composite) ﬁelds exhibit a leading factor of N , so that if the 2-point function is normalized, the higher truncated functions are also suppressed by inverse powers of N , and the limit is again a generalized free ﬁeld [26].1 Another obvious way to obtain a generalized free ﬁeld is to restrict a free Klein-Gordon ﬁeld of mass M in 1+4 dimensions to the 1+3-dimensional √ hypersurallen-Lehmann weight is dρ(m2 ) = dm2 / m2 − M 2 , face x4 = 0. The resulting K¨ supported at m2 ≥ M 2 . Finally, the AdS-CFT correspondence associates with the free Klein-Gordon ﬁeld on d + 1-dimensional Anti-deSitter space-time a Gaussian conformal ﬁeld in d-dimensional Minkowski space-time whose scaling dimension ∆ = d2 + ν depends √ on the Klein-Gordon mass M through the parameter ν = 12 d2 + 4M 2 . Its 2point function proportional to (−(x − x )2 )−∆ is a superposition of all masses with K¨ allen-Lehmann weight dρ(m2 ) = dm2 m2ν (cf. Sect. 4). As a consequence of the continuous superposition of masses, there is no Lagrangean and no canonical stress-energy tensor associated with a generalized free ﬁeld. The ﬁrst purpose of this article (entirely unrelated to AdS) is the construction of a non-canonical stress-energy tensor (Sect. 3). This stress-energy tensor turns out to be more singular than a Wightman ﬁeld, but it fulﬁlls the requirements as a density for the generators of (global) space-time symmetries. If smeared with a 1 On the other hand, the limit W of 1/N times the connected functional for ﬁnite N , W N = log ZN , is ﬁnite and non-Gaussian. But W does not deﬁne a quantum ﬁeld theory of its own because it violates positivity (unitarity). The signiﬁcance of this quantity is that N · W gives the asymptotic behaviour of the large N expansion of WN .

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test function, it has ﬁnite expectation values but inﬁnite ﬂuctuations in almost every state of the Hilbert space (including the vacuum). Technically speaking, it is a quadratic form on the Wightman domain, rather than an unbounded operator. Its commutator with ﬁeld operators, however, is well deﬁned as an operator. Our construction of the stress-energy tensor relies on the fact that on the vacuum Hilbert space of a generalized free ﬁeld ϕ there exists a large class of mutually local Wightman ﬁelds (including ϕ itself, hence relatively local w.r.t. ϕ) which is much larger [11, 20, 29] than the class of Wick polynomials of ϕ. They form what is known as the Borchers class [5, 27] of ϕ. These ﬁelds can be expressed in terms of ϕ (and its Wick polynomials) by the use of highly nonlocal (pseudo) diﬀerential operators or convolutions, while still satisfying local commutativity with ϕ (i.e., the commutator vanishes at spacelike separation). This fact illustrates the distinction between the algebraic notion of local commutativity (Einstein locality) underlying the concept of the Borchers class, and a notion of “expressibility in terms of local operations”. E.g., in perturbation theory the Lagrangean interaction density should be local in the former sense in order to ensure locality of the interacting ﬁeld. The second prominent issue of this article is the “holographic” identiﬁcation of a quantum ﬁeld on AdS (the free Klein-Gordon ﬁeld φ) in the Borchers class of the boundary generalized free ﬁeld ϕ (Sect. 4). More precisely, AdS is regarded conveniently as a warped product of Minkowski space-time with R+ (whose coordinate we call z > 0). Then for every ﬁxed value z, φ(z, ·) is a Wightman ﬁeld ϕz (·) on Minkowski space-time in the Borchers class of ϕ, obtained from ϕ by a non-local (pseudo) diﬀerential operator involving a z-dependent Bessel function. Local commutativity of the Klein-Gordon ﬁeld φ on AdS implies that the ﬁelds ϕz and ϕz in Minkowski space-time satisfy a certain “bonus locality” (local commutativity at ﬁnite timelike distance). We shall explicitly derive this property as a consequence of the speciﬁc non-local operations relating the AdS ﬁeld to the boundary ﬁeld, invoking a nontrivial identity for Bessel functions. The canonical stress-energy tensor of the Klein-Gordon ﬁeld on AdS is identiﬁed as a z-dependent generalized Wick product of the boundary ﬁeld. Integrating this ﬁeld over z, yields the singular stress-energy tensor of the generalized free ﬁeld mentioned above. This complies with the fact that the canonical AdS stress-energy tensor is a density in a Cauchy surface of AdS, while the stress-energy tensor for the generalized free ﬁeld on the boundary is a density in a time zero plane of Minkowski space. With these ﬁndings, we want to point out that generalized free ﬁelds are rather well behaved Wightman ﬁelds, which moreover are “closer” to interacting quantum ﬁelds than free Klein-Gordon ﬁelds. It might be advantageous to perform a perturbation around a generalized free ﬁeld, which has already the correct 2-point function of the interacting ﬁeld, while the perturbation only aﬀects the higher truncated correlations. As was noticed implicitly, e.g., in [4], and systematically analyzed in [15], the perturbative approach to the AdS-CFT correspondence may be understood

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as a perturbation around a canonical ﬁeld on AdS with subsequent restriction to the boundary. The interaction part of the action is an integral over AdS of some Wick polynomial in the AdS ﬁeld. Expressing the latter in terms of the limiting generalized free ﬁeld on the boundary, and performing the (regularized) z-integral, one obtains (at least formally) a Lagrangean density on the boundary which is a generalized Wick polynomial of the boundary generalized free ﬁeld. In view of this observation, the present work is also considered as a starting point for a perturbation theory of the generalized free ﬁeld with generalized Wick polynomials as interactions, which includes the perturbative AdS-CFT correspondence as a special case. In the last, somewhat tentative section, we point out the relation between the existence of relatively local ﬁelds beyond the Wick polynomials, and the violation of the time-slice property (primitive causality [24]) and “Haag duality” for generalized free ﬁelds in Minkowski space-time. These issues are discussed in terms of the von Neumann algebras of localized observables associated with a quantum ﬁeld [23]. Although logically unrelated to AdS-CFT, they may be nicely understood in terms of geometric properties of AdS and its boundary, using the above holographic interpretation. The discussion also exhibits a slight but important diﬀerence between the present holographic picture and the “algebraic” notion of holography [33].

2 Generalized free fields Let Md = (Rd , ηµν ) denote d ≥ 2-dimensional Minkowski space-time, and V+ the open forward light-cone (in momentum space). We consider a hermitian scalar generalized free ﬁeld [27, Chap. 2.6] on Md with K¨ allen-Lehmann weight on R+ .

dρ(m2 ) = dm2 It has the form

dd k [a(k)e−ikx + a+ (k)eikx ]

ϕ(x) =

(2.1)

V+

in terms of creation and annihilation operators [a(k), a+ (k )] = (2π)−(d−1) δ d (k − k ),

[a, a] = 0 = [a+ , a+ ].

(2.2)

It is deﬁned on the Fock space H over the 1-particle space H1 = L2 (V+ , dd k), identifying L2 (V+ , dd k) f

≡

(2π)

d−1 2

dd k f (k)a+ (k)Ω ∈ H1 . V+

(2.3)

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H1 , and hence H, is equipped with the obvious unitary positive-energy representation of the Poincar´e group with generators2 d−1 Pµ = (2π) dd k a+ (k) kµ a(k), (2.4) V+

Mµν = (2π)d−1

dd k a+ (k) i(kν V+

∂ ∂ − kµ ν ) a(k), ∂k µ ∂k

(2.5)

such that i[Pµ , ϕ(x)] = ∂µ ϕ(x),

(2.6)

i[Mµν , ϕ(x)] = (xµ ∂ν − xν ∂µ )ϕ(x). The generalized free ﬁeld ϕ is a local ﬁeld because its commutator reads −(d−1) [ϕ(x), ϕ(x )] = (2π) dd k e−ik(x−x ) − eik(x−x ) = V+ dm2 (2π)−(d−1) dd k δ(k 2 − m2 ) e−ik(x−x ) − eik(x−x ) = R+ V+ = dm2 ∆m (x − x )

(2.7)

(2.8)

R+

where ∆m is the commutator function of the free Klein-Gordon ﬁeld of mass m.3

2.1

Relatively local generalized free fields and generalized Wick products

Our ﬁrst observation is that on the same Hilbert space, we can deﬁne ϕh (x) = dd k h(k 2 ) [a(k)e−ikx + a+ (k)eikx ]

(2.9)

V+

with h any smooth polynomially bounded real function on R+ (called “weight function”). These are again hermitian scalar ﬁelds on H, satisfying (2.6) and (2.7). Moreover, all ϕh are local and mutually local ﬁelds, because their commutators [ϕh1 (x), ϕh2 (x )] = dm2 h1 (m2 )h2 (m2 ) ∆m (x − x ) (2.10) R+

2 The reader might be worried about the meaning of derivatives of a(k). The expressions (2.4), (2.5) as well as (2.17), (2.22) below are understood in the distributional sense, i.e., after application of an integral dd ka+ (k)Xa(k) to a 1-particle vector of the form (2.3) the diﬀerential operator X is found acting on the smearing function f ∈ L2 (V+ , dd k), and likewise for n-particle vectors. Thus, (2.4), (2.5), (2.17), (2.22) are the “second quantizations” of the corresponding diﬀerential operators on L2 (V+ , dd k). We shall not discuss here the precise domains on which these hermitian generators are (essentially) self-adjoint. 3 The Klein-Gordon ﬁelds themselves are not present in the theory, though, because square integrable functions in H1 cannot have sharp mass.

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vanish at spacelike distance irrespective of the functions hi . By inspection of the 2-point functions Ω, ϕh1 (x)ϕh2 (x )Ω = dm2 h1 (m2 )h2 (m2 ) Wm (x − x ), (2.11) R+

one sees that each ϕh is a generalized free ﬁeld with K¨allen-Lehmann weight dm2 h(m2 )2 . In fact, the weight function h need not be smooth as long as dm2 h(m2 )2 is a polynomially bounded measure. If the weight function h is a polynomial, then ϕh = h(−)ϕ

(2.12)

is just a derivative of ϕ, and ϕh (f ) = ϕ(h(−)f ) where the support of h(−)f equals (a subset of) the support of f . Hence h(−) is a local operation. But if h is not a polynomial, then h(−) may be tentatively deﬁned on f by multiplication of the Fourier transform fˆ(k) with any function of k 2 which coincides with h on R+ (all giving the same ﬁeld operator ϕ(h(−)f )). This is a highly non-local operation which does not preserve supports. Likewise, one may formally read (2.9) as a convolution in x-space [20] ˇ − y)ϕ(y) dd x H(x (2.13) ϕh (x) = Md

with the distributional inverse Fourier transform of any function H(k) which equals h(k 2 ) on V+ . E.g., if h is analytic, √ h(−) and H(k) may be deﬁned as power series; but as the example of h(z) = cos z exempliﬁes, the inverse Fourier transform of h(k 2 )fˆ(k) or H(k) may not exist due to the rapid growth of h(k 2 ) at negative k 2 . Therefore, expressions like (2.12) or (2.13) in the general case should not be taken literally. These are suggestive ways of rewriting the deﬁnition (2.9), indicating a non-local operation on ϕ which yet gives rise to a local ﬁeld. The above construction of ﬁelds satisfying local commutativity with ϕ and among themselves can be extended to Wick products [11, 20, 29]. The expressions 2 d d k1 dd k2 h(k12 , k22 ) (: ϕ :)h (x) = V+

V+

: [a(k1 )e−ik1 x + h.c.][a(k2 )e−ik2 x + h.c.] :

(2.14)

deﬁne Wightman ﬁelds, relatively local with respect to ϕ and ϕh and mutually local among each other, for every (smooth) polynomially bounded real symmetric function h on R+ × R+ . Formally, they may be represented as point-split limits of the form (: ϕ2 :)h (x) = lim h(−, − ) ϕ(x)ϕ(x ) − Ω, ϕ(x)ϕ(x )Ω . (2.15) x →x

The smoothness of the functions h may be considerably relaxed. While we refer to [11] for details, we point out that for (: ϕ2 :)h to be a Wightman ﬁeld, h2

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ought to be at least a measurable function on R+ × R+ : otherwise (: ϕ2 :)h (f ) fails to be an operator with the vacuum vector in its domain. This can be seen easily from the 2-point function Ω, (: ϕ2 :)h (x)(: ϕ2 :)h (x )Ω = =2 dm21 dm22 h(m21 , m22 )2 Wm1 (x − x )Wm2 (x − x ).

(2.16)

R+

It is clear how this construction generalizes to higher Wick polynomials, and also to multi-local ﬁelds such as (: ϕ(x1 )ϕ(x2 ) :)h . All these ﬁelds satisfy local commutativity among each other with respect to their arguments in spite of the non-local operations involved. It is crucial that the weight functions h depend only on the squares of the four-momenta, since general functions of the components k µ would spoil local commutativity. It is also clear that the construction can be as well applied to Wick polynomials of derivatives ∂µ . . . ∂ν ϕ of the generalized free ﬁeld. Generalized Wick polynomials belong to the Borchers class of the generalized free ﬁeld consisting of the relatively local Wightman ﬁelds deﬁned on the same Hilbert space. With suitable speciﬁcations of the functions h involved, they exhaust the Borchers class [29, 11]. They are natural candidates for perturbative interactions, e.g., in causal perturbation theory [17, 9, 14].

2.2

Conformal symmetry

The 1-particle Hilbert space H1 , and hence the Fock space H, carry also a natural representation of the group of dilations with generator D = (2π)d−1 dd k a+ (k) 2i (k · ∂k ) + (∂k · k) a(k). (2.17) V+

Under this representation, the generalized free ﬁelds ϕh transform according to U (λ)ϕh (x)U (λ)∗ = ϕhλ (λx)

(2.18)

d

where hλ (m2 ) = λ 2 h(λ2 m2 ). In particular, the generalized free ﬁelds with homogeneous weight functions mν , ϕ(∆) = (−)ν/2 ϕ

with

ν ≡ ∆−

d 2

transform like scale-invariant ﬁelds of scaling dimensions ∆ =

(2.19) d 2

+ ν:

U (λ)ϕ(∆) (x)U (λ)∗ = λ∆ ϕ(∆) (λx),

(2.20)

i[D, ϕ(∆) (x)] = (xµ ∂µ + ∆)ϕ(∆) (x).

(2.21)

or in inﬁnitesimal form

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U (λ) scales the momenta (2.4) and commutes with the Lorentz transformations (2.5). Hence it extends the representation of the Poincar´e group to a representation of the Poincar´e-dilation group, also denoted by U . It is well known, that the scale-invariant 2-point functions of these ﬁelds are in fact conformally covariant, extending the ﬁelds to a suitable covering of Minkowski space-time CMd [30]. This means in particular that the unitary representation U of the Poincar´e-dilation group extends to a unitary representation U (∆) of the conformal covering group. Under U (∆) , the ﬁeld ϕ(∆) transforms as a conformally covariant scalar ﬁeld of scaling dimension ∆. Unlike those of the Poincar´e-dilation group, the generators of the special conformal transformations depend on the parameter ∆ = d2 + ν and are explicitly given on H1 by Kµ(∆) = (2π)d−1 dd k +

a (k) such that

V+

∂ ∂ kµ ∂ ∂ kµ − (k · ∂k ) µ − µ (∂k · k) + ν 2 2 ∂k α ∂kα ∂k ∂k k

a(k),

i[Kµ(∆) , ϕ(∆) (x)] = 2xµ (x · ∂) − x2 ∂µ + 2∆xµ ϕ(∆) (x).

(2.22)

(2.23)

We emphasize that although the ﬁelds ϕ(∆) are for all values of ∆ deﬁned on the same Hilbert space (the common Fock space H for all generalized free ﬁelds ϕh ), they are conformally covariant with respect to diﬀerent representations U (∆) of the conformal group on the same Hilbert space. These representations coincide only on the Poincar´e-dilation subgroup. U (∆) does not implement a geometrical point transformation of ϕ(∆ ) , ∆ = ∆, nor of ϕh in general.

3 The stress-energy tensor The purpose of this section is to ﬁnd a stress-energy tensor Θµν (x) for the generalized free ﬁeld (2.1) which has the properties of a local and covariant conserved tensor density for the generators of the Poincar´e group. It should thus satisfy

∂ µ Θµν = 0,

(3.1)

dd−1 x Θ0ν = Pν ,

(3.2)

dd−1 x (xµ Θ0ν − xν Θ0µ ) = Mµν ,

(3.3)

Θµν (x) = Θνµ (x),

[Θµν (x), ϕh (x )] = 0

(3.4) 2

((x − x ) < 0).

(3.5)

With an ansatz of the form (2.14), including derivatives of ϕ, we ﬁnd the solution (3.6) Θµν = : ∂µ ϕ∂ν ϕ − 12 ηµν (∂α ϕ∂ α ϕ + ϕϕ) : δ .

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The generalized Wick product (: · · · :)δ is understood as (: · · · :)h with the choice of the weight “function” h(m21 , m22 ) = δ(m21 , m22 ) ≡ δ(m21 − m22 )θ(m21 ).

(3.7)

This singular choice cannot be avoided due to the requirement (3.2), since the spatial integral over a generalized Wick square such as (2.14) can only enforce equality of the spatial components of the momenta of the creation and annihilation operators, k1 = k2 , while the representation (2.4) of the total momentum operator requires k1µ = k2µ . But with this singular weight function h, the 2-point function (2.16) involving h2 becomes highly divergent, hence the stress-energy tensor has inﬁnite ﬂuctuations in the vacuum state. Smearing Θµν with a test function does not give an operator whose domain contains the vacuum vector. Thus the stressenergy tensor is not a Wightman ﬁeld. But Θµν (f ) is a quadratic form on the Wightman domain of ϕh , i.e., its matrix elements with vectors from that domain are ﬁnite (more precisely: are continuous functionals of the test function f ). To prove this, only the ﬁniteness of Ω, Θµν (f )ϕh1 (f1 )ϕh2 (f2 )Ω, Ω, ϕh1 (f1 )Θµν (f )ϕh2 (f2 )Ω,

(3.8)

Ω, ϕh1 (f1 )ϕh2 (f2 )Θµν (f )Ω needs to be checked since every matrix element of Θµν (f ) on this domain is a sum of terms of either of these forms with ﬁnite coeﬃcients. Explicit evaluation of the above matrix elements, which are all of the form dd k1 fˆ1 (±k1 )h1 (k12 ) dd k2 fˆ2 (±k2 )h2 (k22 ) δ(k12 − k22 ) V+

V+

×P (k1 , k2 )fˆ( ± k1 ± k2 )

(3.9)

with P some polynomial, exhibits their ﬁniteness and continuity with respect to f , for arbitrary test functions fi and arbitrary weight functions hi . Θµν (f ) being a quadratic form on the Wightman domain of ϕh , its commutator with ϕh (g) is a priori well deﬁned as a quadratic form. It turns out to be in fact an operator on the Wightman domain. It vanishes if the supports of f and g are spacelike separated. The formula (3.6) for Θµν is uniquely determined by the requirements (3.1– 5), up to addition of a multiple of (∂µ ∂ν − ηµν )(: ϕ2 :)δ . We note that Θµν is not traceless, nor can it be made traceless by such an addition. Thus it does not provide a density for the generator of the dilations (2.17), nor for the conformal transformations (2.22).

4 Application to AdS-CFT The conformal group SO(2, d) of d-dimensional Minkowski space-time coincides with the group of isometries of d + 1-dimensional anti-deSitter space-time ADSd+1 .

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In fact, ADSd+1 has a conformal boundary which is a twofold covering4 of the Dirac compactiﬁcation CMd of Minkowski space-time, such that the AdS group restricted to the boundary acts like the conformal group on CMd . One-parameter subgroups with future-directed timelike tangent vectors in AdS (“time evolutions”) have future-directed timelike tangent vectors in CMd . Hence, the respective (AdS and conformal) notions of “positive energy” for the unitary representations of SO(2, d) coincide. It was shown in [19] that the scalar Klein-Gordon ﬁeld on ADSd+1 can be canonically quantized on a 1-particle space which carries a positive-energy representation of the AdS group. Our aim is to ﬁnd the explicit relation between these scalar Klein-Gordon ﬁelds φ on ADSd+1 (parameterized by a parameter ν > −1 such that the Klein2 Gordon mass equals M 2 = ν 2 − d4 ) and the generalized free ﬁeld on Minkowski space-time Md (characterized by its scaling dimension ∆ = d2 + ν). Both ﬁelds are deﬁned on the same Fock space H over the 1-particle space H1 (2.3), carrying the same unitary positive-energy representation U (∆) of SO(2, d) under which both ﬁelds transform covariantly in the respective (AdS or conformal) sense. We shall work in the convenient chart of AdS given by Poincar´e coordinates xM ≡ (z ∈ R+ , xµ ∈ Md ), in which the metric takes the form ds2 = gMN dxM dxN = z −2 · (ηµν dxµ dxν − dz 2 ),

(4.1)

i.e., it is a “warped product” of d-dimensional Minkowski space-time Md by R+ , or in the terminology of [7], AdS has a foliation by Md . The chart is given as follows. We ﬁx a pair e± of lightlike vectors in R2,d , e+ · e− = 12 , and a basis eµ of the subspace orthogonal to e± , with eµ · eν = ηµν . Then ξ = z −1 · (xµ eµ + e− + (z 2 − xµ xµ )e+ )

(4.2)

fulﬁlls ξ · ξ = 1. This chart covers PADSd+1 except for the hypersurface ξ · e+ = 0 which formally corresponds to z = ∞. The corresponding chart (xµ ∈ Md ) of CMd , parameterizing the lightlike rays 2,d in R by ζ = R · (xµ eµ + e− − xµ xµ e+ ), (4.3) is Minkowski space-time Md ⊂ CMd . This chart misses out the hypersurface of compactiﬁcation points “at inﬁnity” of Md , consisting of the lightlike rays orthogonal to e+ , namely the rays R · (λe+ + xµ eµ ), xµ xµ = 0. In CMd , these are the points at lightlike distance from ω ≡ R · e+ (the compactiﬁcation point “at spacelike inﬁnity” of Md ). 4 We

denote by ADSd+1 the quadric ξ · ξ = 1 in R2,d (signature (+, +, −, . . . , −)), and by PADSd+1 its quotient by the antipodal identiﬁcation ξ ↔ −ξ. The conformal boundary of this quotient is the Dirac compactiﬁcation CMd of Minkowski space-time whose points are the lightlike rays ζ = R · n, n · n = 0, in R2,d . The ﬁeld theories discussed below in general are deﬁned on covering spaces of the respective manifolds.

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ζ(xµ ) is the boundary point approached by ξ(z, xµ ) as z → 0. Thus the Poincar´e chart meets the boundary exactly in Minkowski space-time. From this discussion, we conclude that the chart (z, xµ ) is mapped onto itself only by the stabilizer subgroup in SO(2, d) of the ray R+ ·e+ in R2,d . This subgroup has the form (4.4) (SO(1, d − 1) × R+ ) Rd . Here, SO(1, d − 1) Rd is the stabilizer group of the vector e+ . SO(1, d − 1) preserves also e− and transforms the basis eµ like the Lorentz group, while Rd takes e− → e− + aµ eµ − aµ aµ e+ , eµ → eµ − 2aµ e+ . Hence the stabilizer subgroup of e+ preserves the coordinate z = (2ξ · e+ )−1 and acts on xµ like the Poincar´e group. The remaining factor R+ in (4.4) scales e± → λ±1 e± and preserves eµ , hence it takes (z, xµ ) to (λz, λxµ ) and acts on the boundary z = 0 like the dilations. We shall refer to these subgroups of SO(2, d) as Poincar´e and dilation subgroups also in the AdS context. Thus, the Poincar´e chart of AdS is preserved by the Poincar´edilation group of Minkowski space-time. The remaining elements of SO(2, d) induce rational transformations of the coordinates (z, xµ ), such as (z, xµ ) → (z, xµ − bµ (x2 − z 2 ))/(1 − 2(b · x) + b2 (x2 − z 2 )), b ∈ Md , restricting to the special conformal transformations of the boundary z = 0.

4.1

The Klein-Gordon field on AdS

We ﬁx any value ν > −1 and set ∆ = The Klein-Gordon ﬁeld on AdS

d 2

+ ν and M 2 = ∆(∆ − d) = ν 2 −

(g + M 2 )φ = (−z 1+d ∂z z 1−d ∂z + z 2 η + M 2 )φ = 0

d2 4 .

(4.5)

has been quantized with an AdS-invariant vacuum state, e.g., in [3, 19]. Its 2-point function can be displayed in the form [7] d 1 2 Ω, φ(z, x)φ(z , x )Ω = 2 (zz ) dm2 Jν (zm)Jν (z m) Wm (x − x ) R+ √ √ d dd k Jν (z k 2 )Jν (z k 2 ) e−ik(x−x ) . (4.6) = (2π)−(d−1) 12 (zz ) 2 V+

Here, Jν is the Bessel function solving Bessel’s diﬀerential equation ((u∂u )2 + u2 )Jν (u) = ν 2 Jν (u),

(4.7) √ d and z 2 Jν (z k 2 )e±ikx (k ∈ V+ ) are the plane-wave solutions of the Klein-Gordon equation (4.5). We note that, depending on the integrality of the parameter ∆ = d2 + ν, the quantum ﬁeld φ is in general deﬁned on a covering space of ADSd+1 [7, 19]. This complicates the analysis, but the complications precisely match the complications arising in the corresponding conformal QFT which is deﬁned on a covering space

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of CMd . We shall limit ourselves to the Klein-Gordon ﬁeld on the Poincar´e chart (z, xµ ), and correspondingly to boundary ﬁelds in Minkowski space-time. We also note that for |ν| < 1, the two possible signs of ν give rise to inequivalent covariant quantizations [8] (in fact, an interpolating one-parameter family [7]) of the Klein-Gordon ﬁeld of the same mass. The commutator functions derived from (4.6) are the same for both signs of ν [7], hence both quantum ﬁeld theories have the same local structure. φ satisﬁes the canonical equal-time commutation relation between the ﬁeld and its canonical momentum π = z 1−d ∂0 φ [φ(z, x), π(z , x )]|x0 =x0 = iδ d−1 ( x − x )δ(z − z ),

(4.8)

as can be veriﬁed from (4.6), using the fact that Wm (x − x ) satisﬁes canonical commutation relations on Md , and using Hankel’s identity

∞

t dt Jν (tu)Jν (tu ) = u−1 δ(u − u ),

(4.9)

0

which expresses the completeness of the plane-wave solutions involved in the integral (4.6).

4.2

Expression in terms of generalized free fields

By comparison of (4.6) with (2.11), we conclude that φ(z, x) can be identiﬁed with φ(z, x) =

√1 2

z

d 2

√ dd k Jν (z k 2 )[a(k)e−ikx + a+ (k)eikx ].

(4.10)

V+

This is of the form φ(z, x) = ϕhz (x)

(4.11)

with hz (m2 ) =

√1 2

d

z 2 Jν (zm)

(4.12)

i.e., for each value of z, φ(z, ·) is one of the generalized free ﬁelds considered in Sect. 2.1. 2−ν From the power law behaviour of Jν (u) ≈ Γ(ν+1) uν (1 + O(u2 )) at small arguments, one obtains lim z −∆ φ(z, x) =

z→0

1

2−ν− 2 Γ(ν+1)

ϕ(∆) (x),

(4.13)

i.e., the generalized free ﬁeld ϕ(∆) is the boundary limit of the Klein-Gordon ﬁeld on AdS. This agrees with the results discussed in [7] and [15].

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Identification of the representations of SO(2, d)

The 2-point function (4.6) is AdS-invariant, i.e., it depends only on the AdSinvariant “chordal distance” (the distance within R2,d ) (−(x−x )2 +(z−z )2 )/2zz . It follows that the representation of the AdS group deﬁned by U (KG) (g) φ(z, x)Ω := φ(g(z, x))Ω

(g ∈ SO(2, d))

(4.14)

is unitary on the 1-particle space H1 , and hence on H. It implements the covariant transformation law U (KG) (g)φ(z, x)U (KG) (g)∗ = φ(g(z, x)).

(4.15)

We want to show that this representation of SO(2, d) coincides with the representation U (∆) of the conformal group, constructed on the same Hilbert space by the extension of the scale-invariant ﬁeld ϕ(∆) on Md to the conformally invariant ﬁeld on CMd (cf. Sect. 2.2). Eq. (4.15), restricted to the boundary z = 0 where g acts like the conformal group on x, shows that U (KG) (g) implements the same conformal point transformation of the limiting ﬁeld ϕ(∆) as U (∆) (g). More speciﬁcally, consider the inﬁnitesimal form of the AdS transformation law (4.15) for the relevant subgroups, i[Pµ(KG) , φ(z, x)] = ∂µ φ(z, x), (KG) i[Mµν , φ(z, x)] = (xµ ∂ν − xν ∂µ )φ(z, x),

i[D(KG) , φ(z, x)] = (z∂z + xµ ∂µ )φ(z, x), i[Kµ(KG) , φ(z, x)] = 2xµ (z∂z + (x · ∂)) + (z 2 − x2 )∂µ φ(z, x).

(4.16)

In the limit z → 0 according to (4.13), the right-hand sides of (4.16) turn into (2.6), (2.7), (2.21), (2.23), respectively. Thus, the inﬁnitesimal generators of the respective subgroups coincide. We conclude that the two representations U (KG) and U (∆) of SO(2, d) coincide, cf. [12]. With the help of the representation U (KG) = U (∆) , the AdS ﬁeld φ and the boundary ﬁeld ϕ(∆) extend to the respective covering spaces of PADSd+1 and CMd . We emphasize once more that this does not apply for the other boundary ﬁelds ϕh .

4.4

“Holographic” interpretation

Combining (4.11) and (2.19), we ﬁnd φ(z, x) = where

√1 2

z ∆ jν (−z 2 ) ϕ(∆) (x),

jν (u2 ) = u−ν Jν (u)

(4.17)

(4.18)

is a (polynomially bounded) convergent power series in u2 . This is an explicit expression for the Klein-Gordon ﬁeld on AdS in terms of its limiting generalized

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free ﬁeld on the boundary. This “holographic” relation5 is possible with the help of non-local (pseudo) diﬀerential operators of the kind discussed in Sect. 2. Obviously, the fact that ϕh = h(−)ϕ are mutually local ﬁelds for arbitrary functions h, ensures that φ(z, x) and φ(z , x ) written in the form (4.11) or (4.17) commute if (x − x )2 < 0. But this is less than locality on AdS, which requires that the commutator must vanish even if (x − x )2 < (z − z )2 . Of course we know that φ is a local ﬁeld on AdS, so the stronger local commutativity of generalized free ﬁelds ϕhz (x) and ϕhz (x ) at ﬁnite timelike Minkowski distance must be true. One can understand the origin of this “bonus locality” for the generalized free ﬁelds involved. Evaluating the commutator function according to (2.10), gives an integral over three Bessel functions (because ∆m at timelike distance (x−x )2 = τ 2 d−2 is also given by a Bessel function (m/τ ) 2 J 2−d (mτ )) of the form 2

∞

I(a, b, c) =

u1−µ du Jµ (au)Jν (bu)Jν (cu)

(4.19)

0

with a2 = (x − x )2 , b = z, c = z and µ = 2−d 2 . This integral can be found, e.g., in [35, Sect. 13·46(1)], where it is shown to vanish if a2 < (b − c)2 . Thus we precisely ﬁnd local commutativity on AdS. It is the speciﬁc form of the Bessel functions in (4.17), (4.18) which is able to ensure locality in a higher-dimensional space-time. This remark should make it clear that deﬁning φ(z, x) = ϕhz (x) with any suitable family of functions hz (m2 ), depending on a parameter z and solving a suitable diﬀerential equation with respect to z, may well produce a quantum ﬁeld on a higher-dimensional space-time solving some equation of motion, but this ﬁeld will in general not satisfy local commutativity.6

4.5

The stress-energy tensor

The Klein-Gordon ﬁeld on AdS has a canonical covariantly conserved stress-energy tensor given by AB 1 DA φDB φ − M 2 φ2 ) : . ΘKG MN (z, x) = : DM φDN φ − 2 gMN (g

(4.20)

Because of the special form of the AdS metric, the covariant tensor continuity equation gives rise to ordinary continuity equations for the Minkowski components ΘKG Mν , ν = z, g MN ∂N (z 1−d ΘKG (4.21) Mν ) = 0. 5 Our use of the term “holography” does not quite match the one originally suggested by ’t Hooft [25], namely the reduction in the bulk of degrees of freedom of a QFT, ascribed to gravitational eﬀects in the presence of a horizon. We rather allude to the enhancement on the boundary of degrees of freedom necessary and suﬃcient to “encode” a non-gravitational QFT in the bulk. 6 Such constructions were proposed in [32].

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Thus z 1−d ΘKG 0ν are densities of conserved quantities, which are the generators of the Poincar´e subgroup of SO(2, d), ∞ Pµ = dzz 1−d dd−1 x ΘKG x), (4.22) 0µ (z, 0, 0 ∞ dzz 1−d dd−1 x (xµ ΘKG x) − xν ΘKG x)). (4.23) Mµν = 0ν (z, 0, 0µ (z, 0, 0

In these integrals, we may express φ in terms of the generalized free ﬁeld ϕ by means of (4.11), thus introducing two z-dependent (Bessel) weight functions, and perform the z-integration. Because of the term g MN ∂M φ∂N φ involving zderivatives, a partial integration becomes necessary after which Bessel’s diﬀerential equation can be used to eliminate all derivatives of the Bessel functions. After performing these steps on the Minkowski components of ΘKG µν , one ends up with a generalized Wick product of (derivatives of) ϕ, whose weight function is the result of the z-integration over the Bessel functions: ∞ α 1 dz z 1−d ΘKG (4.24) µν (z, x) = : ∂µ ϕ∂ν ϕ − 2 ηµν (∂α ϕ∂ ϕ + ϕϕ) : h 0

where h(m21 , m22 )

=

1 2

∞

z dz Jν (zm)Jν (zm ).

(4.25)

0

Once more using Hankel’s identity (4.9) which in this case plays the role of an orthonormality relation, the integral can be performed, giving h(m21 , m22 ) = δ(m21 − m22 ). We have thus exactly reproduced the singular stress-energy tensor found in Sect. 3, ∞

Θµν (x) =

dz z 1−d ΘKG µν (z, x).

(4.26)

0

But the origin of its singular weight function appears in an entirely new light: it is the result of the “holographic” projection of AdS onto its boundary. Some cutoﬀ in the z-integral would smoothen the resulting weight function (4.25). The smoothened stress-energy tensor would still act as a density for generators which generate the correct transformation laws on those AdS ﬁelds which are causally disconnected from the AdS region where the cutoﬀ is eﬀective. In terms of the corresponding generalized free ﬁelds on the boundary, these are generalized free ﬁelds within a restricted region and with a restricted set of weight functions hz .

5 Local algebras At ﬁrst sight, our “holographic” result of Sect. 4.4 does not quite agree with the algebraic analysis of AdS-CFT in [33]. Let us sketch the situation.

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According to (4.11), the AdS ﬁelds at the point (z, x) are expressed in terms of Minkowski space-time ﬁelds at the point x. Hence boundary observables (smeared ﬁelds) localized in a region K ⊂ Md encode all AdS observables localized in the region V (K) = π −1 (K) ⊂ ADSd+1 where π denotes the projection (z, x) → x. We may call this feature “projective holography”. On the other hand, the analysis called “algebraic holography” in [33] is based on the identiﬁcation ACFT (K) = AAdS (W )

(W = W (K))

(5.1)

of the local algebras ACFT (K) of boundary observables localized in “double-cones” d (the conformal transforms of K0 = {(xµ ) : | x| < 1−|x0 |} in the covering K ⊂ CM space of the Dirac compactiﬁcation of Minkowski space-time), with local algebras AKG (W ) of AdS observables localized in the “wedge” regions W = W (K) ⊂ √ µ DSd+1 (the AdS transforms of W0 = {(z, x ) : z 2 + x2 < 1 − |x0 |} in the A covering space of AdS). The wedge W (K) is the causal completion of the boundary region K. The map W = W (K) is a bijection which preserves inclusions, takes d into causal complements in A DSd+1 , and is compatible causal complements in CM with the respective actions of the covering group of SO(2, d). For K ⊂ Md , the wedge region W (K) extends only to ﬁnite “depth” z into AdS and is strictly smaller than V (K), which extends to z = ∞ and contains points causally disconnected from W (K). Hence AAdS (V (K)) is strictly larger than AAdS (W (K)), and the two notions of “holography” cannot be equivalent. We shall show how this apparent conﬂict is resolved, although the discussion should by no means be considered as rigorous. We shall deliberately ignore most of the technical subtleties involved in the passage between the Wightman axiomatic formulation of QFT (in terms of ﬁelds, which are unbounded-operator valued distributions) and the Haag-Kastler [23] algebraic formulation (in terms of localized observables, which are bounded operators). But we are conﬁdent that our argument captures correctly the essential features concerning the “size” of von Neumann algebras of local observables associated with generalized free ﬁelds and with free ﬁelds on AdS. The general idea for the passage from ﬁelds to local algebras is to deﬁne for any open space-time region O the von Neumann algebra A(O) := {φ(f ) : supp f ⊂ O}

(5.2)

where X stands for the algebra of bounded operators on the given Hilbert space which commute with (the closures of) all elements of X. By von Neumann’s density theorem, the double commutant A(O) is the weak closure of the bounded functions of the unbounded smeared ﬁeld operators (such as exp iφ(f ) if φ(f ) is self-adjoint). Obviously, the algebras increase as the regions increase (“isotony”). Although the underlying ﬁelds are local, it is less trivial [13] that local algebras of the form (5.2) associated with spacelike separated regions mutually commute (“locality”). A

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covariant transformation law of the ﬁelds such as (4.15) involves the corresponding transformation of the support of test functions, hence the local algebras (5.2) transform in the obvious sense under conjugation with the unitary representatives of the group (“covariance”). One may imagine that the ﬁelds can be recovered from the algebras by taking suitably regularized limits of elements of algebras associated with regions shrinking to a point, see e.g., [18]. Together with the group of covariance which involves the time evolution, the algebraic data determine the quantum ﬁeld theory.

5.1

“Algebraic” vs. “projective holography”

Applying the prescription (5.2) to the free Klein-Gordon ﬁeld of mass M on AdS, we obtain local algebras AKG (O),

(O ⊂ A DSd+1 ).

(5.3)

Applying the same prescription to the single generalized free ﬁeld ϕ(∆) on the conformal completion of Minkowski space-time, we obtain local algebras A∆ (K),

d ). (K ⊂ CM

(5.4)

Applying it to the the entire family of generalized free ﬁelds ϕh with arbitrary weight functions h on Minkowski space-time, we obtain local algebras Atot (K),

(K ⊂ Md ).

(5.5)

We have the rather obvious inclusions for K ⊂ Md A∆ (K) ⊂ AKG (W (K)) ⊂ AKG (V (K)) ⊂ Atot (K),

(5.6)

of which the ﬁrst reﬂects the limit (4.13), the second is isotony, and the last reﬂects (4.11). We shall show that in fact A∆ (K) = AKG (W (K))

(5.7)

AKG (V (K)) = Atot (K) = Adual ∆ (K).

(5.8)

are proper subalgebras of

In this formula, the “dual completion” [34] is deﬁned as c Adual ∆ (K) := A∆ (K )

(5.9)

where K c is the causal complement of K within Md . Fields which are relatively local to the given ﬁeld, are among the generators of the dual completion. Roughly speaking, the dual completion is the algebraic counterpart of the Borchers class in

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Wightman quantum ﬁeld theory. By a general theorem [10], conformal invariance ensures “conformal duality” A∆ (K) = A∆ (K )

(5.10)

d ; but if K is a double-cone where K is the causal complement of K within CM d c within M , then K is strictly smaller than K , and hence Adual ∆ (K) is expected to be strictly larger than A∆ (K) (violation of Haag duality). (5.7) is the holographic identiﬁcation of AKG (W ) with A∆ (K) in the sense of (5.1). It is deﬁned globally, and covariant with respect to SO(2, d). On the other hand, (5.8) shows that the projective notion of holography pertains to Atot (K) = Adual ∆ (K) instead. Projective holography is deﬁned only with respect to the chosen chart (z, xµ ), and is covariant only under the Poincar´e-dilation subgroup. Before we prove (5.7) and (5.8), we note that Atot (K) does not depend on the parameter ν specifying the scaling dimension of the ﬁeld ϕ(∆) and the mass of the corresponding Klein-Gordon ﬁeld φ. Hence, (5.7) can only be true, if AKG (V (K)) does not change if the generating Klein-Gordon ﬁeld φ of mass M 2 is replaced by φ of mass M 2 . Indeed, for diﬀerent values ν, ν , we have by (4.10) and Hankel’s formula (4.9) ∞ φ(z, x) = z dz Kνν (z, z )φ (z , x) (5.11) 0 d ∞ with the kernel Kνν (z, z ) = (z/z ) 2 0 m dm Jν (zm)Jν (z m). Since this kernel acts on the z coordinate only, it takes test functions supported in V (K) onto test functions supported in V (K), and hence AKG (V (K)) = AKG (V (K)). Let us now turn to (5.7). We invoke a general theorem [1, 6] of Wightman QFT. Let O be a double-cone and T a timelike hypersurface passing through the apices of O. Then, in order to generate A(O) as in (5.2), rather than smear the ﬁeld in O it suﬃces to smear the ﬁeld and all its normal derivatives along O ∩ T . In the present case, O is an AdS wedge W , T is the boundary, and O ∩ T is the corresponding boundary double-cone K. The normal derivatives of φ(z, x) are of the form limz→0 ∂zN z −∆ φ(z, x). By (4.17), and because jν is a power series in −z 2 , these derivatives vanish if N is odd and are proportional to n ϕ(∆) (x) if N = 2n. With (ϕ)(f ) = ϕ(f ) we conclude that AKG (W ) is in fact generated by ϕ(∆) (f ), supp f ⊂ K. This justiﬁes our claim (5.7). Now we turn to (5.8). Because the ﬁelds generating Atot (K) are relatively local with respect to ϕ(∆) , we know that AKG (V (K)) ⊂ Atot (K) ⊂ A∆ (K c ) ≡ Adual ∆ (K). The claim is that equality holds. A∆ (K c ) is generated by all A∆ (J) where J are double-cones in Md spacelike separated from K. Hence its commutant is the intersection, running over the same set of J, of algebras A∆ (J) = A∆ (J ) (by (5.10)) = AKG (W (J )) (by (5.7)). Now, J is spacelike separated from K and belongs to Md iﬀ J contains K and the point ω = R · e+ of CMd (spacelike inﬁnity of Md , cf. Sect. 4), and iﬀ the wedge W (J )

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contains W (K) and has ω as a boundary point (ω ∈ ∂W ). Thus, AKG (W ). A∆ (K c ) =

631

(5.12)

W ⊃W (K) ∂W ω

We may choose K = K0 . The x0 = 0 Cauchy surface of V (K0 ) is C0 = {(z, 0, x) : z > 0, x2 < 1}. Let We be the wedges with Cauchy surface Ce = {(z, 0, x) : z > 0, ( e · x) > −1}, e ∈ Rd−1 , e2 = 1. Each We contains W (K0 ) and has ω as a boundary point, and every wedge which contains W (K0 ) and has ω as a boundary point, contains some We . Thus, by isotony, the intersection of algebras in (5.12) may be taken over We , A∆ (K c ) = AKG (We ). (5.13) e

Now, because of the Klein-Gordon equation, the ﬁeld φ(z, x) is expressible in terms of its Cauchy data at x0 = 0, hence AKG (We ) = AKG (Ce ) and AKG (V (K0 )) = AKG (C0 ) (time-slice property [24], see Sect. 5.2 below). The latter algebras are generated by the canonical x0 = 0 Klein-Gordon ﬁelds φ and π (cf. Sect. 4.1) smeared over the respective regions of the Cauchy surface. Thus (5.8) is reduced to the claim AKG (Ce ) = AKG (C0 ). (5.14) e

The independence of Cauchy data associated with disjoint regions entails [2] (5.15) AKG (Ce ) = AKG Ce . e

e

Thus (5.8) is a consequence of the geometric fact Poincar´e and dilation covariance for all K ⊂ Md .

5.2

e e C

= C0 for K = K0 , and by

Time-slice property

Finally, we consider the validity of the “time-slice property” (also called “primitive causality” [24], or “weak additivity” in [33]) A(O) = A(C)

(5.16)

if O is the causal completion of its Cauchy surface C. A(C) is generated by the ﬁelds and their time derivatives smeared over C. This property holds for canonical free ﬁelds [24], and we have just used it in the case of the Klein-Gordon ﬁeld on AdS. The time-slice property does not hold for generalized free ﬁelds (with h ﬁxed) [24]. E.g., for the above double-cone K0 the Cauchy surface is the ball B0 = {(x0 = 0, x) : x2 < 1}. Smearing the generalized free ﬁeld ϕ(∆) together with its time derivatives over this surface, tests only the boundary limits of the corresponding

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canonical x0 = 0 Klein-Gordon ﬁelds φ and π and their z-derivatives at {(z = 0, x0 = 0, x) : x2 < 1}. Since this set does not constitute a Cauchy surface for the wedge W (K0 ), the generalized free ﬁeld smeared over the Cauchy surface of K0 generates only a proper subalgebra of A∆ (K0 ) = AKG (W (K0 )). This violation of the time-slice property is a necessary feature of algebraic holography [33]. We want to show that the time-slice property is restored for the dual completion Adual ∆ (K) = Atot (K): Atot (K) = Atot (C).

(5.17)

Again we may choose K = K0 . The algebra Atot (K0 ) equals AKG (V (K0 )) by (5.8) and hence is generated by the canonical x0 = 0 Klein-Gordon ﬁelds φ and π smeared over the Cauchy surface C0 of V (K0 ). By (4.11), such smearing are smearing of x0 = 0 generalized free ﬁelds ϕh (0, x) and their time derivatives over the Cauchy surface B0 of K0 . Hence the latter generate Atot (K0 ). This justiﬁes (5.17). It is natural to consider this restoration of the time-slice property as being related to the existence of the singular stress-energy tensor (3.6) which is in some weak technical sense associated with the algebras Atot (K) but not with A∆ (K), and whose integral over a Cauchy surface generates the causal time evolution. Even though the stress-energy tensor is not itself a Wightman ﬁeld, it is not too singular to have this desirable dynamical consequence for the structure of the local algebras associated with the Wightman ﬁelds of the theory.

6 Conclusion We have studied generalized free ﬁelds from a general perspective and established, in spite of the non-canonical nature of these ﬁelds, the existence of a stress-energy tensor which serves as a density for the generators of the Poincar´e group as in the canonical framework. We have pointed out that this stress-energy tensor is a mathematical object which is more singular than a Wightman ﬁeld, but can be obtained as a certain limit of generalized Wick products. We have then studied the free ﬁeld AdS-CFT-correspondence in the light of the previous results on generalized free ﬁelds. In particular, we have given an explicit “holographic” formula expressing the Klein-Gordon ﬁeld on AdS in terms of generalized free ﬁelds on the boundary. We have identiﬁed the above stressenergy tensor for generalized free ﬁelds as an integral “along the z-axis of AdS” over the canonical Klein-Gordon stress-energy tensor ΘKG µν (z, ·) on AdS. These results should be useful as a starting point for a perturbation theory of the AdS-CFT-correspondence. If the AdS ﬁeld is perturbed by a local interaction Lagrangean density LI (φ) on AdS, it is naively expected that the eﬀect on the conformal ﬁeld (generalized free ﬁeld) on the boundary is that of a Lagrangean perturbation by the integral “along the z-axis of AdS” of LI (φ(z, ·)). This integral, akin to (4.24), is again of the form of a (regular or singular) generalized

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Wick product of the boundary ﬁeld. Further analysis of this issue will be pursued elsewhere. Acknowledgments. This work was supported in part by the Deutsche Forschungsgemeinschaft and by the Erwin Schr¨ odinger International Institute (ESI), Vienna. The authors gratefully acknowledge interesting discussions with E. Br¨ uning, C.J. Fewster, K. Fredenhagen, and S. Hollands.

References [1] H. Araki, A generalization of Borchers’ theorem, Helv. Phys. Acta 36, 132–139 (1963). [2] H. Araki, A lattice of von Neumann algebras associated with the quantum theory of a free Bose ﬁeld, J. Math. Phys. 4, 1343–1362 (1963). [3] S.J. Avis, C.J. Isham, D. Storey, Quantum ﬁeld theory in anti-de Sitter spacetime, Phys. Rev. D 18, 3565–3576 (1978). [4] T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal ﬁeld theory, arXiv:hep-th/9808016. ¨ [5] H.-J. Borchers, Uber die Mannigfaltigkeit der interpolierenden Felder zu einer kausalen S-Matrix, Nuovo Cim. 15, 784–794 (1960). ¨ [6] H.-J. Borchers, Uber die Vollst¨andigkeit lorentzinvarianter Felder in einer zeitartigen R¨ohre, Nuovo Cim. 19, 787–793 (1961). [7] M. Bertola, J. Bros, U. Moschella, R. Schaeﬀer, A general construction of conformal ﬁeld theories from scalar anti-de Sitter quantum ﬁeld theories, Nucl. Phys. B 587, 619–644 (2000) [=hep-th/9908140]; M. Bertola, J. Bros, V. Gorini, U. Moschella and R. Schaeﬀer, Decomposing quantum ﬁelds on branes, Nucl. Phys. B 581, 575–603 (2000). [8] P. Breitenlohner, D.Z. Freedman, Stability in gauged extended supergravity, Ann. Physics 144, 249–281 (1982). [9] R. Brunetti, K. Fredenhagen, Microlocal analysis and interacting quantum ﬁeld theories: renormalization on physical backgrounds, Commun. Math. Phys. 208, 623–661 (2000). [10] R. Brunetti, D. Guido and R. Longo, Modular structure and duality in conformal quantum ﬁeld theory, Commun. Math. Phys. 156, 201–220 (1993). [11] E. Br¨ uning, A new class of Wick powers of generalized free ﬁelds, J. Math. Phys. 25, 3064–3075 (1984). [12] V.K. Dobrev, Intertwining operator realization of the AdS/CFT correspondence, Nucl. Phys. B 553, 559–582 (1999). [13] W. Driessler and J. Fr¨ ohlich, The reconstruction of local observable algebras from the Euclidean Green’s functions of a relativistic quantum ﬁeld theory, Ann. Poincar´e Phys. Theor. 27, 221–236 (1977).

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[14] M. D¨ utsch and K. Fredenhagen, Algebraic quantum ﬁeld theory, perturbation theory, and the loop expansion, Commun. Math. Phys. 219, 5–30 (2001). [15] M. D¨ utsch, K.-H. Rehren, A comment on the dual ﬁeld in the AdS-CFT correspondence, Lett. Math. Phys. 62, 171–184 (2002). [16] H. Epstein, On the Borchers class of a free ﬁeld, Nuovo Cim. 27, 886–893 (1963). [17] H. Epstein, V. Glaser, The role of locality in perturbation theory, Ann. Inst. H. Poincar´e A 19, 211–295 (1973). [18] K. Fredenhagen, V. Hertel, Local algebras of observables and pointlike localized ﬁelds, Commun. Math. Phys. 80, 555–561 (1981). [19] C. Fronsdal, Elementary particles in a curved space. II, Phys. Rev. D 10, 589–598 (1974). [20] O.W. Greenberg, Generalized free ﬁelds and models of local ﬁeld theory, Ann. Physics 16, 158–176 (1961). [21] O.W. Greenberg, Heisenberg ﬁelds which vanish on domains of momentum space, J. Math. Phys. 3, 859–856 (1962). [22] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428, 105–114 (1998). [23] R. Haag, D. Kastler, An algebraic approach to quantum ﬁeld theory, J. Math. Phys. 5, 848–861 (1964). [24] R. Haag, B. Schroer, Postulates of quantum ﬁeld theory, J. Math. Phys. 3, 248–256 (1962). [25] G. ’t Hooft, Dimensional reduction in quantum gravity, in: Salam-Festschrift, A. Ali et al. (eds.), World Scientiﬁc 1993, pp. 284–296, arXiv:gr-qc/9310026. [26] G. ’t Hooft, Large N , arXiv:hep-th/0204069. [27] R. Jost, The General Theory of Quantized Fields, AMS, Providence, RI, 1965. [28] G. K¨ allen, On the deﬁnition of the renormalization constants, Helv. Phys. Acta 25, 417–434 (1952); ¨ H. Lehmann, Uber Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder, Nuovo Cim. [9] 11, 342 (1954). [29] A.L. Licht, A generalized asymptotic condition, Ann. Physics 34, 161–186 (1965); and Thesis, Maryland 1963, as referred to in: A.S. Wightman, Introduction to some aspects of the relativistic dynamics of quantized fields, Sect. II, Carg`ese lectures 1964, ed. M. L´evy (Gordon and Breach, New York, 1967). [30] M. L¨ uscher, G. Mack, Global conformal invariance in quantum ﬁeld theory, Commun. Math. Phys. 41, 203–234 (1975). [31] J.M. Maldacena, The large N limit of superconformal ﬁeld theories and supergravity, Adv. Theor. Math. Phys. 2, 231–252 (1998).

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[32] M. Mintchev, Local ﬁelds on the brane induced by nonlocal ﬁelds in the bulk, Class. Quant. Grav. 18, 4801–4812 (2001). [33] K.-H. Rehren, Algebraic holography, Ann. Henri Poincar´e 1, 607–623 (2000); K.-H. Rehren, Local quantum observables in the AdS-CFT correspondence, Phys. Lett. B 493, 383–388 (2000). [34] J.E. Roberts, Net cohomology and its application to field theory, in: Quantum Fields – Algebras, Processes, L. Streit (ed.), Springer 1980, pp. 239–268. [35] G.N. Watson: A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press 1958 (2nd edition). [36] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253–291 (1998). Michael D¨ utsch and Karl-Henning Rehren 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 15/10/02, accepted 24/06/03

Ann. Henri Poincar´e 4 (2003) 637 – 659 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040637-23 DOI 10.1007/s00023-003-0142-8

Annales Henri Poincar´ e

An Indefinite Metric Model for Interacting Quantum Fields on Globally Hyperbolic Space-Times Hanno Gottschalk and Horst Thaler Abstract. We consider a relativistic ansatz for the vacuum expectation values (VEVs) of a quantum ﬁeld on a globally hyperbolic space-time which is motivated by certain Euclidean ﬁeld theories. The Yang-Feldman asymptotic condition w.r.t. an “in”-ﬁeld in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum ﬁelds with indeﬁnite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum ﬁelds is analyzed by construction of the “out”-ﬁelds and calculation of the scattering matrix. A new combined eﬀect of non-trivial quantum scattering and non-stationary gravitational forces is described for this model, as quasi-free “in”- ﬁelds are scattered to “out”-ﬁelds which form a non-quasi-free representations of the CCR. The asymptotic condition, on which the construction is based, is veriﬁed for the concrete example of de Sitter space-time.

1 Introduction The interest in quantum ﬁelds in curved space-times stems from the very physical question how a curved geometry combines with quantum eﬀects. The particle production observed in the case of non-stationary background gravitation (related to the Hawking eﬀect [14, 27]) is possible since the state of the system restricted to diﬀerent space time regions gives in general rise to diﬀerent representations which account for the particle production. In the present work we investigate a similar eﬀect for a class of interacting indeﬁnite metric quantum ﬁeld models on globally hyperbolic space-times. The models, originating from Euclidean quantum ﬁeld theory (QFT) using Poisson ﬁelds [2, 3], in the case of Minkowski space-time give rise to solutions to the modiﬁed Wightman axioms of Morchio and Strocchi [17] which exist in spacetime dimensions higher than three and show a non-trivial scattering behavior, see [1, 2]. It turns out that a similar construction can be carried out on quite general globally hyperbolic manifolds (Sections 2 and 3) without directly deriving them from Euclidean models, which however remain a source of “inspiration”. The theories’ vacuum expectation values fulﬁll the requirements of locality, invariance under time-orientation preserving isomorphisms and Hermiticity (Section 4). They thus give rise to a GNS-like representation of the ﬁeld algebra on an non-degenerate inner product space and hence an “indeﬁnite metric” QFT (see Appendix A). The

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restriction of this representation to the sub-algebras generated by incoming and outgoing ﬁelds can then be investigated through algebraic methods. Our emphasis is put on a scattering theory based on Yang-Feldman equations [28] which also unveils the way how a curved geometry inﬂuences the character of a free ﬁeld when it travels through space-time and at the same time undergoes a quantum-mechanical scattering. Preparing the state for the algebra of “in”-ﬁelds in a quasi-free representation, existence of “out”-ﬁelds ﬁelds hinges upon the dispersion properties of the fundamental solutions of the Klein-Gordon equation and necessitates the formulation of a so-called dispersion condition whose fulﬁllment may depend not only on the manifold structure but also on the dimension. This allows the calculation of the matrix elements between “in”- and “out”ﬁelds which, as in the ﬂat case [1], expose non-trivial scattering. When observing the matrix elements of only the “out”-ﬁelds it turns out that they describe a representation of the CCR which in general is rather diﬀerent from that of the “in”-ﬁelds (Section 5) . In particular, “in”-ﬁelds in quasi-free representations are scattered to “out”-ﬁelds in non-quasi-free representations. What leads to this eﬀect is a combination of non-trivial quantum scattering and time-dependent gravitational forces. It cannot be reduced to the conceptually similar, but mathematically diﬀerent change of representations related to the Hawking-eﬀect [14, 27]. “In”- and “out”-representations are shown to be equivalent in the case of static space-times, where a spectrum condition can be formulated (Section 6). As a concrete example, we verify the dispersion condition for de Sitter spacetime (Section 7). The scope of this paper is to communicate observations made in the case of our comparatively simple model. At this point it is natural to put forward the question, whether the observed eﬀects also play a rˆ ole in the case of more physically motivated perturbative constructions [9, 25, 6] of interacting quantum ﬁelds on non-stationary globally hyperbolic space-times. Our conjectured answer to this question is “yes”. The ﬁndings of this paper can be related with eﬀects in ﬁrst order: φn :-perturbation theory on a non-stationary globally hyperbolic manifold using the calculus of sectorized Feynman graphs of A. Ostendorf [18] and O. Steinmann. [20, 21]. In fact, up to diﬀerent initial conditions (the Feynman rules of [18, 20, 21] on a non-stationary space-time manifold do not lead to a quasifree “in”- or “out”-state) our n-point functions coincide with the evaluation of the “star”-graph with one vertex and n legs, which is the ﬁrst order contribution to the connected n-point function. This indicates that non-quasi-free representations of the CCR have a natural place in interacting QFT on curved space-time and might enhance the recent interest in this topic [16].

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2 The relativistic ansatz We want to construct Wightman functions on a d-dimensional Lorentzian manifold1 (M, g) which is globally hyperbolic and thus is time-oriented by deﬁnition. For x, y ∈ M, we say that x and y, x = y, are light-like/time-like/space-like separated, if they can be connected by a light-like/time-like curve2 or if there is no such curve, respectively. That x is space-like to y is expressed by the symbol x ⊥ y. For x and y light-like or time-like separated and x later than y (in the time-direction ﬁxed on M) we write x ≥ y. By V¯x± we denote the open forward/backward light-cone with base-point x, i.e., V¯x± = {y ∈ M : y ≥ (≤)x}. We next introduce the fundamental functions following [12]. Let Gr and Ga be the retarded/advanced fundamental solution of the Klein-Gordon operator ( + m2 ), i.e., Gr/a are real-valued distributions such that suppy Gr/a (f, y) ⊆ ∪x∈supp f V¯x∓ , f ∈ D(M) = C0∞ (M), and 2 2 f h dx , f, h ∈ D(M), (1) Gr/a (( + m )f, h) = Gr/a (f, ( + m )h) = M

with dx the canonic volume form associated with g. These conditions determine Gr/a uniquely [12]. We also note that Gr (f, h) = Ga (h, f ). Next we deﬁne the antisymmetric commutator function D(f, h) as D(f, h) = Gr (f, h) − Ga (f, h) .

(2)

Obviously, D fulﬁlls the Klein-Gordon equation in both arguments, i.e., D((

+ m2 )f, h) = D(f, (

+ m2 )h) = 0.

(3)

Furthermore, D(f, h) = 0 for supp f ⊥ supp h as a consequence of the support properties of Gr/a . Let D+ be a (complex-valued) distribution in D(M × M, C) such that + ImD = D, (4) D+ (( + m2 )f, h) = D+ (f, ( + m2 )h) = 0 and furthermore D− (f, h) = D+ (h, f ) = D+ (f¯, ¯h), i.e., the real part of D+ is symmetric. We furthermore demand that D+ is invariant under isometric diﬀeomorphisms preserving the time orientation, D+ (fα , hα ) = D+ (f, h), α ∈ G ↑ (M, g), fα (x) = f (α−1 (x)). As this property automatically is fulﬁlled for the imaginary part D of D+ , this is only a condition on the real part of D+ . Lastly, we demand that D+ (f, y) is a measurable function in y and D+ (f¯, f ) ≥ 0 ∀f ∈ D(M). For a discussion on the existence of a (not necessarily G ↑ (M, g)-invariant) D+ see [27, Chapter 4.2] – for D+ with the Hadamard property, D+ (f, y) is a measurable function in y ∀f ∈ D(M, C), cf. [27, Chapter 4.6]. Lastly, the G ↑ (M, g) invariance 1 i.e., 2 By

the metric g carries signature (1, −1, . . . , −1). a curve c : [0, 1] → M s.t. g(c (s), c (s)) = 0 / > 0 ∀s ∈ [0, 1].

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of at least one such Hadamard state can be justiﬁed from the physical belief that a “good” vacuum state has maximal symmetry properties. This is, e.g., true for Minkowski- and de Sitter space-time (see Section 7) and for static space-times the uniqueness of states fulﬁlling the spectral condition implies at least invariance under time-translations (see Section 6). Under these conditions, the two-point function D+ , via second quantization, gives rise to a G ↑ (M, g)-covariant representation of the canonic commutation relations (CCR) on a Hilbert space H through free ﬁelds, cf. (11)–(13) below. It is a well-known and fundamental fact for quantum ﬁeld theory that in the absence of the spectral property and for a possibly “too small” symmetry group G ↑ (M, g) the real part of D+ is not uniquely ﬁxed and thus on curved space-time there are many non-equivalent admissible representations of the CCR [27]. We now write down the equations for the truncated Wightman functions Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 T of our model. The Wightman functions, or vacuum expectation values, are then given in terms of truncated Wightman functions via T Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 = Ψ0 , φ(fj1 ) · · · φ(fjl )Ψ0

(5) I∈P (n) {j1 ,...jl }∈I

where P (n) is the collection of all partitions of {1, . . . , n} into disjoint, nonempty subsets {j1 , . . . , jl } where j1 < . . . < jl . A comment concerning the use of symbols like Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 is in order. Until Section 5 they only stand for distributions in D(M×n , C) . Once the Hermiticity condition is veriﬁed for these distributions in Sections 4 and 5, one can apply the GNS-construction on inner product spaces, see Appendix A, and one a posteriori veriﬁes that Ψ0 has a proper mathematical meaning as the GNS-vacuum, φ(f ) (and also the incoming and outgoing ﬁelds φin/out (f )) as operator-valued distributions on the non-degenerate inner product space and Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 as vacuum expectation values w.r.t. the non-degenerate inner product ., . . Anticipating this standard construction, we use this notation from the beginning. Let Ψ0 , Ψ0 = 1, Ψ0 , φ(f )Ψ0 T = 0 and Ψ0 , φ(f )φ(h)Ψ0 = b2 D+ (f, h) T

∀f, h ∈ D(M, C)

(6)

for some b2 > 0. For the higher order truncated Wightman functions containing one “current” entry j(f ) = φ(( + m2 )f ) we set T

Ψ0 , φ(f1 ) · · · φ(fk−1 )j(fk )φ(fk+1 ) · · · φ(fn )Ψ0

= bn

k−1 l=1

D− (fl , fk )

n

D+ (fl , fk ), fl ∈ D(M, C), l = 1, . . . , n. (7)

l=k+1

Again, bn ∈ R is some arbitrary constant. In the next step we ﬁx the Wightman function with two (or more) “current” entries T

Ψ0 , φ(f1 ) · · · φ(fk−1 )j(fk )φ(fk+1 ) · · · φ(fr−1 )j(fr )φ(fr+1 ) · · · φ(fn )Ψ0 = 0, (8)

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i.e., any truncated vacuum expectation value containing two (or more) current operators j vanishes. Before solving the above ansatz by specifying initial conditions in the next section, we would like to brieﬂy sketch from Euclidean QFT which motivates the Equations (6)–(8). We suppose that there exists a Riemannian manifold obtained as the analytic continuation of M to purely imaginary Euclidean time. We consider an Euclidean ﬁeld theory on the Riemannian manifold which is the solution of the stochastic partial diﬀerential equation (∆ denotes the Laplacian) −∆ϕ + m2 ϕ = η ,

(9)

where η is some noise ﬁeld with a mixed Gauss-Poisson distribution [2]. The noise ﬁeld η can thus be seen as the Euclidean analogue of the current j. A proper choice of η [3] then leads to a solution ϕ which in the ﬂat case has Schwinger (moment) functions which can be analytically continued to real relativistic time [2]. The solutions in this case coincide with the solution given in the next section. To be just a little bit more detailed, let us emphasize that Equation (8) on the Euclidean side is entailed by the fact that truncated correlations of noise ﬁelds vanish at the non-coinciding points – and only such points would matter for a bona ﬁde analytic continuation. The remaining two equations, (6) and (7) then can be directly traced back to the analytic continuation of Euclidean two-point functions of the random ﬁeld model. If D± are these analytic continuations depending on the Euclidean time-ordering, (7) is the relativistic analogue of a related Euclidean equation describing the coupling of ϕ to the noise-ﬁeld η. The constants b2 and bn depend on the probability distribution of η, in particular bn = 0, n ≥ 3, if η is purely Gaussian.

3 Construction of Wightman functions If one could uniquely invert the operator + m2 , the equation (7) would uniquely determine the truncated vacuum expectation values of the ﬁeld φ, however this is not the case. To get the vacuum expectation value with φ(fk ) instead of j(fk ) = φ(( + m2 )fk ) it is necessary to specify the initial conditions for the ﬁeld φ(x). Our choice to do this is to specify initial conditions for large times x0 → ∓∞ and to postulate that for such asymptotic times the local ﬁeld φ(x) converges to free incoming or outgoing ﬁelds φin/out (x). The adequate technical formulation is given by the Yang-Feldman equations [28]

=

Ψ0 , φ(f1 ) · · · φ(fk−1 )φ(fk )φ(fk+1 ) · · · φ(fn )Ψ0 T Ψ0 , φ(f1 ) · · · φ(fk−1 )φin/out (fk )φ(fk+1 ) · · · φ(fn )Ψ0 T

+

Ψ0 , φ(f1 ) · · · φ(fk−1 )j(Gr/a fk )φ(fk+1 ) · · · φ(fn )Ψ0 T ,

(10)

for k = 1, . . . , n and n ∈ N, fl ∈ D(M, C), Gr/a fk (y) = Gr/a (fk , y). Clearly, Gr/a fk ∈ D(M, C) for fk = 0 which can be seen as an “infra-red” problem, cf. the discussion preceding Condition 3.1 below.

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From (5)–(6) and (10) we immediately get that the choice T Ψ0 , φa (f )φb (h)Ψ0 = b2 D+ (f, h) for a, b = in/loc/out

(11)

for the truncated two-point function is a uniquely given solution which agrees with the postulates which have been set up so far. Let us next consider the problem for the truncated n-point functions, n ≥ 3. Here it is also important to note that one can only ﬁx the initial or the ﬁnal behavior of φ, but not both, as this would over-determine the system. Here we use the conventions that (10) thus for the in-case is part of the ansatz whereas for the out-case it is a statement which we have to verify for the solution we give. Likewise, we have to postulate the characteristic properties of a free ﬁeld for φin , namely that free ﬁelds fulﬁll the CCR and the Klein-Gordon equation. But we also have to specify a representation for the “in”-ﬁelds, as in the absence of the spectral condition invariance, equations of motion and commutation relations do not suﬃce to ﬁx the representation uniquely. Our choice for the representation of the “in”-ﬁeld is (11) for the two-point function and T Ψ0 , φin (f1 ) · · · φin (fn )Ψ0 = 0 for n ≥ 3,

(12)

hence we want the “in”-ﬁeld to be in a quasi-free representation [27]. Once the GNS-like construction of Appendix A has been carried through and φin (f ) is realized as an operator-valued distribution on the indeﬁnite metric state space, (12) together with (11) immediately implies φin ((

+ m2 )f ) = 0,

[φin (f ), φin (h)] = ib2 D(f, h)

(13)

whereas the analogous statement for φout has to be proven3. Before we proceed on the basis of the above assumptions, we have to discuss whether the asymptotic condition (10) makes sense at all. The existence of free asymptotic ﬁelds cannot be expected for an arbitrary Lorentzian manifold M. If we, e.g., consider a Lorentzian manifold of the form M = R × Σ with Σ compact and metric ds2 = dt2 − dσ 2 , where dσ 2 is a Riemannian metric on Σ, then we have neither a natural dispersion of wave-packets in non-compact space (for suﬃciently high dimension d of the space-time M) nor a dispersion which is due to the expansion of the space-time at asymptotic times. We therefore need a criterion on our manifold (M, g) which implies that either (or both) of the above dispersion eﬀects is strong enough to guarantee the asymptotic condition (10) with φin/out free ﬁelds. Such dispersion is most conveniently formulated in terms of the fundamental function D+ which determines all the other fundamental functions: √ a simple re-deﬁnition of ﬁeld strengths φ → (1/ b2 )φ one obtains the usual normalization of the CCR. 3 By

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Condition 3.1 (i) On the manifold (M, g) there exists a D+ as described in Section 2 that fulﬁlls the n-point dispersion condition M |D+ (f, x)|n dx < ∞ ∀f ∈ D(M, C) for n ≥ 3 or n ≥ 4. (ii) Furthermore, if fl → 0 in D(M, C) then D+ (fl , x) → 0 ∀x ∈ M and ∃ a measurable function F (x) ≥ |D+ (fl , x)| ∀x ∈ M, l ∈ N s.t. M F (x)|D+ (f, x)|n−1 dx < ∞, ∀f ∈ D(M, C). If in our models b3 = 0, we in general also have a non-trivial decay of a single “particle”4 , and in this case we also have to demand Condition 3.1 to hold for n = 3. It is clear from the condition for D+ (f, x) that one or several functions D+ (f, x) can be replaced by other fundamental functions D− (f, x), D(f, x) and also Gr/a (f, x) as the latter outside of the causal closure of the compact support of f behave like D+ (f, x) or are equal to zero. For 4-dimensional Minkowski space-time Condition 3.1 (n = 3) follows from the decay behavior of fundamental functions in arbitrary directions, see, e.g., [19], and for de Sitter space-time we will verify Condition 3.1 in Section 7. Let us thus resume the construction of Wightman functions for n ≥ 3. By the Yang-Feldman equations for the in-case, we can replace one local ﬁeld by a “in”-ﬁeld and a current, e.g., φ(f1 ) T Ψ0 , φ(f1 )φin (f2 ) · · · φin (fn )Ψ0

=

Ψ0 , φin (f1 ) · · · φin (fn )Ψ0

T

T Ψ0 , j(Gr f1 )φin (f2 ) · · · φin (fn )Ψ0 n = bn Gr (f1 , y) D+ (fl , y) dy . (14) +

M

l=2

Here we used (12) and we exploited the fact that by the Yang-Feldman equations (10) the diﬀerence between the local and the “in”-ﬁeld is given by a current in order to replace “in”-ﬁelds by local ﬁelds according to the equations (8) and we ﬁnally evaluated the vacuum expectation value containing one current and local ﬁelds by (7). By Condition 3.1 the integral of the right-hand side of (14) converges. Also, the right-hand side of (14) is a distribution in D(M×n , C) . To see this, we let one of the test functions fl go to zero in D(M, C). From Condition 3.1 (ii) it then follows that the right-hand side of (14) converges to zero by Lebesgue’s theorem of dominated convergence. By induction, we can now calculate Ψ0 , φ(f1 ) · · · φ(fk )φin (fk+1 ) · · · φin (fn ) T Ψ0 using (14) and the same arguments as in the ﬁrst step. Continuing in this 4 Such a reaction is forbidden on Minkowski space-time due to energy-momentum conservation, but on general space-times the possibility of such a reaction has to be taken into account.

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way, we get after k = n steps the Wightman functions of the local ﬁelds: T

Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 = bn

n k−1 k=1

M l=1

n

D− (fl , y)Gr (fk , y)

D+ (fl , y) dy ,

l=k+1

(15) where again Condition 3.1 assures that the integrals in (15) exist and deﬁne a distribution in D(M×n , C) . It also does not matter in which order we replace the φin in (12) by local ﬁelds, as this only changes the order of summation in (15). We have thus derived Theorem 3.2 Let Condition 3.1 be fulﬁlled. Given the ansatz (6), (7), (8) and the asymptotic condition (10) for a free “in”-ﬁeld in a quasi-free representation (cf. (12)–(13)), there exists a unique solution for the (truncated) Wightman functions of the local ﬁeld given by (11) and (15).

4 Properties of Wightman functions In this section we verify the fundamental properties of the Wightman functions constructed in Section 3. Theorem 4.1 The Wightman functions constructed in Section 3 fulﬁll the properties of Hermiticity, G ↑ (M, g)-invariance and locality. We start with the proof of the ﬁrst property, Hermiticity: T Ψ0 , φ(f1 ) · · · φ(fn )Ψ0 T = Ψ0 , φ(f¯n ) · · · φ(f¯1 )Ψ0 .

(16)

For n = 2 (16) follows from the properties of D+ . For larger n ∈ N, this relation can be easily veriﬁed from (15) as complex conjugation of the right-hand side exchanges D± (f¯l , y) with D∓ (fl , y) and Gr (f¯k , y) with Gr (fk , y). After re-ordering the sum, we then see that the complex conjugate is just the right-hand side of (15) with the reversed order of the arguments. Let next α ∈ G ↑ (M, g). To verify invariance, we ﬁrst note that Gr (fα , hα ) = Gr (f, h) and analogous identities hold for D± . Hence, T

Ψ0 , φ(f1,α ) · · · φ(fn,α )Ψ0

n n k−1 = bn D− (fl,α , y)Gr (fk,α , y) D+ (fl,α , y) dy k=1

= bn

M l=1

n k−1 k=1

M l=1

l=k+1

D− (fl , α−1 (y))Gr (fk , α−1 (y))

n

(17)

D+ (fl , α−1 (y)) dy.

l=k+1

As dy is α−1 -invariant, the right-hand side coincides with the right-hand side of (15) which establishes invariance under G ↑ (M, g).

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It remains to verify locality. By explicit calculation we get T

Ψ0 , φ(f1 ) · · · [φ(fk ), φ(fk+1 )] · · · φ(fn )Ψ0

k−1 = bn D− (fl , y)[Gr (fk , y)D+ (fk+1 , y) + D− (fk , y)Gr (fk+1 , y) M l=1

n

− Gr (fk+1 , y)D+ (fk , y) − D− (fk+1 , y)Gr (fk , y)]

= ibn

k−1 M l=1

D+ (fl , y) dy

l=k+2

D+ (fl , y)[Gr (fk , y)D(fk+1 , y) − D(fk , y)Gr (fk+1 , y)]

n

D+ (fl , y) dy .

(18)

l=k+2

For all y ∈ M one of the following cases holds true: (I) V¯y+ ∪ V¯y− neither intersects supp fk nor supp fk+1 : In this case D(fk , y) = D(fk+1 , y) = 0 and the term in the brackets [. . .] on the right-hand side of (18) vanishes. (II) y ∈ supp fk : From supp fk ⊥ supp fk+1 it follows that D(fk+1 , y) = Gr (fk+1 , y) = 0 and the bracket [. . .] in (18) again vanishes. (III) V¯y− and V¯y+ intersect supp fk : As in (II) one concludes y ⊥ supp fk+1 and [. . .] = 0. (IV) Only V¯y− intersects supp fk and y ∈ supp fk : Then supp fk ⊥ supp fk+1 ⇒ supp fk+1 ∩ V¯y+ = ∅ hence Gr (fk , y) = Gr (fk+1 , y) = 0 and again [. . .] = 0. (V) Only V¯y+ intersects supp fk and y ∈ supp fk : One replaces the expressions Gr in the bracket with Ga via adding a term [D(fk+1 , y)D(fk , y) − D(fk , y) × D(fk+1 , y)] = 0, cf. (2), and then concludes as in (IV) that Ga (fk , y) = Ga (fk+1 , y) = 0 ⇒ [. . .] = 0. Hence the bracket [. . .] in (18) vanishes for all y ∈ M. This establishes Theorem 4.1. Without pretending to be mathematically rigorous, we want to indicate, why in general positivity of the Wightman functions cannot be expected. We give an argument similar to the Jost-Schroer theorem [22] in the Minkowski case. Let us assume for a moment that positivity holds. One then gets the vacuum representation of the algebra of local ﬁelds through the well-known Wightman reconstruction theorem [22]. Let us furthermore assume that the vacuum is separating for the local ﬁelds – rather general suﬃcient conditions which imply this (“Reeh-Schlieder property”) can be found in [23]. It then follows from (5) that j(f )Ψ0 , j(f )Ψ0 = 0, ∀f ∈ D(M, C), hence j(f )Ψ0 = 0 and j(f ) = 0. From the Yang-Feldman equations (10) one then gets φ = φin in contradiction with (15).

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5 Scattering behavior In this section we determine the (non-trivial) scattering behavior of the theory and the properties and representation of the outgoing ﬁeld. Solving the Yang-Feldman equations (10) for the “out”-ﬁeld, we obtain for a1 , . . . , an =in/loc/out: T

=

Ψ0 , φa1 (f1 ) · · · φan (fn )Ψ0

k−1 n n bn D− (fl , y)Gr (fk , y) D+ (fl , y) dy k=1:ak =loc

+

bn

n k=1:ak =out

M l=1

k−1 M l=1

l=k+1

D− (fl , y)D(fk , y)

n

D+ (fl , y) dy .

(19)

l=k+1

Here all integrals exist, cf. Condition 3.1. Just as in Section 4 one can show that (19) is Hermitean and invariant under G ↑ (M, g). As we shall show below, locality holds for each of the ﬁelds – “in”, “loc” and “out”-ﬁeld – separately, but of course not jointly. The equation (19) in particular determines the (truncated) scattering(S-)matrix elements T in φ (f¯k ) · · · φin (f¯1 )Ψ0 , φout (fk+1 ) · · · φout (fn )Ψ0 k n n i − + − = bn D (fl , y) D (fl , y) dy − D (fl , y)dy . (20) 2 M M l=1

l=k+1

l=1

Here we used Hermiticity of the ﬁelds (see below and Appendix A) and D = (−i/2)(D+ − D− ) to simplify (19) in this special case using “telescope” cancellations. Next we want to discuss questions concerning the algebraic properties of “in”- and especially “out”-ﬁelds. (5), (11) and (19) give the collection of mixed non-truncated vacuum expectation values of “in”-, “loc”- and “out”-ﬁelds, called the form factor functional [1]. One can now use the standard GNS-construction on inner product spaces to obtain a representation of the algebra generated from “in”- “loc”- and “out”-ﬁelds on some non-degenerate inner product space (V, ., . ) with a speciﬁc GNS-“vacuum” vector Ψ0 ∈ V. This gives a precise mathematical meaning to the ﬁelds φin (f ), φ(f ), φout (f ) as operator-valued distributions on V (the domain of deﬁnition is the entire space) and justiﬁes writing the left-hand side of (19) as a truncated vacuum expectation value w.r.t. the inner product ., . . For the details we refer to the Appendix A. That the incoming ﬁeld φin fulﬁlls the Klein-Gordon equation and the CCR was part of our ansatz, cf. (13). If the scattering deﬁned in our model is reasonable, the same properties should also hold for the “out”-ﬁeld: Theorem 5.1 The outgoing ﬁeld φout fulﬁlls the Klein-Gordon equation and the CCR on the entire state space V.

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It is rather easy to verify the Klein-Gordon equations for φout : If we apply the Klein-Gordon operator + m2 in (19) for ak =out to fk , then the argument ( + m2 )fk in the ﬁrst sum on the right-hand side stands in one of the functions D+ or D− and in the second sum on the right-hand side in one of the functions D+ , D− or D. As all these fundamental functions fulﬁll the Klein-Gordon equation, cf. (3) and (4) the assertion follows for the truncated n-point functions, n ≥ 3. But it also holds for the truncated two-point function b2 D+ as it fulﬁlls this equation in both arguments. If we now go over from truncated to non-truncated vacuum expectation values, the test function fk occurs in exactly one truncated n-point function and hence the vacuum expectation value vanishes if the Klein-Gordon operator is applied. This proves Ψ, φout (( + m2 )f )Φ = 0 ∀Ψ, Φ ∈ V. As the inner product on V is non-degenerate, φout (( + m2 )f )Φ = 0 ∀Φ ∈ V follows. By deﬁnition, this means φout + m2 φout = 0. Next we prove the CCR. The following lemma, connecting CCR and truncation on a general level, is needed: Lemma 5.2 For the CCR for φout to hold it is necessary and suﬃcient that for n ≥ 3, k = 1, . . . , n − 1, a1 , . . . , ak−1 , ak+2 , . . . , an ∈ {in, loc, out}, f1 , . . . , fn ∈ D(M, C) arbitrary T Ψ0 , φa1 (f1 ) · · · [φout (fk ), φout (fk+1 )] · · · φan (fn )Ψ0 = 0. (21) The proof of Lemma 5.2 can be found in Appendix B. We thus have to verify the suﬃcient condition (21). If we calculate the left-hand side of this equation using (19) we obtain (up to a sign) the same expression as on the right-hand side of (18) with Gr replaced by D. But then it follows from the expression in the brackets [· · · ] on the right-hand side of (18) that after this replacement the expression vanishes identically for arbitrary test functions fk , fk+1 (with not necessarily space-like separated support). This proves Theorem 5.1. Having proven the main features of the free ﬁeld for the “out”-ﬁeld, it remains to investigate the representation of the CCR given by the “out”-ﬁeld. The truncated two-point function for φin and φout coincides, cf. (11). For the truncated n-point functions, n ≥ 3, we however ﬁnd similarly as in (20) through “telescope” cancellations for real-valued test-functions f1 , . . . , fn ∈ D(M, R) n

n T i out out + − Ψ0 , φ (f1 ) · · · φ (fn )Ψ0 = bn D (fl , y) dy − D (fl , y) dy 2 M M l=1 l=1 n

− = bn Im D (fl , y) dy . (22) M l=1

Hence, the representations of the CCR given by the ﬁeld φin and the one given by φout are unitary equivalent5 only if the right-hand side of (22) vanishes. Suﬃcient 5 In the restrictive sense that there exists a linear isometry (w.r.t. ., .) V from the “in”space generated by application of the “in”-ﬁelds to the vacuum to the related “out”-space s.t.

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conditions for this will be given in the next section – they apply to the case of static space-times. Hence, a non-vanishing of the right-hand side of (22) can be seen as a consequence of the interaction of a non-static space-time (a time-dependent classical gravitation) with the quantum scattering due to non-vanishing truncated n-point functions, n ≥ 3, leading to a non-Fock and not quasi-free representation for the “out”-ﬁeld.

6 The case of static space-times In this section we consider the special case of M being a static space-time possessing a global foliation M = R×Σ, equipped with the metric ds2 = α(x)dt2 −dσ 2 , where α ∈ C ∞ (Σ) is the “lapse”-function and dσ 2 is a Riemannian metric on Σ. In the described situation time translations form a one parameter group of symmetries and the fundamental functions, e.g., D+ only depend on the diﬀerences of time arguments: D+ (x, y) = D+ (x0 − y 0 , x, y). We then deﬁne the Fourier transform6 + + ˆ −1/2 ˆ ˆ D (f, h1 , h2 ) = D (f , h1 , h2 ), f (E) = (2π) e−itE f (t) dt, (23) R

f ∈ S(R), h1 , h2 ∈ D(Σ, C) with S(R) the space of complex-valued Schwartz funcˆ −, G ˆ r/a of the tions on R and D(Σ, C) = C0∞ (Σ, C). The Fourier transforms D remaining fundamental functions are deﬁned analogously and the Fourier transform of vacuum expectation values (19) is deﬁned by taking the Fourier transform in each time argument tl = x0l , l = 1, . . . , n. We sometimes suppress the test functions h1 , h2 ∈ D(Σ, C) if they do not matter in a speciﬁc argument. Let {U(t)}t∈R be the unitary (w.r.t. the indeﬁnite inner product ., . ) representation of the one parameter group of time translations, cf. Theorem 4.1 and Appendix A. In such a situation spectral conditions can be formulated as follows: Condition 6.1 (i) D+ fulﬁlls the spectral condition with spectral gap > 0 if the Fourier transˆ+ ∩ ˆ + (E), vanishes for E < , i.e., supp D form in the time variable, D (−∞, ) = 0. (ii) The indeﬁnite metric QFT over (M, g) constructed in Section 5 fulﬁlls the spectral condition if ∀Φ, Ψ ∈ V Ψ, U(t)Φ f (−t) dt = 0 , ∀f ∈ S(R), supp fˆ ∩ (−∞, 0) = 0. (24) R

Condition 6.1 (i) ﬁxes D+ uniquely, cf. [27, Chapter 3.3]. V Ψ0 = Ψ0 and V φin V −1 = φout , cf. [22]. In [27] the notion is used in the larger sense that not necessarily V Ψ0 = Ψ0 . 6 Throughout this section we assume that all fundamental functions and all vacuum expectation values (19) are tempered distributions in the time arguments s.t. Fourier transforms are well deﬁned as Fourier transforms of tempered distributions. This property of course has to be veriﬁed for given (M, g) and D + .

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The spectral condition (ii) means that the spectrum of the generator of time translations, i.e., the spectrum of the Hamiltonian, is bounded from below with lowest eigenvalue zero (assumed by the vacuum). The spectral condition (i) means that this is only true for the space of “one particle states”, i.e., the sub-space of V obtained by applying once the local ﬁeld to the vacuum. Thus, (i) is weaker than (ii). In our model we however also have that (i) implies (ii): Theorem 6.2 Let (M, g) be a static space-time and let D+ fulﬁll the spectral condition 6.1 (i). Then, the spectral condition is fulﬁlled for the entire theory, cf. Condition 6.1 (ii). Furthermore, the representations of the CCR generated by “in”- and “out”-ﬁelds are unitary equivalent. We ﬁrst prove the spectral property (24). By the same methods as in the ﬂat case [22] one can show that the spectral condition 6.1 (ii) is equivalent to the spectral condition for the vacuum expectation values in the sense that the support of their in the time arguments lies in the set {(E1 , . . . , En ) ∈ nFourier transform n Rn : l=1 El = 0, l=r El ≥ 0, r = 2, . . . , n}. Furthermore, this spectral condition for the vacuum expectation values is equivalent to the spectral condition for the truncated vacuum expectation values, cf. [5, 7]. For the two-point function this is just Condition 6.1 (i). We therefore only have to verify this support property for the expressions (19). Let us Fourier transform any term in the ﬁrst sum on the right-hand side of (19). Suppressing hl ∈ D(Σ, C), y ∈ Σ arguments and the Σ · · · dy integration, the result is up to a constant bn /2π k−1

ˆ − (El ) G ˆ r (Ek ) D

l=1

n l=k+1

n ˆ + (El ) δ( D El ).

(25)

l=1

Note that the product nof distributions in (25) is well deﬁned by Condition 3.1. For r = k + 1, . . . , n, l=r El > 0 on the support of the above expression as each ¯ 1, h ¯ 2 ), ˆ + (f¯θ , h ˆ − (f, h1 , h2 ) = D El > 0, l = r, . . . , n. Let thus r ≤ k. We note that D ˆ − ∩ (−, ∞) = ∅. fθ (E) = f (−E), f ∈ S(R), h1 , h2 ∈ D(Σ, C), and thus supp D In the support of the distribution (25) we thus have El < 0, l = 1, . . . , k − 1 and r−1 n consequently l=r El = − l=1 El > 0. The terms in the second sum on the right-hand side of (19) can be treated ˆ r . This analogously, as in the above argument we did not need any properties of G establishes the spectral condition. In order to prove unitary equivalence of the CCR representations for “in”and “out”-ﬁelds, we have to show that the right-hand side of (22) vanishes. Taking the Fourier transform in the time arguments of the term in the brackets in (22) yields n n − ˆ D (El ) δ( El ) = 0 , (26) l=1

l=1

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as the delta function and the product have disjoint support. Here again we suppressed a factor 1/2π, arguments hl ∈ D(Σ, C), y and dy -integration over Σ. This proves the theorem. By the same argument we also get that in the static case only the ﬁrst term in the curly brackets on the right-hand side of (20) gives a non-vanishing contribution. In particular, this applies to the case of Minkowski space-time, where we recover the same scattering behavior as in [1]. The ﬁrst term on the right-hand side of (20) in energy-momentum space then just gives the on-shell and energymomentum conservation term (up to a constant) describing somehow the “simplest possible” non-trivial scattering behavior. Another immediate consequence of Theorem 6.2 follows from the fact that the distribution D+ is positive, hence “in”- and “out” ﬁelds create positive Fock representations: Corollary 6.3 Under the conditions of Theorem 6.2 one gets that the restriction of the inner product ., . to the spaces V in/out ⊆ V generated by repeated application of the asymptotic ﬁelds φin/out to the vacuum is positive semi-deﬁnite.

7 Verifications for de Sitter space-time In this section we want to consider de Sitter spaces as a concrete example of curved space-times. The choice of the Sitter spaces is particularly interesting in the respect that Condition 3.1 sensitively depends on the dimension, see Theorem 7.1. Only for dimensions d ≥ 6 Condition 3.1 can be veriﬁed for all orders of the Wightman functions. In lower dimensions it may well depend on the order, as for example, for dimension 4 the third order does not exist. This is an infrared problem which has also been observed by Tagirov [25] in the context of : φ3 : theory. In the present case this could be repaired by simply choosing b3 = 0. Note that the Sitter spaces have spheres as Cauchy surfaces and this compactness at space-like distances hinders dispersion. On the other hand the volume of de Sitter spaces increases rather fast when moving along the time-like direction, which may facilitate dispersion. A more careful treatment given below than shows that these eﬀects really seem to be responsible for whether Condition 3.1 holds or does not hold. The choice of de Sitter spaces is also convenient for the discussion in as far as there is a preferred vacuum, the so-called Euclidean or Bunch-Davies vacuum [10]. It is the distinguished one, which is selected from other choices by the demand of covariance and the Hadamard condition, see, e.g., [4, 8]. Note that de Sitter spaces are maximally symmetric, i.e., the dimension of the symmetry group is maximal, hence covariance is a natural axiom to be imposed on the Wightman functions. The Hadamard condition then selects the particular two-point function which has the same singular behavior at light-like distances as the two-point function of the free ﬁeld in Minkowski space.

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Below we shall prove the following Theorem 7.1 Given the d-dimensional de Sitter space-time Xd and the BunchDavies vacuum D+ on it, Condition 3.1 (case n ≥ 3) holds if (dn−2n−2d)/2 > −1. In order to investigate Condition 3.1, we shall use essentially the results given in [11, 25], where an orthonormal mode expansion is used for the representation of the two-point function. Let us recall some basic features. Given the (d + 1)dimensional Minkowski space (Md+1 , g) with metric g = gµν dxµ dxν , where gµν = diag(+1, −1, . . . , −1). The d-dimensional de Sitter space (Xd , g) is deﬁned as

Xd := x ∈ Md+1 | gµν xµ xν = (x0 )2 − (x1 )2 − · · · − (xd )2 = −r2 , r > 0, equipped with the metric g = ι∗ g, where ι∗ denotes the pull-back with respect to the embedding ι : Xd → Md+1 . The eigenmodes are calculated in “global” coordinates, which in fact are given by the diﬀeomorphisms (− π2 , π2 ) × Sd−1 −→ Xd κd : (27) (τ, α) −→ (r tan τ, rα/ cos τ ). The pull-back of the volume form dV d on Xd is then given by κ∗d dV d = cos−d τ dτ dΩ, where dΩ is the volume form on the sphere Sd−1 . In these coordinates the KleinGordon equation ( + m2 )ϕ = 0 reads d (28) cos τ ∂τ cos2−d τ ∂τ − cos2 τ ∆Sd−1 + m2 ϕ = 0, m = mr. Solutions to (28) can be found by separation of variables. Setting ϕ(τ, α) = T (τ )Ξ(α) gives (29) (∆Sd−1 + κ2 )Ξ = 0, and

d (30) cos τ ∂τ cos2−d τ ∂τ − κ2 cos2 τ + m2 T = 0, ∞ where κ2 is the separation constant. Note that L2 (Sd−1 ) s=0 Hs (Sd−1 ), where s d−1 the subspaces H (S ) are spanned by the spherical harmonics (Ξsl )0≤l≤h(s,d) of degree s, h(s, d) = dimHs (Sd−1 ), which at the same time are eigensolutions of (29) with eigenvalues −κ2 = −s(s + d − 2). Moreover, the quasi-regular representation direct sum Q : SO(d) → L2 (Sd−1 ), given by Q(k)(f )(α) = f (k −1 α), splits into a ∞ of unitary irreducible representations Qs on Hs (Sd−1 ), such that Q = s=0 Qs . If s s s the quv mean the matrix elements of Q with respect to the bases (Ξl )0≤l≤h(s,d) , then the following relation holds [26, p. 470] s (k), Ξsl (α) = (dimHs (Sd−1 ))1/2 ql0

(31)

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where α = ken , k ∈ SO(d), en = (0, 0, . . . , 1) being the invariant vector with respect to the subgroup SO(d − 1), so that Sd−1 SO(d)/SO(d − 1). Pairs of linearly independent solutions to (30) are given by [11] Tp± (τ ) = cos(d−2)/2 τ u± p (τ ), p = s + (d − 2)/2, where u± p (τ ) =

1 Γ(p + µ)Γ(p − µ + 1)e±ipτ F (µ, 1 − µ, p + 1; e±iτ /(2 cos τ )), p! µ = 1/2(1 − 1 − 4m2 ) − u+ p = up ,

(32)

(33)

(34)

F being the hypergeometric function. We thus get the following system of solutions − + s − s ϕ+ pl (x) = Tp (τ )Ξl (α), and ϕpl (x) = Tp (τ )Ξl (α), p = s + (d − 2)/2,

s ∈ N0 , 0 ≤ l ≤ h(s, d).

(35)

In terms of these solutions the two-point function is expressed as7 iD+ (x1 , x2 ) =

∞

Tp− (τ1 )Tp+ (τ2 )Ξsl (α1 )Ξsl (α2 )

s=0 0≤l≤h(s,d) (d−2)/2

= cos

(d−2)/2

τ1 cos

τ2

∞

(36) + 1/2(d−2) u− (cos(α1 , α2 )), p (τ1 )up (τ2 )A(s, d)Cs

s=0

with

Ξsl (α1 )Ξsl (α2 ) = A(s, d)Cs1/2(d−2) (cos(α1 , α2 )),

(37)

0≤l≤h(s,d) 1/2(d−2)

where Cs is a Gegenbauer polynomial, A(s, d) = (2s + (d − 2))Γ(1 + 1/2(d − 2))/(2π 1+1/2(d−2) (d − 2)), and (α1 , α2 ) denotes the angle between α1 and α2 . The convergence of the series (36) has to be understood in the weak topology of D(Xd × Xd , C) . We will see that D+ (x1 , x2 ) is even an element in the subspace L(D(Xd , C); Cb (Xd , C)), the space of continuous linear mappings from D(Xd , C) to the Banach space of complex-valued bounded continuous functions on Xd , equipped with the supremum norm. This means that applying (or smearing with) a test function the series will not only converge to a function in Cb (Xd , C), but the result will depend continuously on the chosen test functions. At the same time Condition 3.1 will be veriﬁed. So let us smear (36) with an f ∈ D(X, C) in one argument, say x1 . In order to have control on the summation we need the following asymptotic formulas [13, Ch. 2.2.2 and 1.18]

and 7 Our

iD +

F (µ, 1 − µ, p + 1, e±iτ /(2 cos τ )) = 1 + O(1/p),

(38)

(p!)−2 Γ(p + µ)Γ(p − µ + 1) = 1/p + O(1/p2 ).

(39)

corresponds Tagirov’s

−D + ,

cf. [25].

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(38) holds on every compact interval [a, b] ⊂ (− π2 , π2 ). Using (38) and (39) in (36), we need to investigate the series D[a,b] (τ2 , α2 ) = 1[a,b] (τ2 ) cos(d−2)/2 τ2 ∞ × (1/p + O(1/p2 ))Ξsl (α1 )Ξsl (α2 ) s=0 0≤l≤h(s,d)

π d−1 (− π 2 , 2 )×S

(40)

× e−i(d−2)τ1 /2 e−isτ1 eisτ2 ei(d−2)τ2 /2 f (τ1 , α1 ) cos(−d−2)/2 τ1 dτ1 dΩ(α1 ). If we consider the similar series expansion without the terms 1/p + O(1/p2 ), ∞ ˜ D(τ2 , α2 ) = Ξsl (α1 )Ξsl (α2 ) s=0 0≤l≤h(s,d)

π d−1 (− π 2 , 2 )×S

(41)

× e−i(d−2)τ1 /2 e−isτ1 eisτ2 ei(d−2)τ2 /2 f (τ1 , α1 ) cos(−d−2)/2 τ1 dτ1 dΩ(α1 ), then up to the factors ei(d−2)τ1 /2 , e−i(d−2)τ2 /2 it represents a sub-series of the harmonic expansion of the function f˜(τ1 , α1 ) := f (τ1 , α2 ) cos(−d−2)/2 τ1 , f˜ ∈ D(S1 ×Sd−1 , C), if we identify S1 R/(2πZ+π) and extend f˜ to [−π, π]×Sd−1 by setting it equal zero outside its support. Using the relation (31) we may regard this modiﬁed expression as a series on the compact Lie group S1 × SO(d). Before proceeding, we recall the following facts from harmonic analysis on Lie groups. ˆ denote the equivalence classes of Let a compact Lie group K be given and let K ˆ are ﬁniteirreducible unitary representations of K. The representatives Uλ , λ ∈ K λ dimensional with dimension denoted d(λ). We may write uij (k) for the matrix of Uλ (k), k ∈ K, after having chosen some basis. According to the theorem of Peter-Weyl any f ∈ L2 (K) has the following series or harmonic expansion in the L2 -sense [15, Theorem 26.40], f=

d(λ)

d(λ)(f, uλij )uλij ,

(42)

ˆ i,j=1 λ∈K

where (f, uλij ) = K f uλij dk. The integration is performed with respect to the Haar measure dk and the bar means complex conjugation. (42) remains true when switching to complex conjugates. For functions f ∈ C ∞ (K, C) this statement can be sharpened a lot [24, Theorem 1]. Let w denote the dimension of the maximal toral subgroup of K and let dim K = w + 2γ. For f ∈ C ∞ (K, C) the series (42) converges absolutely and uniformly with the estimate d(λ) ˆ i,j=1 λ∈K

d(λ)|(f, uλij )| uλij ∞ ≤ ∆lK f 2 (N 2 |λ|2γ−4l )1/2 , ∀2l > w/2 + γ, ˆ λ∈K

(43) where ·∞ denotes the supremum norm, N is a constant, ∆K is the Laplacian on K and ·2 denotes the L2 (K)-norm. |λ| stands for the norm of the highest weight corresponding to λ.

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Now the series of the absolute values of (41) can be estimated by the corresponding sub-series of absolute values of the harmonic expansion of f˜ on the Lie group K = S1 ×SO(d), which makes (41) converge absolutely and uniformly with estimate (43). This in turn implies absolute uniform convergence of (40) together with the bound (44) |D[a,b] (τ2 , α2 )| ≤ cos(d−2)/2 τ2 Cf (τ2 , α2 ), where Cf is a continuous function on S1 × SO(d − 1). But for every x2 = (τ2 , α2 ) we have τ2 ∈ [a, b] for appropriate − π2 < a < b < π2 , so we get |D+ (f, x2 )| ≤ cos(d−2)/2 τ2 Cf (τ2 , α2 ), ∀x2 ∈ Xd .

(45)

On the other hand, if a sequence (fl )l≥0 converges to zero in the topology of C ∞ (S1 × SO(d), C), then one easily establishes the convergence to zero of the corresponding sequence (f˜l )l≥0 in the same topology. The topology of C ∞ (S1 × SO(d), C) is equivalent to the topology generated by the seminorms pj (f ) = j ∆S1 ×SO(d) (f ) , hence we may deduce by (43) that ∞

lim Cfl ∞ = 0 .

l→∞

Using (45) we may conclude that the expression |D+ (f, (τ2 , α2 ))|n cos−d τ2 dτ2 dΩ(α2 )

(46)

(47)

π d−1 (− π 2 , 2 )×S

exists, if (dn − 2n − 2d)/2 > −1. Due to (46) also Condition 3.1 (ii) is fulﬁlled with F (x) = supn∈N Cfn ∞ cos(d−2)/2 τ2 . This proves Theorem 7.1.

A

The GNS-construction on an inner product space

Let Dext. , the extended Borchers algebra, be the free tensor algebra generated by Dext. = D(M, C3 ), i.e., Dext. =

∞ ext. ⊗n ext. ⊗0 , D = C. D

(48)

n=0

The addition on Dext. is componentwise and the multiplication is given by the tensor product. The involution on Dext. is given by the operation (f1 ⊗ · · ·⊗ fn )∗ = f¯n ⊗ · · · ⊗ f¯1 , fl ∈ Dext. , where the bar stands for complex conjugation. As we want to use this unital, involutive algebra to represent “in”-, “loc”- and “out”ﬁelds, the three components of Dext. = D(M, C3 ) are labeled “in” for the ﬁrst component, “loc” for the second and “out” for the third. Then our collection of mixed vacuum expectation values obtained from (5), (11) and (19) generates

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an Hermitean functional (called the form factor functional F ) on the extended Borchers algebra D ext. through F (f ) = f0 Ψ0 , Ψ0 +

∞

a1 an Ψ0 , φa1 (fn,1 ) · · · φan (fn,n )Ψ0 , (49)

n=1 a1 ,...,an ∈{in,loc,out}

where f ∈ Dext. , f = (f0 , . . . , fk , 0, . . .), f0 ∈ C, fn = f1,n ⊗ · · · ⊗ fn,n , fl,n ∈ Dext. , and we use the normalization Ψ0 , Ψ0 = 1. Let L = L(F ) = {f ∈ D ext. : F (g ⊗ f ) = 0 ∀g ∈ D ext. }. Then L is a left-ideal in Dext. , i.e., f ∈ L ⇒ g ⊗ f ∈ L ∀g ∈ D ext. . In particular, L is a (complex) vector space and we can deﬁne the quotient vector space V = Dext. /L. Let [f ] = f + L denote the rest class of f in V. Then, [f ], [g] = F (f ∗ ⊗ g)

(50)

gives a well-deﬁned and non-degenerate inner product ., . on V. To see this, we note that by the deﬁnition of L the right-hand side of (50) does not depend on the choice of g ∈ [g]. By Hermiticity, F (f ∗ ⊗ g) = F (g ∗ ⊗ f ), the same applies to f ∈ [f ]. Also, if [g], [f ] = 0 ∀[g] ∈ V it follows that [f ] = L, which proves the non-degeneracy. The “vacuum state” Ψ0 , so far just a suggestive notation, is now identiﬁed with the GNS-vacuum [(1, 0, . . .)] ∈ V. Likewise, we want to identify the local and asymptotic “ﬁelds” φ and φin/out with operator-valued distributions acting on V. As L is a left-ideal, we obtain a left-action of Dext. on V through f · [g] = [f ⊗ g] ∀ f ∈ Dext. , [g] ∈ V. Thus, every element of D ext. can be identiﬁed with an operator on V. This, in particular, applies to Dext. ⊆ Dext. . Let now f ∈ D, i.e., f = (0, f ext. , 0, . . .), f ext. ∈ D(M, C3 ) such that only the ﬁst (“in”) / second (“loc”) / third (“out”) component of f ext. is diﬀerent from zero and let this component be given by f ∈ D(M, C). We then deﬁne φin/loc/out (f )Ψ = f · Ψ, ∀Ψ ∈ V. This rigorously deﬁnes φin/loc/out (f ) (in the text we suppress the superscript “loc” for the local ﬁeld). Furthermore, by Hermiticity of F , the ﬁelds φin/loc/out are Hermitean w.r.t. ., . , i.e., Ψ, φin/loc/out (f )Φ = φin/loc/out (f¯)Ψ, Φ ∀Φ, Ψ ∈ V and f ∈ D(M, C). Lastly, we want to construct a representation U of the orthochonous symmetry group G ↑ (M, g). As D ext. as a unital tensor algebra is generated by Dext. , it follows that V is the linear span of vectors generated by repeated application of φin , φloc and φout to the vacuum Ψ0 . To deﬁne U it is thus enough to set U(α)φin/loc/out (f )U−1 (α) = φin/loc/out (fα ) and U(α)Ψ0 = Ψ0 ∀α ∈ G ↑ (M, g). This is well deﬁned, as the action of the symmetry group on Dext. , maps L into itself, as a consequence of the invariance of F under such transformations, F (f ) = F (f α ). The invariance of the vacuum expectation values then also implies that U is a unitary representation U∗ = U−1 where the adjoint is taken w.r.t. the inner product ., . .

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B Proof of Lemma 5.2 Let us prove that (21) implies the CCR. We ﬁrst use the cluster expansion (5) for the following vacuum expectation value Ψ0 , φa1 (f1 ) · · · [φout (fk ), φout (fk+1 )] · · · φan (fn )Ψ0  T  = Ψ0 , φaj1 (fj1 ) · · · φajl (fjl )Ψ0

I∈P (n)

{j1 ,...jl }∈I

−

{j1 ,...jl }∈I

 T a a Ψ0 , φ j1 (fj1 ) · · · φ jl (fjl )Ψ0  .

(51)

Here we ﬁxed ak , ak+1 =out and we deﬁned jr = k +1 if jr = k, jr = k if jr = k +1 and jr = jr else. We can divide the partitions I of (1, . . . , n) into three classes:

1) k and k + 1 belong to diﬀerent sets A and A in the partition I. Then there exists exactly one partition I which is identical to I with the exception that k and k + 1 are exchanged8 . The two terms in (51) belonging to I and I then cancel and hence the sum over all partitions in this class gives zero. 2) k and k + 1 are in the same set A = {q1 , . . . , k, k + 1, . . . , qr } of the partition I and A contains more than two elements, i.e., r ≥ 3. The summand in (51) belonging to such a partition is equal to T Ψ0 , φaq1 (fq1 ) · · · [φout (fk ), φout (fk+1 )] · · · φaqr (fqr )Ψ0 T × Ψ0 , φaj1 (fj1 ) · · · φajl (fjl )Ψ0 .

(52)

{j1 ,...jl }∈I\A

By (21) the contribution from the partitions of this class also vanishes. 3) k and k + 1 are in the same set of the partition I and the set contains only these two elements. The sum over all partitions in this class yields T Ψ0 , [φout (fk ), φout (fk+1 )]Ψ0 T a a Ψ0 , φ j1 (fj1 ) · · · φ jl (fjl )Ψ0 × =

I∈P (n−2) {j1 ,...,jl }∈I ib2 D(fk , fk+1 ) Ψ0 , φa1 (f1 ) · · · φak−1 (fk−1 )φak+2 (fk+2 ) · · · φan (fn )Ψ0 .

Here we used the notation jr = jr if jr < k and jr = jr + 2 if jr ≥ k.

(53)

From 1)–3) it follows that the left-hand side of (51) is equal to the right-hand side of (53). As all states in V are generated by repeated application of “in”-, 8 We assumed that Ψ , φin/loc/out (f )Ψ T = 0, i.e., partitions with A = {k} and A = {k+1} 0 0 for which an I coincides with I give a zero contribution.

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“loc” and “out”-ﬁelds to the vacuum, this equality implies Ψ, [φout (f ), φout (h)] − ib2 D(f, h)Φ = 0 ∀Φ, Ψ ∈ V. The non-degeneracy of ., . on V now implies [φout (f ), φout (h)] = ib2 D(f, h). This proves the suﬃciency part of the lemma. As we only need this part, we only sketch the necessity: If the CCR hold, then the left-hand side of (51) is equal to the right-hand side of (53). That this implies (21) follows from the cluster expansion (5) by induction over n taking into account that the partitions in class 1) above do not contribute to (51). Acknowledgments. Discussions with S. Albeverio, W. Junker, F. Ll´edo and V. Moretti were very helpful to bring a minimum of order into the materials presented here. H. G. likes to thank for ﬁnancial support via D.F.G. project “Stochastic analysis and systems with inﬁnitely many degrees of freedom”. H. T. gratefully acknowledges the ﬁnancial support through the European TMR fellowship and the SFB 256.

References [1] S. Albeverio, H. Gottschalk, Scattering theory for quantum ﬁelds with indefinite metric, Commun. Math. Phys. 216, 491–513 (2001). [2] S. Albeverio, H. Gottschalk, J.-L. Wu, Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions, Rev. Math Phys. Vol. 8, No. 6, 763–817, (1996). [3] S. Albeverio, H. Gottschalk, J.-L. Wu, SPDEs leading to local, relativistic vector ﬁelds with indeﬁnite metric and non-trivial S-matrix, Proc. Trento Conf. on “Stochastic analysis”, Trento 2000, eds. G. da Prato and L. Tubaro, M. Dekker 2002. [4] B. Allen, Vacuum states in de Sitter space, Phys. Rev. D 32, 3136–3149 (1985). [5] H. Araki, On the asymptotic behaviour of vacuum expectation values at large spacelike separations, Ann. Phys. 11, 260–274 (1960). [6] N.D. Birrell, P.C.W. Davies, Quantum ﬁelds in curved space, Cambridge Monographs on mathematical physics, Cambridge University press, 1984. [7] N.N. Bogoliubov, A.A. Logunov, A.I. Ossak, I.V. Todorov, General principles of quantum ﬁeld theories, Kluwer Acad. Publ., 1990. [8] J. Bros, U. Moschella, Two-point functions and quantum ﬁelds in de Sitter universe, Rev. Math. Phys. 8, 327–391 (1996). [9] R. Brunetti, K. Fredenhagen, Microlocal analysis and interacting quantum ﬁeld theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208, 623–661 (2000).

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[10] T.S. Bunch, P.C.W Davies, Quantum ﬁeld theory in de Sitter space: renormalization by point-splitting, Proc. Roy. Soc. London Ser. A 360, 117–134 (1978). [11] E.A. Chernikov, N.A. Tagirov, Quantum theory of scalar ﬁeld in de Sitter space-time, Ann. Inst. H. Poincar´e, Sect. A 9, 109–141 (1968). [12] J. Dimock, Algebras of local observables on a manifold, Commun. Math. Phys. 77, 219–228 (1980). [13] A. Erd´elyi, editor: Higher transcendental functions, Vol. 1 and 2, Bateman manuscript project. New York: McGraw-Hill 1953. [14] S. Hawking, Particle production by black holes, Comm. Math. Phys. 43, 199– 220 (1975). [15] E. Hewitt, K.A. Ross, Abstract harmonic analysis II. Die Grundlehren der mathematischen Wissenschaften, Band 152. New-York/Berlin: Springer Verlag 1970. [16] S. Hollands and W. Ruan, The state space of perturbative quantum ﬁeld theory in curved spacetimes, Ann. H. Poincar´e 3, 635–657 (2002). [17] G. Morchio, F. Strocchi, Infrared singularities, vacuum structure and pure phases in local quantum ﬁeld theory, Ann. Inst. H. Poincar´e, 33, 251–282 (1980). [18] A. Ostendorf, Feynman rules for Wightman functions, Ann. Inst. H. Poincar´e 40, 273–290 (1984). [19] D. Ruelle, On the asymptotic condition in quantum ﬁeld theory, Helv. Phys. Acta 35, 147–163, (1962). [20] O. Steinmann, Perturbation theory of Wightman functions, Commun. Math. Phys. 152, 627–645 (1993). [21] O. Steinmann, Perturbative quantum electrodynamics and axiomatic ﬁeld theory, Springer Berlin/Heidelberg/N.Y., 2000. [22] R.F. Streater, A. S. Wightman, PCT, spin, statistics and all that. New York: Benjamin 1964. [23] A. Strohmaier, R. Verch, M. Wollenberg, Microlocal analysis of quantum ﬁelds on curved space-time: Analytic wavefront sets and Reeh-Schlieder theorems, Journ. Math. Phys. 43 No. 11, 5514–5530 (2002). [24] M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka J. Math. 8, 33–47 (1971).

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[25] E.A. Tagirov, Consequences of ﬁeld quantization in de Sitter type cosmological models, Ann. Phys. 76, 561–579 (1973). [26] N.J. Vilenkin, Special functions and the theory of group representations. Translations of mathematical monographs, Vol. 22. American mathematical society, Providence, R. I. 1968. [27] R.M. Wald, Quantum ﬁeld theory in curved space-time and black hole thermodynamics, Chicago Univ. Press 1993. [28] C.N. Yang, D. Feldman, The S-matrix in the Heisenberg representation, Phys. Rev. 79, 972–987 (1950).

Hanno Gottschalk and Horst Thaler Institut f¨ ur angewandte Mathematik Rheinische Friedrich-Wilhelms-Universit¨ at Wegelerstr. 6 D-53115 Bonn Germany email: [email protected] email: [email protected] Communicated by Klaus Fredenhagen submitted 03/05/02, accepted 18/07/03

Ann. Henri Poincar´e 4 (2003) 661 – 683 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040661-23 DOI 10.1007/s00023-003-0143-7

Annales Henri Poincar´ e

Modified Wave Operators for Maxwell-Schr¨ odinger Equations in Three Space Dimensions Akihiro Shimomura

Abstract. We study the scattering theory for the Maxwell-Schr¨ odinger equations under the Coulomb gauge and the Lorentz gauge conditions in three space dimensions. These equations belong to the borderline between the short range case and the long range one. We prove the existence of modified wave operators for those equations for small scattered states with no restriction on the support of the Fourier transform of them.

1 Introduction We study the scattering theory for the Maxwell-Schr¨ odinger equations in three space dimensions. We consider those equations under two gauge conditions, that is, the Coulomb gauge and the Lorentz one. In the present paper, we construct modiﬁed wave operators to the Maxwell-Schr¨odinger equations with above two gauge conditions for small scattered states without any restrictions on the support of the Fourier transform of them. The Maxwell-Schr¨odinger system in the Coulomb gauge is as follows:  1 2 2    i∂t u = − 2 (∇ − iA) u + g(|u| )u, 2A = P(Im(u(∇ − iA)u)),    ∇ · A = 0,

(MS-C)

where 1 g(|u| ) ≡ 4π 2

1 2 ∗ |u| u = (−∆)−1 |u|2 , |x| 2 ≡ ∂t2 − ∆,

P ≡ 1 − ∇∆−1 ∇·, and Im ζ denotes the imaginary part of a complex number ζ. Here u and A are complex- and R3 -valued unknown functions of (t, x) ∈ R × R3 , respectively. The last equation in the system (MS-C) is called the Coulomb gauge condition.

662

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The Maxwell-Schr¨odinger system in the Lorentz gauge is as follows:  1 2    i∂t u = − 2 (∇ − iA) u + φu,    2φ = |u|2 , (MS-L)   2A = Im(u(∇ − iA)u),     ∂t φ + ∇ · A = 0. Here u, φ and A are complex-, real- and R3 -valued unknown functions of (t, x) ∈ R × R3 , respectively. The last equation in the system (MS-L) is called the Lorentz gauge condition. A large amount of work has been devoted to the asymptotic behavior of solutions for the nonlinear Schr¨ odinger equation and for the nonlinear wave equation. We consider the scattering theory for systems centering on the Schr¨ odinger equation, in particular, Klein-Gordon-Schr¨ odinger, Wave-Schr¨ odinger and MaxwellSchr¨ odinger systems. In the scattering theory for the linear Schr¨ odinger equation, (ordinary) wave operators are deﬁned as follows. Assume that for a solution of the free Schr¨ odinger equation with given initial data φ, there exists a unique time global solution u for the perturbed Schr¨ odinger equation such that u behaves like the given free solution as t → ∞. (This case is called the short range case, and otherwise we call the long range case.) Then we deﬁne a wave operator W+ by the mapping from φ to u|t=0 . In the long range case, ordinary wave operators do not exist and we have to construct modiﬁed wave operators including a suitable phase correction in their deﬁnition. For the nonlinear Schr¨ odinger equation, the nonlinear wave equation and systems centering on the Schr¨ odinger equation, we can deﬁne the wave operators and introduce the modiﬁed wave operators in the same way (for the nonlinear Schr¨ odinger and wave equation, see the references mentioned above, and for systems, see [3, 12, 14, 16]). According to linear scattering theory, it seems that the systems (MS-C) and (MS-L) in three space dimensions belongs to the borderline between the short range case and the long range one, because the solution of the three-dimensional free Maxwell equation decays like t−1 in L∞ , as t → ∞. The two-dimensional Klein-Gordon-Schr¨odinger system and the three-dimensional Wave-Schr¨ odinger system also belong to the same case. There exist some results of the long range scattering for nonlinear equations and systems. Ozawa [11] and Ginibre and Ozawa [2] proved the existence of modiﬁed wave operators in the borderline case for the nonlinear Schr¨ odinger equation in one space dimension and in two and three space dimensions, respectively. Those results have been extended to the Klein-Gordon-Schr¨ odinger system in two space dimensions by Ozawa and Tsutsumi [12], to the Wave-Schr¨ odinger system in three space dimensions by Ginibre and Velo [3] and to the Maxwell-Schr¨ odinger system under the Coulomb gauge condition in three space dimensions by Ginibre and Velo [4] and by Tsutsumi [16]. In all results of systems mentioned above, the Maxwell-Schr¨ odinger system etc., the authors assumed a restriction on the support of the Fourier transform

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of the scattered state φ of the Schr¨ odinger part. (Note that in Ginibre and Velo [3, 4], no size restriction on for data is assumed.) More precisely, the restriction supp φˆ ⊂ {ξ ∈ R3 : |ξ| ≥ 1 + ε} ∪ {ξ ∈ R3 : |ξ| ≤ 1 − ε} for some ε > 0 was supposed for the Wave-Schr¨odinger and Maxwell-Schr¨ odinger system in three space dimensions (see Ginibre and Velo [3] and Tsutsumi [16]). Roughly speaking, the reason why they assumed that condition is as follows. Their proof is based on improved decay estimates of the interaction term which take into account the diﬀerence between the propagation property of the solution to the Schr¨ odinger and wave equations. The property of ﬁnite propagation speed and the Huygens principle for the three-dimensional wave equation imply the following time decay estimate for the solution v0 to the free wave equation: v0 (t)L∞ (|x|≥(1+ε)t) + v0 (t)L∞ (|x|≤(1−ε)t) = Oε,N (t−N ), for any ε, N > 0. This yields an improved time decay estimate of the L2 -norm of the cross term u0 v0 , where u0 is the solution ˆ of the free Schr¨ odinger equation, u0 (t)v0 (t)L2 (R3 ) ∼ t−3/2 φ(·/t)v 0 (t)L2 (R3 ) = −3/2 ˆ ˆ (φ(·/t)v0 (t)L2 (|x|≥(1+ε)t) + φ(·/t)v0 (t)L2 (|x|≤(1−ε)t) ) = Oε,N (t−N ) as t → t ∞ for any N > 0. (cf. u0 (t)v0 (t)L2 (R3 ) = O(t−3/2 ) with no restriction on the support of the Fourier transform of data). Recently, in [14] and [13], the author has proved the existence of wave operators for the two-dimensional Klein-Gordon-Schr¨ odinger system and the threedimensional Wave-Schr¨ odinger system with the Yukawa type interaction, respectively, for small scattered states without any restrictions on the support of their Fourier transform. The proof for the Klein-Gordon-Schr¨ odinger sytem is mainly based on the construction of suitable second approximations [u2 , v2 ] of the solution to that system so that (i∂t + 12 ∆)u2 − u0 v0 and (∂t2 − ∆ + 1)v2 + |u0 |2 decay faster than u0 v0 and −|u0 |2 as t → ∞, respectively, and that the Cook-Kuroda method is applicable. Here u0 and v0 are the solutions of the free Schr¨ odinger and Klein-Gordon equation, respectively. Furthermore combining idea of [3] with that of [13], Ginibre and Velo [5] have proved the existence of modiﬁed wave operators for the three-dimensional Wave-Schr¨ odinger equation with restrictions on neither size of the scattered states nor the support of their Fourier transform. In this paper, we construct modiﬁed wave operators to the system (MS-C) and (MS-L) in three space dimensions for small scattered states with no restriction on the support of their Fourier transform. Our main idea of proof is as follows. We begin with the system (MS-C), that is, the Coulomb gauge case. First we determine C C an asymptotic proﬁle for the Maxwell part AC 0 + A1 , where A0 is the free wave C solution and A1 is a suitable second correction term of the asymptotic proﬁle. Secondly, we determine the asymptotic proﬁle uC odinger component. a for the Schr¨ Since the time decay estimate of AC 1 is not suﬃcient to prove the existence of C 2 C ordinary wave operators, that is, the long range eﬀect (AC 1 · (x/t) − g(|ua | ))ua C appears, we introduce the modiﬁed free dynamics ua of the Dollard type for the C C 2 C Schr¨ odinger equation such that (i∂t + 12 ∆)uC a + (A1 · (x/t) − g(|ua | ))ua decays C C 2 C faster than (A1 · (x/t) − g(|ua | ))ua by using the method of phase correction. C On the other hand, another eﬀect AC 0 · (x/t)ua appears. In general, this decays

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slowly as t → ∞ and brings diﬃculty. Furthermore, since all the derivatives with respect to x of the function (x/t) · AC 0 decay as fast as itself, we cannot apply the C method of phase correction to (x/t) · AC 0 ua . Fortunately in the Coulomb gauge C C case A0 · (x/t)ua decays faster than in general case, since 2AC 0 = 0 and ∇ · C C AC = 0 imply 2(x · A ) = 0, that is, x · A is also a solution for the free wave 0 0 0 equation under the Coulomb gauge condition. We next explain the idea of the proof for the system (MS-L), that is, the Lorentz gauge case. As in the case of the Coulomb gauge, we ﬁrst determine an asymptotic proﬁle for the Maxwell part L L L L L L L (φL 0 + φ1 , A0 + A1 ), where (φ0 , A0 ) are the free wave solution and (φ1 , A1 ) is a suitable second correction term of the asymptotic proﬁle. We construct a modiﬁed free proﬁle uL p of the Dollard type by the method of phase correction in order L L to overcome the diﬃculty coming from the long range eﬀect uL p (A1 · (x/t) − φ1 ), which appears later, as in the case of Coulomb gauge. (uL p is a principal term of the asymptotic proﬁle for the Schr¨ odinger component.) Finally, we note that since all the derivatives with respect to x of the function ∇ · AL 0 decay as fast as itself, we cannot apply the method of phase correction to the slowly decaying L term (i/2)∇ · AL 0 up which appears later. To overcome this diﬃculty, we construct odinger a suitable second correction term uL r of the asymptotic proﬁle for the Schr¨ L r part such that (i∂t + 12 ∆)uL r − (i/2)∇ · A0 up decays faster than (i/2)∇ · A0 u1 so that the Cook-Kuroda method is applicable, as in [13, 14, 15]. Before stating our main result, we introduce some notations. Notations. We use the following symbols: ∂0 = ∂t =

∂ , ∂t

∂ α = ∂xα = ∂1α1 ∂2α2 ∂3α3

∂j =

for t ∈ R and x = (x1 , x2 , x3 ) ∈ R3 . Let q

3

L ≡ L (R ) =

ψ : ψLq =

for j = 1, 2, 3,

for a multi-index α = (α1 , α2 , α3 ),

∇ = (∂1 , ∂2 , ∂3 ),

q

∂ ∂xj

∆ = ∂12 + ∂22 + ∂32 , 1/q |ψ(x)| dx < ∞ for 1 ≤ q < ∞, q

R3

L∞ ≡ L∞ (R3 ) = {ψ : ψL∞ = ess. supx∈R3 |ψ(x)| < ∞} . We denote the set of rapidly decreasing functions on R3 by S. Let S be the set of tempered distributions on R3 . For w ∈ S , we denote the Fourier transform of w by w. ˆ For w ∈ L1 (Rn ), w ˆ is represented as −n/2 w(x)e−ix·ξ dx. w(ξ) ˆ = (2π) Rn

For s, m ∈ R, we introduce the weighted Sobolev spaces H s,m corresponding to the Lebesgue space L2 as follows: H s,m ≡ {ψ ∈ S : ψH s,m ≡ (1 + |x|2 )m/2 (1 − ∆)s/2 ψL2 < ∞}.

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We also denote H s,0 by H s . For 1 ≤ p ≤ ∞ and a positive integer k, we deﬁne the Sobolev space Wpk corresponding to the Lebesgue space Lp by Wpk ≡

  

ψ ∈ Lp : ψWpk ≡

∂ α ψLp

|α|≤k

  <∞ . 

Note that for a positive integer k, H k = W2k and the norms · H k and · W2k are equivalent. For s > 0, we deﬁne the homogeneous Sobolev spaces H˙ s by the completion of S with respect to the norm wH˙ s ≡ (−∆)s/2 wL2 .

(1.1)

If s < 0, we set H˙ s ≡ {w ∈ S : (−∆)s/2 w ∈ L2 }. Then H˙ s is a Banach space with the norm (1.1) for s > 0. On the other hand, H˙ s is a semi-normed space with the semi-norm (1.1) for s < 0. Let Y and Z be two Banach spaces with the norms · Y and · Z , respectively. We denote wY ∩Z ≡ wY + wZ , for w ∈ Y ∩ Z. Then Y ∩ Z is a Banach space with the norm · Y ∩Z . We set for t ∈ R, it

U (t) ≡ e 2 ∆ , ω ≡ (−∆)1/2 . We denote various constants by C and so forth. They may diﬀer from line to line, when it does not cause any confusion. We state our result for the case of the Coulomb gauge. C ˙C C Let (uC + , A+ , A+ ) be given scattered states, where u+ is complex-valued, and C C 3 ˙C A+ and A˙ + are R -valued. Throughout this paper, we assume that AC + and A+ satisfy the divergence free condition, that is, ˙C ∇ · AC + = ∇ · A+ = 0.

(1.2)

Our result for the Coulomb gauge case is as follows. 6,7 5,3 4,3 , AC , A˙ C and Theorem 1.1. Let uC + ∈H + ∈ H + ∈H C ˙C δ ≡ uC + H 6,7 + A+ H 5,3 + A+ H 4,3

(1.3)

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˙C be suﬃciently small. Assume that AC + and A+ satisfy the condition (1.2). Then there exists a unique solution (u, A) of the equation (MS-C) satisfying 1

u∈

C k ([0, ∞); H 3−2k ),

k=0

A ∈ C([0, ∞); H˙ 1 ∩ H˙ 3 ), sup t≥2

t (log t)2

1

∂t A ∈

2

C k ([0, ∞); H 2−k ),

k=0

∂tj u(t) − ∂tj uC D (t)H 3−2j

j=0

∞

+ t

u(s) −

8/3 uC D (s)W42

3/8

3/2 t C sup A(t) − AC ˙ 1 ∩H ˙2 0 (t) − A1 (t)H 2 t≥2 (log t) +

2

< ∞,

ds

j C ∂tj A(t) − ∂tj AC 0 (t) − ∂t A1 (t)H 2−j

j=1

t C + A(t) − AC ˙3 0 (t) − A1 (t)H (log t)2 +

3

j C ∂tj A(t) − ∂tj AC 0 (t) − ∂t A1 (t)H 3−j

< ∞.

j=1

Here AC 1

C −1 sin ωt)A˙ C AC 0 (t, x) ≡ ((cos ωt)A+ )(x) + ((ω + )(x),

is the solution of the following linear equation: x C x 2 1 2 C C ∂t A1 − ∆A1 = 3 P ˆ u z(t)2 , t t + t C AC ˙ 1 ∩H ˙ 3 + ∂t A1 (t)H 2 → 0, 1 (t)H

(1.4)

(1.5) (1.6)

as t → +∞, where z ∈ C ∞ (Rt ; R) such that z(t) = 0 for |t| ≤ 1/2, z(t) = 1 for |t| ≥ 1, and we deﬁne i|·|2

C

− 2t −iS (t,−i∇) C e u+ )(x) uC D (t, x) ≡(U (t)e 2 C x x i|x| 1 e 2t −iS (t, t ) , u ˆC = (it)3/2 + t

where 2 S C (t, x) ≡ (g(|ˆ uC + | ))(x) log t − x ·

for t ≥ 1 and x ∈ R3 . A similar result holds for negative time.

1

t

AC 1 (s, sx) ds,

(1.7)

(1.8)

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Remark 1.1. The function AC 0 is a unique solution of the Cauchy problem for the free wave equation 2 ∂t A − ∆A = 0, A(0, x) = AC ∂t A(0, x) = A˙ C + (x), + (x). Furthermore it follows from (1.2) that AC 0 satisﬁes the Coulomb gauge condition: ∇ · AC 0 = 0. Under the assumptions of Theorem 1.1, there exists a unique solution AC 1 of the linear ﬁnal value problem (1.5)–(1.6) (see Lemma 2.3 below). C We determine the phase function S C such that (i∂t + 12 ∆)uC D + (A1 · (x/t) − C 2 C C C 2 C g(|uD | ))uD decays more rapidly than (A1 · (x/t) − g(|uD | ))uD , as t → ∞. The modiﬁed free proﬁle uC D is called the Dollard type. For details, see Section 2 in Ginibre and Ozawa [2]. The function uC D (t) is asymptotically equivalent to the modiﬁed free proﬁle C −iS C (t,−i∇) U (t)e−iS (t,−i∇) uC for the Schr¨ odinger equation, that is, uC + D (t) − U (t)e C 2 u+ → 0 in L , as t → ∞ (see, e.g., Ginibre and Ozawa [2]). Let C ˙C C C ˙C V C ≡ {(uC + , A+ , A+ );u+ H 6,7 + A+ H 5,3 + A+ H 4,3 ≤ δ, ˙C ∇ · AC + = ∇ · A+ = 0},

where δ is deﬁned by (1.3). The following corollary is an immediate consequence of Theorem 1.1. Corollary 1.1. For the equation (MS-C), the modiﬁed wave operator W+C : C ˙C C (uC + , A+ , A+ ) → (u(0), A(0), ∂t A(0)) is well deﬁned on V , where (u, A) is the solution to the equation (MS-C) obtained in Theorem 1.1. Similarly the modiﬁed wave operator W−C for negative time is also well deﬁned on V C . Hereafter we assume that δ deﬁned by (1.3) satisﬁes 0 < δ ≤ 1. Next we state our result for the case of the Lorentz gauge. L ˙L L ˙L L Let (uL + , φ+ , φ+ , A+ , A+ ) be given scattered states, where u+ is complexL L L L 3 ˙ ˙ valued, φ+ and φ+ are real-valued, and A+ and A+ are R -valued. Throughout ˙L L ˙L this paper, we assume that φL + , φ+ , A+ and A+ satisfy the condition L φ˙ + + ∇ · AL + = 0, (1.9) L ∆φ + ∇ · A˙ L = 0. +

+

Our result for the Lorentz gauge case is as follows. 6,7 5,2 ˙ L 5,3 4,3 Theorem 1.2. Let uL , φL , φ+ ∈ H 4,2 , AL , A˙ L , + ∈ H + ∈ H + ∈ H + ∈ H −1 L 0,2 −1 ˙ L 0,2 and ω A+ ∈ H , ω A+ ∈ H L L ˙L ˙L η ≡uL + H 6,7 + φ+ H 5,2 + φ+ H 4,2 + A+ H 5,3 + A+ H 4,3 + ω −1 AL H 0,2 + ω −1 A˙ L H 0,2 +

+

(1.10)

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L ˙L ˙L be suﬃciently small. Assume that φL + , φ+ , A+ and A+ satisfy the condition (1.9). Then there exists a unique solution (u, φ, A) of the equation (MS-L) satisfying 1

u∈

C k ([0, ∞); H 3−2k ),

k=0

φ ∈ C([0, ∞); H˙ 1 ∩ H˙ 3 ), A ∈ C([0, ∞); H˙ 1 ∩ H˙ 3 ), sup t≥2

t (log t)2

1

∂t φ ∈ ∂t A ∈

2 k=0 2

C k ([0, ∞); H 2−k ), C k ([0, ∞); H 2−k ),

k=0

∂tj u(t) − ∂tj uL D (t)H 3−2j

j=0

∞

+ t

u(s) −

8/3 uL D (s)W42

3/8

3/2 t L sup φ(t) − φL ˙ 1 ∩H ˙2 0 (t) − φ1 (t)H 2 t≥2 (log t) +

2

< ∞,

ds

j L ∂tj φ(t) − ∂tj φL 0 (t) − ∂t φ1 (t)H 2−j

j=1

+ +

t L φ(t) − φL ˙3 0 (t) − φ1 (t)H (log t)2

3

j L ∂tj φ(t) − ∂tj φL 0 (t) − ∂t φ1 (t)H 3−j

< ∞.

j=1

3/2 t L sup A(t) − AL ˙ 1 ∩H ˙2 0 (t) − A1 (t)H (log t)2 t≥2 +

2

∂tj A(t)

−

∂tj AL 0 (t)

−

∂tj AL 1 (t)H 2−j

j=1

+ +

t L A(t) − AL ˙3 0 (t) − A1 (t)H (log t)2

3

∂tj A(t)

−

∂tj AL 0 (t)

−

∂tj AL 1 (t)H 3−j

< ∞.

j=1

Here L −1 sin ωt)φ˙ L φL 0 (t, x) ≡ ((cos ωt)φ+ )(x) + ((ω + )(x), L L −1 ˙ A0 (t, x) ≡ ((cos ωt)A+ )(x) + ((ω sin ωt)AL + )(x),

(1.11) (1.12)

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L φL 1 and A1 are the solutions of the following ﬁnal value problems of the linear equations: 1 L x 2 L ˆ+ ∂t2 φL (1.13) z(t)2 , 1 − ∆φ1 = 3 u t t L φL t → +∞, (1.14) ˙ 1 ∩H ˙ 3 + ∂t φ1 (t)H 2 → 0, 1 (t)H

and

1 x L x 2 u ˆ z(t)2 , t3 t + t + ∂t AL t → +∞, 1 (t)H 2 → 0,

L ∂t2 AL 1 − ∆A1 =

AL ˙3 ˙ 1 ∩H 1 (t)H

(1.15) (1.16)

respectively, where z ∈ C ∞ (Rt ; R) such that z(t) = 0 for |t| ≤ 1/2, z(t) = 1 for |t| ≥ 1, and we deﬁne i|·|2

L

− 2t −iS (t,−i∇) L e u+ )(x) uL D (t, x) ≡(U (t)e 2 x i|x| −iS L (t, x ) 1 t , e 2t uˆL = (it)3/2 + t

where L

S (t, x) ≡

1

t

L (φL 1 (s, sx) − x · A1 (s, sx)) ds,

(1.17)

(1.18)

for t ≥ 1 and x ∈ R3 . A similar result holds for negative time. L Remark 1.2. The functions φL 0 and A0 are unique solutions of the Cauchy problems for the free wave equations 2 ∂t φ − ∆φ = 0, φ(0, x) = φL (x), ∂t φ(0, x) = φ˙ L (x) +

and

+

∂t2 A − ∆A = 0, A(0, x) = AC + (x),

∂t A(0, x) = A˙ C + (x),

L respectively. Furthermore it follows from (1.9) that φL 0 and A0 satisfy the Lorentz gauge condition: L ∂t φL 0 + ∇ · A0 = 0.

Under the assumptions of Theorem 1.2, there exist unique solutions φL 1 and of the linear ﬁnal value problems (1.13)–(1.14) and (1.15)–(1.16), respectively (see Lemma 3.2 below). As in the case of the Coulomb gauge case, we determine the phase function L L L L S L such that (i∂t + 12 ∆)uL D + (A1 · (x/t) − φ1 )uD decays more rapidly than (A1 · L L L (x/t) − φ1 )uD , as t → ∞. The function uD (t) is asymptotically equivalent to L the modiﬁed free proﬁle U (t)e−iS (t,−i∇) uL odinger equation, that is, + for the Schr¨ L −iS L (t,−i∇) L 2 uD (t) − U (t)e u+ → 0 in L , as t → ∞. AL 1

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Let L ˙L L ˙L V L ≡ {(uL + , φ+ , φ+ , A+ , A+ ); L L ˙L ˙L uL + H 6,7 + φ+ H 5,2 + φ+ H 4,2 + A+ H 5,3 + A+ H 4,3 + ω −1 AL H 0,2 + ω −1 A˙ L H 0,2 ≤ η, +

L φ˙ L + + ∇ · A+ = 0,

+

˙L ∆φL + + ∇ · A+ = 0},

where η is deﬁned by (1.10). The following corollary is an immediate consequence of Theorem 1.2. Corollary 1.2. For the equation (MS-L), the modiﬁed wave operator W+L : L ˙L L ˙L L (uL + , φ+ , φ+ , A+ , A+ ) → (u(0), φ(0), ∂t φ(0), A(0), ∂t A(0)) is well deﬁned on V , where (u, φ, A) is the solution to the equation (MS-C) obtained in Theorem 1.2. Similarly the modiﬁed wave operator W−L for negative time is also well deﬁned on V L . Hereafter we assume that η deﬁned by (1.10) satisﬁes 0 < η ≤ 1.

2 Proof of Theorem 1.1 In this section, we prove Theorem 1.1. C Let (uC a , Aa ) be a given asymptotic proﬁle of the equation (MS-C), namely an approximate solution for that equation as t → ∞. We introduce the following functions: 1 C2 C C C C 2 C RSC ≡ LuC a − iAa · ∇ua + |Aa | ua − g(|ua | )ua , 2 C C C C 2 C RM ≡ 2AC a − P(Im(ua ∇ua ) − Aa |ua | ), where

1 (2.1) L ≡ i∂t + ∆, 2 C The functions RSC and RM are diﬀerences between the left-hand sides and the righthand sides in the ﬁrst and the second equations of the system (MS-C) substituted (u, A) = (ua , Aa ), respectively. In view of Section 3 in Tsutsumi [16], we have the following proposition. Proposition 2.1. Assume that there exists a constant δ > 0 such that C uC a (t)H 3 + ∂t ua (t)H 1 ≤ δ , C −3/2 3 + ∂t u (t)W 1 ≤ δ (1 + t) uC , a (t)W∞ a ∞ C −1 3 + ∂t A (t)W 1 ≤ δ (1 + t) , AC a (t)W∞ a ∞

(log(2 + t))2 , (1 + t)2 log(2 + t) ≤ δ (1 + t)5/2

RSC (t)H 3 + ∂t RSC (t)H 1 ≤ δ C C RM (t)H 2 + ∂t RM (t)H 1

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for t ≥ 0, and that δ > 0 is suﬃciently small. Then there exists a unique solution for the equation (MS-C) satisfying u∈

1

C k ([0,∞);H 3−2k ),

k=0

A ∈ C([0,∞); H˙ 1 ∩ H˙ 3 ), sup t≥2

t (logt)2

1

2

∂t A ∈

C k ([0,∞);H 2−k ),

k=0

∂tj u(t) − ∂tj uC a (t)H 3−2j +

j=0

t

∞

3/8 8/3 u(s) − uC (s) ds < ∞, a W2 4

3/2 2

t j j C C 2−j (t) + ∂ A(t) − ∂ A (t) sup A(t) − A ˙ 1 ∩H ˙2 H t t a a H 2 t≥2 (logt) j=1 +

3

t j j C C 3−j (t) + ∂ A(t) − ∂ A (t) A(t) − A < ∞. ˙3 H t t a a H (logt)2 j=1

C We construct an asymptotic proﬁle (uC a , Aa ) for the equation (MS-C) which satisfy the assumptions of Proposition 2.1. We ﬁrst consider an asymptotic proﬁle AC a for the Maxwell part. We deterC C C C mine it of the form AC a = A0 + A1 , where A0 is deﬁned by (1.4) and A0 will be determined later. We recall the time decay estimates of solutions for the free wave equation. (see, e.g., Section 1 in Klainerman [8]):

Lemma 2.1. In three space dimensions, there exists a constant C > 0 such that for t ≥ 0, (cos ωt)f L∞ ≤ C(f H 2 + f W12 )(1 + t)−1 ≤ Cf H 2,2 (1 + t)−1 , ω −1 (sin ωt)f L∞ ≤ C(f H 1 + f W11 )(1 + t)−1 ≤ Cf H 1,2 (1 + t)−1 . We begin with time decay estimates of the solution AC 0 for the free wave of the free wave equation equation. We can easily see that if the solution AC 0 C satisﬁes the Coulomb gauge condition ∇ · AC = 0, then x · A 0 0 is also a solution of C )| = x·A and (∂t (x·AC the free wave equation with initial data (x·AC 0 t=0 + 0 ))|t=0 = C ˙ x · A+ . According to Lemma 2.1, this implies the following: ˙C Lemma 2.2. Assume that AC + and A+ satisfy the Coulomb gauge condition (1.2). Then there exists a constant C > 0 such that for t ≥ 0, C C C 3 + ∂t A (t)W 1 + x · A (t)W 3 + ∂t (x · A (t))W 1 AC 0 (t)W∞ 0 0 0 ∞ ∞ ∞ C C −1 ˙ ≤C(A H 5,3 + A H 4,3 )(1 + t) +

≤Cδ(1 + t)−1 , where δ > 0 is deﬁned in (1.3).

+

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Next we choose a second correction term AC 1 of the asymptotic proﬁle of the Maxwell part. We consider the ﬁnal value problem (1.5)–(1.6). The following lemma is obtained as in Lemma 2.4 in Tsutsumi [16]. Lemma 2.3. There exists a unique solution AC 1 of the equation (1.5) satisfying C A1 ∈ C([0, ∞); H˙ 1 ∩ H˙ 7 ), ∂t AC 1 ∈

6

C j ([0, ∞); H 6−j ),

j=0

∂tj ∂xα AC 1 (t)L2

2 ≤ C(1 + t)−(k−1/2) uC + H 6,7

j+|α|=k

for t ≥ 0 and k = 1, . . . , 7, and

C 2 −1−k ∂tj ∂xα AC ≤ Cδ 2 (1 + t)−1−k 1 (t)L∞ ≤ Cu+ H 6,7 (1 + t) j+|α|=k

for t ≥ 0 and k = 0, . . . , 5, where δ > 0 is deﬁned in (1.3). We choose the function C C AC a ≡ A0 + A1

as the asymptotic proﬁle for the Maxwell part. We next construct an asymptotic proﬁle uC odinger part. a of the Schr¨ Lemma 2.3 implies the time decay estimate of the interaction term (U (t)uC +) C 2 −1 C 2 ((x/t) · AC (t) − g(|U (t)u | ) = O(t ), where A is the second term of the L 1 + 1 asymptotic proﬁle of the Maxwell part introduced above. This means that the existence of ordinary wave operators is not expected. As in [2], [11] and [13], we have to construct modiﬁed free proﬁle uC a of the Dollard type by the method of C C C 2 phase correction such that LuC + u (A a a 1 · (x/t) − g(|ua | )) decays faster than C C C 2 ua (A1 · (x/t) − g(|ua | )). Let S C be the real-valued function deﬁned in (1.8). The following lemma immediately follows from Lemma 2.3. Lemma 2.4. Let j + l ≤ 5. There exists a constant C > 0 such that for t ≥ 1,

log t log t 2 (1 + | · |)−j−1 ∂xα ∂tj S C (t)L∞ ≤ CuC ≤ Cδ 2 j , + H 6,7 j t t |α|≤l

where δ > 0 is deﬁned in (1.3). uC D,

Let uC D be the function deﬁned in (1.7). To avoid the singularity at t = 0 of we introduce the following function uC a: C uC a (t, x) ≡uD (t, x)z(t) i|x|2 C x 1 C x e 2t −iS (t, t ) z(t), u ˆ = + 3/2 t (it)

where z ∈ C ∞ (Rt ; R) such that z(t) = 0 for |t| ≤ 1/2, z(t) = 1 for |t| ≥ 1.

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C 3 Remark 2.1. Since z(t) = 1 for |t| ≥ 1, uC a (t, x) = uD (t, x) for t ≥ 1 and x ∈ R .

odinger part. We choose the function uC a as an asymptotic proﬁle for the Schr¨ We introduce the following linear operators: J = J(t) ≡ x + it∇, (M (t)f )(x) ≡ e

i|x|2 /2t

f (x), x 1 (D(t)f )(x) ≡ . f 3/2 t (it)

(2.2) (2.3) (2.4)

Remark 2.2. It is well known that U (t) = M (t)D(t)F M (t), JM D = iM D∇, where F is the Fourier transform. Using these relations, we see that −1 (JuC Mx e−iS a )(t, x) =z(t)(U (t)(M (t)

=M (t)D(t)[∇(e−iS =

C

C

(t,−i∇) C u+ ))(x)

(t,·) C uˆ+ )]

x z(t) i|x|2 /2t−iS C (t,x/t) C C x C x ∇S t, u ˆ + i∇ˆ u , e + + t t t (it)3/2

where Mx is the multiplication operator by the function x. Using Lemma 2.4 and noting Remark 2.2, we obtain the following lemma (see, e.g., Lemma 3.2 in [13]). Lemma 2.5. There exists a constant C > 0 such that for t ≥ 0, C C uC a (t)H 3 + ∂t ua (t)H 1 ≤ Cu+ H 6,7 ≤ Cδ, C C (JuC a )(t)H 3 + ∂t (Jua )(t)H 1 ≤ Cu+ H 6,7 log(2 + t) ≤ Cδ log(2 + t), C C −3/2 3 + ∂t u (t)W 1 ≤ Cu H 6,7 (1 + t) uC ≤ Cδ(1 + t)−3/2 , a (t)W∞ a + ∞ 1

j C x C C 2 C ∂t Lua (t) + AC 1 (t) · ua (t) − g(|ua (t)| )ua (t) t H 3−2j j=0

(log(2 + t))2 (1 + t)2 (log(2 + t))2 ≤Cδ 2 , (1 + t)2 2 ≤CuC + H 6,7

where δ > 0 is deﬁned in (1.3).

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C From the deﬁnitions of uC a and Aa , we see that

1 C C C 2 C RSC =LuC a + (x · A1 )ua − g(|ua | )ua t 1 1 C2 C 1 C C C − AC a · Jua + |Aa | ua + (x · A0 )ua , t 2 t 1 C C C C 2 RM Re(uC =P Ju ) + A |u | . a a a a t C are as follows. The estimates of RSC and RM

Lemma 2.6. There exists a constant C > 0 such that for t ≥ 0, (log(2 + t))2 , (1 + t)2 log(2 + t) ≤ Cδ , (1 + t)5/2

RSC (t)H 3 + ∂t RSC (t)H 1 ≤ Cδ C C RM (t)H 2 + ∂t RM (t)H 1

where δ > 0 is deﬁned in (1.3). Proof. By Lemmas 2.2, 2.3 and 2.5, we have RSC (t)H 3 + ∂t RSC (t)H 1

1 C C C 2 C ≤C ∂tj {LuC a (t) + A1 (t) · (x/t)ua (t) − g(|ua (t)| )ua (t)}H 3−2j j=0 C C C 3 (Ju )(t)H 3 + ∂t A (t)W 1 (Ju )(t)H 1 + t−1 (AC a (t)W∞ a a a ∞ C C 2 C 1 ∂t (Ju )(t)H 1 ) + A (t) + AC 3 (Jua )(t)H 3 a (t)W∞ a a W∞ C C C 2 C 1 A (t)W 1 (Ju )(t)H 1 + A (t) + ∂t AC 1 ∂t (Jua )(t)H 1 a (t)W∞ a a a W∞ ∞ C C C 3 u (t)H 3 + ∂t (x · A )(t)W 1 u (t)H 1 + t−1 (x · AC 0 (t)W∞ a 0 a ∞ C 1 ∂t u (t)H 1 ) + x · AC 0 (t)W∞ a

≤ Cδ

(log(2 + t))2 , (1 + t)2

C C RM (t)H 2 + ∂t RM (t)H 1 C C C 2 (Ju )(t)H 2 + ∂t u (t)W 1 (Ju )(t)H 1 ≤C[t−1 (uC a (t)W∞ a a a ∞ C C C C 1 ∂t (Ju )(t)H 1 ) + A (t)W 2 u (t)W 2 u (t)H 2 + uC a (t)W∞ a a a a ∞ ∞ C C C C C 1 u (t)W 1 u (t)H 1 + A (t)W 1 u (t)W 1 ∂t u (t)H 1 ] + ∂t AC a (t)W∞ a a a a a ∞ ∞ ∞

≤Cδ

log(2 + t) . (1 + t)5/2

The proof of this lemma is complete.

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We assume that δ > 0 deﬁned by (1.3) is suﬃciently small. In view of LemC mas 2.2, 2.3, 2.5 and 2.6, we see that our asymptotic proﬁle (uC a , Aa ) satisﬁes the assumptions of Proposition 2.1 for δ = Cδ, where C > 0 is a constant independent of δ and t. Theorem 1.1 follows immediately from Proposition 2.1 and Remark 2.1. The proof of Theorem 1.1 is complete.

3 Proof of Theorem 1.2 In this section, we prove Theorem 1.2. L L Let (uL a , φa , Aa ) be a given asymptotic proﬁle of the equation (MS-L), namely an approximate solution for that equation as t → ∞. We introduce the following functions: i 1 L2 L L L L L L L RSL ≡ LuL a − iAa · ∇ua − (∇ · Aa )ua + |Aa | ua − φa ua , 2 2 L L 2 ≡ 2φL RE a − |ua | , L L L L 2 L ≡ 2AL RM a − Im(ua ∇ua ) − Aa |ua | ),

(3.1) (3.2) (3.3)

L L where L is deﬁned by (2.1). The functions RSL , RE and RM are diﬀerences between the left-hand sides and the right-hand sides in the ﬁrst, the second and the third equations of the system (MS-C) substituting (u, φ, A) = (ua , φa , Aa ), respectively. In view of Section 4 in [13] and Section 3 in [16], we have the following proposition. Since in fact, Tsutsumi did not use the Coulomb gauge condition ∇ · A = 0 in Section 3 of [16], and the equation (MS-L) has no interaction term of a product of φ and A, the methods in [13] and [16] are applicable to our problem.

Proposition 3.1. Assume that there exists a constant η > 0 such that L uL a (t)H 3 + ∂t ua (t)H 1 ≤ η , L −3/2 3 + ∂t u (t)W 1 ≤ η (1 + t) uL , a (t)W∞ a ∞ L −1 3 + ∂t φ (t)W 1 ≤ η (1 + t) φL , a (t)W∞ a ∞ L −1 3 + ∂t A (t)W 1 ≤ η (1 + t) , AL a (t)W∞ a ∞

RSL (t)H 3 + ∂t RSL (t)H 1 ≤ η

(log(2 + t))2 , (1 + t)2

L L RE (t)H 2 + ∂t RE (t)H 1 ≤ η (1 + t)−5/2 , L L RM (t)H 2 + ∂t RM (t)H 1 ≤ η

for t ≥ 0, and that η > 0 is suﬃciently small.

log(2 + t) (1 + t)5/2

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Then there exists a unique solution for the equation (MS-L) satisfying u∈

1

k=0

C k ([0, ∞); H 3−2k ),

φ ∈ C([0, ∞); H˙ 1 ∩ H˙ 3 ),

∂t φ ∈

2

C k ([0, ∞); H 2−k ),

k=0

A ∈ C([0, ∞); H˙ 1 ∩ H˙ 3 ),

∂t A ∈

2

C k ([0, ∞); H 2−k ),

k=0

sup t≥2

1 t ∂ j u(t) − ∂tj uL a (t)H 3−2j (log t)2 j=0 t ∞ 3/8 8/3 L + u(s) − ua (s)W 2 ds < ∞, t

4

3/2 2

t j j L L sup ∂t φ(t) − ∂t φa (t)H 2−j φ(t) − φa (t)H˙ 1 ∩H˙ 2 + 2 t≥2 (log t) j=1

3

t j j L L + ∂t φ(t) − ∂t φa (t)H 3−j φ(t) − φa (t)H˙ 3 + < ∞. (log t)2 j=1

3/2 2

t j j L L 2−j (t) + ∂ A(t) − ∂ A (t) sup A(t) − A ˙ 1 ∩H ˙2 H t t a a H 2 t≥2 (log t) j=1 +

3

t j j L L (t) + ∂ A(t) − ∂ A (t) 3−j A(t) − A < ∞. ˙3 H t t a a H (log t)2 j=1

As in the case of the Coulomb gauge, we construct an asymptotic proﬁle L L , φ (uL a a , Aa ) for the equation (MS-L) which satisfy the assumptions of Proposition 3.1. L We ﬁrst consider an asymptotic proﬁle (φL a , Aa ) for the Maxwell part. We L L L L L L L determine it as of the form (φa , Aa ) = (φ0 + φ1 , AL 0 + A1 ), where φ0 and A0 L L are deﬁned by (1.11) and (1.12), respectively, and φ0 and A0 will be determined later. L We begin with time decay estimates of the solutions φL 0 and A0 . We can easily L L see that if the solutions φ0 and A0 of the free wave equation satisfy the Lorentz L L L gauge condition ∂t φL 0 +∇·A0 = 0, then x·A0 −tφ0 is also a solution of the free wave L L L L equation with initial data (x · A0 − tφ0 )|t=0 = x · AL + and (∂t (x · A0 − tφ0 ))|t=0 = L x · A˙ L + − φ+ . According to Lemma 2.1, this fact implies the following:

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L ˙L ˙L Lemma 3.1. Assume that φL + , φ+ , A+ and A+ satisfy the condition (1.9). Then there exists a constant C > 0 such that for t ≥ 0, L L ˙ L H 4,2 )(1 + t)−1 3 + ∂t φ (t)W 1 ≤C(φ H 5,2 + φ φL 0 (t)W∞ 0 + + ∞

≤Cη(1 + t)−1 , L L −1 ˙L 3 + ∂t A (t)W 1 ≤C(A H 5,2 + A AL 0 (t)W∞ 0 + + H 4,2 )(1 + t) ∞

≤Cη(1 + t)−1 , L L L 3 + ∂t (x · A (t) + tφ (t))W 1 x · AL 0 (t) + tφ0 (t)W∞ 0 0 ∞ L L L ˙ 3,2 5,2 4,2 ≤C(φ H + A H + A H )(1 + t)−1 +

+

−1

≤Cη(1 + t)

+

,

where η > 0 is deﬁned in (1.10). L Next we choose a second correction term (φL 1 , A1 ) of the asymptotic proﬁle of the Maxwell part. We consider the ﬁnal value problems (1.13)–(1.14) and (1.15)– (1.16). The following lemma is obtained as in Lemma 2.4 in Tsutsumi [16].

Lemma 3.2. There exists a unique solution φL 1 of the equation (1.13) satisfying ˙1 ˙7 φL 1 ∈ C([0, ∞); H ∩ H ), ∂t φL 1 ∈

6

C j ([0, ∞); H 6−j ),

j=0

∂tj ∂xα φL 1 (t)L2

2 ≤ C(1 + t)−(k−1/2) uL + H 5,7

j+|α|=k

for t ≥ 0 and k = 1, . . . , 7, and

L 2 −1−k ∂tj ∂xα φL ≤ Cη 2 (1 + t)−1−k 1 (t)L∞ ≤ Cu+ H 5,7 (1 + t)

j+|α|=k

for t ≥ 0 and k = 0, . . . , 5. Similarly there exists a unique solution AL 1 of the equation (1.15) satisfying ˙1 ˙7 AL 1 ∈ C([0, ∞); H ∩ H ), ∂t AL 1 ∈

j+|α|=k

6

C j ([0, ∞); H 6−j ),

j=0

∂tj ∂xα AL 1 (t)L2

2 ≤ C(1 + t)−(k−1/2) uL + H 6,7

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for t ≥ 0 and k = 1, . . . , 7, and

L 2 −1−k ∂tj ∂xα AL ≤ Cη 2 (1 + t)−1−k 1 (t)L∞ ≤ Cu+ H 6,7 (1 + t) j+|α|=k

for t ≥ 0 and k = 0, . . . , 5. Here η > 0 is deﬁned in (1.10). We choose the functions L L L L L (φL a , Aa ) = (φ0 + φ1 , A0 + A1 )

as the asymptotic proﬁle for the Maxwell part. We next construct an asymptotic proﬁle uL odinger part. We a of the Schr¨ L L L L determine it as of the form uL = u + u . u and u are principal and remainder a p r p r L L odinger terms of ua , respectively. Namely, up is a modiﬁed free solution for the Schr¨ L equation and uL is a second correction term which decays faster than u as t → ∞. r p From the deﬁnition (3.1) of RSL , we have 1 i L L L L L L L RSL =LuL p + (x · A1 )up − φ1 up + Lur − (∇ · A0 )up t 2 1 1 i L L L L − AL JuL + (x · AL 0 − tφ0 )up − (∇ · A0 )ur t a p t 2 1 L2 L L L − iAa · ∇uL r − |Aa | ua − φa ur . 2

(3.4)

L L L According to Lemmas 3.1 and 3.2, it is expected that (1/t)(x · AL 1 )up , φ1 up and L (i/2)(∇ · AL 0 )up decay most slowly in the right-hand side in the equality (3.4). L We begin with a principal term uL p of an asymptotic proﬁle ua for the Schr¨ odinger component. Lemma 3.2 implies the time decay estimate of the interL L −1 L ), where φL action term (U (t)uL + )((x/t) · A1 (t) − φ1 (t))L2 = O(t 1 and A1 are the second terms of the asymptotic proﬁle of the Maxwell part introduced above. This means that the existence of ordinary wave operators is not expected. As in the case of the Coulomb gauge, we have to construct modiﬁed free proﬁle uL p of the L C L +u (A ·(x/t)−φ Dollard type by the method of phase correction such that LuL p p 1 1) L L (A · (x/t) − φ ). decays faster than uL p 1 1 Let S L be the real-valued function deﬁned in (1.18). The following lemma immediately follows from Lemma 3.2.

Lemma 3.3. Let j + l ≤ 5. There exists a constant C > 0 such that for t ≥ 1,

2 (1 + | · |)−j−1 ∂xα ∂tj S L (t)L∞ ≤ CuL + H 6,7

|α|≤l

where η > 0 is deﬁned in (1.10).

log t log t ≤ Cη 2 j , j t t

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Let uL D be the function deﬁned in (1.17). To avoid the singularity at t = 0 of we introduce the following function uL p: L uL p (t, x) ≡uD (t, x)z(t) i|x|2 L x 1 L x e 2t −iS (t, t ) z(t), u ˆ = + 3/2 t (it)

where z ∈ C ∞ (Rt ; R) such that z(t) = 0 for |t| ≤ 1/2, z(t) = 1 for |t| ≥ 1. L 3 Remark 3.1. Since z(t) = 1 for |t| ≥ 1, uL p (t, x) = uD (t, x) for t ≥ 1 and x ∈ R .

Let J, M and D be the operators deﬁned by (2.2)–(2.4). Remark 3.2. As in the case of the Coulomb gauge (see Remark 2.2), we have the following relation. −1 Mx e−iS (JuL p )(t, x) =z(t)(U (t)(M (t)

=M (t)D(t)[∇(e−iS =

L

L

(t,−i∇) L u+ ))(x)

(t,·) L u ˆ+ )]

x z(t) i|x|2 /2t−iS L (t,x/t) L L x L x ∇S t, u ˆ + i∇ˆ u , e + + t t t (it)3/2

where Mx is the multiplication operator by the function x. As in the Coulomb gauge case, using Lemma 3.3 and noting Remark 3.2, we obtain the following lemma. Lemma 3.4. There exists a constant C > 0 such that for t ≥ 0, L L uL p (t)H 3 + ∂t up (t)H 1 ≤ Cu+ H 6,7 ≤ Cη, L L (JuL p )(t)H 3 + ∂t (Jup )(t)H 1 ≤ Cu+ H 6,7 log(2 + t) ≤ Cη log(2 + t), L L −3/2 3 + ∂t u (t)W 1 ≤ Cu H 6,7 (1 + t) ≤ Cη(1 + t)−3/2 , uL p (t)W∞ p + ∞ 1

j L x L L L ∂t Lup (t) + AL u (t) · (t) − φ (t)u (t) 1 p 1 p 3−2j t H j=0

(log(2 + t))2 (1 + t)2 (log(2 + t))2 ≤Cη 2 , (1 + t)2 2 ≤CuL + H 6,7

where η > 0 is deﬁned in (1.10). The time decay estimate ∇ · A0 (t)L∞ = O(t−1 ) is not suﬃcient to prove Theorem 1.2 directly by the Cook-Kuroda method. Unfortunately, since all the derivatives with respect to x of the function ∇·AL 0 decay as fast as itself, we cannot

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apply the method of phase correction (as in Lemma 3.4) to the slowly decaying L L term (i/2)∇ · AL 0 up . Therefore we next construct the second correction term ur L L of the asymptotic proﬁle of the Schr¨ odinger part such that Lur − (i/2)up ∇ · A0 L decays faster than (i/2)up ∇ · A0 as t → ∞ as in [13, 14, 15]. We consider the second correction term uL r of the following form: L uL r (t, x) = up (t, x)∇ · V (t, x),

where V is the following R3 -valued function: ˙ V (t, x) = ((cos ωt)Q)(x) + ((ω −1 sin ωt)Q)(x). We determine R3 -valued functions Q and Q˙ of x ∈ R3 . We ﬁrst note the following identity: 1 1 L(wz) = w ∆z + zLw + (−iJw · ∇z + iwP z), (3.5) 2 t where w and z are complex- and real-valued functions of x ∈ R3 , respectively, P ≡ t∂t + x · ∇ and J is the operator deﬁned by (2.2). It is well known that if z solves the free wave equation, then so does P z because 2P = (P +2)2, and that if w is an approximate solution of the free Schr¨ odinger equation, then so is Jw because JL − LJ = 0. Noting these facts and putting w = uL p and z = ∇ · V , we expect that the most L is (1/2)u slowly decaying part of LuL r p ∆∇ · V . Now we set −2 L Q(x) ≡ −i(−∆)−1 AL A+ (x), + (x) = −iω −1 ˙ L −2 ˙ L ˙ Q(x) ≡ −i(−∆) A+ (x) = −iω A+ (x),

so that the equality i 1 L up ∆∇ · V = uL ∇ · AL 0 2 2 p L L L L holds. Then it is expected that LuL r − up ∇ · A0 decays faster than up ∇ · A0 as t → ∞. From the equality (3.5), we have i L 1 L L L L LuL r − up ∇ · A0 = ∇ · V Lup + (−iJup · ∇(∇ · V ) + iup P ∇ · V ). 2 t

(3.6)

Remark 3.3. It is well known that ˙ P ∇ · V (t, ·) = (cos ωt)(Mx · ∇(∇ · Q)) + (ω −1 sin ωt)((1 + Mx · ∇)∇ · Q), where Mx is the multiplication operator by the function x. By Lemmas 2.1, 3.3, 3.4, the deﬁnition of uL r , the equation (3.6) and the inequality f gH k ≤ Cf H k gW∞ k for positive integer k, we have the following lemma.

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Lemma 3.5. There exists a constant C > 0 such that for t ≥ 0, L −3/2 uL , r (t)H 3 + ∂t ur (t)H 1 ≤ Cηt L −5/2 3 + ∂t u (t)W 1 ≤ Cηt uL , r (t)W∞ r ∞ 1 2

j 2 (log(2 + t)) ∂t LuL (t) − i (∇ · AL (t))uL (t) ≤ Cη , r 0 p 3−2j 2 (1 + t)2 H j=0

where η > 0 is deﬁned in (1.10). We choose an asymptotic proﬁle for the Schr¨ odinger part L L uL a = up + ur . L L Finally we derive estimates RSL , RE and RM . Recall the deﬁnitions (3.2) and L L L (3.3) of RE and RM . It follows from deﬁnitions of AL 1 and up that L L 2 L RE = −2Re(uL p ur ) − |ur | ,

(3.7)

1 L L L L L L = Re(uL RM p Jur ) − Im(ur ∇up + up ∇ur t L 2 + uL r ∇ur ) + Aa |ua | .

(3.8)

From the equalities (3.4), (3.7), (3.8), Lemmas 3.1, 3.2, 3.4 and 3.5, we obtain the following lemma exactly in the same way as in the proof of Lemma 2.6. In L particular, recall that x · AL 0 + tφ0 is a solution of the free wave equation under the Lorentz gauge condition (see also Lemma 3.1). Lemma 3.6. There exists a constant C > 0 such that for t ≥ 0, RSL (t)H 3 + ∂t RSL (t)H 1 ≤ Cη

(log(2 + t))2 , (1 + t)2

L L RE (t)H 2 + ∂t RE (t)H 1 ≤ Cη(1 + t)−5/2 , L L (t)H 2 + ∂t RM (t)H 1 ≤ Cη RM

log(2 + t) , (1 + t)5/2

where η > 0 is deﬁned in (1.10). We assume that η > 0 deﬁned by (1.10) is suﬃciently small. In view of L L Lemmas 3.1, 3.2, 3.4, 3.5 and 3.6, we see that our asymptotic proﬁle (uL a , φa , Aa ) satisﬁes the assumptions of Proposition 3.1 for η = Cη, where C > 0 is a constant independent of η and t. Theorem 1.2 follows immediately from Proposition 3.1, Remark 3.1 and Lemma 3.5. The proof of Theorem 1.2 is complete.

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Acknowledgments. The author would like to express his deep gratitude to Professor Jean Ginibre for his valuable remarks [1] which pointed out the estimates of x · AC 0 in Lemma 2.2 so that the second correction term for the Schr¨ odinger component is not neended in the Coulomb gauge case. Furthermore in the Lorentz gauge case, he also simpliﬁed the explanation about the choice of the second correction term for the Schr¨ odinger component in the preliminary version of the paper by pointing out the equation (3.5). The author would also like to thank Professor Yoshio Tsutsumi for his helpful comments and constant encouragement. Finally he is grateful to Professors Shu Nakamura and Kenji Yajima for their constant encouragement.

References [1] J. Ginibre, Unpublished note. [2] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys. 151, 619–645 (1993). [3] J. Ginibre and G. Velo, Long range scattering and modiﬁed wave operators for the Wave-Schr¨ odinger system, Ann. Henri Poincar´e 3, 537–612 (2002). [4] J. Ginibre and G. Velo, Long range scattering and modiﬁed wave operators for the Maxwell-Schr¨ odinger system I. The case of vanishing asymptotic magnetic ﬁeld, Preprint. [5] J. Ginibre and G. Velo, Long range scattering and modiﬁed wave operators for the Wave-Schr¨ odinger system II, Preprint. [6] Y. Guo, K. Nakamitsu and W. A. Strauss, Global ﬁnite energy solutions of the Maxwell-Schr¨ odinger system, Comm. Math. Phys. 170, 181–196 (1995). [7] N. Hayashi and T. Ozawa, Modiﬁed wave operators for the derivative nonlinear Schr¨ odinger equations, Math. Ann. 298, 557–576 (1994). [8] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33, 43–101 (1980). [9] K. Nakamitsu and M. Tsutsumi, The Cauchy problems for the coupled Maxwell-Schr¨ odinger equations, J. Math. Phys. 27, 211–216 (1986). [10] K. Nakamitsu and M. Tsutsumi, Global existence of solutions to the Cauchy problems for the coupled Maxwell-Schr¨ odinger equations in two space dimensions, in “Physical Mathematics and Nonlinear Partial Diﬀerential Equations”, J.H. Light-bourne and S.M. Rankin (eds.), New York: Marcel Dekker 1988.

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[11] T. Ozawa, Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Comm. Math. Phys. 139, 479–493 (1991). [12] T. Ozawa and Y. Tsutsumi, Asymptotic behavior of solutions for the coupled Klein-Gordon-Schr¨ odinger equations, preprint, RIMS-775 (1991); Adv. Stud. Pure Math. 23, 295–305 (1994). [13] A. Shimomura, Modiﬁed wave operators for the coupled Wave-Schr¨ odinger equations in three space dimensions, Discrete Contin. Dyn. Syst., to appear. [14] A. Shimomura, Wave operators for the coupled Klein-Gordon-Schr¨ odinger equations in two space dimensions, Funkcial Ekvac., to appear. [15] A. Shimomura, Scattering theory for Zakharov equations in three space dimensions with large data, Preprint, UTMS 2003-1. [16] Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Maxwell-Schr¨ odinger equations in three space dimensions, Comm. Math. Phys. 151, 543–576 (1993). [17] K. Yajima, Existence of solutions for Schr¨ odinger evolution equations, Comm. Math. Phys. 110, 415–426 (1987). Akihiro Shimomura Department of Mathematics Gakushuin University 1-5-1 Mejiro, Toshima Tokyo 171-8588 Japan email: [email protected] Communicated by Vincent Rivasseau submitted 14/11/02, accepted 29/04/03

Ann. Henri Poincar´e 4 (2003) 685 – 712 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040685-28 DOI 10.1007/s00023-003-0144-6

Annales Henri Poincar´ e

An Averaging Theorem for Quasilinear Hamiltonian PDEs Dario Bambusi Abstract. We study the dynamics of Hamiltonian quasilinear PDEs close to elliptic equilibria. Under a suitable nonresonance condition we prove an averaging theorem according to which any solution corresponding to smooth initial data with small amplitude remains very close to a torus up to long times. An application to quasilinear wave equations in an n-dimensional paralleliped is given.

1 Introduction In this paper we study the dynamics of Hamiltonian partial diﬀerential equations of quasilinear type (in the sense of Kato [Kat75, Kat85]). In particular we concentrate on small amplitude solutions. Assuming (a) that the frequencies of small oscillation fulﬁll a new nonresonance condition and (b) a natural estimate on the Lyapunov exponents of the system, we will construct inﬁnitely many approximate integrals of motion. We will deduce that solutions corresponding to smooth initial data remain O(M ) close to a ﬁnite-dimensional torus up to times O(−1 ), where is the norm of the initial datum and M is an arbitrary integer. The abstract theorem is then applied to a quasilinear wave equation in an ndimensional parallelepiped (see Equation 3.1 below). In particular we show that the abstract nonresonance condition is fulﬁlled for almost all values of the parameters. In order to verify the assumption on the Lyapunov exponent we use Kato’s theory with the addition of quantitative estimates of the constants. The proof of the abstract theorem is accomplished in four steps. The ﬁrst one consists in making a Galerkin cutoﬀ; that is in approximating the inﬁnitedimensional Hamiltonian system by a ﬁnite-dimensional one. The second consists in applying to the approximating system a classical recursive algorithm [Whi16, Che24a, Che24b, Gio88] which construct some integrals of motion. It is well known that such an algorithm gives rise to divergent series. Truncating the construction at a given order one obtains objects (the “integrals”) having small Poisson Brackets with the (ﬁnite-dimensional) Hamiltonian. Thirdly we show that the Poisson Brackets of the “integrals” with the complete inﬁnite-dimensional Hamiltonian are very small in a small ball of a phase space of smooth functions. This is not enough to ensure that the integrals remain approximatively constant along the solutions; indeed one still has to ensure that the norm of the solution remains small up to the times we are interested in. This will be obtained by adding an assumption on the Lyapunov exponents of the system. Using this assumption we are able to make

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the forth step, i.e., to prove that, up to the times we are interested in, the integrals remain approximately constant. We point out that in semilinear models the assumption on Lyapounov exponents is automatic, while it can be veriﬁed using Kato’s theory in quasilinear models. The construction of the approximate integrals is quite delicate since one has to take into account the dependence on the dimension of the truncated system of all the constants entering in the estimates. This is obtained by using a precise computation of the constants due to Giorgilli [Gio88]. As far as we know, the main result of the present paper constitutes the ﬁrst averaging type theorem applicable to quasilinear equations, namely equations in which the nonlinearity contains as many derivatives as the linear part. We also emphasize that this is important in view of physical applications like magnetohydrodynamics or elastodynamics, where the equations are quasilinear. The main limitation of the result of the present paper rests in its time of validity. Indeed known results for ﬁnite-dimensional systems and for some semilinear partial diﬀerential equations (see [Bou00, Bam03, Bou96, BG03]) describe the dynamics over a time scale much longer than the O(−1 ) that we obtain, precisely over a time scale of order −M with arbitrary M . On the other hand these results with longer time scales require a nonresonance condition that is stronger than ours. Such a stronger condition is only exceptionally satisﬁed in more than one space dimension. A further result applicable to quite general semilinear models in arbitrary space dimension was obtained by Kuksin[Kuk89], but the time scale covered by that paper is also of order O(−1 ). Moreover Kuksin’s result allows to describe only the dynamics corresponding to initial data in which the energy is initially concentrated on a ﬁnite (and ﬁxed) number of linear modes. Finally we recall the papers [SV87, Pal96, Kro89, MS02] where Galerkin truncations have been used together with averaging techniques in order to study the dynamics of nonlinear PDEs. In particular the work [MS02] had a great inﬂuence on the present paper. Related results can be found in [Sha85, Cra96, Bam99a, BN98]. Plan of the paper. In Section 2 we state the main results of the paper. In Section 3 we give the application to the nonlinear wave equation. In Section 4 we recall the algorithm of construction of the integrals of motion in ﬁnite-dimensional Hamiltonian systems. In Section 5 we prove the main result of the paper. In Section 6 we show that the diophantine condition we use is fulﬁlled with probability one in the wave equation. In Section 7 we use Kato’s theory to verify the assumptions on Lyapunov exponents in the case of nonlinear wave equation. Finally in the appendix we prove a result on the geometry of level surfaces of the approximate integrals. Acknowledgment. I thank Antonio Giorgilli for some interesting discussions and for explaining me the key ideas leading to the proof of Proposition 5.4. I also thank Alberto Arosio for some discussions on Kato’s theory. This work was developed under the partial support of Gruppo Nazionale di Fisica Matematica (INdAM).

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2 Abstract results 2.1

The approximate integrals

Consider a Hamiltonian system of the form H(p, q) =

ωj

j≥1

p2j + qj2 + P (p, q) 2

(2.1)

where {ωj } is a sequence of strictly positive quantities and (pj , qj ) are conjugated variables. We will denote by z ≡ {(pj , qj )}j≥1 a phase point and by XH (z) ≡ ∂H ∂H (z), ∂p (z) the (formal) Hamiltonian vector ﬁeld of a function H. − ∂q j j

To deﬁne precisely the phase space consider the Hilbert space 2s of the sequences {xj }j≥1 such that 2

xs :=

j 2s x2j < ∞ ,

j≥1

and denote Ps := 2s × 2s . If z ≡ (p, q) ∈ Ps is a phase point, we will denote by 2 2 zs := ps + qs , its norm and by Bs (R) the open ball of radius R in Ps . For any positive (large) N denote by ω (N ) := (ω1 , . . . , ωN ) the truncation of length N of the frequency vector. Having ﬁxed a positive (large) r∗ , we assume: H1) (Nonresonance) There exist α and γ > 0 such that for any N one has |ω (N ) · k| ≥

γ , ∀k ∈ ZN , 0 < |k| ≤ r∗ + 2 Nα

where |k| := |k1 | + · · · + |kN |.

Remark 2.1. Consider the case where ωj = j d1 +

σj , j d2

σj ∈ [0, 1] ,

d1 , d2 ≥ 0

one can show that H1 is fulﬁlled if the parameters σj are chosen in a set of probability one with respect to the product measure. To ensure that H0 (p, q) :=

j≥1

ωj

p2j + qj2 2

describes the linearization of the system at the origin we assume

(2.2)

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H2) There exist s∗ and a neighbourhood of the origin U ⊂ Ps∗ , a positive C and ν ≥ 1 such that ν+2 |P (z)| ≤ C zs∗ , ∀z ∈ U. Then we need some smoothness of the Hamiltonian and of its vector ﬁeld, so we assume that H3) One has H ∈ C r∗ +3 (U).

H4) There exists a real d with the following property: for any positive s ≥ s∗ there exists a neighbourhood of the origin Us+d ⊂ Ps+d such that XH ∈ C r∗ +2 (Us+d , Ps ). Remark 2.2. Actually it is enough that the above assumptions are fulﬁlled for an inﬁnite unbounded set of values of s. Remark 2.3. From H2–H4 it follows that, in a ball of radius in Ps+d , one has XP s |P | ν . |H0 | XH0 s In order to ensure the applicability of the ﬁnite-dimensional algorithm we assume H5) The nonlinearity is reversible, namely one has P (−p, q) = P (p, q).

Denote the linear actions by Ij := (p2j + qj2 )/2. In the following we will always denote by C a positive (usually large) constant whose value can change from line to line. Moreover we deﬁne the constants β := s∗ + 2 + α ,

s := 3r∗ β + 2β + α + d .

and, for any small , we will consider the integer number 1

N = N () := [− 4β ] , Theorem 2.4. Fix M ≥ 4, and assume that H1–H5 hold with some r∗ > 43 M − 3 then there exists a positive ∗ , such that, for any 0 < < ∗ 1) there exist N approximate integrals of motion {Φ(j) }N j=1 , which are analytic in Bs∗ (∗ ), and fulﬁll sup H; Φ(j) (z) ≤ C z∈Bs∗ +s ()

sup z∈Bs∗ ()

(j) Φ (z) − Ij (z) ≤ C2 1/4 .

(2.3)

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2) For any s ≥ s∗ there exist s and Cs such that one has sup H; I (s,N ) (z) ≤ Cs M , ∀ < s z∈Bs+s ()

where I (s,N ) :=

j>N

j 2s Ij .

If we assume also local well-posedness of the dynamics then we can add some dynamical properties of the solutions. So, consider the Hamilton equations of motions (2.4) z˙ = XH (z) , and assume: H6) (Local well-posedness) ∀s ≥ s∗ there exists s > 0 with the following property: ∀z0 ∈ Bs+d (s+d ) there exist T > 0 and a unique solution z ∈ C 0 ([0, T ]; Ps+d) ∩ C 1 ([0, T ], Ps )

of 2.4 with z0 = z(0). In the following, given z0 ∈ Bs∗ (∗ ), we will denote by Tz0 := w ∈ Ps : Φ(j) (w) = Φ(j) (z0 ) , j = 1, . . . , N , Ij (w) = 0 ∀j > N

the level surface of the Φ(j) ’s through z0 . In the appendix we will prove that Tz0 is an N -dimensional smooth torus. Moreover, given an initial datum z0 ∈ Bs (), we deﬁne the corresponding escape time τ,s (z0 ) from Bs () as the inﬁmum of the times for which the solution is outside the ball Bs (). In the theorem we are going to state there are some free parameters, namely M1 , M2 , M3 . They are only subjected to the constraints M1 + M3 = M , 2M2 + M3 + 2 s+1 β ≤ M. Theorem 2.5. If in addition to the assumptions of Theorem 2.4, one assumes also that H6 holds, then, ∀s ≥ s∗ there exists ∗s such that the following holds true. Consider an initial datum z0 ∈ Bs+s () with < ∗s , then one has |Φ(j) (t) − Φ(j) (0)| ≤ (s,N ) (t) − I (s,N ) (0) ≤ I ds (z(t), Tz0 ) ≤ for all the times t such that |t| ≤ min

CM1 , j = 1, . . . , N CM1 CM2

1 , τ (z ) 2,s+s 0 CM3

where ds (·; ·) denotes the distance in Ps .

(2.5)

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Remark 2.6. From trivial considerations on the unperturbed equation one would obtain the same conclusion with M1 + M3 ≤ ν + 1 and 2M2 + M3 ≤ ν + 1. In the C ∞ nonresonant case such quantities can be made arbitrarily large. So Theorem 2.5 is an averaging theorem up to any ﬁnite order.

2.2

Quasilinear equations

We are now going to bound from below the escape time in the case of quasilinear systems. We begin with some notations and deﬁnitions. Let X and Y be Hilbert spaces, with Y densely embedded in X. The space of bounded linear operators from Y to X will be denoted by B(Y, X). We assume the Hamilton equations of 2.1 to be quasilinear, i.e., that K1) (Quasilinearity) There exists s∗ , and, for any s ≥ s∗ a positive Rs+d , a map Bs+d (Rs+d ) z → A(z) ∈ B(Ps+d , Ps ) and a map g : Bs+d (Rs+d ) → Ps+d such that the Hamilton equations of H take the form (2.6) z˙ = A(z)z + g(z) ≡ XH (z) . The key assumption concerns the properties of the linearized ﬂow. For any function ζ ∈ C 0 ([0, T ], Ps+d ) ∩ C 1 ([0, T ], Ps ) we will consider the family of linear operators Aζ (t) := A(ζ(t)) and the corresponding linear time-dependent diﬀerential equation z˙ = Aζ (t)z .

(2.7)

We recall that, under suitable assumptions there exists a corresponding evolution operator U (t, s), strongly continuous as a map in B(X, X) and also as a map in B(Y, Y ). The operator U (t, s) is deﬁned by the property that the solution of 2.7 fulﬁlling z(s) = z0 is given by z(t) = U (t, s)z0 . We assume: K2) (Linear estimate) There exists ν ≥ 1 such that, for any s ≥ s∗ , any > 0, any T > 0 and any function ζ ∈ C 0 ([0, T ], Ps+d ) ∩ C 1 ([0, T ], Ps ) fulﬁlling ˙ (2.8) sup ζ(t)s+d + sup ζ(t) ≤ t∈[0,T ]

t∈[0,T ]

s

the evolution operator U (t, s) associated to Equation 2.7 exists and fulﬁlls the estimate ν sup U (t, τ )2 →2 ≤ M eβ T , (2.9) 0≤t≤τ ≤T

s+d

s+d

with some constants M, β independent of ζ, T, .

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K3) g is of class C 1 (Us+d , Ps+d ) and has a zero of order ν +1 at the origin, namely ν+1

g(z)s ≤ Cs zs

.

Remark 2.7. In the semilinear case where the linear operator A is independent of z, one has A = XH0 . It follows that assumption K2 is automatically true. Remark 2.8. In the true quasilinear case one can use Kato’s theory [Kat75, Kat85] to verify assumptions K2. This is what we will do in the application to the nonlinear wave equation. Theorem 2.9. Under the same assumption of Theorem 2.4, assume also K1–K3, then there exists s and, for any s ≥ s∗ there exists a constant s such that the following holds true. For any initial datum z0 fulﬁlling := z0 s+s ≤ s ,

(2.10)

along the corresponding solution one has |Φ(j) (t) − Φ(j) (0)| ≤ (s,N ) (t) − I (s,N ) (0) ≤ I ds (z(t), Tz0 ) ≤

¯

CM1 , j = 1, . . . , N C

¯1 M ¯

CM2

(2.11) (2.12) (2.13)

for all times

¯2 = ¯ 1 = M − ν, M where M

M−ν 2

1 |t| ≤ Cν

. − s+1 β

(2.14)

Remark 2.10. The strongest limitation of the present result rests in its time of validity. Indeed, in the ﬁnite-dimensional case and also in the case of semilinear equations in one space dimension [Bam03], a similar description of the dynamics has been proved to hold up to times of order −M for any M . Due to results of the kind of [KM80] we expect the time 2.14 to be optimal for quasilinear models. However the nonresonance condition H1 is not satisﬁed by the model considered in [KM80], so it is not a counterexample to the result of the present paper. We think it would be very interesting to ﬁnd an example satisfying the assumptions of Theorem 2.9 but developing singularities over a the time scale −ν . Remark 2.11. The theory of [Bam03] can be extended to some semilinear equations in higher space dimension (see [BG03]) like the nonlinear Schr¨ odinger. The main property needed to obtain such an extension is a nonresonance condition stronger than H1. However such a condition is not fulﬁlled in general high-dimensional examples like the nonlinear wave equation.

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3 Application to quasilinear wave equations Having ﬁxed a diophantine1 vector a = (a1 , . . . , an ) ∈ Rn with ai > 0, consider √ the n-dimensional parallelepiped R with sides of length π/ ai , namely

π n R := x ≡ (x1 , . . . , xn ) ∈ R : 0 < xi < √ ai and the nonlinear wave equation utt − ∆u + mu − bij (u, ∇u)∂i ∂j u + g(u, ∇u) = 0 , u∂R = 0 ,

x∈R

(3.1)

where we used the summation convention for the indices i, j = 1, . . . , n, we denoted by ∇u ≡ (∂1 u, . . . , ∂n u) the derivatives of u with respect to the space variables, and bij , g are functions of class C ∞ . Moreover we assume that the system is Hamiltonian, i.e., that there exists a C ∞ function W = W (u, s1 , . . . , sn ) such that bij (u, s1 , . . . , sn ) =

∂ 2W (u, s1 , . . . , sn ) , ∂si ∂sj

g=

∂2W ∂W − ∂k u , ∂u ∂u∂sk

so that the Hamiltonian function of the system is given by 2 2 ∇u mu2 v H(u, v) = + + + W (u, ∇u) dn x , 2 2 2 R

(3.2)

where v = u˙ is the momentum conjugated to u, and we denoted by . the Euclidean norm of a vector of Rn . Assume that W has a zero of order ν + 2 at the origin, namely that ν+2

|W (u, s)| ≤ C (|u| + s)

ν≥1.

,

(3.3)

Expand u in Fourier series in the space variable, namely write u(x, t) =

j≡(j1 ,...,jn )

uj (t)

n

1/4

ai

√ sin( ai ji xi ) ,

(3.4)

i=1

with ¯ n := k ≡ (k1 , . . . , kn ) ∈ Z ¯ n : ki ≥ 1 , i = 1, . . . , n , j ≡ (j1 , . . . , jn ) ∈ N 1 We

recall that it means that there exist a positive γ ¯ and a real τ¯ such that |a1 k1 + · · · + an kn | ≡ |a · k| ≥

γ1 , |k|τ¯

∀k ∈ Zn \ {0}

and that diophantine vectors form a set of full measure in Rn .

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so that 3.1 is converted into the inﬁnite system ¯n u ¨j + ωj2 uj = fj (u) , j ∈ N where f is of order at least ν + 1 in u, and ωj := µj + m , µj := a1 j12 + a2 j22 + · · · + an jn2 .

(3.5)

We come to the nonlinear problem. To ﬁt the abstract scheme we enumerate the eigenvalues µj of the Laplacian by integers indices j ∈ N in such a way that the µj ’s form an increasing sequence. This establishes a 1 to 1 correspondence ¯ n and N. Deﬁne pj := √ωj u˙ j , qj := uj /√ωj . Then it is easy to see that between N the space Ps is isomorphic to the space Fs of the functions (u, v) ∈ H s˜+1 ⊕ H s˜ with s˜ = ns/2, fulﬁlling the compatibility conditions s˜ (−∆)j u∂Ω = 0 , 0 ≤ j ≤ , (3.6) 2 s˜ + 1 (−∆)j v ∂Ω = 0 , 0 ≤ j ≤ −1 . (3.7) 2 We will consider only values of s such that s˜ is integer; in view of Remark 2.2 this is possible. Concerning the validity of the diophantine type condition H1 we have the following Theorem 3.1. Fix b > 0, then there exists a subset J ⊂ [0, b] of measure b such that, if m ∈ J , then the frequencies ωj fulﬁll the assumption H1. This theorem will be proved in Section 6. Assumptions H2 and H3 are a trivial consequence of Sobolev embedding theorems, and also H6 is automatically true. To fulﬁll the smoothness assumptions of the vector ﬁeld we assume that the potential W is even in each of the arguments, namely W ((−)c0 u,(−)c1 s1 ,...,(−)cn sn ) = W (u,s1 ,...,sn ) ,

∀(c0 ,c1 ,...,cn ) ∈ (Z2 )n+1 , (3.8)

from which in particular it follows ν ≥ 2. To verify assumptions H4, H6 and K1–K3 it is convenient to proceed as follows: First rescale the domain in order to transform it into a standard cube with sides of length π; then identify the space Fs with the space A of the functions of class H s˜+1 (Tn ) × H s˜(Tn )) (where T = R/2πZ) which are skewsymmetric in each variable, namely that fulﬁll u((−)c1 x1 , . . . , (−)cn xn ) = (−)c1 +···+cn u(x1 , . . . , xn ) . Clearly one has Fs A ∩ (H s˜+1 (Tn ) × H s˜(Tn )). Due to the symmetry properties that 3.8 induces on the nonlinearity, the subspace A is invariant under the

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dynamics of 3.1 in Tn . So the validity of H4 is a simple consequence of Sobolev inequalities. The same is true for K1 and K3. Local existence and uniqueness (i.e., H6) was proven by Kato (see [Kat85] Theorem 14.3). Theorem 3.2. Assume W ∈ C ∞ and 3.8, then property K2 holds. For the proof see Section 7. So the general theory applies and in particular one has Theorem 3.3. For any positive large M there exists a set J ⊂ [0, b] of measure b such that, for any m ∈ J there exists a positive constant s , and for any s ≥ s∗ a constant s , such that the following holds true. Assume that the initial datum z0 ≡ (u0 , v0 ) fulﬁlls := z0 s+s ≤ s (3.9) then along the corresponding solution of 3.1 one has ds (z(t), Tz0 ) ≤ CM

for

|t| ≤

1 Cν

(3.10)

where Tz0 is a ﬁnite-dimensional smooth torus (recall that ν is the constant appearing in 3.3). We conclude by some remarks on the possibility of extending the result to more general domains; to this and there are two kinds of diﬃculties. The ﬁrst one is related to the veriﬁcation of the nonresonance condition. Following the proof of Theorem 3.1 one can show that if the Dirichlet eigenvalues µj of the Laplacian in the considered domain fulﬁll the gap estimate µj+1 − µj > C/j δ

(3.11)

with a given δ, then the nonresonance condition H1 is fulﬁlled for most values of the parameter m. However we do not know whether there exist domains diﬀerent form the parallelepiped in which the gap condition is fulﬁlled. Moreover in the case of space dimension 2 Schnirelman proved that 3.11 is never true for smooth convex domains (see [Kuk94, Laz93]). The second point concerns smoothness of the solutions. Indeed it is well known that there are some compatibility conditions which are necessary and suﬃcient for the smoothness of solutions of the wave equation (which in turn is essential for our approach). We have no clear ideas on how to deal with such conditions in the case of general domains.

4 The classical algorithm Here we recall the classical algorithm of construction of the approximate integrals of motion and a result by Giorgilli [Gio88] that we need.

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Consider a Hamiltonian system with N degrees of freedom, of the form Hk (4.1) H= k≥0

where Hk is a homogeneous polynomial of degree k + 2 in the phase variables and H0 is of the form 2.2. Let Ij = (p2j + qj2 )/2 be an action of the linear system, we aim to construct an integral of motion of the form Φ(j) := Ij + l≥1 Φl with Φl a homogeneous polynomial of degree l + 2. Clearly Φ(j) has to fulﬁll the equation H; Φ(j) = 0. Inserting the Taylor expansions of H and of Φ and equating terms of the same degree we get the recursive system LH0 Φl = Ψl , l ≥ 1 ,

LH0 := {H0 ; . }

(4.2)

where Ψl is determined by Ψ1

=

{I; H1 } (4.3)

Ψl

=

l−1

{Φj ; Hl−j } + {I; Hl } .

j=1

So, provided one is able to solve the homological Equation 4.2, the formal integral of motion can be computed. In order to solve the homological equation we remark that at the l-th step 4.2 appears as a linear equation in the ﬁnite-dimensional space of the polynomials of degree l+2. So, if one is able to diagonalize the linear operator LH0 , then Equation 4.2 is easily solved. To diagonalize LH0 introduce the variables 1 ξj := √ (qj − ipj ) , 2

1 ηj := √ (qj + ipj ) , (4.4) 2 ωl ξl ηl (and the symwhich give the unperturbed Hamiltonian the form H0 = plectic form becomes idξl ∧ dηl ), so that one has LH0 ξ j η k = [i(k − j) · ω] ξ j η k ,

jN k1 kN ξ j η k ≡ ξ1j1 . . . ξN η1 . . . ηN ,

i.e., the monomial ξ j η k form the basis of the space of polynomials on which LH0 is diagonal. So, 4.2 is easily solvable provided Ψl has no component on the kernel of LH0 . In turn this can be easily veriﬁed recursively2 in the case where Hk (p, q) is even in p as we assumed. As is well known the series deﬁning the integral of motion are in general divergent, so, following [Gio88], one has to stop the construction at a given order and 2 Indeed, a component on the kernel of L H0 is always even in the momenta p, while from the construction of Hk such a function always turns out to be either independent of p or skewsymmetric in it (see [Gio88]).

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to estimate the Poisson bracket of the so-obtained function with the Hamiltonian. To this end remark that, deﬁning Φ

(j,r∗ )

:= Ij +

r∗

Φl ,

l=1

one has

˙ (j,r∗ ) = H; Φ(j,r∗ ) = O(|(p, q)|r∗ +3 ) Φ

(4.5)

where we denoted for simplicity Φ0 := Ij . Finally, in the ﬁnite-dimensional case, one can obtain a long time estimate of the solution as follows. First repeat the construction for all j’s, then using the approximate constant of motion as Lyapunov functions in order to control that the domain of construction of such functions is not left by the solution up to the times one is interested in. For the application to inﬁnite-dimensional systems we have to keep into account the dependence of all the constants on the number N of degrees of freedom. To this end we recall a quantitative result by Giorgilli [Gio88]. We ﬁrst have to introduce some notations. Fix j Rl ≡ (R1 , . . . , RN ) with Rj > 0 and consider a polynomial f (p, q) = j,l fjl p q , then the size of f is measured by the norm |fjl |Rj+l . f R := i,l

To control the small denominators assume that there exists a non-increasing se∗ quence {αs }rs=1 such that |ω · k| ≥ αs for k ∈ ZN , 0 < |k| ≤ s + 2 .

(4.6)

Finally consider the complex neighbourhood of the origin ∆ρR := (p, q) ∈ C2N : (|pl |2 + |ql |2 )1/2 ≤ ρRl and denote

Λ := (min Rl )−1 . l

Theorem 4.1 (Giorgilli). Consider a Hamiltonian system of the form 4.1, even in the momenta p, and assume that, for a given R ∈ RN + there exist constants h > 0 and E > 0 such that Hk R ≤ hk−1 E for k ≥ 1; assume that the linear frequencies fulﬁll 4.6. Then, for any integer 0 < r ≤ r∗ there exist N truncated (l) integrals Φ(l,r) = Il + rs=1 Φs such that H; Φ(l,r) is a power series starting with terms of degree at least r + 3. Moreover, for z ∈ ∆ρR and ρ < 1/h, one has the bounds 24E 3 (l,r) − Il (z) < ρ [1 − (σr ρ)r ](1 − σr ρ)−1 (4.7) Φ α1 (4.8) H; Φ(l,r) (z) < Cr ρr+3 (1 − hρ)−2 ,

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where σ1

=

σr

=

Cr

=

1 1 (r + 1)! r−1 8 r 12Λ2 E + hα1 , r>1, 9 l=2 αl r (r + 2)! 8 r 8E 12Λ2 E + hα1 , r≥1. 9 l=1 αl

(4.9)

5 Proof of the abstract results Expand the perturbation P in Taylor series up to order r∗ + 2, P =

r∗

Pl + R∗

l=1

where Pl are homogeneous polynomial of degree l + 2 and R∗ is the remainder. We will denote r∗ Pl . H∗ := H0 + l=1

Remark 5.1. The polynomials Pl are analytic functions, and therefore, introducing the complexiﬁcation PsC∗ of Ps∗ , one has that there exist constants R∗ , Cl such that l+2 zs∗ ≤ R∗ . |Pl (z)| ≤ Cl zs∗ , ∀z ∈ PsC∗ To make the Galerkin cutoﬀ ﬁx a large N that will eventually be related to , introduce the projector ΠN deﬁned by ΠN (p, q) = (p1 , . . . , pN , q1 , . . . , qN ) , (N )

and put H (N ) := H0

+

r∗ l=1

(N )

H0 One has

(5.1)

Hl , with

:= H0 ◦ ΠN ,

Hl := Pl ◦ ΠN .

H = H (N ) + (H∗ − H (N ) ) + R∗ .

Lemma 5.2. For any s ≥ s∗ there exist a constant C and a domain Us such that, for any r ≥ 0, and any positive integer N , one has r +2

∗ XR∗ (z)s ≤ C zs+d ,

∀z ∈ Us+d (5.2)

XH −H (N ) (z) ≤ C z s+r+d , ∗ s Nr

∀z ∈ Us+d+r .

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Ann. Henri Poincar´e

Proof. First remark that 1l − ΠN s+r→s =

1 . Nr

(5.3)

and that XH∗ −H (N ) (z) = [XH∗ (z) − XH∗ (ΠN z)] + ( 1l − ΠN )XH∗ (ΠN z) , By H4, using the norm of the diﬀerential to estimate the Lipschitz constant of XH∗ , the ﬁrst square bracket is estimated by sup dXH∗ (z)s+d→s 1l − ΠN s+d+r→s+d zs+d+r ≤

z∈Us+d

C zs+r+d . Nr

The second term is trivially estimated using 5.3. The estimate of XR∗ is obtained by applying the Lagrange estimate of the remainder of the Taylor expansion. We apply now Theorem 4.1 to H (N ) . Lemma 5.3. There exist N truncated analytic integrals Φ(l,r∗ ) = Il + the Hamiltonian system H (N ) . Moreover for any z ∈ Ps∗ such that z∞ :=

sup { p2k + qk2 } ≤

k=1,...,N

the following estimates hold (l,r∗ ) − Il (z) Φ H (N ) ; Φ(l,r∗ ) (z)

r∗

(l)

s=1

Φs for

1 CN β+α

(5.4)

< C(z∞ N a )3 N α a 3

< C (z∞ N )

z∞ N

β r∗

(5.5) .

where a = s∗ + 2, β := a + α. Proof. To compute Giorgilli’s norm of a function f remark that, if f (z) = (k being a multiindex) one has 1 ∂ |k| f fk = , k! ∂z k z=0

k! := k1 !k2 ! . . . kN !

and that, deﬁning Rj∗ := R/j s∗ +1 , with a positive R, one has ∆R∗ ⊂ Bs∗ (2R) , where we considered complex balls.

k

fk z k

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An Averaging Theorem for Quasilinear Hamiltonian PDEs

Applying the Cauchy inequality to the polynomial Hl (z) =

699

k

Hl,k z k one

gets |Hl,k | ≤

1 1 sup |Hl | ≤ (R∗ )k ∆R∗ (R∗ )k ≤

sup |Hl |

Bs∗ (2R)

1 CR|k| ≤ C(N s∗ +1 )|k| . (R∗ )k

We apply Giorgilli’s theorem in the domain ∆R with Rj := R for all j’s. Summing over the indices and taking into account that there are at most (2N )l distinct polynomials of degree l in 2N variables one gets Hl R ≤ C(RN s∗ +2 )3 (RN s∗ +2 )l−1 which allows to choose E = C(RN s∗ +2 )3 and h = RN s∗ +2 . Compute now the various constants of Giorgilli’s theorem; provided α ≥ 2a, one has σr∗ = CRN 2α+a and Cr∗ (RN a+α )r∗ (RN a )3 . Finally take ρ := z∞ /R. This gives the result. From now on we will always denote simply by Φ(l) the approximate integrals (l,r∗ ) Φ . Proof of Theorem 2.4. Compute the Poisson bracket of Φ(l) with the Hamiltonian: it is given by H (N ) ; Φ(l) + H∗ − H (N ) ; Φ(l) + R∗ ; Φ(l) = H (N ) ; Φ(l) + dΦ(l) XH∗ −H (N ) + dΦ(l) XR∗ . By 5.5, using the Cauchy inequality one has, with a suitable deﬁnition of C, (l) sup dΦ (z) ≤ CN β+α z∞ , z ∞ ≤C/N β+α

from which, using Lemma 5.2 and Equation 5.2 one has r∗ 3 (z∞ N a ) H; Φ(l) (z) ≤ C z∞ N β +

C r∗ +2 zs+r+d N β+α zs∗ + Cs zs+d N β zs∗ Nr −1/4

choose now N such that N β = zs∗

(remark that the s∗ norm controls the ∞

norm), then the argument of the ﬁrst bracket is smaller than z3/4 s∗ , and the last term turns out to be much smaller than the ﬁrst one. We choose now r in such a way that the ﬁrst two terms are of the same order of magnitude. This leads to the choice r := 3r∗ β + 2β + α , s := r + d

and to the thesis.

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To prove that the level surfaces of the approximate integrals are tori and that the distance from such tori is controlled by the diﬀerence between the values of the integrals one has to pay attention to the fact that the action Il is singular on the hypersurface pl = ql = 0. We have the following Proposition 5.4. There exists η∗ > 0 such that the relations z∞ (j) z ) Φ (z) − Φ(j) (¯

< 2 η ,

with < imply

¯ z ∞ <

< ,

∀j = 1, . . . , N ,

1 CN 3s∗ +6+α

and

ΠN (z) = ΠN (¯ z) = 0

η < η∗

d∞ (¯ z , Tz ) < Cη 1/2 .

For the proof see the appendix. Proof of Theorem 2.5. The only thing to be proved is the last estimate 2.5. To obtain the estimate in the s norm remark that one has ΠN zs ≤ N s+1 ΠN z∞ . We come now to the proof of Theorem 2.9. To this end the key lemma is the following one Lemma 5.5. Assume K1–K3, then there exist C, ∗ such that for any initial datum fulﬁlling (5.6) z0 s+d ≤ /C , < ∗ the existence time of the corresponding solution of 2.6 is larger than 1/(Cν ) and moreover one has the estimate sup 0≤t≤1/Cν

z(t)s+d ≤ .

(5.7)

Proof. Let z be the solution of the Cauchy problem for 2.6, then by the standard continuation argument it can be continued at least until z(t)s+d <

(5.8) holds. Let T¯ be the ﬁrst time at which 5.8 is violated, then one has z(T¯ )s+d = . Denote g z (t) := g(z(t)). So, z(t) fulﬁlls the “linear” equation z˙ = Az (t)z + g z (t),

0 ≤ t ≤ T¯

(5.9)

where the estimate 2.9 holds until time T¯ . By Theorem 2 of [Kat75] (which follows from the formula of variation of constants) the solution of 5.9 satisﬁes the estimate ν ¯ = z(T¯)s+d ≤ M eβ T z0 s+d + Cν+1 T¯ which, provided z0 s+d / is small enough, implies T¯ > 1/Cν with a suitable C.

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Proof of Theorem 2.9. Deﬁne ˜ := C zs+s with the constant of 5.6, then by Lemma 5.5 one has z(t)s+s ≤ ˜. Thus, from Theorem 2.4 (item 3) it follows (j) M1 , Φ (t) − Φ(j) (0) ≤ C˜

|t| ≤

1 . ˜ν

deﬁning := zs+s and redeﬁning the constants one has the thesis.

6 Proof of Theorem 3.1 For k ∈ ZN \ {0} with 0 < |k| ≤ r∗ + 2 deﬁne γ Rk (γ, α) := m ∈ I : ω (N ) · k < α . N We will estimate the measure of the union of such sets. First one has Lemma 6.1. The eigenvalues µj fulﬁll the gap condition µj+1 − µj ≥

C , jδ

∀j ≥ 1

(6.1)

with δ = 4¯ τ /n, τ¯ being the exponent in the diophantine condition for a. Proof. Since a is a diophantine vector one has γ1 2 2 |µk − µj | = al (kl − jl ) ≥ 2 , ( l |kl − jl2 |)τ¯ l but

|kl2 − jl2 | < C k2 j2 ≤ C [inf (ai )]−2 µj µk ≤ Ck 2/n j 2/n

l

where k and j are the integer indices corresponding to k and j respectively, and we used the well-known estimate C1 j 2/n ≤ µj ≤ C2 j 2/n .

(6.2)

Since the minimum of the diﬀerence is realized for k = j + 1, one has |µk − µj | ≥

C j 4¯τ /n

All what we will prove from now on will be based only on the gap estimate 6.1. Moreover we will label the frequencies only by the scalar indices.

702

D. Bambusi

Lemma 6.2. For any K ≤ N , equal than N ; then one has ωj (1) ωj (2) . dω (1) dωj(2) j . dm dm . . . . . . dK−1 ω (1) dK−1 ω (2) j j . dmK−1

dmK−1

Ann. Henri Poincar´e

let j (1) , . . . , j (K) be K diﬀerent indices smaller or dωj(K) dm C . ≥ K 2 (δ+2/n)−2K/n . N . dK−1 ωj(K)

. .

ωj (K)

. . . . . . . .

(6.3)

dmK−1

Proof. First remark that, by explicit computation one has dj ωi (−)j+1 (2j − 3)! = . dmj 2j−2 (j − 2)!2j (µi + m)j− 12

(6.4)

Substituting 6.4 in the left-hand side of 6.3 we get the determinant to be estimated. To obtain the estimate factorize from the l-th column the term (µj (l) + m)1/2 , and (2j−3)! from the j-th row the term 2j−2 (j−2)!2j . Forgetting the inessential powers of −1, we obtain that the determinant to be estimated is given by 1 1 1 . . . 1 xj (1) xj (2) xj (3) . . . xj (K)   K K−1 x2j (1) x2j (2) x2j (3) . . . x2j (K) (2j − 3)!  . . . . . . . ωj (l)  j−2 j 2 (j − 2)!2 . . . . . . . j=1 l=1 . . . . . . . K−1 K−1 K−1 x (1) xK−1 x . . . x (2) (3) (K) j

j

j

j

where we denoted by xj := (µj + m)−1 ≡ ωj−2 . The last determinant is a Vandermond determinant whose value is given by (xj (l) − xj (k) ) . (6.5) l
Now one has 1 1 |µi − µj | C C − ≥ > δ+2/n δ+2/n |xi − xj | = = µi + m µj + m ωi2 ωj2 (min {i, j})δ ωi2 ωj2 i j (6.6) Denoting yi := i−(δ+2/n) , 6.5 is estimated by ! K l−1 l−1 K K K l−1 (l−1) Cyi(l) yi(N ) = C l=2 yi(N ) = C yiK(l) , (6.7) yi(l) l=2 N =1

l=2

from which the thesis immediately follows.

N =1

l=1

The rest of the proof follows very closely the proof of Theorem 6.16 of [Bam03]. We repeat the main steps for completeness.

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Lemma 6.3 (Proposition of Appendix B in [BGG85]). Let u(1) , . . . , u(K) be K (i) independent vectors with u ≤ 1. Let w ∈ RK be an arbitrary vector, then 1 there exist i ∈ [1, . . . , K], such that |u(i) · w| ≥

w1 det(u(i) ) , K 3/2

where det(u(i) ) is the determinant of the matrix formed by the components of the vectors u(i) . For the proof see [BGG85]. Corollary 6.4. Let w ∈ RN be a vector with only K components diﬀerent from zero, namely those with index i(1) , . . . , i(K) ; assume K ≤ N , K ≤ r∗ + 2. Then, for any m ∈ I there exists, an index i ∈ [0, . . . , K − 1] such that i w · d ω (m) ≥ C w1 (6.8) i dm Nρ where ω is the frequency vector, and ρ = K 2 (δ + n2 ) −

2K n

+ n − 1/n.

Proof. The proof is obtained by considering the vectors  i d ω di ω   d1i ω dm if dm i i > 1 dmi (i) u := di ω  di ω  if dm i ≤ 1 dmi

(6.9)

and applying Lemma 6.3. For more details see [Bam03], proof of Corollary 6.10. Lemma 6.5 (Lemma 8.4 of [Bam99b]). Let g : I → R be N times diﬀerentiable, and assume that (1) ∀m ∈ I there exists i ≤ K − 1 such that

di g dmi (m)

>d

(2) there exists A such that |g (i) (m)| ≤ A ∀m ∈ I, and ∀i with 1 ≤ i ≤ K. Deﬁne Ih := {m ∈ I : |g(m)| ≤ h} , then

A |Ih | ≤ 2(2 + 3 + · · · + r + 2d−1 )h1/K |I| d For the proof see [Bam99b] and [XYQ97]. By combining Lemma 6.5 and Lemma 6.4 we get

Lemma 6.6. For any choice of k one has |Rk (γ, α)| ≤ |I|C with ς = −2ρ + α/K.

γ 1/K Nς

(6.10)

704

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Ann. Henri Poincar´e

Proof of Theorem 3.1. By 6.10 one has & (2N )r∗ +3 |I|Cγ 1/K |Rk (γ, α)| < , Rk (γ, α) ≤ Nς k

k

which, provided α is chosen large enough, is bounded uniformly with respect to N . Taking the union over γ one obtains the thesis.

7 Proof of Theorem 3.2 First of all we introduce scaled variables x ˜i = Equation 3.1 is converted into

√ ai xi , and u˜ by u = ˜ u, so that

utt − aij ∂i ∂j u + mu = ν g (u, ∇u)

(7.1)

where we omitted tildes, and used the notations aij = ai δij + ν bij ; in what follows we will omit the index from g, aij and bij . We consider Equation 7.1 in Tn ; moreover we will assume that the initial datum has zero average and that the vector ﬁeld lives invariant the space of functions with zero average. Since we are interested in the case of Dirichlet boundary condition this is not restrictive. We will prove the theorem by considering directly Equation 7.1 using the spaces Y = H s+1 × H s and X := H s × H s−1 . Thus the unbounded part of 7.1 is ∂t ζ = A(ζ)ζ with ζ ≡ (u, v), and 0 1l A(ζ) := , A(u) := aij (u, ∇u)∂i ∂j − m A(u) 0 Having ﬁxed w ∈ C 0 (I; H s+1 ) ∩ C 1 (I; H s ) with I := [0, T ], we have to show that Aw (t) := A(w(t)) fulﬁlls assumption K2. We begin by estimating the stability constants of the operator family Aw (t) in X. To this end it is useful to make reference to the second order equation u¨ = Aw (t)u

(7.2)

where Aw (t) := aij (w(t), ∇w(t))∂i ∂j −m. We remark that since w(t) has the property 2.8, one has bij ∈ C 0 (I; H s ) ∩ C 1 (I; H s−1 ) with bounded norms. The proof will be based on Proposition 9.3 of [Kat85], so ﬁrst of all we recall its statement. Proposition 7.1 (Proposition 9.3 of [Kat85]). Assume that: (i a) H and H have structures of real Hilbert spaces, with inner products . t and . t depending on t ∈ I, such that the norms induced by them are equivalent to the standard norms H and H , respectively. These inner

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products are Lipschitz continuous in t, in the sense that |u, vt − u, vs | ≤ c|t − s| uH vH , and similarly for . t . (i b) {A(t)} is a family of (unbounded) linear operators in H, such that A(t) = AS (t)+ AR (t), where AS (t) is selfadjoint and uniformly positive deﬁnite with respect to inner product . t , while AR (t) is uniformly bounded (with bound K, say) from H to H. Moreover D(AS (t)) ⊂ H and AS (t)u, vt = u, vt Then the family

(u ∈ D(AS (t)), v ∈ H ) .

A(t) :=

0 A(t)

(7.3)

1l 0

is a stable family in H ⊕ H with stability constants M , β := 2c + K, where M 1/2 depends on the relation between the equivalent norms H and , 0 . Lemma 7.2. The evolution operator of the Equation 2.7 fulﬁlls the estimate sup

0≤s≤t≤T

U (s, t)X,X ≤ M eC

ν

T

with constants M, C independent of T and . Proof. We deﬁne the spaces H := H s−1 , and H := H s . The standard scalar product in H s−1 is deﬁned by 1/2 u, vH ≡ u, vH s−1 ≡ (Ds−1 u)(Ds−1 v) , D := (−∆) and the integral is taken over the whole torus. The standard scalar product in H is deﬁned in the same way with s in place of s − 1. Deﬁne the scalar product , t := , H (independent of t), and the operator AS AS u := −D−(s−1) ∂j ajk ∂k (Ds−1 u) + mu . Then , t is deﬁned by 7.3. It is easy to see that the operator AS is uniformly elliptic (just use positivity of aij ). We will study the relation of this operator with A˜ := ∂j (ajk ∂k ) + m, who in turn is related to Aw (t) by ˜ = Au + (∂j ajk )(∂k u) . Au Remark that the second term deﬁnes an operator whose norm, as an operator from H to H, is bounded by a constant times ν . We now compare the operators A˜ and AS . For simplicity we restrict to the case where s − 1 = 2r, i.e., s is odd3 . One has

˜ Ds−1 = A˜ + D−(s−1) ∂j [ajk ∂k , Ds−1 ] . AS = A˜ + D−(s−1) A, 3 The

7.3.

general case can be dealt with using the operators Si defined in the proof of Lemma

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Ann. Henri Poincar´e

So, it is enough to compute [ajk ∂k , (∂l ∂l )r ]u (summation convention!). To this end consider r (∂l ∂l ) (ajk ∂k u) = ∂l1 ∂l1 . . . ∂lr ∂lr (ajk ∂k u) which is the sum of terms of the form |α| |β| ∂ ∂ cα ajk u ∂xα ∂xβ where α and β are multiindices fulﬁlling |α|+|β| = 2r. The only term with |β| = 2r is the one where the Laplacians act entirely on u so it disappears when computing the commutator. It is easy to see that all the other terms deﬁne operators which are bounded as operators from H to H, provided s > [n/2]. And moreover the norms of such operators are bounded by a constant times ν . Applying the remaining two operators (namely D−(s−1) and ∂j ) one has that A = AS + AR , with AR : H → H having norm bounded by a constant times ν . Finally, it is very easy to see that

| u, vt − u, vs | ≤ Cν uH s vH s |t − s| . So, Proposition 7.1 gives the result. To estimate the linear ﬂow in Y we construct an isomorphism S : Y → X such that SAw S −1 = Aw + ν B with an operator B : X → X uniformly bounded. We deﬁne D 0 S := . (7.4) 0 D Lemma 7.3. One has

SAw S −1 = Aw + ν B

with B uniformly bounded as an operator from X to X Proof. One has SAw S −1 =

0 DAw D−1

1l , 0

so it is enough to prove that DAw D−1 = Aw + ν B with B : H s → H s−1 uniformly bounded. To prove this fact remark that ν B = [D, Aw ]D−1 = [D, ν bij ∂i ∂j ]D−1 so the result is equivalent [D, bij ∂i ∂j ] : H s+1 → H s−1 continuously. But [D, bij ∂i ∂j ]u = [D, bij ]∂i ∂j u ,

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so it is enough the prove that ∀b ∈ H s one has [D, b] : H s−1 → H s−1 continuously. To prove this fact we compute the commutator in the Fourier space. Denoting by a “hat” the (space) Fourier coeﬃcients of a function one has ˆbk−l u ˆbk−l l u {[D, b]u}∧ (k) = k ˆl − ˆl (7.5) l

=

l

ˆbk−l u ˆl (k − l)

(7.6)

l

The last bracket can be rewritten in the form k1 + l1 kn + ln k12 + · · · + kn2 − l12 − · · · − ln2 = (k1 − l1 ) + · · · + (kn − ln ) . k + l k + l k + l Deﬁne now the bilinear operators Si by ∧

{Si (b, u)} (k) :=

l

then one has [D, b]u =

ki + li ˆbk+l u , ˆl k + l

Si (∂i b, u) .

i

We prove now that Si : H s−1 × H s−1 → H s−1 continuously, and from this the thesis will immediately follow. Since Si is bilinear it is enough to prove that Si (v, u)H s−1 ≤ C vH s−1 uH s−1 . One has Si (v, u)2H s−1

= ≤

k k

2(s−1)

ki +li ˆk−l uˆl k + l lv

2

ki +li 2 2 ( l |ˆ vk−l u ˆl |) supk,l k + l = C k k2(s−1) ( l |ˆ vk−l | |ˆ ul |)2 .

k k

2(s−1)

(7.7)

uk |eik·x , and similarly for v + . Then one has u+ H s−1 = Deﬁne now u+ (x) := k |ˆ uH s−1 , and therefore the square root of the last line of 7.7 is given by C u+ v + H s−1 ≤ C u+ H s−1 v + H s−1 = C uH s−1 vH s−1

(with a diﬀerent C), which is the thesis. Then the estimate (3.3) of [Kat75] gives Corollary 7.4. The evolution operator of Equation 2.7 fulﬁlls the estimate sup

0≤s≤t≤T

U (s, t)Y,Y ≤ M eC

with constants M, C independent of T and .

ν

T

708

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Ann. Henri Poincar´e

8 Appendix In this appendix we will prove Proposition 5.4. We will work R2N . In order to keep into account the dependence of all we will do on N it is worth to scale the coordinates, namely to deﬁne coordinates (˜ p, q˜) by ˜ p=p,

˜ q=q

and to consider the integrals 1 ˜ (l) (˜ Φ p, q˜, ) := 2 Φ(l) (˜ p, ˜ q) . From now to the end of the appendix we will use only these new variables and integrals, so will omit tildes. The role of small parameter will be played by µ := N 3a+α . Lemma 8.1. There exist analytic coordinates (ξl , ηl ) deﬁned in the ball (ξl , ηl ) < 1/C In which one has Φ(l) =

ξl2 + ηl2 3 + O(|ξl , ηl | ) , 2

(ξ, η) − (p, q)∞ ≤ Cµ .

The big O is uniform with respect to µ. Proof. Write

p2l + ql2 + µgl (p, q, µ) , 2 by the ﬁrst of the estimates 5.5 the functions gl are smooth and bounded provided µ is small enough. Denote by pˆl the set of variables Φ(l) (p, q) =

pˆl := (p1 , . . . , pl−1 , pl+1 , . . . , pN ) and similarly for qˆl . First remark that, provided µ is small enough, there exists a surface §l where the function Φ(l) , considered as a function of pl , ql has a minimum (just use the implicit function theorem applied to the diﬀerential of Φ(l) ). Moreover such a surface is analytic and has equation pl = µGlp (ˆ pl , qˆl , µ) ,

ql = µGlq (ˆ pl , qˆl , µ) .

Introduce new coordinates by xl := pl − µGlp (ˆ pl , qˆl , µ) ,

yl := ql − µGlq (ˆ pl , qˆl , µ) .

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Using the implicit function theorem one can see that, provided µ < µ∗ with a suitable µ∗ , this coordinate system is well deﬁned in a neighbourhood (in the .∞ norm) of the origin (independent of µ). In these coordinates one has ' ( 3 xl , yˆl ) (xl , yl ) + O(|xl , yl | ) Φ(l) (x, y) = Bl (ˆ ( ' xl , yˆl ) is positive deﬁnite. It follows that there exists where the quadratic form Bl (ˆ a rotation Rl (ˆ xl , yˆl , µ) in the plane xl , yl which diagonalize such a quadratic form. Moreover, for µ = 0 the transformations Rl reduce to the identity. Introduce the coordinates ' ( xl , yˆl , µ) (xl , yl ) . (ξl , ηl ) = Rl (ˆ By implicit function theorem they are well deﬁned in a neighbourhood of the origin and reduce the integrals to the form Φ(l) =

λ1l (ξˆl , ηˆl )ξl2 + λ2l (ξˆl , ηˆl )ηl2 + O(|ξl , ηl |3 ) . 2

scaling the variables one has the thesis.

Lemma 8.2. There exist C1 , C2 and η∗ > 0 with the following properties: let z ∈ R2N and z¯ ∈ R2N be in a ball of radius smaller than 1/C1 and fulﬁll z )| < η , |Φj (z) − Φj (¯

with η < η∗

then d∞ (Tz , z¯) ≤ C2 η 1/2 (recall that we are using rescaled coordinates). Proof. Introduce the coordinates of Lemma 8.1. Since they are analytic they are also Lipschitz and therefore, in such coordinates distances are estimated by a constant times the original distance. Introduce now polar coordinate ρj , φj in the planes ξj , ηj . Then one has Φ(j) = ρ2j + ρ3j fj (ρ, φ) , with bounded and smooth fj ’s. So it is possible to use the contraction mapping principle to solve the above system with respect to the ρ2j obtaining ρ2j := Φ(j) + (Φ(j) )3/2 f˜j (Φ, φ) . One has d∞ (¯ z , Tz ) = sup |ρj − ρ¯j | j

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with an obvious notation. Fixing j and omitting it, we have ¯ ¯ +Φ ¯ 3/2 f˜(Φ, ¯ φ) |ρ − ρ¯| = Φ + Φ3/2 f˜(Φ, φ) − Φ ¯ ¯ +Φ ¯ 3/2 f˜j (Φ, ¯ φ) Φ + Φ3/2 f˜j (Φ, φ) − Φ = . ¯ ¯ +Φ ¯ 3/2 f˜(Φ, ¯ φ) Φ + Φ3/2 f˜(Φ, φ) + Φ Now, the function Φ + Φ3/2 f˜j (Φ, φ) is Lipschitz in Φ, and therefore the numerator ¯ Write Φ ¯ = Φ + δ, then the denominator is estimated by a constant times Φ − Φ. is monotonically increasing function of Φ, and therefore it has its minimum for Φ √ = 0. Evaluating at such a point one gets that it is larger than a constant times δ. From this the thesis easily follows. Going back to the nonrescaled variables one has the thesis of Proposition 5.4.

References [Bam99a] D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schr¨ odinger equation, Math. Z. 130, 345–387 (1999). [Bam99b]

, On long time stability in Hamiltonian perturbations of nonresonant linear PDEs, Nonlinearity 12, 823–850 (1999).

[Bam03]

, Birkhoﬀ normal form for some nonlinear PDEs, Commun. Math. Phys. 234, 253–285 (2003).

[BG03]

D. Bambusi, B. Grebert Forme normale pour NLS en dimension quelconque, CRAS (2003).

[BGG85] 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, 1–25 (1985). [BN98]

D. Bambusi and N.N. Nekhoroshev, A property of exponential stability in nonlinear wave equation near the fundamental linear mode, Physica D 122, 73–104 (1998).

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J. Bourgain, Construction of approximative and almost-periodic solutions of perturbed linear Schr¨ odinger and wave equations, Geometric and Functional Analysis 6, 201–230 (1996).

[Bou00]

, On diﬀusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80, 1–35 (2000).

[Che24a] T.M. Cherry, On integrals developable about a singular point of a Hamiltonian system of diﬀerential equations, Proc. Camb. Phil. Soc. 22, 325–349 (1924).

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, On integrals developable about a singular point of a Hamiltonian system of diﬀerential equations, II, Proc. Camb. Phil. Soc. 22, 510–533 (1924).

[Cra96]

W. Craig, Birkhoﬀ normal form for water waves, Mathematical problems in the theory of water waves (J.C. Saut F. Dias, J.M. Ghidaglia, ed.), vol. 200, AMS, 1996.

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A. Giorgilli, Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium, Ann. Inst. Henri Poincar´e 48, 423–439 (1988).

[Kat75]

T. Kato, Quasi-linear equations of evolution, with applications to partial diﬀerential equations, Lect. Notes Math. (New York), vol. 448, Springer, 1975.

[Kat85]

, Abstract diﬀerential equations and nonlinear mixed problems, Scuola Normale Superiore, Pisa, 1985.

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S. Kleinerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure and Appl. Math. 33, 241–263 (1980).

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M.S. Krol, On Galerkin-averaging method for weakly nonlinear wave equations, Math. meth. in the appl. Sci. 11, 649–664 (1989).

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S.B. Kuksin, An averaging theorem for distributed conservative systems and its application to Von Karman’s equation, PMM U.S.S.R. 53, 150– 157 (1989).

[Kuk94]

, KAM-theory for partial diﬀerential equations, Proceedings of the First European Congress of Mathematics, vol. 2, Birkh¨ auser, 1994, pp. 123–157.

[Laz93]

V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions., Springer-Verlag, Berlin, 1993.

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K. Matthies and A. Scheel, Exponential averaging for Hamiltonian evolution equations, Trans. Amer. Math. Soc. 355, 747–773 (2002).

[Pal96]

H. Pals, The Galerkin-averaging method for the Klein-Gordon equation in two space dimensions, Nonlinear Anal. TMA 27, 841–856 (1996).

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J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure and Appl. Math. 38, 685–696 (1985).

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A.C.J. Stroucken and F. Verhulst, The Galerkin-averaging method for nonlinear, undamped continuous systems, Math. meth. in the appl. Sci. 335, 520–549 (1987).

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E.T. Whittaker, On the adelphic integral of the diﬀerential equations of dynamics, Proc. Roy. Soc. Edinburgh, Sect. A 37, 95–109 (1916).

[XYQ97] J. Xu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226, 375–387 (1997). Dario Bambusi Universit` a di Milano Dipartimento di Mathematica Via Saldini 50 I-20133 Milano Italy email: [email protected] Communicated by Eduard Zehnder submitted 29/10/02, accepted 12/05/03

Ann. Henri Poincar´e 4 (2003) 713 – 738 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040713-26 DOI 10.1007/s00023-003-0145-5

Annales Henri Poincar´ e

Van Hove Hamiltonians – Exactly Solvable Models of the Infrared and Ultraviolet Problem Jan Derezi´ nski Abstract. Quadratic bosonic Hamiltonians with a linear perturbation are studied. Depending on the infrared and ultraviolet behavior of the perturbation, their properties are described from the point of view of spectral and scattering theory.

1 Introduction Some of the simplest exactly solvable models in classical and quantum physics are quadratic Hamiltonians with a linear perturbation. Following [Sch] we will call them van Hove Hamiltonians. They arise in classical and quantum ﬁeld theory in many contexts. For example, the Hamiltonian of electrodynamics with prescribed external charges is a van Hove Hamiltonian. In the case of ﬁnite degrees of freedom, an appropriate translation in phase space transforms a van Hove Hamiltonian into a purely quadratic Hamiltonian. In the quantum case this translation can be implemented by a unitary Bogolubov transformation. If the phase space is inﬁnite-dimensional, the situation becomes more complicated, because the above-mentioned Bogolubov transformation can become “improper” (not implementable by a unitary operator). Still, as it is well known, one can fully analyze the properties of van Hove Hamiltonians even in the case of the inﬁnite-dimensional phase space. These properties are quite interesting, both physically and mathematically. In fact, some types of van Hove Hamiltonians can be viewed as exactly solvable toy models of renormalization, both in the infrared and ultraviolet regime. Depending on the assumptions on the perturbation, one can distinguish several types of these Hamiltonians with distinct properties. Let us describe some of the results of our paper in a somewhat informal language. We will restrict ourselves to the quantum case, since the classical case is very similar. (In the main part of the paper a diﬀerent, more compact notation is used; moreover, both the classical and quantum case is treated.) Let (K, dk) be a space with a measure. (We use the notation dk to denote an arbitrary measure, not necessarily the Lebesgue measure.) Let K k → a∗ (k) and K k → a(k) be the corresponding bosonic creation and annihilation operators. Let K k → h(k) be an almost everywhere positive measurable function, describing the dispersion relation. Finally, suppose that K k → z(k) is a complex function describing the interaction. By a van Hove Hamiltonian we will mean a

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self-adjoint operator given by the formal expression H := h(k)a∗ (k)a(k)dk + z(k)a∗ (k)dk + z(k)a(k) dk + c,

(1.1)

where c is an arbitrary (maybe inﬁnite) constant. It turns out that if h(k)<1

|z(k)|2 dk +

|z(k)|2 h(k)≥1 h(k)2 dk

< ∞.

(1.2)

then properly interpreted (1.1) deﬁnes a family of self-adjoint operators. There are two natural choices of c in (1.1). The ﬁrst choice is c = 0. It is possible if we assume that h(k)<1

|z(k)|2 dk +

h(k)≥1

|z(k)|2 h(k) dk

< ∞,

(1.3)

Under this condition we can deﬁne the van Hove Hamiltonian of the ﬁrst kind: HI := h(k)a∗ (k)a(k)dk + z(k)a∗ (k)dk + z(k)a(k) dk. (1.4) The second choice is c = h(k)<1

|z(k)|2 h(k) dk.

|z(k)|2 h(k) dk

+

It is possible if

|z(k)|2 h(k)≥1 h(k)2 dk

< ∞.

(1.5)

Under this condition we can deﬁne the van Hove Hamiltonians of the second kind: z(k) z(k) HII := h(k) a∗ (k) + a(k) + dk. (1.6) h(k) h(k) The conditions (1.3) and (1.5) are true at the same time iﬀ

|z(k)|2 dk < ∞, h(k)

(1.7)

and then the two types of van Hove Hamiltonians are well deﬁned and diﬀer by a constant: |z(k)|2 HII = HI + dk. (1.8) h(k) Next we would like to describe various types of behavior of van Hove Hamiltonians. We will consider separately the case when the dispersion relation is separated away from zero and the case of a bounded dispersion relation. In the former case the infrared problem is absent and we can study the ultraviolet problem in its pure form. In the latter case the ultraviolet problem is absent and all the diﬃculties are due to the infrared problem. We will see that one can distinguish 3 × 3 = 9 distinct classes of van Hove Hamiltonians.

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Ultraviolet problem

Assume that for some 0 < c0 we have c0 ≤ h(k). We allow h to be unbounded. (In quantum ﬁeld theory this behavior is typical for massive particles.) Under this assumption, we can always deﬁne HII . We can distinguish three cases: (1) |z(k)|2 dk < ∞. This is the most regular case. The perturbation is operator bounded. HI and HII are well deﬁned by the Kato-Rellich theorem [Kato, RS2]. One can deﬁne a “dressing operator” z(k) ∗ z(k) U := exp − a (k)dk + a(k)dk (1.9) h(k) h(k) which intertwines HII with the quadratic Hamiltonian: HII = U h(k)a∗ (k)a(k)dkU ∗ .

(1.10)

Because of (1.10), the Hamiltonians have a ground state. |z(k)|2 |z(k)|2 dk = ∞. (2) h(k) dk < ∞, This is also quite a regular case. The only diﬀerence with (1) is the fact that the perturbation is only form bounded and one needs to use the KLMN theorem to deﬁne HI or HII [Kato, RS2]. |z(k)|2 2 (3) |z(k)| h(k)2 dk < ∞, h(k) dk = ∞. The operator HI is not deﬁned. This follows from the fact that the “counterterm” in (1.8) is inﬁnite. But the dressing operator (1.9) is well deﬁned, which can be used to deﬁne HII . Note that all the diﬃculty stems from the ultraviolet behavior of the interaction. Thus this case provides an example of “the ultraviolet renormalization”: one has to subtract an inﬁnite counterterm from HI to deﬁne a Hamiltonian.

1.2

Infrared problem

Let us assume that the function k → h(k) is bounded, but we allow h to have arbitrarily small positive values. (This is typical for zero-mass particles with an ultraviolet cut-oﬀ.) Then the operator HI is always well deﬁned. We can distinguish the following three cases: (1) |z(k)| h(k)2 dk < ∞. Again, this is the most regular case. The perturbation is operator bounded, both HI and HII are well deﬁned by the Kato-Rellich theorem [Kato, RS2]. One can deﬁne the dressing operator (1.9) and the Hamiltonians have a ground state.

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|z(k)|2 < ∞, h(k)2 dk = ∞. In this case, the perturbation is operator bounded, but the dressing operator is not deﬁned. HI , and consequently also HII , have no ground states, but are bounded from below. This is the case where the infrared problem manifests itself. In quantum ﬁeld theory this case is typical for massless particles (e.g., photons) interacting by a linear term with an external classical source of nonzero charge, as noted by Kibble [Ki]. One can say that the vacuum “escapes from the Hilbert space”, “soft photons are present in every state in the Hilbert space”. The dressing operator (at least in the usual sense) is no longer well deﬁned. |z(k)|2 (3) |z(k)|2 dk < ∞, h(k) dk = ∞. This is the case where the infrared problem is the most severe. In order to deﬁne HI one cannot use the Kato-Rellich nor the KLMN theorem. Instead one needs Nelson’s commutator theorem [RS2]. HI is unbounded from below and the dressing operator is not well deﬁned. The operator HII is not deﬁned at all.

(2)

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|z(k)| h(k) dk

Scattering theory

Van Hove Hamiltonians are also interesting from the point of view of scattering theory. It turns out that the usual formalism of scattering theory, used in the context of Schr¨ odinger operators and described for instance in [RS3], does not apply in the case of van Hove Hamiltonians. Instead, one needs to use some other versions to scattering theory. We have in fact a choice of at least two approaches. One of them is based on replacing the usual deﬁnition of wave operators with a deﬁnition that uses the so-called Abelian limit [Ya]. In order to deﬁne unitary wave operators one has to perform the so-called “renormalization of the wave function”. This approach breaks down if we have the infrared problem (unless one is willing to divide by zero). The second approach to scattering for van Hove Hamiltonians is based on the notion of asymptotic ﬁelds. It goes back to the so-called LSZ formalism. In this approach there is no need for renormalization and the infrared problem is manifested by the non-Fock property of asymptotic ﬁelds. An essentially the same formalism works in a much more complicated context, e.g., [HK, DG1, DG2]. Both approaches lead essentially to the same wave operator equal to the dressing operator. Unfortunately, the scattering operator turns out to be equal to one – so physically, scattering theory of van Hove Hamiltonians turns out to be trivial.

1.4

Remarks about the literature

Many of the ideas of this paper are contained in the literature in one form or another.

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The analysis of the ultraviolet problem can be found, e.g., in the book of Berezin [Be], Section III.7.4 and [Sch], following earlier papers [vH, EP, To, GS]. The understanding of the infrared problem can be traced back to the paper by Bloch and Nordsieck [BN], and was then analyzed in a series of papers of Kibble [Ki]. The infrared problem is related to the so-called coherent state representations, which were discussed already in [Frie]. The infrared problem was also studied for some models similar to the van Hove Hamiltonian in [Mi, Fr] and recently in [AHH], Lemma 4.3. In recent literature one can ﬁnd analysis of operators similar to but more complicated than van Hove Hamiltonians – under the names of the spin-boson, the Pauli-Fierz or the non-relativistic QED Hamiltonian, see, e.g., [DG1, BFS1, DJ]. In fact, one can view van Hove Hamiltonians as a special, exactly solvable subclass of Pauli-Fierz Hamiltonians. We believe that it is useful to study van Hove Hamiltonians to gain intuition about properties of Pauli-Fierz Hamiltonians. In mathematical literature, scattering theory based on asymptotic ﬁelds applied to models similar to the van Hove model goes back to [KM1, KM2, HK], and was recently developed in [DG1, DG2]. In particular, in [KM2], Theorem 3.1, scattering for the van Hove Hamiltonian (without the infrared problem) is studied. We could not ﬁnd a complete treatment of van Hove Hamiltonians in the literature. We think that a careful analysis of van Hove Hamiltonian helps to understand some of the concepts of quantum ﬁeld theory. It is also an instructive exercise in the theory of unbounded operators.

2 Notation 2.1

Diﬀerentiation in a Hilbert space

Let W be a Hilbert space with the scalar product (·|·). Let Dom G be a subset of W and G : Dom G → R. Let w0 ∈ Dom G and w ∈ W. We will say that the derivative of G at w0 in the direction of w exists iﬀ there exists > 0 such that {w0 + tw : |t| < } ⊂ Dom G and there exists d G(w0 + tw) =: w∇G(w0 ). dt t=0 We will say that G is diﬀerentiable at w0 iﬀ D := {w ∈ W : w∇G(w0 ) exists } is a dense linear subspace of W and D w → w∇G(w0 ) is a bounded linear functional. If this is the case, then the gradient of G at w0 is denoted by ∇G(w0 ) ∈ W and deﬁned by Re(w|∇G(w0 )) = w∇G(w0 ), w ∈ W. We will say that a map α : W → W preserves the scalar product iﬀ (α(w1 ) − α(w2 )|α(w3 ) − α(w4 )) = (w1 − w2 |w3 − w4 ),

w1 , . . . , w4 ∈ W.

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Unbounded operators

Let h be a self-adjoint operator on W with h ≥ 0. sph will denote the spectrum of h. Let Dom h denote its domain and (Dom h)∗ the space of bounded antilinear functionals on Dom h. If w1 ∈ Dom h and w2 ∈ (Dom h)∗ , then we will write (w1 |w2 ) for the action of w2 on w1 . Clearly, W can be embedded in a natural way in (Dom h)∗ . The operators h and h1/2 extend to bounded maps from W to (Dom h)∗ . We denote by hW and h1/2 W respectively the images of these extensions. Clearly, the algebraic sums W + h1/2 W, h1/2 W + hW and W + hW are well deﬁned subspaces of (Dom h)∗ . (Note, in particular, that W + hW coincides with (Dom h)∗ .)

2.3

Fock spaces ◦

[RS2, BR] If X is a vector space, then Γs (X ) will denote the algebraic symmetric Fock space over X , that means the space of ﬁnite linear combinations of symmetric tensor products of elements of X . Ω will denote the vacuum. If W is a Hilbert space, then Γs (W) will denote the (complete) Fock space, ◦

that is the completion of Γs (W). If u is a contraction on W, then Γ(u) denotes the contraction on Γs (W) that on the n particle sector equals u⊗n . If h is a self-adjoint operator, then dΓ(h) denotes the self-adjoint operator that on the n particle sector equals h ⊗ 1⊗(n−1) + · · · + 1⊗(n−1) ⊗ h.

Abusing the notation, Let b be a sesquilinear form on W with the domain W. we will use the symbol dΓ(b) to denote the sesquilinear form on Γs (W) with the ◦

that on the n particle sector equals b ⊗ 1⊗(n−1) + · · · + 1⊗(n−1) ⊗ b. domain Γs (W)

2.4

Creation and annihilation operators

The notion of creation and annihilation operators is standard in the context of Fock spaces and can be found, e.g., in [BR, RS2]. Nevertheless, we will need slight generalizations of these concepts.

⊂ W. Let w be an antilinear form on W with the domain Domw = W ◦

We deﬁne the annihilation operator w(a) as an operator with the domain Γs (W) satisfying √

w(a) z ⊗n := n(w|z)z ⊗(n−1) , z ∈ W. ◦

(Note that vectors of the form z ⊗n span Γs (W).) The operator w(a) is closable iﬀ w ∈ W. If this is the case, we denote its closure by the same symbol. Its adjoint is called the creation operator and denoted by w(a∗ ) := w(a)∗ .

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◦

If f ∈ Γs (Z), then f (a∗ ) and f (a) have the obvious meaning as polynomials ∗ in a and a. For instance z ⊗n (a∗ ) = (z(a∗ ))n and z ⊗n (a) = (z(a))n . We also introduce the ﬁeld operator 1 φ(w) := √ (w(a∗ ) + w(a)), 2

(2.11)

which is self-adjoint on Dom w(a). We also introduce the Weyl operators W (w) := eiφ(w) .

(2.12)

If w ∈ W, then the annihilation operator w(a) is not closable. In this case the creation operator w(a∗ ) is not even densely deﬁned and is not of much use. On the other hand, we can deﬁne the “creation form”, denoted by w(a∗f ): ◦

× Γs (W) (Φ, Ψ) → (Φ|w(a∗f )Ψ) := (w(a)Φ|Ψ). Γs (W) Note that w(a∗f ) is a diﬀerent object from w(a∗ ). In what follows, however, we will sometimes abuse the notation and we will write w(a∗ ) instead of w(a∗f ).

3 Van Hove Hamiltonians In Subsections 3.1 and 3.2 we consider the classical dynamics α generated by a Hamiltonian G equal to a quadratic polynomial. In the remaining subsections we discuss their quantum counterparts – the dynamics β and its Hamiltonian H.

3.1

Classical dynamics

Suppose that h is a positive operator on W. We will assume that Kerh = {0}. Let z ∈ W + hW .

(3.13)

For w ∈ W, t ∈ R, we deﬁne αt (w) := eith w + (eith − 1)h−1 z. It is easy to see that R t → αt is a 1-parameter group of transformations preserving the scalar product. Let us note the following property of the dynamics α. Let p1 , p2 be two complementary orthogonal projections commuting with h. For i = 1, 2 let Wi := Ranpi . For w ∈ W, set wi := pi w. Set hi := pi h, treated as a self-adjoint operator on Wi . Let αi be the dynamics on Wi deﬁned by hi , zi . Then the dynamics α splits: αt (w1 , w2 ) = (αt1 (w1 ), αt2 (w2 )). In particular, we can take p1 := 1[0,1] (h), p2 := 1]1,∞[ (h).

(3.14)

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Then h1 is bounded and h2 is bounded from below by a positive constant. In the case of h1 , the ultraviolet problem is absent, but the infrared problem can show up. In the case of h2 we have the opposite situation: the infrared problem is absent, but we can face the ultraviolet problem.

3.2

Classical Hamiltonian

The dynamics α preserves the symplectic form Im(·|·). Therefore, we should expect that α possesses a Hamiltonian G, that is a function satisfying d t α (w)) = Re(w1 |∇G(αt (w))), w1 ∈ W, Im(w1 | dt

(3.15)

or equivalently, d t α (w) = i∇G(αt (w)). (3.16) dt In this subsection we will construct such Hamiltonians. (Note, however, that in general these Hamiltonians will not be everywhere deﬁned.) First we deﬁne Hamiltonians of α in two special cases. If z ∈ W + h1/2 W, then we deﬁne Dom GI := Dom h1/2 and for w ∈ Dom GI we set 1 GI (w) := ((w|hw) + (z|w) + (w|z)) . 2 We will say that GI is the Hamiltonian of α of the ﬁrst kind. If z ∈ h1/2 W + hW, then we deﬁne Dom GII := {w ∈ W : h1/2 w + h−1/2 z ∈ W}, and for w ∈ Dom GII we set GII (w) :=

1 1/2

h w + h−1/2 z 2 . 2

We will say that GII is the Hamiltonian of α of the second kind. Clearly, both GI and GII are well deﬁned iﬀ z ∈ h1/2 W, and then 1 GII = GI + (z|h−1 z). 2 If z is any functional satisfying (3.13) we split the dynamics α into two parts α1 and α2 , as explained at the end of Subsection 3.1. We can then deﬁne the Hamiltonian G1,I for the dynamics α1 and the Hamiltonian G2,II for the dynamics α2 . We will say that a function G is a Hamiltonian of α iﬀ its domain equals Dom G1,I ⊕ Dom G2,II and it is equal to G(w1 , w2 ) := G1,I (w1 ) + G2,II (w2 ) + c, where c ∈ R is arbitrary. Note that, in general, (W + h1/2 W) ∪ (h1/2 W + hW) is strictly smaller than W + hW. Therefore, for some z, the dynamics α is well deﬁned but neither the Hamiltonian of the ﬁrst kind GI nor the Hamiltonian of the second kind GII are well deﬁned.

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Theorem 3.1 Let z ∈ W + hW and let G be a Hamiltonian for αt . Then the following statements are true: (1) The function G is diﬀerentiable at w ∈ W iﬀ hw + z belongs to W. We then have ∇G(w) = hw + z. The dynamics t → αt (w) is diﬀerentiable wrt t iﬀ hαt (w) + z ∈ W. We have d t α (w) = i(hαt (w) + z), dt which can be written in the form (3.16). Note also that αt leaves Dom G invariant and G is constant along the trajectories. (2) 0 belongs to Dom G iﬀ z ∈ W + h1/2 W. We have G = GI iﬀ GI (0) = 0. (3) G is bounded from below iﬀ z ∈ h1/2 W + hW. We have G = GII iﬀ inf G = 0. (4) G has a minimum iﬀ z ∈ hW. This minimum is at −h−1 z, and then GII (w) =

3.3

1 w + h−1 z|h(w + h−1 z) . 2

Quantum dynamics

Let h, z be as above. Assume (3.13), that is z ∈ W + hW. Deﬁne the family of unitary operators on Γs (W) −1 V (t) := Γ(eith ) exp (1 − e−ith )h−1 z(a∗ ) − (1 − eith )h z(a) . Deﬁne for B ∈ Γs (W) β t (B) := V (t)BV (t)∗ .

(3.17)

It is easy to check that β is a 1-parameter group of ∗-automorphisms of B(Γs (W)) continuous in the strong operator topology. In order to make the relationship with the classical dynamics more clear, one can note that β t (w(a∗ )) = ((eith − 1)h−1 z|w) + (eith w)(a∗ ). Let W = W1 ⊕ W2 , as in Subsection 3.1. Then we have the identiﬁcation Γs (W) = Γs (W1 ) ⊗ Γs (W2 ). The dynamics β factorizes β t (B1 ⊗ B2 ) = β1t (B1 ) ⊗ β2t (B2 ), B1 ∈ B(Γs (W1 ), B2 ∈ B(Γs (W2 ).

(3.18)

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Quantum Hamiltonian

We say that a self-adjoint operator H is a Hamiltonian of the dynamics β iﬀ β t (B) = eitH Be−itH .

(3.19)

Clearly, (3.19) deﬁnes H only up to an additive constant. For z ∈ W + h1/2 W, deﬁne UI (t) := eiIm(h

−1

z|eith h−1 z)−it(z|h−1 z)

V (t).

For z ∈ h1/2 W + hW, deﬁne UII (t) := eiIm(h

−1

z|eith h−1 z)

V (t).

We easily check that both UI (t) and UII (t) are 1-parameter strongly continuous unitary groups. Therefore, there exist self-adjoint operators HI and HII such that UI (t) = eitHI ,

UII (t) = eitHII .

Clearly, both HI and HII are Hamiltonians of β. They are both well deﬁned iﬀ z ∈ h1/2 W, and then HII = HI + (z|h−1 z). A Hamiltonian of β will be called a van Hove Hamiltonian. The operator HI will be called the van Hove Hamiltonian of the ﬁrst kind and HII will be called the van Hove Hamiltonian of the second kind. Theorem 3.2 Let z ∈ W + hW. Then the following statements are true: (1) There exist Hamiltonians of β. (2) Let H be a Hamiltonian of β. Ω belongs to Dom|H|1/2 (the form domain of H) iﬀ z ∈ W + h1/2 W. Under this condition H = HI iﬀ (Ω|HΩ) = 0. (3) The operator H is bounded from below iﬀ z ∈ h1/2 W + hW. Under this condition H = HII iﬀ inf H = 0. (4) The operator H has a ground state (inf H is an eigenvalue of H, where inf H denotes the inﬁmum of the spectrum of H) iﬀ z ∈ hW. Then we can deﬁne the “dressing operator” −1 U := exp −h−1 z(a∗ ) + h z(a) , and HII = U dΓ(h)U ∗ .

(3.20)

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Proof of (1). We split the dynamics β as in (3.18) with the projections p1 , p2 given by (3.14). Then we can deﬁne the Hamiltonian H1,I for β1 and the Hamiltonian H2,II for β2 . We set H := H1,I ⊗ 1 + 1 ⊗ H2,II . 2 Easy calculations show that, at least formally, HI = dΓ(h) + z(a∗ ) + z(a), ∗

HII = dΓ(h) + z(a ) + z(a) + (z|h

(3.21) −1

z).

(3.22)

Below we will make these formulas precise. Remark 3.3 By the spectral theorem, one can ﬁnd a measure space (K, dk) such that W is isomorphic to L2 (K, dk) and h is a multiplication operator by a mea∗ surable function K k → h(k). Then we ∗can introduce K k →∗ a (k), a(k) ∗ and write z(k)a (k)dk, z(k)a(k), h(k)a (k)a(k)dk instead of z(a ), z(a) and dΓ(h). We used this notation in the introduction. For example, Condition (1.2) of the introduction, 2 |z(k)|2 dk + h(k)≥1 |z(k)| h(k)2 dk < ∞, h(k)<1 corresponds to the condition z ∈ W + hW. The notation involving the operator-valued measures a∗ (k) and a(k) is very common and often convenient, but it depends on a non-canonical choice of the measure space K, and therefore we do not use it.

3.5

3 types of the ultraviolet problem

In this subsection we describe van Hove Hamiltonians without the infrared problem. Theorem 3.4 Assume that h is a self-adjoint (possibly unbounded) operator bounded from below by a positive constant. Then the dressing operator U and the van Hove Hamiltonian of the second kind HII are well deﬁned. HII possesses a unique ground state at 0. Moreover, we can distinguish 3 cases: (1) Let z ∈ W. Then z(a∗ ) + z(a) is a dΓ(h)-bounded operator with the inﬁnitesimal bound. HI and HII are self-adjoint on Dom dΓ(h) and can be deﬁned by the formulas (3.21), (3.22) and by the Kato-Rellich theorem. (2) Let z ∈ h1/2 W\W. Then z(a∗ )+z(a) is not an operator. Instead of z(a∗ )+z(a) we should consider ◦

the form with the domain Γs (Dom z) equal to z(a∗f ) + z(a). This form is dΓ(h)-form bounded with the inﬁnitesimal bound. The operators HI and HII are bounded from below and their form domains equal Dom dΓ(h)1/2 . They can be deﬁned by the formulas (3.21), (3.22) and by the KLMN theorem.

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(3) Let z ∈ hW\h1/2 W. Then the form z(a∗f ) + z(a) is not dΓ(h)-form bounded. HI is not deﬁned.

3.6

3 types of the infrared problem

In this subsection we describe van Hove Hamiltonians without the ultraviolet problem. Theorem 3.5 Assume that h is a bounded positive operator. Then the formula (3.21) deﬁnes the operator HI as an essentially self-adjoint operator on Dom dΓ(h+1) by Nelson’s commutator theorem. Moreover, we can distinguish the following three cases: (1) Let z ∈ hW Then z(a∗ ) + z(a) is dΓ(h)-bounded with the inﬁnitesimal bound. The operators HI and HII are self-adjoint on Dom dΓ(h) and they can be deﬁned by the formulas (3.21), (3.22) and the Kato-Rellich theorem. They have ground states and the dressing operator U is well deﬁned. (2) Let z ∈ h1/2 W\hW Then z(a∗ ) + z(a) is dΓ(h)-bounded with the inﬁnitesimal bound. Again, the operators HI and HII are self-adjoint on Dom dΓ(h) and they can be deﬁned by the formulas (3.21), (3.22) and the Kato-Rellich theorem. They are bounded from below but have no ground state and the dressing operator U is not deﬁned. (3) Let z ∈ W\h1/2 W. Then z(a∗ ) + z(a) is not dΓ(h)-form bounded. HI is not bounded from below and the operator HII is not deﬁned at all. In the following subsections we will show various elements of the above theorems.

3.7

Relative form boundedness of ﬁeld operators

Proposition 3.6 Let h be a positive operator on W and z ∈ h1/2 W. Then (1)

z(a)Ψ 2 ≤ (z|h−1 z)(Ψ|dΓ(h)Ψ). (2) z(a∗f ) + z(a) is form bounded wrt dΓ(h) with the inﬁnitesimal bound. More precisely, for any > 0, we have |(Ψ|(z(a∗f ) + z(a))Ψ)| ≤ (z|h−1 z)(Ψ|dΓ(h)Ψ) + −1 Ψ 2 .

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Proof. (1) If z is an antilinear functional on W with the domain Domz, then Dom z × Dom z (w1 , w2 ) → (w1 |z)(z|w2 ) deﬁnes a sesquilinear form, that we will denote by |z)(z|. Note that the following inequality is true: |z)(z| ≤ (z|h−1 z)h, (3.23) Clearly,

z(a)Ψ 2 = (Ψ|dΓ (|z)(z|) Ψ) . (3.23) implies

dΓ (|z)(z|) ≤ (z|h−1 z)dΓ(h).

(2)

±(Ψ|(z(a∗f ) + z(a))Ψ) = ±2Re Ψ|z(a)Ψ) ≤ 2 Ψ

z(a)Ψ ≤ −1 Ψ 2 + z(a)Ψ 2 . 2

Corollary 3.7 If z ∈ h1/2 W, then both (3.21) and (3.22) (with z(a∗ ) replaced by z(a∗f )) deﬁne by the KLMN theorem the self-adjoint operators HI and HII with the form domains Dom dΓ(h)1/2 .

3.8

Relative boundedness of ﬁeld operators

Proposition 3.8 Let h be a positive operator on W and z ∈ h1/2 W ∩ W. Then (1)

(z(a∗ ) + z(a))Ψ 2 ≤ 4(z|h−1 z)(Ψ|dΓ(h)Ψ) + 2 z 2 Ψ 2 . (2) z(a∗ )+z(a) is bounded wrt dΓ(h) with the inﬁnitesimal bound. More precisely, for any > 0, we have

(z(a∗ ) + z(a))Ψ 2 ≤ 2(z|h−1 z) dΓ(h)Ψ 2 + 2−1 (z|h−1 z) + 2 z 2 Ψ 2 . Proof. (2) follows immediately from (1). To see (1) we compute using Proposition 3.6 (1):

(z(a∗ ) + z(a))Ψ 2

≤ 2 z(a∗ )Ψ 2 + z(a)Ψ 2 = 4 z(a∗ )Ψ 2 + 2 z 2 Ψ 2 ≤ 4(z|h−1 z)(Ψ|dΓ(h)Ψ) + 2 z 2 Ψ 2 . 2

Corollary 3.9 If z ∈ W ∩ h1/2 W, then both (3.21) and (3.22) deﬁne by the KatoRellich theorem the self-adjoint operators HI and HII with the domains Dom dΓ(h).

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The inﬁmum of van Hove Hamiltonians

Proposition 3.10 Assume z ∈ W + h1/2 W. Then the operator HI satisﬁes inf HI = −(z|h−1 z), spHI = −(z|h−1 z) + sp dΓ(h). Proof. We drop I from HI . It is suﬃcient to assume that 0 is not an isolated point in the spectrum of h.

(3.24)

This in particular implies that sp dΓ(h) = [0, ∞[. Thus we need to show that spH = [−(z|h−1 z), ∞[. Step 1) Setting = ±(z|h−1 z)−1 in Proposition 3.6 (1) with we get dΓ(h) + z(a∗ ) + z(a) ≥ −(z|h−1 z).

(3.25)

Step 2) For any n deﬁne W n := 1[ n1 ,∞[ (h)W, hn := 1[ n1 ,∞[ (h)h, z n := 1[ n1 ,∞[ (h)z, and the operators on Γs (W n ), H n := dΓ(hn ) + z n (a∗ ) + z n (a) and U n := exp −1 (−h−1 z n (a∗ ) + h z n (a)). Then U n∗ H n U n = dΓ(hn ) − (z n |h−1 z n ). Clearly, inf dΓ(hn ) = 0, hence inf H n = −(z n |h−1 z n ), spH n = −(z n |h−1 z n ) + spdΓ(hn ).

(3.26)

Step 3) Likewise, deﬁne Wn := 1[0, n1 [ (h)W, hn := 1[0, n1 [ (h)h, zn := 1[0, n1 [ (h)z, and the operator on Γs (Wn ), Hn := dΓ(hn ) + zn (a∗ ) + z n (a). We have Ω ∈ DomHn and (Ω|Hn Ω) = 0. Hence, inf Hn ≤ 0. (3.27) Arguing as in Step 1) we get Hn ≥ −(zn |h−1 zn ).

(3.28)

Step 4) Γs (W) can be identiﬁed with Γs (Wn ) ⊗ Γs (W n ) and H = Hn ⊗ 1 + 1 ⊗ H n .

(3.29)

Therefore, inf H = inf Hn + inf H n , spH = spHn + spH n .

(3.30)

Step 5) If t > 0, then by (3.24), we will ﬁnd a sequence tn ∈ sp dΓ(hn ) such that tn → t. By Step 4), inf Hn + tn ∈ sp H. But by Step 3), inf Hn → 0. By the closedness of spH, t ∈ sp H. 2

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Essential self-adjointness of van Hove Hamiltonians

Proposition 3.11 Suppose that z ∈ W. Then HI is essentially self-adjoint on Dom dΓ(1 + h). Proof. First assume that h is bounded. We apply Nelson’s commutator theorem with the comparison operator B := 1 + dΓ(1 + h) [RS2]. In fact,

(z(a∗ ) + z(a))Ψ ≤ c N Ψ ≤ c BΨ) ,

dΓ(h)Ψ ≤ BΨ . Moreover, [B, (z(a∗ ) + z(a))] = (1 + h)z(a∗ ) + (1 + h)z(a). Hence |(Ψ|[B, (z(a∗ ) + z(a))]Ψ)| ≤ (Ψ|BΨ). Hence H is essentially self-adjoint on DomB. Next consider an arbitrary h. As described in Subsection 3.3, we can split W = W1 ⊕ W2 , where W1 = Ran1[0,1] (h). We can deﬁne the operator HI on ◦

◦

◦

Γs (W) = Γs (W1 ) ⊗ Γs (W2 ) and it splits as HI = HI,1 ⊗ 1 + 1 ⊗ HI,2 , with HI,i = dΓ(hi ) + zi (a∗ ) + z i (a). We proved above that HI,1 is essentially self-adjoint on Dom dΓ(h1 + 1). By Corollary 3.9, HI,2 is self-adjoint on Dom dΓ(h2 ). This implies that HI is essentially ◦

◦

self-adjoint on Dom dΓ(h1 + 1)⊗Dom dΓ(h1 ), which is dense in Dom dΓ(h + 1). (⊗ denotes the algebraic tensor product). 2

3.11

Absence of a ground state

Let us recall the following well-known result about coherent states:

is a dense subspace of W and f is an antilinear Theorem 3.12 Suppose that W

Ψ ∈ Domh(a) and

functional on W. Let Ψ ∈ Γs (W), for any w ∈ W, h(a)Ψ = (h|f )Ψ. Then the following is true: (1) If f ∈ W, then Ψ is proportional to exp(f (a∗ ) − f (a))Ω. (2) If f ∈ W, then Ψ = 0.

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wn−1 (a) · · · w 1 (a)Ψ ∈ Proof. By induction we show that for w1 , . . . , wn ∈ W, Domw n (a) and w n (a) · · · w 1 (a)Ψ = (w1 |f ) · · · (wn |f )Ψ. This implies (w1 (a∗ ) · · · wn (a∗ )Ω|Ψ) = (w1 |f ) · · · (wn |f )(Ω|Ψ). In particular,

(3.31)

(w|Ψ) = (w(a∗ )Ω|Ψ) = (w|f )(Ω|Ψ), w ∈ W.

is dense in W we see that (Ω|Ψ)f is a bounded functional Using the fact that W on W, hence it belongs to W. Thus either f ∈ W or (Ω|Ψ) = 0. In the latter case, (3.31) implies that Ψ = 0. 2 Proposition 3.13 HI has a ground state iﬀ z ∈ hW. Proof. If z ∈ hW, then we can deﬁne the dressing transformation and U Ω is a ground state of HI . Suppose that z ∈ hW. We use the notation of the proof of Proposition 3.10. Let Ψ be a ground state of H. Then it is also a ground state of Hn ⊗ 1 and of 1 ⊗ H n. Being a ground state of 1 ⊗ H n, it must be equal to Ψn ⊗ U n Ω. Therefore, ∞

:= W n for w ∈ W n=1

w(a)Ψ = (w|h−1 z)Ψ.

is dense in W. By Theorem 3.12, this means that either h−1 z ∈ W or But W Ψ = 0. 2

4 Scattering theory 4.1

The usual formalism

The most common setup for scattering theory starts with a pair of self-adjoint operators H0 and H. The wave operators Ω± are deﬁned by the formulas Ω± := s− lim eitH e−itH0 . t→±∞

(4.32)

Note that Ω± are automatically isometric and Ω± H0 = HΩ± .

(4.33)

The scattering operator is deﬁned as S := Ω+∗ Ω− . It satisﬁes S = w−

lim

t+ ,t− →∞

eit+ H0 e−i(t+ +t− )H eit− H0

(4.34) (4.35)

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and commutes with H0 . If we have RanΩ+ = RanΩ− , then S is unitary.

4.2

Wave operators deﬁned by the Abelian limit – general formalism

The limits in (4.32) often do not exists. This happens, for instance, if H0 has an eigenvector, which is not an eigenvector of H. This is the case of many models of quantum ﬁeld theory, whose free vacuum (the unique eigenvector of H0 ) is often diﬀerent from the interacting vacuum (the unique eigenvector of H). Nevertheless, sometimes even in this situation some kind of a scattering theory can be developed. In this subsection we will describe one of possible approaches to scattering theory, which, as we will see, works in the case of van Hove Hamiltonians. One can argue that this approach, or some variation, is implicit in most textbook presentations of QFT. Again, we start with a pair of self-adjoint operators H0 and H. We suppose that there exists the Abelian limit ∞ Ω± := s− lim 2 e±itH e∓itH0 e−2t dt. (4.36) ur ↓0

0

± Note that there is no guarantee that Ω± ur are isometric. One knows only that Ωur ± are contractions. One can easily see that Ωur have the intertwining property: ± Ω± ur H0 = HΩur .

(4.37)

We will call Ω± ur the “unrenormalized wave operators”. Of course, it may happen that Ω± = 0. ur Deﬁne the “renormalization of wave function operator” ± Z ± := Ω±∗ ur Ωur .

It is easy to see hat Z ± commutes with H0 . Assume that KerZ ± = {0}. Then we deﬁne the “renormalized wave operators” ± ± −1/2 . Ω± rn := Ωur (Z )

Note that Ω± rn are isometric and have the intertwining property: ± Ω± rn H0 = HΩrn .

(4.38)

The unrenormalized scattering operator is deﬁned as − Sur := Ω+∗ ur Ωur .

(4.39)

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Note that it can be also obtained as the following weak limit: ∞ ∞ Sur = w− lim 4− + dt− dt+ eit+ H0 e−i(t− +t+ )H eit− H0 e−2(− +2+ )t . − ,+ ↓0

0

0

Sur is a contraction that commutes with H0 . We can deﬁne the renormalized scattering operator as − Srn := (Z + )−1/2 Sur (Z − )−1/2 = Ω+∗ rn Ωrn .

Srn also commutes with H0 and if − RanΩ+ rn = RanΩrn ,

then it is unitary. We will see in the following subsection that van Hove Hamiltonians provide an example where the formalism based on the Abelian limit is applicable.

4.3

Wave operators for the van Hove Hamiltonian

Let z ∈ h1/2 W + hW. Let H = HII be the van Hove Hamiltonian of the second kind and H0 := dΓ(h). It is easy to see that in the case of the van Hove Hamiltonian, the limit in (4.32) does not exist unless H = H0 . Hence the construction of wave operators cannot be based on the approach described in Subsection 4.1. We will show, however, that the formalism of Subsection 4.2 works for van Hove Hamiltonians if h has an absolutely continuous spectrum. It will turn out that the two renormalized wave operator coincide with one another and are equal to the dressing operator U . The operators Z± =: Z coincide and are just constants. The renormalized scattering operator equals identity. If z ∈ hW, then Z = 0. This is one of the manifestations of the infrared problem. All these statements are described in the following theorem. Theorem 4.1 Suppose that h has absolutely continuous spectrum and z ∈ h1/2 W + hW. Then Ω± ur exists and 1/2 Ω± U, ur = Z

where Z = 0 iﬀ z ∈ hW, and then

Sur = Z,

Z = e− h

−1

z 2

.

Z = (Ω|U Ω)2 .

We can then deﬁne the renormalized operators, which are equal Ω± rn = U,

Srn = 1.

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Proof. Let us assume that z ∈ hW. (The general case can be obtained by the limiting argument.) Set g := h−1 z. eitH e−itH0

= U eitH0 U ∗ e−itH0 = U exp(−eith g(a∗ ) + e−ith g(a)) 1

(4.40)

2

= U e− 2 g exp(−eith g(a∗ )) exp(e−ith g(a)). Let Pm be the projection onto the states with ≤ m particles. Suppose that Pm Ψ = Ψ. Now ∞ 2 0 e−2t exp(−eith g(a∗ )) exp(e−ith g(a))Ψdt = Ψ ∞ +2 0 e−2t exp(−eith g(a∗ )) − 1 Ψdt (4.41) ∞ −2t exp(−eith g(a∗ )) exp(e−ith g(a)) − 1 Ψdt. +2 0 e The norm of the third term can be estimated from above by ∞ 2 e−2t exp(−eith g(a∗ ))Pm

exp(e−ith g(a)) − 1 Ψ dt. 0

Clearly, exp(−eith g(a∗ ))Pm is bounded uniformly in time. Besides, exp(e−ith g(a)) − 1 Ψ → 0, by the absolute continuity of h and the Riemann-Lebesgue lemma. Therefore, the third term of (4.41) goes to zero. The second term equals ∞

2

n=1

=

∞ n=1

∞ 0

n

ith ⊗n ∗ e−2t (−1) (a )Ψdt n! (e g)

(−1)n n! 2 (2

−1 ⊗n

− idΓ(h))

g

∗ (a )Ψ.

(4.42)

Note that the nth term on the right goes to zero as 0 and can be estimated by

(m + 1) · · · (m + n)

g n Ψ . (4.43) n! The series with elements (4.43) is convergent. Hence by the dominated convergence theorem, (4.42) goes to zero as 0. This shows that, for a ﬁnite particle Ψ, the left-hand side of (4.41) goes to 2 Ψ. By density, we can extend this to all Ψ ∈ Γs (W).

4.4

Asymptotic ﬁelds – general formalism

There exists an alternative approach to scattering in quantum ﬁeld theory. Instead of starting from wave operators, one looks at the limits of certain observables in

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the interaction picture. There are various forms of this approach, some of them go under the name of the LSZ formalism, see e.g., [Schwa]. Let us present the abstract framework of one of the versions of this approach developed in [HK] and used, e.g., in [DG2]. Suppose that H is a self-adjoint operator on the Fock space Γs (W) and h is a self-adjoint operator on W. Assume that for some subspace W1 ⊂ W there exists s− lim eitH W (e−ith w)e−itH =: W ± (w),

w ∈ W1 .

t→±∞

Then

W1 w → W ± (w) ∈ U (Γs (W))

(4.44)

are two representations of Canonical Commutation Relations (CCR), that means i

W ± (w1 )W ± (w2 ) = e− 2 Im(w1 |w2 ) W ± (w1 + w2 ). Moreover, they satisfy eitH W ± (w)e−itH = W ± (eith w). Suppose that the representations (4.44) are unitarily equivalent to the Fock representation, which means that there exist unitary operators Ω± such that W ± (w) = Ω± W (w)Ω±∗ .

(4.45)

Then the operators Ω± are deﬁned up to a phase factor. They are called wave operators. The scattering operator is deﬁned as S := Ω+∗ Ω− . Again, the scattering operator is deﬁned up to a phase factor. Suppose that both the formalism of Subsection 4.2 and of Subsection 4.4 apply. One can ask whether the renormalized wave operators Ω± rn , deﬁned as in Subsection 4.2, and the wave operator Ω± deﬁned in this section coincide up to a phase factor. In general, there seems to be no guarantee for this to hold. Nevertheless we will see that this is true in the case of van Hove Hamiltonians.

4.5

Asymptotic ﬁelds for van Hove Hamiltonians

The formalism of asymptotic ﬁelds works very well in the case of van Hove Hamiltonians. Theorem 4.2 Let h have an absolutely continuous spectrum, 0 ≤ β ≤ 1 and z ∈ h1−β W + hW. Let w ∈ Dom h−β . (1) There exists the norm limit lim eitH W (e−ith w)eitH = W (w)ei2Re(w|h

|t|→∞

−1

z)

=: W as (w).

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(2) Dom h−β w → W as (w) is a regular representation of CCR. (3) This representation is unitarily equivalent to the Fock representation iﬀ z ∈ hW, and then W as (w) = U W (w)U ∗ , where U is the dressing operator. Proof.

eitH W (e−ith w)eitH = W (w) exp i2Re(w|(1 − e−ith )h−1 z) .

Now h−1 z ∈ h−β W + W and w ∈ Dom h−β . Hence lim|t|→∞ (w|e−ith h−1 z) = 0 by the Riemann-Lebesgue lemma. This proves (1). 2

5 Examples 5.1

Harmonic oscillators

In this section we describe van Hove Hamiltonians in a somewhat more concrete setting, typical for physical applications. We will restrict ourselves to the classical case, since it is parallel to the quantum case. We will describe a system of harmonic oscillators with a linear perturbing potential. Up to now, we assumed that our system is described by phase space W. There was no need to introduce the conﬁguration space. For a system of harmonic oscillators it is however natural to start from a conﬁguration space, which will be described by a real Hilbert space X with the scalar product denoted by the dot. The preliminary phase space is X ⊕ X . It has the structure of a symplectic space with the symplectic form (x1 , ξ1 )ω(x2 , ξ2 ) = x1 · ξ2 − x2 · ξ1 .

(5.46)

Note, however, that we will have to take a slightly diﬀerent phase space. Let r denote a positive operator on X and q is a linear functional on X (possibly unbounded and not everywhere deﬁned). A system of harmonic oscillators with a linear perturbing potential is described by the (classical) Hamiltonian G(x, ξ) =

1 1 |rx|2 + |ξ|2 + q · x, 2 2

deﬁned for x Dom r ∩ Dom q, ξ ∈ X . It is easy to see that X ⊕ X is not an appropriate space for the Hamiltonian G. It is natural to replace it by the space W := r−1/2 X ⊕ r1/2 X . We keep the symplectic form (5.46) We equip W with the complex structure i(x, ξ) := (−r−1 ξ, rx).

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We can view W as a complex Hilbert space: (w1 |w2 ) = r1/2 x1 · r1/2 x2 + r−1/2 ξ1 · r−1/2 ξ2 + ix1 · ξ2 − ix2 · ξ1 , w1 = (x1 , ξ1 ), w2 = (x2 , ξ2 ). Note that the symplectic form (5.46) is the imaginary part of the scalar product. Introduce z := (r−1/2 q, 0) and a positive self-adjoint operator h on W deﬁned by h := r ⊕ r. Then we can rewrite G as G(w) =

1 (w|hw) + (z|w) + (w|z) . 2

Note that the infrared and ultraviolet conditions expressed in terms of q instead of z have their power shifted by 1/2. More precisely, z ∈ hα W iﬀ q ∈ r1/2+α X .

5.2

Scalar massless ﬁeld theory

Suppose that X = L2 (Rd ), r = |i∇|. Then the Hamiltonian 12 |rx|2 + 12 |ξ|2 describes the so-called scalar massless ﬁeld theory. After taking the Fourier transformation, the operator r becomes the multiplication by |ξ|, where ξ is the momentum variable. Suppose that we add a linear perturbation given by q ∈ S(Rd ). After taking the Fourier transformation we get qˆ ∈ S(Rd ) and we see that the ultraviolet problem is absent. The infrared problem will depend on whether qˆ(0) equals zero or not. qˆ(0) equals the integral of q over the whole conﬁguration space. Since in some physical examples q can be interpreted as the density of the charge, we will call qˆ(0) the total charge. Note that if qˆ(0) = 0, then |ˆ q (ξ)| = O(|ξ|) around zero. Concerning the type of the infrared behavior, we easily get the following table (the number in the round brackets corresponds to the part of Theorem 3.5): Dimension of conﬁguration space

Nonzero total charge

Zero total charge

d=1

Hamiltonian undeﬁned

(2)

d=2

(3)

(1)

d=3

(2)

(1)

d≥4

(1)

(1)

Remark 5.1 As we see from the table, in dimension 3, in the nonzero charge case we get the infrared behavior of type (2). Thus the Hamiltonian is bounded from below, but the ground state is absent. This is the type of the infrared problem widely discussed in the literature [Ki]. Some authors say, however, that the type (2) behavior is an artifact of the model and disappears if one takes a more physical Hamiltonian. In fact, in [BFS2]

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it is proven that the (ultraviolet cut-oﬀ ) Hamiltonian of QED with some types of a potential (e.g., the Coulomb potential) possesses a ground state. This is related to the fact that in above considerations we considered a scalar ﬁeld, whereas photons in QED have spin one and are coupled to the charge by the minimal coupling prescription.

6 Time-dependent van Hove Hamiltonians In this section we describe a certain class of strongly continuous dynamics on the Fock space. One can say that these are the dynamics generated by a timedependent family of van Hove Hamiltonians. Let R t → g t ∈ W be a continuous vector-valued function and R t → ut ∈ U (W) be a strongly continuous function with values in unitary operators. We assume that g 0 = 0 and u0 = 1. Set V (t) := Γ(ut ) exp ig t (a∗ ) + ig t (a) . For A ∈ B(Γs (W)) we set β t (A) := V (t)AV (t)∗ . Note that

β t (w(a∗ )) = ut w(a∗ ) + i(g t |w).

Theorem 6.1 (1) V (t) is a strongly continuous family of unitary operators on Γs (W) such that V (0) = 1. (2) β t is a pointwise strongly continuous family of ∗-automorphisms of B(Γs (W)) such that β 0 is the identity. (3) V (t) is the distinguished implementation of β t in the following sense: if V˜ (t) is a family of unitary operators such that β t (A) = V˜ (t)AV˜ (t)∗ and (Ω|V˜ (t)Ω) > 0, then V˜ (t) = V (t). One can ask what is the time-dependent generator of V (t). To answer this question we proceed formally, without worrying about the exact meaning of various objects involved in our formulas. Suppose that the dot denotes the temporal derivative. It is easy to check the following identities: t ∗ t t d igt (a∗ )+igt (a) = 2i Im(g˙ t |g t ) + ig˙ t (a∗ ) + ig˙ (a) eig (a )+ig (a) , dt e d t dt Γ(u )

= dΓ(u˙ t ut∗ )Γ(ut ).

Therefore, 1 d V (t) = i Im(g˙ t |g t ) + ut g˙ t (a∗ ) + ut g˙t (a) − idΓ(u˙ t ut∗ ) V (t) . dt 2

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Now suppose that t → z t is a family of vectors and t → ht is a family of self-adjoint operators. Suppose that ut is the solution of d t u = iht ut , dt and g t :=

t

u0 = 1;

us∗ z s ds,

0

t σ t := 12 Im (z t |ut us∗ z s )ds. 0

Then

d V (t) = iH(t)V (t), dt

where

H(t) := dΓ(ht ) + z t (a∗ ) + z t (a) + σ t .

Thus, at least on a formal level, V (t) is generated by van Hove Hamiltonians. Acknowledgments. The author would like to acknowledge useful discussions with C. G´erard and S. DeBi`evre. He is also grateful to the referee for the remarks, which helped him to make some improvements in the paper. His research was partly supported by the Postdoctoral Training Program HPRN-CT-2002-0277 and the grant SPUB127 funded by KBN. A part of this work was done during his visit to the Aarhus University supported by MaPhySto funded by the Danish National Research Foundation.

References [Ar]

A. Arai, A note on scattering theory in non-relativistic quantum electrodynamics, J. Phys. A: Math. Gen. 16, 49–70 (1983).

[AHH]

A. Arai, M. Hirokawa, F. Hiroshima, On the absence of eigenvectors of Hamiltonians in a class of massless quantum ﬁeld models without infrared cutoﬀ, J. Funct. Anal. 168, 470–497 (1999).

[BFS1]

V. Bach, J. Fr¨ ohlich, I. Sigal, Quantum electrodynamics of conﬁned nonrelativistic particles, Adv. Math. 137, 299 (1998).

[BFS2]

V. Bach, J. Fr¨ ohlich, I. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation ﬁeld, Commun. Math. Phys. 207, 249 (1999).

[Be]

F. Berezin, The Method of Second Quantization, second edition, Nauka, 1986.

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[BN]

F. Bloch, A. Nordsieck, Phys. Rev. 52, 54 (1937).

[BR]

O. Brattelli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. I and II, Springer, Berlin (1981).

[DG1]

J. Derezi´ nski, C. G´erard, Asymptotic completeness in quantum ﬁeld theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11, 383 (1999).

[DG2]

J. Derezi´ nski, C. G´erard, Spectral and scattering theory of spatially cutoﬀ P (ϕ)2 Hamiltonians, Comm. Math. Phys. 213, 39–125 (2000).

[DJ]

J. Derezi´ nski, V. Jakˇsi´c, Spectral theory of Pauli-Fierz operators, Journ. Func. Analysis, 241–327 (2001).

[EP]

S. Edwards, P.E. Peierls, Field equations in functional form, Proc. Roy. Soc. Acad. 224, 24–33 (1954).

[Frie]

K.O. Friedrichs, Mathematical aspects of quantum theory of ﬁelds, New York 1953.

[Fr]

J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons an massless, scalar bosons, Ann. Inst. H. Poincar´e 19, 1–103 (1973).

[GS]

O.W. Greenberg, S.S. Schweber, Clothed particle operators in simple models of quantum ﬁeld theory, Nuovo Cimento 8, 378–406 (1958).

[HK]

R. Høegh-Krohn, Asymptotic limits in some models of quantum ﬁeld theory, I, J. Math. Phys. 9, 2075–2079 (1968).

[KM1]

Y. Kato, N. Mugibayashi, Regular perturbation and asymptootic limits of operators in quantum ﬁeld theory, Prog. Theor. Phys. 30, 103–133 (1963).

[KM2]

Y. Kato, N. Mugibayashi, Regular perturbation and asymptootic limits of operators in ﬁxed source theory, Prog. Theor. Phys. 31, 300–310 (1964).

[Ki]

T.W.B. Kibble, J. Math. Phys. 9, 15 (1968), Phys. Rev. 173, 1527 (1968); Phys. Rev. 174, 1882 (1968); Phys. Rev. 175, 1624 (1968).

[Kato]

T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin (1976).

[Mi]

O. Miyatake, On the non-existence of solution of ﬁeld equations in quantum mechanics, J. Inst. Polytech. Osaka City Univ. Ser. A Math. 2, 89–99 (1952).

[RS2]

M. Reed, B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness, London, Academic Press (1975).

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M. Reed, B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, London, Academic Press (1978).

[Schwa] A.S. Schwarz, Mathematical foundations of quantum ﬁeld theory, Atomizdat 1975, Moscow (Russian). [Sch]

S.S. Schweber, Introduction to nonrelativistic quantum ﬁeld theory, Harper and Row 1962.

[To]

S. Tomonaga, On the eﬀect of the ﬁeld reactions on the interaction of mesotrons and nuclear particles I, Progr. Theor. Phys. 1, 83–91 (1946).

[vH]

L. van Hove, Les diﬃcult´es de divergences pour un mod`ele particulier de champ quantiﬁ´e, Physica 18, 145–152 (1952).

[Ya]

D. Yafaev, Mathematical Scattering Theory, AMS.

Jan Derezi´ nski Department of Mathematical Methods in Physics Warsaw University Ho˙za 74 PL-00-682, Warszawa Poland email: [email protected] Communicated by Ch. G´erard submitted 06/12/02, accepted 17/03/03

Ann. Henri Poincar´e 4 (2003) 739 – 793 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040739-55 DOI 10.1007/s00023-003-0146-4

Annales Henri Poincar´ e

Return to Equilibrium for Pauli-Fierz Systems Jan Derezi´ nski and Vojkan Jakˇsi´c Abstract. We study ergodic properties of Pauli-Fierz systems – W ∗ -dynamical systems often used to describe the interaction of a small quantum system with a bosonic free field at temperature T ≥ 0. We prove that, for a small coupling constant uniform as the positive temperature T ↓ 0, a large class of Pauli-Fierz systems has the property of return to equilibrium. Most of our arguments are general and deal with mathematical theory of Pauli-Fierz systems for an arbitrary density of bosonic field.

1 Introduction A quantum system is often described by a W ∗ -algebra M with a σ-weakly continuous group of automorphisms t → τ t . The pair (M, τ ) is called a W ∗ -dynamical system and τ a W ∗ -dynamics. We say that the system (M, τ ) has the property of return to equilibrium if there exists a normal state ω on M such that for all normal states φ and A ∈ M, lim φ(τ t (A)) = ω(A).

|t|→∞

Such ω is obviously unique and τ -invariant. Physical intuition suggests the following quasitheorem. Quasitheorem Suppose that (M, τ ) describes a quantum system that is: (1) inﬁnitely extended; (2) a localized perturbation of a thermal equilibrium system; (3) suﬃciently regular; (4) suﬃciently generic. Then (M, τ ) has the property of return to equilibrium. Conditions (1) and (3) are idealizations necessary to prove sharp mathematical results. In particular, it is well known that ﬁnite volume (conﬁned) quantum systems do not return to equilibrium. Condition (2) is related to the issue of stability of equilibrium states (see [BR2] and references therein). It is expected on physical grounds, and in some circumstances it can been proven, that if (M, τ ) describes a localized perturbation of a physical system away from thermal equilibrium, then there are no normal τ -invariant states (see Subsections 3.6 and 7.9).

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Concerning (4), some assumptions are necessary to prevent the existence of internal symmetries which would lead to an artiﬁcial multiplicity of τ -invariant normal states. In our paper the conditions of this type will be called eﬀective coupling conditions and they will be generically satisﬁed. In this paper we will study a class of quantum systems which are commonly used to describe the interaction of a “small” quantum system, often called an “atom”, with a “bosonic reservoir”. We will call them Pauli-Fierz systems [PF]. They arise in physics as simpliﬁed versions of the non-relativistic QED. We note that in the literature the name “Pauli-Fierz Hamiltonians” appears in a number of diﬀerent (although closely related) contexts. Our deﬁnition of Pauli-Fierz systems is consistent with our previous work [DG, DJ1]. Our main result is a precise formulation of the conditions described in the “quasitheorem” and a proof that under these conditions Pauli-Fierz systems have the property of return to equilibrium. Results closely related to ours can be found in [BFS2, JP2, M] and we will discuss them in Subsection 1.2. The rest of this section is devoted to an informal discussion of our main results. In our paper the small system is described by a ﬁnite-dimensional Hilbert space K and a Hamiltonian K. The bosonic reservoir is described by a pair (Z, h) where Z and h are the Hilbert space and the energy operator of a single boson. We will always assume that h ≥ 0. Physically, the bosons can be interpreted as phonons or photons. The interaction between the small system and the reservoir is speciﬁed by a form-factor λv, where v ∈ B(K, K ⊗ Z) and λ is a real coupling constant which controls the strength of the interaction. Our main results hold for suﬃciently small nonzero values of λ. The data (K, K, Z, h, v) determine the basic Pauli-Fierz Hamiltonian, which is deﬁned as the self-adjoint operator H = K ⊗ 1 + 1 ⊗ dΓ(h) + λV on the Hilbert space H = K ⊗ Γs (Z), where Γs (Z) is the bosonic Fock space over the 1-particle space Z and the interaction term V is the ﬁeld operator associated to the form-factor v. Thus we obtain the W ∗ -dynamical system (1.1) B(H), eitH · e−itH . The W ∗ -dynamical system (1.1) is however not our main object of study. We are interested in a family of W ∗ -dynamical systems that arise as thermodynamical limits of (1.1) and which describe Pauli-Fierz systems with non-zero radiation ﬁeld density. Apart from (K, K, Z, h, v), these systems are parameterized by a positive operator (the radiation density operator) ρ on Z commuting with h. We call them Pauli-Fierz systems at density ρ. To describe such systems one needs to use the so-called Araki-Woods representations of CCR [AW, BR2]. In typical cases, for instance if ρ has some continuous spectrum, the corresponding W ∗ -algebras are of

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type III. The system (1.1) corresponds to the density ρ = 0 and its W ∗ -algebra B(H) is of type I. A special class of radiation densities is given by Planck’s law at inverse temperature β, that is ρβ = (eβh − 1)−1 . Such densities describe a system close to thermal equilibrium at temperature 1/β. A large part of our paper is not restricted to the thermal case and deals with an arbitrary radiation density. These results are useful in the study of non-equilibrium theory of Pauli-Fierz systems. For shortness, in the remaining part of the introduction we restrict ourselves to the case of thermal densities. The main object of our study is a 1-parameter family of W ∗ -dynamical systems (Mβ , τβ )

(1.2)

AW ∗ where Mβ = B(K) ⊗ MAW β,l , and Mβ,l is the Araki-Woods W -algebra corresponding to the density ρβ . The dynamics is deﬁned in a canonical way given the data (K, K, Z, h, v) and the inverse temperature β ∈]0, ∞]. For β = ∞ the system (1.2) coincides with (1.1). Under the conditions used in our paper the W ∗ -dynamical systems (1.2) are non-equivalent for distinct β. The Pauli-Fierz systems (1.2) considered in our paper satisfy the ﬁrst two conditions of the “quasitheorem”. They describe an inﬁnitely extended system (this is expressed in particular by the fact that h has continuous spectrum). Since the radiation density of the bosonic ﬁeld is given by the Planck law, the system is near thermal equilibrium. The information on the W ∗ -dynamics τβ is conveniently encoded by a certain self-adjoint operator Lβ called the Liouvillean. The operator Lβ is canonically deﬁned in the standard representation of Mβ . For β < ∞ (positive temperatures), under quite broad conditions one can show the existence of a (τβ , β)-KMS vector, which is an eigenvector of Lβ with a zero eigenvalue. This result was proven in [DJP] and is based on an extension of the well-known result of Araki [Ar, BR2]. For β = ∞ (the zero temperature) in many circumstances one can show that the Pauli-Fierz Hamiltonian H has a ground state [AH, BFS1, Ge, Sp2, Sp3]. The ground state gives rise to an eigenvector of the corresponding Liouvillean L∞ with a zero eigenvalue. For β < ∞, the return to equilibrium can be deduced from spectral properties of Lβ . In particular, the return to equilibrium follows if Lβ has no singular spectrum except for a nondegenerate eigenvalue at zero. For a W ∗ -dynamical system (M, τ ) with M being a type I factor, the return to equilibrium never holds (unless the algebra is 1-dimensional). Therefore, the Pauli-Fierz systems with β = ∞ do not have the return to equilibrium property. In the literature [HS1, FGS], one can ﬁnd a related property called the relaxation to a ground state, which in some cases can be proven for zero temperature systems. Note, however, that to prove this property one needs to consider appropriate C ∗ -dynamical systems, whereas we always consider W ∗ -dynamical systems.

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The regularity assumption on Pauli-Fierz systems that we make is based on the ideas and results of [JP1, JP2]. This method consists in “gluing” together negative and positive frequencies of the bosons, which allows to deﬁne a “conjugate operator” – the generator of translations in the spectral variables. In the original approach of [JP1, JP2] the analyticity of the form-factor with respect to the translations was assumed and the Liouvillean was studied using the complex deformation method. Here, we assume only the diﬀerentiability of a suﬃciently high order and study the spectrum using the Mourre theory developed for this purpose in our previous paper [DJ1]. The Mourre theory allows us to treat Pauli-Fierz systems more eﬃciently, especially at low temperatures. To express our eﬀective coupling assumptions we use some simple algebraic conditions derived from the so-called Fermi Golden Rule, which describes how to compute eigenvalues and resonances to the second order. In particular, it can be used to predict which eigenvalues will disappear after the interaction is “switched on”. The information obtained by the Fermi Golden Rule can be conveniently encoded in the so-called Level Shift Operator Γβ – an object that plays a crucial role in our paper. The Liouvillean of a Pauli-Fierz system in the absence of interaction has a large kernel (of dimension at least dim K). After the interaction is “switched on”, the dimension of the kernel of Lβ is at least one. Our aim is to show that there are no other eigenvectors of Lβ for small nonzero λ. For small nonzero λ and all β ∈]0, ∞[, that is for the whole range of positive temperatures, analysis of the Level Shift Operator Γβ gives a single condition that on the formal level indicates the absence of the singular spectrum of Lβ except for a nondegenerate eigenvalue at zero. In order to check this condition one constructs a certain (ﬁnite-dimensional) ∗-algebra N ⊂ B(K) which depends only on (K, K, Z, h, v) and not on the inverse temperature β. Our positive temperature effective coupling assumption is that N = C1. The result of [DJ1] provides a rigorous method to show that the above assumption together with a suﬃcient regularity of the form-factor imply the desired spectral properties of Lβ , and hence imply the return to equilibrium. The result described so far is, however, not uniform in the temperature. Our main goal is to show that under suitable conditions Pauli-Fierz systems have the property of return to equilibrium uniformly in the temperature. This requires a detailed analysis of the zero temperature case, which is in many respects diﬀerent from that of positive temperatures. Analysis of the Level Shift Operator Γ∞ yields natural eﬀective coupling conditions under which one should expect that for a suﬃciently small nonzero λ the Pauli-Fierz Hamiltonian H has no singular spectrum except for a nondegenerate ground state. These conditions involve the nondegeneracy of the unperturbed ground state and the strict positivity of a certain auxiliary operator. The result of [DJ1] gives a rigorous proof that these conditions together with a suﬃcient regularity of the form-factor imply the desired properties of Pauli-Fierz Hamiltonians. As an immediate consequence, under the

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same conditions zero temperature Liouvilleans have no singular spectrum except possibly for a nondegenerate eigenvalue at zero. If the zero temperature and the positive temperature eﬀective coupling assumptions hold and if the form-factor is suﬃciently regular, then we can establish return to equilibrium uniformly in the temperature. More precisely, we show that for any β0 > 0 there is λ0 > 0 such that for 0 < |λ| < λ0 and β ∈ [β0 , ∞[ the Pauli-Fierz Liouvillean Lβ has no singular spectrum except for a nondegenerate zero eigenvalue. It follows that under the same conditions the system (Mβ , τβ ) has the property of return to equilibrium and this is the main result of our paper. We emphasize that the above-mentioned eﬀective coupling conditions are important ingredients of our approach. These conditions are optimal if one considers only the 2nd order perturbation theory. They are quite simple algebraically and it is perhaps surprising that a single eﬀective coupling condition applies to all positive temperatures. Similar results can be given if we consider radiation densities that are not necessarily given by the Planck law. For instance, we show that if the small system interacts with two bosonic reservoir at distinct temperatures, then generically the coupled system has no normal time-invariant states.

1.1

Organization of the paper

In Section 2 we brieﬂy review the deﬁnitions and results of the theory of W ∗ algebras needed in our paper. In particular, we quote the results of [DJP]. In Section 3 we give a simpliﬁed presentation of our main results. To make the paper more accessible, in this section we restrict ourselves to the case of a scalar massless ﬁeld. This section is not used in the remaining part of the paper, where a more general class of models is considered and a diﬀerent notation is used. Section 3 serves as a quick introduction to our results and allows to compare them easily with the results existing in the literature. In Section 4 we introduce the notation and review some basic facts which we will need in the paper. In Section 5 we introduce Pauli-Fierz operators and review their properties following [DJ1]. In Section 6 we introduce Pauli-Fierz systems at density ρ, compute PauliFierz Liouvilleans Lρ and study their properties. The main technical results of this section concern the structure of the Level Shift Operator Γρ of the Liouvillean Lρ . In Section 7 we discuss thermal Pauli-Fierz systems. In Subsection 7.4 we give conditions under which thermal Pauli-Fierz systems have the property of return to equilibrium for a ﬁxed inverse temperature. The result uniform in the temperature is described in Subsection 7.7. As we have already mentioned, the main regularity assumption our method requires concerns the gluing condition of [JP1, JP2, DJ1]. In Section 8 we discuss the gluing condition in the context of scalar and vector massless bosons.

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Comparison with the literature

Hamiltonians similar to those considered in our paper appear frequently in the physics literature, see, e.g., [PF]. In the recent years there has been a revival of interest in rigorous results about these operators, starting with such papers as [BFS1, DG, HS2, JP1, Sk]. From the technical point of view, the results of our paper concern mainly spectral properties of a certain class of Pauli-Fierz operators. A large part of the literature on spectral analysis of Pauli-Fierz operators can be divided into two classes. The ﬁrst uses the generator of translations as the main tool and the second uses the generator of dilations. In the context of Pauli-Fierz systems, the generator of translations was used for the ﬁrst time in [JP1, JP2], where it was applied to deform analytically positive temperature Pauli-Fierz Liouvilleans. These papers also contain the ﬁrst proof that thermal Pauli-Fierz systems have the property of return to equilibrium. The inﬁnitesimal version of this method based on the Mourre theory was developed in [DJ1]. That paper was a technical preparation for the present paper. In fact, in the introduction to [DJ1] we roughly described the applications contained in this paper (without, however, giving exact conditions). The generator of translations is also the main tool of an interesting paper by Merkli [M], which is devoted to the proof of return to equilibrium in the mean. This paper is based on the technique of a “modiﬁed conjugate operator” originally due to H¨ ubner and Spohn [HS2] and elaborated later in [BFSS]. The results of [M] are closely related to ours. One of the diﬀerences is that Merkli studies return to equilibrium in the mean and he does not show the absence of singular continuous spectrum for Pauli-Fierz Liouvilleans. His proof is based on the virial theorem, whereas the method of [DJ1] is based on the limiting absorption principle. Merkli’s main result is not uniform in the temperature. In the context of Pauli-Fierz systems with positive mass the generator of dilations was used for the ﬁrst time in [OY]. In the zero temperature massless case it was used ﬁrst in [BFS1]. In [BFS2] the generator of dilations was used to study return to equilibrium of Pauli-Fierz systems uniformly in the temperature. A distinctive feature of the papers [BFS1, BFS2] is the so-called renormalization group technique, which in this context is meant to describe an iterative procedure based on the Feshbach method, used to control the spectrum of Pauli-Fierz operators. The results of [BFS2] resemble closely ours and Merkli’s. Strictly speaking, however, the conditions of [BFS2] are not comparable to ours and one can ﬁnd interactions which can be treated with one method and not by the other. With regard to the infrared singularity, the conditions of [BFS2] for the uniform in temperature return to equilibrium are somewhat less restrictive than ours. There is a vast body of literature dealing with Pauli-Fierz systems in the Van Hove weak coupling limit s = λ2 t, λ ↓ 0, with s ﬁxed (see, e.g., [Da, Sp2]). In this limit one obtains an irreversible Markovian dynamics on the algebra of the small system. The generator of this dynamics is sometimes called the Davies

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generator. The Level Shift Operator, which arises through the Fermi Golden Rule for the Liouvillean and is one of the main tools of our paper, is similar to the Davies generator in many respects. Let us stress, however, that they are diﬀerent operators. The Davies generator describes the evolution of observables and always has a zero eigenvalue. On the other hand, the Level Shift Operator describes the shift of eigenvalues and resonances of the Liouvillean and often does not have a zero eigenvalue. In the thermal case, however, these operators are closely related, see [DJ2]. The eﬀective coupling assumptions for return to equilibrium used in our paper are diﬀerent from those found in the literature. To our knowledge they are simpler and less restrictive – in fact, they are optimal in the context of the 2nd order perturbation theory. They are based on a detailed algebraic analysis of the Level Shift Operator for Pauli-Fierz Liouvilleans that seems to appear for the ﬁrst time in the literature. Somewhat similar eﬀective coupling conditions for return to equilibrium of quantum Markovian semigroups were given in [Fr, Sp2]. One of the consequences of our method is a relatively simple proof of the nondegeneracy of the ground state of Pauli-Fierz Hamiltonians under certain regularity and eﬀective coupling assumptions. The other proofs in the literature use Perron-Frobenius type arguments and are restricted to positivity preserving interactions [BFS1, Sp1, Sp3]. The only exception that we know is the proof based on the “renormalization group” contained in [BFS1]. The ﬁrst result about existence of KMS states for Pauli-Fierz systems goes back to [FNV] where the spin-boson system was considered. It was also proven in [BFS2] under a more restrictive infrared condition than that of our paper. Our result about a system coupled to several reservoirs at diﬀerent temperatures can be compared with recent works on non-equilibrium quantum statistical physics [JP3, Ru]. Note that these papers use C ∗ -dynamical systems rather than W ∗ -dynamical systems and look for stationary states that are not normal.

2 Algebraic preliminaries In this section we review some elements of the theory of W ∗ -algebras needed in our paper. For more details we refer the reader to [DJP], and also [BR1, BR2, St]. One of the most important concepts of the modern theory of W ∗ -algebras is the so-called standard representation. We say that a quadruple (π, H, J, H+ ) is a standard representation of a W ∗ -algebra M if π : M → B(H) is a ∗-representation, J is an antiunitary involution on H and H+ is a self-dual cone in H satisfying the following conditions: (1) (2) (3) (4)

Jπ(M)J = π(M) ; Jπ(A)J = π(A)∗ for A in the center of M; JΨ = Ψ for Ψ ∈ H+ ; π(A)Jπ(A)H+ ⊂ H+ for A ∈ M. Every W ∗ -algebra has a unique (up to

unitary equivalence) standard representation.

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The standard representation has several important properties. First, every normal state ω has a unique vector representative in H+ (there is a unique normalized vector Ω ∈ H+ such that ω(A) = (Ω|π(A)Ω)). Secondly, for every W ∗ dynamics τ on M there is a unique self-adjoint operator L on H such that π(τ t (A)) = eitL π(A)e−itL ,

eitL H+ = H+ .

(2.3)

The operator L is called the Liouvillean of the W ∗ -dynamical system (M, τ ). Theorem 2.1 Let ω be a normal state and Ω ∈ H+ its vector representative. Then ω is τ -invariant iﬀ Ω ∈ KerL. Theorem 2.2 (1) The Liouvillean L has no eigenvalues iﬀ the W ∗ -dynamics τ has no normal invariant states. (2) The Liouvillean L has exactly one nondegenerate eigenvalue at zero iﬀ the W ∗ -dynamics τ has a single normal invariant state. (3) Suppose L has no singular spectrum except for a nondegenerate eigenvalue at zero, and that the corresponding eigenstate is separating for M. Then the system (M, τ ) has the property of return to equilibrium. Theorem 2.1 follows easily from (2.3). Theorem 2.2 (3) was proven in [JP1] although similar results can be traced to much older literature (see [BR1, Ja]). We now describe some results concerning perturbation theory of W ∗ dynamical systems. Our presentation follows [DJP]. Let (M, τ ) be a W ∗ -dynamical system and (π, H, J, H+ ) a standard representation of M. Let L be the Liouvillean of τ . Let V be a self-adjoint operator aﬃliated to M. Let us state the following assumption: Assumption 2.A L + π(V ) is essentially self-adjoint on D(L) ∩ D(π(V )) and LV := L + π(V ) − Jπ(V )J is essentially self-adjoint on D(L) ∩ D(π(V )) ∩ D(Jπ(V )J). Theorem 2.3 [DJP] Assume that 2.A holds and set τVt (A) := π −1 eit(L+π(V )) π(A)e−it(L+π(V ) . Then τV is a W ∗ -dynamics on M and LV is the Liouvillean of (M, τV ). Our ﬁnal subject is the perturbation theory of KMS states. We will describe the results of [DJP], which extend the well-known results of Araki [Ar, BR2] valid for bounded perturbations. Let ω be a (τ, β)-KMS state and Ω ∈ H+ its vector representative. We will call Ω a (τ, β)-KMS vector (or a β-KMS-vector for the dynamics τ ). We make the following additional assumption on the perturbation V :

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Assumption 2.B

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e−βπ(V )/2 Ω < ∞.

Theorem 2.4 [DJP] Assume that Assumptions 2.A and 2.B hold. Then (1) Ω ∈ D(e−β(L+π(V ))/2 ) and the vector ΩV := e−β(L+π(V ))/2 Ω is a (τV , β)-KMS vector. (2) Let ωV (A) = (ΩV |π(A)ΩV )/ΩV 2 . Then ωV is a (τV , β)-KMS state on M. Note that if V ∈ M, then Assumptions 2.A and 2.B are automatically satisﬁed, and the above theorems reduce to the well-known results of Araki [Ar, BR2].

3 Simplified presentation of the main results This section gives a self-contained description of simpliﬁed versions of our main results. It will not be used in the remaining part of the paper. The reader who prefers a more complete exposition can skip this section and go directly to Section 4. In this section the 1-particle bosonic space is Z = L2 (Rd ) and the 1-particle energy h is the operator of multiplication by |ξ|, where ξ ∈ Rd describes the momentum. The small system is described by a ﬁnite-dimensional Hilbert space K and a Hamiltonian K. The interaction is described by a measurable operatorvalued function (form-factor) Rd ξ → v(ξ) ∈ B(K).

3.1

Pauli-Fierz system at zero temperature

The Hilbert space of the Pauli-Fierz system at zero temperature is K⊗Γs (L2 (Rd )), where Γs (L2 (Rd )) denotes the symmetric (bosonic) Fock space over the 1-particle space L2 (Rd ). The free Pauli-Fierz Hamiltonian is Hfr := K ⊗ 1 + 1 ⊗ |ξ|a∗ (ξ)a(ξ)dξ, where a∗ (ξ)/a(ξ) are the creation/annihilation operators of bosons of momentum ξ ∈ Rd . We assume that the form-factor satisﬁes (1 + |ξ|−1 )v(ξ)2 dξ < ∞. (3.4) The interaction is given by the operator V := (v(ξ) ⊗ a∗ (ξ) + v ∗ (ξ) ⊗ a(ξ))dξ, and the full Pauli-Fierz Hamiltonian equals H := Hfr + λV, where λ ∈ R. H is self-adjoint on D(Hfr ) and bounded from below.

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We have discussed in [DJ1] how Pauli-Fierz Hamiltonians arise as an approximation to the standard Hamiltonian of the non-relativistic QED. A related discussions can be found in [BFS1]. A simplest non-trivial example of a Pauli-Fierz Hamiltonian is the so-called spin-boson model where K = C2 , K = σz and v(ξ) = σx α(ξ) (σz and σx are the usual Pauli matrices and α ∈ L2 (Rd ) satisﬁes (3.4)).

3.2

Bosonic fields at non-zero density

Assume that the radiation density of the bosonic ﬁeld is described by a positive measurable function ρ(ξ) on Rd . The observables of the bosonic reservoir are then described by the W ∗ -algebra MAW ρ,l , the (left) Araki-Woods algebra at density ρ. This algebra is constructed as follows. It is represented on the Hilbert space Γs (L2 (Rd ) ⊕ L2 (Rd )). The creation/annihilation operators corresponding to the ﬁrst L2 (Rd ) (which describe excitations) are denoted by a∗l (ξ)/al (ξ), and those corresponding to the second L2 (Rd ) (describing holes) are denoted by a∗r (ξ)/ar (ξ). (l/r stand for left/right). MAW ρ,l is generated by the operators of the form exp i

1 1 (f (ξ)(1 + ρ(ξ)) 2 a∗l (ξ) + f (ξ)ρ(ξ) 2 ar (ξ) + hc)dξ ,

where f ∈ L2 (Rd ) satisﬁes

3.3

|f (ξ)|2 ρ(ξ)dξ < ∞.

Pauli-Fierz systems at non-zero density

The Pauli-Fierz algebra at density ρ, Mρ , is deﬁned by Mρ := B(K) ⊗ MAW ρ,l . To deﬁne the dynamics, we need the following assumption:

Assumption 3.A

(1 + |ξ|2 )(1 + ρ(ξ))v(ξ)2 dξ < ∞.

Set Lsemi fr

:= K ⊗ 1 + 1 ⊗

Vρ :=

|ξ|a∗l (ξ)al (ξ) − |ξ|a∗r (ξ)ar (ξ) dξ,

1 1 v(ξ) ⊗ (1 + ρ(ξ)) 2 a∗l (ξ) + ρ(ξ) 2 ar (ξ) dξ + hc, Lsemi := Lsemi + λVρ . ρ fr

(3.5)

Proposition 3.1 Assume that Assumption 3.A holds. Then the operator Lsemi is ρ essentially self-adjoint on D(Lfr ) ∩ D(Vρ ) and semi

τρt (A) := eitLρ is a W ∗ -dynamics on Mρ .

semi

Ae−itLρ

(3.6)

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We will call the W ∗ -dynamical system (Mρ , τρ ) the Pauli-Fierz system at density ρ. In the absence of interaction (λ = 0) we call it a free Pauli-Fierz system. The identity representation Mρ → B(K ⊗ Γs (L2 (Rd ) ⊕ L2(Rd )) will be called the semi-standard representation of the Pauli-Fierz system, to distinguish it from the standard representation described in the next subsection. Similarly, we will the Pauli-Fierz semi-Liouvillean at density ρ. call the operator Lsemi ρ

3.4

Pauli-Fierz systems in standard representation

Let K be a Hilbert space complex conjugate to K. The standard representation of the algebra Mρ is realized on the Hilbert space H := K ⊗ K ⊗ Γs (L2 (Rd ) ⊕ L2 (Rd )), and for B ⊗ C ∈ B(K) ⊗ MAW ρ,l , π(B ⊗ C) = B ⊗ 1K ⊗ C. For the description of the cone Hρ,+ and the modular conjugation J we refer the reader to Subsection 6.6. Note that Vρ is aﬃliated to Mρ and π(Vρ ) =

1 1 v(ξ) ⊗ 1 ⊗ (1 + ρ(ξ)) 2 a∗l (ξ) + ρ(ξ) 2 ar (ξ) dξ + hc,

Jπ(Vρ )J =

1 1 1 ⊗ v(ξ) ⊗ (1 + ρ(ξ)) 2 a∗r (ξ) + ρ(ξ) 2 al (ξ) dξ + hc.

The Liouvillean of the free Pauli-Fierz system is Lfr = K ⊗ 1 ⊗ 1 − 1 ⊗ K ⊗ 1 + 1 ⊗ 1 ⊗ |ξ|a∗l (ξ)al (ξ) − |ξ|a∗r (ξ)ar (ξ) dξ. Set Lρ := Lfr + λπ(Vρ ) − λJπ(Vρ )J. Proposition 3.2 Assume that Assumption (3.A) holds. Then the operator Lρ is essentially self-adjoint on D(Lfr ) ∩ D(π(Vρ )) ∩ D(Jπ(Vρ )J) and is the Liouvillean of the Pauli-Fierz system (Mρ , τρ ). Let us note that from the mathematical point of view Pauli-Fierz Hamiltonians, semi-Liouvilleans and Liouvilleans belong to the class of operators that we call Pauli-Fierz operators. This class of operators has been studied in detail in our previous paper [DJ1].

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Thermal Pauli-Fierz systems

Let β > 0 be the inverse temperature. A Pauli-Fierz system whose radiation density is given by the Planck law ρβ (ξ) = (eβ|ξ| − 1)−1 is called a thermal PauliFierz system at inverse temperature β. Due to the speciﬁc form of the Planck law Assumption 3.A takes a somewhat simpler form and is equivalent to:

Assumption 3.B

(|ξ|2 + |ξ|−1 )v(ξ)2 dξ < ∞.

With a slight abuse of the notation instead of the subscript ρβ we will use β, so Lβ and τβt now stand for Lρβ and τρtβ etc. Using the main result of [DJP] described in Theorem 2.4 (see Theorem 7.3) one can easily show Theorem 3.3 Assume that Assumption 3.B holds. Then for all λ ∈ R and β ∈ ]0, ∞[ the Pauli-Fierz system (Mβ , τβ ) has a unique β-KMS state.

3.6

Main results

In this subsection we state simpliﬁed versions of our main results, described more precisely and proved in Section 7. We use the following notation. sp(K) denotes the spectrum of K and k0 = inf sp(K). The spectral projection of K onto k ∈ R will be denoted by 1k (K) and v k1 ,k2 (ξ) = 1k1 (K)v(ξ)1k2 (K). Obviously, v k1 ,k2 (ξ) = 0 unless k1 , k2 ∈ sp(K). p ∈ R+ denotes the radial coordinate. S d−1 is the d − 1dimensional unit sphere, ω ∈ S d−1 is the angle coordinate and dω is the surface measure on S d−1 . Let F + be the set of positive diﬀerences of eigenvalues of K. (In physical terms, these are the Bohr frequencies of the small system – the energies of photons that can be emitted.) An important role will be played by a certain subset N of bounded operators on K deﬁned as follows: B ∈ B(K) belongs to N iﬀ for almost all ω ∈ S d−1 we have v k−p,k (pω) = v k−p,k (pω)B, p ∈ F +, B k∈sp(K)

B∗

k∈sp(K)

v k−p,k (pω) =

k∈sp(K)

B

k∈sp(K)

lim p−1/2 v k,k (pω) = p↓0

v k−p,k (pω)B ∗ ,

p ∈ F + , (3.7)

k∈sp(K)

k∈sp(K)

lim p−1/2 v k,k (pω)B. p↓0

Obviously, 1 ∈ N. Note also that N is a ∗-algebra invariant wrt eitK · e−itK . We start with a result which does not hold uniformly in the temperature.

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Theorem 3.4 Suppose that Assumption 3.B holds and the following conditions are satisﬁed: (1) ∂p3 p−1+d/2 p1/2 v(pω)2 dpdω < ∞, ∂pj p−1+d/2 v(pω) = (−1)j ∂pj p−1+d/2 v ∗ (pω) , j = 0, 1, 2, ω ∈ S d−1 . p=0

p=0

N = C1.

(2)

Then for any 0 < β < ∞ there exists λ0 (β) > 0 such that for 0 < |λ| < λ0 (β) the Pauli-Fierz Liouvillean Lβ has no singular spectrum except for a simple eigenvalue at zero. Consequently, under the above conditions the system (Mβ , τβ ) has the property of return to equilibrium. Condition (1) is our regularity assumption. Note that it allows for quite singular infrared behavior of the form-factor. For example, assume that v(ξ) is smooth outside of zero and of compact support. Then (1) holds if around zero v(ξ) = v0 |ξ|1−d/2 ,

(3.8)

where v0 ∈ B(K) is self-adjoint. In particular, Theorem 3.4 applies to models derived from QED in the so-called ohmic case (see Subsection 8.1). Condition (2) is our positive temperature eﬀective coupling assumption. Note that it does not depend on the temperature. The next theorem holds uniformly in the temperature. Theorem 3.5 Suppose that Assumption 3.B holds and the following conditions are satisﬁed: (1) p∂p3 p−1+d/2 v(pω)2 dpdω < ∞; p−5+2j ∂pj p−1+d/2 v(pω)2 dpdω < ∞, j = 0, 1, 2; ∂pj p−1+d/2 v(pω) = (−1)j ∂pj p−1+d/2 v ∗ (pω) , j = 0, 1, 2, ω ∈ S d−1 . p=0

p=0

(2) N = C1. (3) dim 1k0 (K) = 1 (the operator K has a nondegenerate smallest eigenvalue). (4) There exists c > 0 such that for all k ∈ sp(K), k = k0 , (v ∗ )k,k−p (pω)v k−p,k (pω)dω ≥ c1k (K). p>0

S d−1

Then for any 0 < β0 < ∞ there exists λ0 > 0 such that for 0 < |λ| < λ0 and β ∈ ]β0 , ∞[ the Pauli-Fierz Liouvillean Lβ has no singular spectrum except for a simple eigenvalue at zero. Consequently, under the above conditions the system (Mβ , τβ ) has the property of return to equilibrium.

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In comparison with Theorem 3.4, in Theorem 3.5 we need two additional eﬀective coupling assumptions (3) and (4). Note also that the regularity assumption (1) of Theorem 3.5 is much stronger than that of Theorem 3.4. For example, assume that v(ξ) is smooth away from zero and of compact support. Then (1) of Theorem 3.5 holds if around zero we have v(ξ) = v0 |ξ|α , where v0 ∈ B(K) and α > (7 − d)/2 (compare with (3.8)). Let us mention that our formalism can be applied to non-thermal radiation densities. For instance, if the small system interacts with several reservoirs at distinct temperatures, each satisfying the conditions of Theorem 3.4, then the Liouvillean has no singular spectrum. Consequently, under these assumptions the Pauli-Fierz system has no normal states.

4 Basic notation and facts 4.1

Miscellanea

We set C+ := {z ∈ C : Imz > 0}. Throughout the paper S cl denotes the closure of 2 1/2 . a set S, so Ccl + = {z ∈ C : Imz ≥ 0}. We will use the shorthand x := (1 + x ) −1 −1 δ(p) denotes the Dirac delta at 0, Pp the principal value of p , and (p+i0)−1 := lim↓0 (p+i )−1 . We will sometimes use the so-called Sochocki formula: (p + i0)−1 = Pp−1 − iπδ(p).

4.2

Operators in Hilbert spaces

Let H be a Hilbert space with the scalar product (Ψ|Φ), Ψ, Φ ∈ H. B(H), B+ (H) and U (H) denote the set of bounded, bounded positive and unitary operators on H. l2 (H) will denote the Hilbert space of Hilbert-Schmidt operators on H with 2 the scalar product (A|B) = Tr(A∗ B). l+ (H) is the set of positive Hilbert-Schmidt operators. If Ψ ∈ H, then |Ψ) and (Ψ| denote respectively the operators C λ → λΨ ∈ H,

H Φ → (Ψ|Φ) ∈ C.

Obviously, (Ψ| := |Ψ)∗ . If Ψ = 1, then |Ψ)(Ψ| is the orthogonal projection onto the subspace spanned by Ψ. sp(A) denotes the spectrum of a closed operator A on H. If Θ is an isolated bounded subset of sp(A) (closed and open in the relative topology of sp(A)), then 1Θ (A) denotes the spectral (Riesz) projection of A onto Θ. If A is self-adjoint and Θ is a Borel subset of R, then 1Θ (A) denotes the spectral projection of A onto Θ. 1p (A) denotes the projection onto the subspace spanned by the eigenvectors of A. 1ac (A) and 1sc (A) := 1 − 1ac (A) − 1p (A) denote respectively the projections onto the absolutely continuous and the singular continuous part of the spectrum of A. spp (A), spac (A), spsc (A) denote respectively

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the point spectrum (the set of eigenvalues), the absolutely continuous spectrum and the singular continuous spectrum of A. If z ∈ sp(A) is an isolated point of sp(A), or A is self-adjoint, we will write 1z (A) instead of 1{z} (A). We denote the real and imaginary part of A ∈ B(H) by AR :=

1 (A + A∗ ), 2

AI :=

1 (A − A∗ ). 2i

Clearly, A = AR + iAI . A is called dissipative if AI ≤ 0.

4.3

Level Shift Operator

In the physics literature, the formulas for computing 2nd order corrections for eigenvalues and especially resonances often go under the name of the Fermi Golden Rule. In this subsection we will introduce an operator, sometimes called the Level Shift Operator, that can be used to formalize the Fermi Golden Rule. Suppose that H is a Hilbert space with a distinguished ﬁnite-dimensional subspace Hv . We set Hv := (Hv )⊥ . We will often use 2 × 2 matrix notation for operators on H = Hv ⊕ Hv . For example, any A ∈ B(H) can be written as

Avv Avv . (4.9) A= Avv Avv Suppose that Lfr is a self-adjoint operator that leaves Hv invariant. Then

0 Lvv fr Lfr = . (4.10) vv 0 Lfr For A ∈ B(Hv ) and e1 , e2 ∈ R we set vv Ae1 ,e2 := 1e1 (Lvv fr )A1e2 (Lfr ).

Let Q be a self-adjoint operator on H such that Qvv = 0 and Qvv is bounded. Let

vv −1 vv ) Q , w(z) := Qvv (z1vv − Lfr

Assume that for all e ∈

sp(Lvv fr )

vv z ∈ sp(Lfr ).

(4.11)

the limit

lim w(e + i )ee =: w(e + i0)ee ↓0

exists and set Γ :=

w(e + i0)ee .

e∈sp(Lvv fr )

We will call Γ the Level Shift Operator associated to the triple (Hv , Lfr , Q).

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Note that Γ is a dissipative operator (in general, Γ is not self-adjoint), Γe1 ,e2 = R R vv vv I I vv 0 for e1 = e2 , Lvv fr Γ = Γ Lfr and Lfr Γ = Γ Lfr . Let us now describe applications of the Level Shift Operator. Assume that Lvv + Qvv is essentially self-adjoint on D(Lvv ) ∩ D(Qvv ). Then we can deﬁne the self-adjoint operator L = Lfr + λQ, where λ ∈ R is a coupling constant. The Level Shift Operator Γ can be used to describe some properties of the operator L for a small coupling constant. First of all, if we make appropriate analyticity assumptions similar to those of [JP1, JP2], then the operator 2 Lvv fr + λ Γ

(4.12)

can be used to predict the approximate location and the multiplicities of eigenvalues and resonances of L for small λ. If we do not make analyticity assumptions, then we cannot deﬁne the notion of resonance. Still, the Level Shift Operator can be used to study eigenvalues of L. In particular, in [DJ1] we proved that for a certain class of Pauli-Fierz operators L and for a small nonzero λ, the operator 2 1e (Γ)(Lvv (4.13) fr + λ Γ) m∈sp(Γ)∩R

predicts the approximate location of eigenvalues of L and that the estimate dim 1p (L) ≤ dim KerΓI gives an upper bound on their multiplicity. These results will be described in Subsection 5.4.

4.4

Space L2 (R)

In this subsection we describe some operators acting on L2 (R). Let r denote the self-adjoint operator of multiplication by the variable in R, rΨ (p) := pΨ(p). Throughout this paper, in the context of the space L2 (R) the generic name for a variable in R will be p. On the other hand, the multiplication operator on L2 (R) by its natural variable will be denoted by r. We denote by s the self-adjoint operator sΨ(p) := −i∂p Ψ(p), and by C(R) the set of all continuous functions on R.

(4.14)

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For p ∈ R, deﬁne an operator πp : C(R) → R by πp f := f (p). It is well known that, for η > 1/2, D(sη ) is a subset of C(R). Hence the operator πp s−η is well deﬁned on L2 (R). The following two results are well known. Proposition 4.1 Let η > 1/2. Then (1) The functional πp s−η is bounded on L2 (R). (2) The map R p → πp s−η ∈ B(L2 (R), C) is continuous. Proposition 4.2 Let n be a positive integer and η > n − 12 . Then the function C+ z → s−η (z − r)−n s−η ∈ B(L2 (R)) extends from C+ to a continuous function on Ccl +. Let η > 1/2. In what follows, the functional πp s−η will be often used in the following context. Let G be a Hilbert space. Obviously, G ⊗ C = G. Hence, we can introduce the family of maps 1G ⊗ πp s−η : G ⊗ L2 (R) → G.

(4.15)

Clearly, the maps (4.15) are bounded and depend continuously on p.

4.5

Space L2 (R, G)

Let G be a Hilbert space (not necessarily separable). We say that a function R p → Ψ(p) ∈ G belongs to L2 (R, G) iﬀ (1) There exists a separable subspace G0 such that Ψ(p) ∈ G0 for all p ∈ R. (2) For any Φ ∈ G, the function R p → (Φ|Ψ(p)) ∈ C is measurable. (3) Ψ(p)2 dp < ∞. Let N (R, G) be the set of all Ψ ∈ L2 (R, G) such that Ψ(p) = 0 for almost all p ∈ R and L2 (R, G) := L2 (R, G)/N (R, G). There exists a unique unitary operator G ⊗ L2 (R) → L2 (R, G)

(4.16)

such that Ψ ⊗ f ∈ G ⊗ L2 (R) is mapped onto p → f (p)Ψ. Let K be another Hilbert space and q ∈ B(K, G ⊗ L2 (R)). Suppose that for some η > 1/2, 1G ⊗sη q ∈ B(K, G ⊗ L2 (R)). Then for p ∈ R we can deﬁne q(p) := 1G ⊗πp q = 1G ⊗πp s−η 1G ⊗sη q ∈ B(K, G). Clearly R p → q(p) ∈ B(K, G)

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is a continuous function. Note that for Ψ ∈ K, the vector qΨ ∈ G⊗L2 (R) can be identiﬁed through (4.16) with the function p → q(p)Ψ in L2 (R, G). For any f ∈ L∞ (R) the following identity holds: ∗ q f (r)q = q ∗ (p)f (p)q(p)dp. (4.17) Note the estimate ∗

q q ≤

4.6

∗

q (p)q(p)dp =

q(p)2 dp.

(4.18)

Conjugate Hilbert spaces

Let K be a Hilbert space. The space K conjugate to K is any Hilbert space with a distinguished antiunitary map K Ψ → Ψ ∈ K.

(4.19)

The map (4.19) is called the conjugation on K. By the Riesz lemma, the map K Ψ → (Ψ| ∈ B(K, C) is an isomorphism between K and B(K, C) = K∗ . The inverse of the map (4.19), which is a conjugation on K, will be denoted by the same symbol. Hence K = K and Ψ = Ψ. If A ∈ B(K), then A ∈ B(K) is deﬁned by K Ψ → A Ψ := AΨ ∈ K. We will often use the identiﬁcation of the set of Hilbert-Schmidt operators l2 (K) with K ⊗ K so that |Φ1 )(Φ2 | ∈ l2 (K) corresponds to Ψ1 ⊗ Ψ2 ∈ K ⊗ K. This identiﬁcation can be sometimes confusing. To avoid misunderstanding we will try to make clear which convention we use at the moment. In particular, let us note that the following identities hold for B ∈ B(K) and C ∈ K ⊗ K l2 (K): B⊗1K C = BC,

1K ⊗B C = CB ∗ .

(4.20)

On the left-hand side C is interpreted as an element of K ⊗ K and on the right as an element of l2 (K). An antiunitary map κ on K such that κ2 = 1 will be called an internal conjugation on K. Note that if we have a ﬁxed internal conjugation κ on K, then K is naturally identiﬁed with K. Therefore, in this case we do not need to introduce K.

4.7

The conjugation

Let K and W be Hilbert spaces. In this subsection we introduce a certain antilinear map from a dense subspace of B(K, K ⊗ W) to a dense subspace of B(K, K ⊗ W).

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Let v ∈ B(K, K ⊗ W). We say that v is conjugable if there exists v ∈ B(K, K ⊗ W) such that for Φ, Ψ ∈ K and w ∈ W, (Φ⊗w | vΨ) = (v Φ|Ψ ⊗ w). If v exists, then it is unique. Remark 4.3 Given an orthonormal basis {wi : i ∈ I} in W, any v ∈ B(K, K⊗W) can be decomposed as v= Bi ⊗ |wi ), (4.21) i∈I

be understood in terms of the strong operator where Bi ∈ B(K) and the sum should ∗ ∗ convergence. Note that v v = B i Bi . It is easy to see that v is conjugable i∈I ∗ iﬀ i∈I Bi Bi is bounded. If this is the case, v :=

Bi∗ ⊗ |wi ).

i∈I

and v ∗ v =

i∈I

Bi Bi∗ .

Proposition 4.4 Suppose that either K or W is ﬁnite-dimensional. Then all v ∈ B(K, K ⊗ W) are conjugable. Moreover, if n := min (dim K)2 , dim W , then √ v ≤ nv.

Proof. Clearly, dim B(K) = (dim K)2 . Therefore, we can choose an orthonormal system {wi } in W with at most n elements such that (4.21) is true. Now v 2 = Bi Bi∗ ≤ Bi Bi∗ = Bi∗ Bi ≤ n Bi∗ Bi = nv2 . i

i

i

i

Remark 4.5 If W and K are inﬁnite-dimensional, then it is easy to ﬁnd an example of v ∈ B(K, K ⊗ W) which is not conjugable. Notation. In what follows, if ρ is an operator on W and v ∈ B(K, K ⊗ W), we will write ρv instead of 1K ⊗ρ v. Proposition 4.6 (1) If v is conjugable, then so is v ; moreover, v = v. (2) If ρ ∈ B(W), then (ρv) = ρv . (3) If B ∈ B(K), then (vB) = B ∗ ⊗1W v .

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Coupling Hilbert-Schmidt operators

Let K and W be Hilbert spaces. In this subsection we describe some notation and identities related to the space K ⊗ K ⊗ W l2 (K) ⊗ W. Let H1 , H2 be Hilbert spaces. If B ∈ B(K) and A ∈ B(K ⊗ H1 , K ⊗ H2 ), we deﬁne ˇ := θ−1 ⊗1H2 B⊗A θ⊗1H1 ∈ B(K ⊗ K ⊗ H1 , K ⊗ K ⊗ H2 ), B ⊗A

(4.22)

where θ : K ⊗ K → K ⊗ K is deﬁned as θ Ψ1 ⊗Ψ2 := Ψ2 ⊗Ψ1 . In other words, if C ∈ B(K), A ∈ B(H1 , H2 ), we set ˇ C ⊗ A := C ⊗ B ⊗ A. B⊗ ˇ “tensoring in the middle”. We will sometimes call the operation ⊗ Tr denotes the trace. In the context of coupled systems Tr will be reserved for the partial trace over the space K. To denote the partial trace over the space W we will use tr. Thus, if C is an operator on K ⊗ W, then trC is an operator on K. The following propositions describe some algebraic properties of tensoring in ˇ and the operation, which we will use in our computations. They the middle ⊗ can be skipped on the ﬁrst reading. Proposition 4.7 Let A ∈ l2 (K), B ∈ l2 (K, K ⊗ W), vl ∈ B(K, K ⊗ W), vr ∈ B(K, K ⊗ W) and suppose that vr is conjugable. Then the following statements hold: ˇ l A = vl A, 1K ⊗v ˇ l∗ B = vl∗ B, 1K ⊗v 1K ⊗v r A = A⊗1W vr ,

(4.23)

1K ⊗v ∗r B = trBvr∗ , where on the left we use the K ⊗ K notation and on the right the l2 (K) notation. Proof. It is suﬃcient to prove the statement for vl = C ⊗ |w) and vr = C ⊗ |w) where C ∈ B(K) and w ∈ W. Then ˇ l = C ⊗ 1K ⊗ |w), 1K ⊗v ˇ l∗ = C ∗ ⊗ 1K ⊗ (w|, 1K ⊗v 1K ⊗v r = 1K ⊗ C ⊗ |w), ∗

1K ⊗v ∗r = 1K ⊗ C ⊗ (w|.

(4.24)

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We may also assume that B = D ⊗ |w0 ) for some w0 ∈ W. Using (4.20) we get ˇ lA 1K ⊗v

= CA ⊗ |w)

= C⊗|w) A,

ˇ l∗ B 1K ⊗v

= C ∗ D (w|w0 )

1K ⊗v r A

= AC ∗ ⊗ |w)

1K ⊗v ∗r B

= DC(w|w0 )

= C ∗ ⊗(w| D⊗|w0 ), = A⊗1W C⊗|w) , ∗ . = tr D⊗|w0 ) C⊗|w)

(4.25)

Proposition 4.8 Let A ∈ l2 (K), ρ ∈ B(W), vl ∈ B(K, K ⊗ W) and vr ∈ B(K, K ⊗ W). Suppose that vl , vr are conjugable. Then ˇ l A = trρvl Avr∗ = vl∗ A⊗ρ∗ vr , 1K ⊗v ∗r ρ 1K ⊗v

(4.26)

ˇ l∗ ρ 1K ⊗v r A = trvl A vr∗ ρ∗ = vl∗ A⊗ρ vr . 1K ⊗v

(4.27)

If vl,1 , vl,2 ∈ B(K, K ⊗ W) are conjugable, then ∗ ∗ ∗ ˇ l,1 ˇ l,2 A = trvl,1 1K ⊗v ρ 1K ⊗v vl,2 A⊗ρ∗ = vl,1 ρvl,2 A.

(4.28)

If vr,1 , vr,2 ∈ B(K, K ⊗ W) are conjugable, then ∗ ∗ 1K ⊗v ∗r,1 ρ 1K ⊗v r,2 A = trA⊗ρ vr,2 vr,1 = Avr,2 ρ∗ vr,1 .

(4.29)

Proof. We will prove only (4.26). First note that ˇ l A = 1K ⊗v r ρ vl A = trρvl Avr∗ . 1K ⊗v ∗r ρ 1K ⊗v

(4.30)

We take vl = Cl ⊗|wl ) and vr = Cr ⊗|wr ) for some Cl , Cr ∈ B(K), wl , wr ∈ W. Then (4.30) is equal to ∗ = tr Cl ⊗|ρwl ) A Cr ⊗(wr | tr Cl ⊗|ρwl ) A Cr ⊗ |wr ) = Cl ACr tr |ρwl )(wr | = Cl ACr (w l |ρ∗ w r ) ∗ = Cl ⊗(w l | A⊗ρ∗ Cr ⊗|w r ) = Cl ⊗ |wl ) A⊗ρ∗ Cr ⊗|wr ) = vl∗ A⊗ρ∗ vr .

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5 Abstract Pauli-Fierz operators In this section we ﬁrst introduce the notation that we will use to describe the interaction of a second-quantized system with another system. Then we introduce the class of Pauli-Fierz operators. We also describe a number of results about these operators contained in the literature, especially in [DJ1], which we will use later on. In this section we look at Pauli-Fierz operators just as a certain class of abstract self-adjoint operators. Only in the next two sections we will put them in the context of W ∗ -dynamical systems.

5.1

Creation/annihilation operators in coupled systems

n Suppose that W is a Hilbert space. Γs (W) = ⊕∞ n=0 Γs (W) will denote the bosonic Fock space over W (see, e.g., [BR2], also [DJ1]). Consider another Hilbert space E. In this subsection we discuss the formalism of the coupled system described by the Hilbert space E ⊗ Γs (W). Let q ∈ B(E, E ⊗ W). The creation and annihilation operators associated to q were introduced in [DG, DJ1]. Since we will use a somewhat diﬀerent notation from [DJ1], we discuss these notions in detail. We deﬁne the creation operator q(a∗ ) as the (unbounded) quadratic form on (W) E ⊗ Γs (W) whose only nonzero matrix elements are between Ψn+1 ∈ E ⊗ Γn+1 s and Ψn ∈ E ⊗ Γns (W), for n ≥ 0, and are equal √ (Ψn+1 |q(a∗ )Ψn ) := n + 1(Ψn+1 | q⊗1⊗n W Ψn ).

The annihilation operator q ∗ (a) is deﬁned as q(a∗ )∗ = q ∗ (a). Note that both q(a∗ ) and q ∗ (a) are closed. Remark 5.1 In [DJ1] q(a∗ ) and q ∗ (a) were denoted a∗ (q) and a(q) respectively. For further reference, let us note the following straightforward facts: Proposition 5.2 (1) Let Ψ ∈ E ⊗ Γns (W). Then the following estimates hold: √ √ q(a∗ )Ψ ≤ n + 1qΨ, q ∗ (a)Ψ ≤ nqΨ. (5.31) (2) If Ψ0 ∈ E ⊗ Γ0s (W), then q(a∗ )Ψ0 = qΨ0 ∈ E ⊗ Γ1s (W).

(3) If Ψ1 ∈ E ⊗ Γ1s (W), then q ∗ (a)Ψ1 = q ∗ Ψ1 ∈ E ⊗ Γ0s (W).

5.2

Pauli-Fierz operators

Let E and W be as above. From now on we will always assume that E is ﬁnitedimensional. Let E be a self-adjoint operator on E and r a self-adjoint operator on W. A self-adjoint operator on H := E ⊗ Γs (W) of the form Lfr := E ⊗ 1 + 1 ⊗ dΓ(r) will be called a free Pauli-Fierz operator.

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For a given q ∈ B(E, E ⊗ W) the Pauli-Fierz interaction is deﬁned by Q = q(a∗ ) + q ∗ (a). It follows from Nelson’s theorem on analytic vectors that Q is essentially selfadjoint on E ⊗ Γﬁn s (W). The operator L := Lfr + λQ, where λ ∈ R, will be called a Pauli-Fierz operator. We know two sets of assumptions that guarantee the self-adjointness of PauliFierz operators. 1

Theorem 5.3 (1) If r ≥ 0 and r− 2 q is bounded, then L is self-adjoint on D(Lfr ). (2) If |r|q is bounded, then L is essentially self-adjoint on D(Lfr ) ∩ D(Q). The proof of (1) can be found in Proposition 5.2 of [DJ1] and the proof of (2) in Proposition 5.1 of [DJ1]. We remark that (1) has been known for a long time, see, e.g., [BFS1]. The part (2) was ﬁrst proven in [JP1].

5.3

Level Shift Operator for Pauli-Fierz operators

We start by the description of a condition which plays a central role in our study. This condition was introduced [JP1, JP2] and was also used in [DJ1, M]. We assume that there exists a Hilbert space G and a unitary operator U : W → L2 (R) ⊗ G such that the operator U rU ∗ is the operator of multiplication by the variable in R. We ﬁx such an operator U and identify W ≡ L2 (R) ⊗ G. We will often make use of the self-adjoint operator s := −i∂p ⊗ 1G introduced already in Subsection 4.4. Let Hv := E ⊗ Γ0s (W) be the distinguished subspace of H := E ⊗ Γs (W). Note that the map E Ψ → Ψ ⊗ Ω ∈ Hv , identiﬁes E with Hv . Likewise, the operator Lfr preserves the subspace Hv and vv Lvv = 0 and Qvv = q. fr is identiﬁed with the operator E on E. Note also that Q For z ∈ C+ set vv −1 vv w(z) := Qvv (z1vv − Lfr ) Q

= q ∗ (z − E ⊗ 1 − 1 ⊗ r)−1 q. The next proposition follows from Proposition 4.2. (It is also a special case of Theorem 6.1 in [DJ1].) Proposition 5.4 Assume that sη q ∈ B(E, E ⊗ W) for some η > 1/2. Then the function C+ z → w(z) extends by continuity to a continuous function on Ccl +.

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Under the condition of this proposition the Level Shift Operator for the triple (E ⊗ Γ0s (W), Lfr , Q) is well deﬁned and is equal to Γ=

(q ∗ )e1 ,e2 (e1 − e2 + i0 − r)−1 q e2 ,e1 ,

(5.32)

e1 ,e2 ∈sp(E)

where q e1 ,e2 := 1e1 (E)⊗1W q 1e2 (E).

5.4

Spectral theory of Pauli-Fierz operators

The following theorem is a consequence of the main results of [DJ1]. Theorem 5.5 Let η > 2, > 0, C > 0 and q ∈ B(E, E ⊗ W) be such that: (a) L is essentially self-adjoint on D(Lfr ) ∩ D(Q) for all λ. (b) sη q ≤ C. (c) ΓI ≤ − (1 − 10 (ΓI )). Then there exists λ0 > 0, which depends on q only through η, and C, such that for 0 < |λ| < λ0 the following holds: (1) spsc (L) = ∅. (2) dim 1p (L) ≤ dim 10 (ΓI ). Remark. If (c) is replaced with the condition ΓI < − , then all the conclusions of the theorem hold under the weaker assumption η > 1. Moreover, in this case 10 (ΓI ) = 0, and we conclude that L has no point spectrum. Proof of Theorem 5.5. By Theorems 6.2, 6.3 and 6.4 of [DJ1], there exists λ0 > 0 such that for 0 < |λ| < λ0 , spsc (L) = ∅ and dim 1p (L) ≤ dim 1R (Γ). Since Γ is a dissipative operator, dim 1R (Γ) ≤ dim 10 (ΓI ) (see Proposition 3.2 of [DJ1]). Hence dim 1p (L) ≤ dim 10 (ΓI ). The proofs of Theorems 6.2, 6.3 and 6.4 yield that the constant λ0 depends only on η, and C. Let us note that in [DJ1] we actually proved much more than what we stated above. The following theorem, adapted from [DJ1], expresses in precise terms the intuition that the operator 1m (Γ)(E + λ2 Γ) m∈sp(Γ)∩R

predicts the approximate location of eigenvalues of L and estimates from above their multiplicity. For x ∈ R and > 0, we set I(x, ) := [x − , x + ]. Theorem 5.6 Suppose that sη q < ∞ for some η > 2, that L is essentially self-adjoint on D(Lfr ) ∩ D(Q) for all λ and let κ = 1 − η −1 . Then there exists λ0 > 0 and α > 0 such that for 0 < |λ| < λ0 , the following holds:

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(1) If e ∈ sp(E) and m ∈ sp(Γee ) ∩ R, then dim 1pI(e+λ2 m, α|λ|2+κ ) (L) ≤ dim 1m (Γee ). (2)

spp (L) ⊂

I(e + λ2 m, α|λ|2+κ ).

e∈sp(E) m∈R∩sp(Γee )

6 Pauli-Fierz systems In this section we consider a certain class of W ∗ -dynamical systems that we call Pauli-Fierz systems. Subsections 6.1–6.4 are devoted to zero temperature Pauli-Fierz systems. From the algebraic point of view they are the simplest class of Pauli-Fierz systems. Their W ∗ -algebras are type I factors – they are just B(H). The generators of their dynamics in the irreducible representation are given by bounded from below PauliFierz operators. Such operators will be called Pauli-Fierz Hamiltonians. A PauliFierz Hamiltonian is completely determined by the data (K, K, Z, h, v), where K is the energy operator of the small system on the Hilbert space K and h is a positive operator describing the boson energy on the 1-particle Hilbert space Z. The form-factor v ∈ B(K, K ⊗ Z) describes the interaction. Subsections 6.5–6.8 are devoted to Pauli-Fierz systems at density ρ, where the radiation density operator ρ is a positive operator on Z commuting with the 1-particle energy operator h. A Pauli-Fierz system at density ρ is uniquely determined by (K, K, Z, h, v, ρ). The case ρ = 0 corresponds to zero temperature systems. There are two representations of Pauli-Fierz systems that we will use. The ﬁrst is somewhat simpler – we will call it the semi-standard representation. The second one, the standard representation, is more complicated, but also more natural from the algebraic point of view. In both representations there are certain distinguished Pauli-Fierz operators that implement the dynamics. In the semi-standard representation this operator . In the standard is called the Pauli-Fierz semi-Liouvillean and is denoted Lsemi ρ representation it is called the Pauli-Fierz Liouvillean and denoted by Lρ . The main results of this section concern the Level Shift Operator for Lρ , denoted Γρ , and are described in Section 6.7. In particular, in Theorem 6.13 we give an algebraic characterization of KerΓIρ , which will later lead to the main eﬀective coupling assumption of our paper.

6.1

Pauli-Fierz Hamiltonians

Throughout this section we assume that K is a self-adjoint operator on a ﬁnitedimensional Hilbert space K, h is a positive operator on a Hilbert space Z and

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v ∈ B(K, K ⊗ Z). The self-adjoint operator Hfr := K ⊗ 1 + 1 ⊗ dΓ(h) on K ⊗ Γs (Z) will be called a free Pauli-Fierz Hamiltonian. The interaction is described by the self-adjoint operator V = v(a∗ ) + v ∗ (a). The operator H := Hfr + λV, where λ ∈ R, is called a Pauli-Fierz Hamiltonian. We will need 1

Assumption 6.A h− 2 v ∈ B(K, K ⊗ Z). By Theorem 5.3, Assumption 6.A implies that H is self-adjoint on D(Hfr ) and bounded from below.

6.2

Gluing of reservoir 1-particle spaces

An important role in our paper will be played by the Hilbert space Z ⊕ Z and the self-adjoint operator r := h ⊕ (−h). The following operators: Z ⊕ Z (z1 , z 2 ) → τ (z1 , z 2 ) := (z 2 , z1 ) ∈ Z ⊕ Z,

(6.33)

Z ⊕ Z (z1 , z 2 ) → (z1 , z 2 ) := (z2 , z 1 ) ∈ Z ⊕ Z,

(6.34)

will be also useful. Note that τ is linear, antilinear, and (z1 , z 2 ) = τ (z1 , z 2 ). The most important assumption that we need is the gluing condition introduced in [JP1] and further elaborated in [DJ1]. Assumption 6.B There exists a Hilbert space G and a unitary U : Z ⊕Z → L2 (R)⊗ G such that U ∗ rU is the operator of multiplication by the variable in R. In what follows we assume that Assumption 6.B holds and we identify Z ⊕ Z with L2 (R) ⊗ G and r with the multiplication operator (rΨ)(p) := pΨ(p). Let us note that Z is identiﬁed with L2 (R+ ) ⊗ G and h with 1[0,∞[ (r)r. Likewise, Z is identiﬁed with L2 (R− ) ⊗ G. Thus (κΨ)(p) := Ψ(−p),

p ∈ [0, ∞[,

(6.35)

deﬁnes an antiunitary map on Z, which satisﬁes κh = hκ. Thanks to κ, the operation can be viewed as a map of B(K, K ⊗ Z) into itself. In the expression (v, 0) below we interpret 0 as an operator from K to K ⊗ Z. Thus, (v, 0) : K → K ⊗ L2 (R) ⊗ G, where we used the identiﬁcations K⊗Z ⊕ K⊗Z K⊗(Z ⊕ Z) K ⊗ L2 (R) ⊗ G.

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This operator can be also written as a function deﬁned for almost all p ∈ R with values in B(K, K ⊗ G): (v, 0)(p) =

v(p),

p > 0;

0,

p ≤ 0.

(6.36)

Let s be as in (4.14). In the following assumption we have η ≥ 0. sη (v, 0) ∈ B(K, K⊗(Z ⊕ Z)).

Assumption 6.C(η)0

This assumption will be used in the next section in the analysis of the Level Shift Operator for Pauli-Fierz Hamiltonians.

6.3

Level Shift Operator for Pauli-Fierz Hamiltonians

In this subsection we will calculate the Level Shift Operator for the triple (K⊗Γ0s (Z), Hfr , V ). We will use the notation introduced in Subsection 4.3. In particular, we recall that v k1 ,k2 = 1k1 (K)⊗1Z v 1k2 (K). Let Fk := {k − k1 : k1 ∈ sp(K)} be the set of allowed transition energies from the level k ∈ sp(K). The set of positive and negative transition energies from k is denoted by Fk+ := Fk ∩ ]0, ∞[ and Fk− := Fk ∩ ] − ∞, 0[ respectively. We also set

F :=

Fk ,

F + :=

k∈sp(K)

Fk+ .

k∈sp(K)

Let Hv := K ⊗ Γ0s (Z) and w(z)

:= V vv (z1vv − Hfrvv )−1 V vv = v ∗ (z − K ⊗ 1 − 1 ⊗ h)−1 v.

Proposition 6.1 Assume that Assumption 6.C(η)0 holds with η > function C+ z → w(z) extends to a continuous function on Ccl +.

1 2.

Then the

Proof. We apply the trick of “gluing non-physical free bosons” [DJ1]. Consider the extended 1-boson space Z ⊕ Z and deﬁne the operators r = h ⊕ (−h) and q = (v, 0). Then, for z ∈ C+ , v ∗ (z − h)−1 v = q ∗ (z − r)−1 q, and the statement follows from Proposition 5.4.

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The Level Shift Operator, Γ, of the triple (K ⊗ Γ0s (Z), Hfr , V ) is equal to

Γ=

Γkk ,

k∈sp(K)

Γkk =

(v ∗ )k,k−p (p + i0 − h)−1 v k−p,k .

p∈Fk

With a slight abuse of notation we set v(p) := (v, 0)(p) (recall (6.36)). Let v k1 ,k2 (p) := 1k1 (K)⊗1G v(p)1k2 (K), (v ∗ )k1 ,k2 (p) := 1k1 (K)v(p)∗ 1k2 (K)⊗1G . The Assumption 6.C(η)0 with η > 1/2 ensures that v(p) is a continuous function (see Subsection 4.5). Hence, in particular, v(0) = 0. Moreover, we have (ΓR )kk

=

(v ∗ )k,k−p P(p − h)−1 v k−p,k

p∈Fk

=

p∈Fk

(ΓI )kk

= −π = −π

(v ∗ )k,k−p (p1 )P(p − p1 )−1 v k−p,k (p1 )dp1 ,

p∈Fk+

(v ∗ )k,k−p δ(p − h)v k−p,k

(6.37)

(v ∗ )k,k−p (p)v k−p,k (p).

p∈Fk+

We have described above ΓI and ΓR in two forms. In the ﬁrst form we use the self-adjoint operator h on Z and a number p ∈ R. Strictly speaking, neither the principal value P(p − h)−1 nor the delta function δ(p − h) are well deﬁned as self-adjoint operators. However, within the context of (6.37), these formulas are well deﬁned by the integral expressions using the representation of v into a direct integral with the ﬁbers v(p). Let k0 denote the ground state energy of K, that is, k0 := inf sp(K). For later reference, we note that the ground states of K belong to the kernel of ΓI : Proposition 6.2 Ran1k0 (K) ⊂ KerΓI . One expects that for a “generic” form-factor v, the kernel of ΓI should coincide with the subspace of ground states of K. This leads to the ﬁrst eﬀective coupling assumption that we will use in our paper. Assumption 6.D Ran1k0 (K) = KerΓI . Our second eﬀective coupling assumption is that the ground state of K is simple: Assumption 6.E dim Ran1k0 (K) = 1.

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Spectral structure of Pauli-Fierz Hamiltonians

In this subsection we formulate the main result of this paper concerning Pauli-Fierz Hamiltonians. It will say that if the interaction v is suﬃciently regular and the eﬀective coupling assumptions 6.D and 6.E hold, then the Pauli-Fierz Hamiltonian H for small nonzero coupling constants has purely absolutely continuous spectrum except possibly for a simple eigenvalue at inf sp(H). We start with the observation that an appropriate regularity of the gluing implies the self-adjointness of H. Theorem 6.3 Assume that Assumption 6.C(η)0 holds with η > 1/2. Then Assumption 6.A holds. Proof. We use the method described in the proof of Proposition 6.1. In particular, we use the operators q and r introduced in the proof of this proposition. By Assumption 6.C(η)0 with η > 12 and Proposition 4.2, for any p ∈ R the operator q ∗ (p + i0 − r)−1 q exists and is bounded. Setting p = 0 gives q ∗ (i0 − r)−1 q = v ∗ h−1 v. 1

Hence, h− 2 v is bounded. Now we deduce spectral information on H.

Theorem 6.4 Suppose that Assumption 6.C(η)0 holds with η > 2 and that Assumptions 6.D and 6.E hold. Then there exists λ0 > 0 such that for 0 < |λ| < λ0 the following holds: (1) dim 1p (H) ≤ 1. (2) spsc (H) = ∅. Proof. We again use the method and the notation of Proposition 6.1. Extend the space K ⊗ Γs (Z) to the space K ⊗ Γs (Z) ⊗ Γs (Z) K ⊗ Γs (Z ⊕ Z). The space K⊗Γs (Z) is identiﬁed with the subspace K⊗Γs (Z)⊗Γ0s (Z). Consider the extended operators Lfr

:= Hfr ⊗ 1 − 1 ⊗ dΓ(h)

Q := V ⊗ 1

K ⊗ 1 + 1 ⊗ dΓ(r), q(a∗ ) + q ∗ (a),

and set L := Lfr + λQ H ⊗ 1 − 1 ⊗ dΓ(h). By Theorem 6.3, H is self-adjoint on D(Hfr ) and therefore L is self-adjoint on D(Lfr ) (see Section 5.2 in [DJ1]). Note also that spp (H) = spp (L),

spsc (H) = spsc (L).

(6.38)

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Clearly, L is a Pauli-Fierz operator such that sη q < ∞ for η > 2. The Level Shift Operator of the triple (K⊗Γ0s (Z ⊕Z), Lfr , Q) is equal (after the obvious identiﬁcation of the Hilbert spaces) to the Level Shift Operator of the triple (K ⊗ Γ0s (Z), Hfr , V ), which we studied in the last subsection. Assumptions 6.D and 6.E yield that dim KerΓI = 1. Therefore, Theorem 5.5 implies that there exists λ0 > 0 such that for 0 < |λ| < λ0 we have dim 1p (L) ≤ 1, spsc (L) = ∅. By (6.38), this implies dim 1p (H) ≤ 1, spsc (H) = ∅. Remark 6.5 For a large class of interactions one can show that dim 1inf sp(H) (H) ≥ 1, namely that Pauli-Fierz Hamiltonian H has a ground state. Results of this kind were proven in [AH, BFS1, Ge, Sp2, Sp3]. If to the assumptions of Theorem 6.4 we add the assumptions of the above references, then we can replace (1) with dim 1p (H) = 1 and spp (H) = inf sp(H).

6.5

Pauli-Fierz systems of density ρ

In this subsection we introduce Pauli-Fierz W ∗ -dynamical systems. They will be the main subject of the remaining part of this section. Let ρ ≥ 0 be an operator commuting with h. It will be called the radiation density. The left Araki-Woods W ∗ -algebra, denoted by MAW ρ,l is deﬁned as the W ∗ -subalgebra of B(Γs (Z ⊕ Z)) generated by the operators z ∈ D(ρ1/2 ), exp i (1 + ρ)1/2 z, ρ1/2 z (a∗ ) + hc , where (z1 , z 2 )(a∗ ) denotes the usual creation operators on Γs (Z ⊕ Z). The PauliFierz algebra at density ρ is deﬁned as Mρ := B(K) ⊗ MAW ρ,l .

(6.39)

Mρ → B(Γs (Z ⊕ Z))

(6.40)

The identity map will be called the semi-standard representation of Mρ . (The bosonic part of (6.40) is already standard, the part involving K is not – hence the name.) Proposition 6.6 Assume that (1 + ρ)1/2 v ∈ B(K, K ⊗ Z). 1

(6.41)

Then ρ 2 v is a bounded operator and the operators 1 1 1 1 (1 + ρ) 2 v, 0 (a∗ ) + 0, v ∗ ρ 2 (a), v ∗ (1 + ρ) 2 , 0 (a) + 0, ρ 2 v (a∗ ), which act on K ⊗ Γs (Z ⊕ Z), are aﬃliated to Mρ .

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Proof. Using Proposition 4.4 and the fact that K has a ﬁnite dimension we see that the boundedness of (1 + ρ)1/2 v implies the boundedness of ((1 + ρ)1/2 v) = (1 + ρ)1/2 v . Next note that ρ1/2 (1 + ρ)−1/2 is bounded. This implies the boundedness of ρ1/2 v . In what follows we assume (6.41) and set 1

1

vρ := ((1 + ρ) 2 v, ρ 2 v ) ∈ B(K, K ⊗ (Z ⊕ Z)). Note that in terms of a direct integral of operators in B(K, K ⊗ G): 1 (1 + ρ) 2 v(p), p > 0, vρ (p) = 1 ρ 2 v (−p), p ≤ 0. Let Vρ

1 1 1 1 := (1 + ρ) 2 v, ρ 2 v (a∗ ) + v ∗ (1 + ρ) 2 , v ∗ ρ 2 (a) = vρ (a∗ ) + vρ∗ (a).

The operator Vρ is essentially self-adjoint on the space of ﬁnite particle vectors. Moreover, it is aﬃliated to Mρ . The free Pauli-Fierz semi-Liouvillean is the self-adjoint operator on K ⊗ Γs (Z ⊕ Z) deﬁned as Lsemi fr

:= K ⊗ 1 + 1 ⊗ dΓ(h ⊕ −h) = K ⊗ 1 + 1 ⊗ dΓ(r).

The full Pauli-Fierz semi-Liouvillean of density ρ is Lsemi := Lsemi + λVρ . ρ fr

(6.42)

Let us formulate the following assumption: Assumption 6.Fρ

(1 + h)(1 + ρ)1/2 v ∈ B(K, K ⊗ Z).

Theorem 6.7 (1) τfrt (A) := eitLfr Ae−itLfr is a W ∗ -dynamics on Mρ . (2) Suppose that Assumption 6.Fρ hold. Then Lsemi is essentially self-adjoint on ρ semi

itLρ t D(Lsemi fr ) ∩ D(Vρ ) and τρ (A) := e

semi

Ae−itLρ

is a W ∗ -dynamics on Mρ .

Proof. Part (1) is obvious. Arguing as in the proof of Proposition 6.6 we see that Assumption 6.F ρ 1 implies that (1 + h)ρ 2 v is bounded. Hence (1 + |r|)vρ is bounded. Therefore, Theorem 5.3 yields that Lsemi is essentially self-adjoint on D(Lsemi ρ fr ) ∩ D(Vρ ). Since Vρ is aﬃliated with Mρ , Theorem 3.3 in [DJP] implies that τρ is a W ∗ dynamics. Mρ , τρ will be called the Pauli-Fierz W ∗ -dynamical system at density ρ.

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Standard representation of Pauli-Fierz systems

Consider the representation π : Mρ → B(K ⊗ K ⊗ Γs (Z ⊕ Z)) deﬁned by ˇ π(A) := 1K ⊗A,

A ∈ Mρ = B(K) ⊗ MAW ρ,l ,

ˇ was introduced in (4.22). Clearly, where ⊗ AW π(B(K) ⊗ MAW ρ,l ) = B(K) ⊗ 1K ⊗ Mρ,l .

Set J := JK ⊗ Γ( ), where JK Ψ1 ⊗ Ψ2 := Ψ2 ⊗ Ψ1 ,

Ψ1 , Ψ2 ∈ K,

(6.43)

and was introduced in (6.34). Note that if A ∈ Mρ , then Jπ(A)J = 1K ⊗ 1K ⊗Γ(τ ) A 1K ⊗Γ(τ ) , where τ was introduced in (6.33). Set 2 Hρ,+ := {AJA B⊗Ω, : A ∈ Mρ , B ∈ l+ (K)}cl .

Proposition 6.8

π, K⊗K⊗Γs (Z⊕Z), J, Hρ,+

is a standard representation of Mρ . Set Lfr := K ⊗ 1 ⊗ 1 − 1 ⊗ K ⊗ 1 + 1 ⊗ 1 ⊗ dΓ(r). Proposition 6.9 Lfr is the standard Liouvillean of the free Pauli-Fierz system (Mρ , τfr ). In what follows we assume (6.41). Note that ˇ ρ π(Vρ ) := 1K ⊗V 1 1 1 1 ˇ (1 + ρ) 2 v, ρ 2 v (a∗ ) + 1K ⊗ ˇ v ∗ (1 + ρ) 2 , v ∗ ρ 2 (a) = 1K ⊗ ˇ ρ (a∗ ) + 1K ⊗v ˇ ρ∗ (a). = 1K ⊗v π(Vρ ) is essentially self-adjoint on K ⊗ K ⊗ Γﬁn s (Z ⊕ Z), aﬃliated to B(K) ⊗ 1K ⊗ , and MAW ρ,l Jπ(Vρ )J := 1K ⊗ 1K ⊗Γ(τ ) Vρ 1K ⊗Γ(τ ) 1 1 1 1 = 1K ⊗ ρ 2 v , (1 + ρ) 2 v (a∗ ) + 1K ⊗ v ∗ ρ 2 , v ∗ (1 + ρ) 2 (a) = 1K ⊗ τ vρ (a∗ ) + 1K ⊗ vρ∗ τ (a). Set Lρ := Lfr + λπ(Vρ ) − λJπ(Vρ )J.

(6.44)

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Proposition 6.10 Suppose that Assumption 6.Fρ holds. Then Lρ is essentially selfadjoint on D(Lfr )∩D(π(Vρ ))∩D(Jπ(Vρ )J) and is the Liouvillean of the Pauli-Fierz system (Mρ , τρ ). Proof. The assumptions imply that ˇ ρ − 1K ⊗τ vρ ) ∈ B(K ⊗ K, K ⊗ K ⊗ (Z ⊕ Z)), (1 + |r|)(1K ⊗v and the essential self-adjointness of Lρ follows from Theorem 5.3. The operator Lfr + λπ(Vρ ) is essentially self-adjoint on D(Lfr ) ∩ D(π(Vρ )) and π(τρt (A)) = eit(Lfr +λπ(Vρ )) π(A)e−it(Lfr +λπ(Vρ )) . Hence all the conditions of Theorem 2.3 are satisﬁed and Lρ is the Liouvillean of (Mρ , τρ ).

6.7

Level Shift Operator for Pauli-Fierz Liouvilleans

We will see in this subsection that the Level Shift Operator of the Liouvillean Lρ has very special algebraic properties. Let us formulate the following family of assumptions parameterized by the radiation density operator ρ and a number η ≥ 0. sη vρ ∈ B(K, K ⊗ (Z ⊕ Z)). Assumption 6.C(η)ρ Note that Assumption 6.C(η)0 , introduced in Subsection 6.2, is the special case of Assumption 6.C(η)ρ for ρ = 0. In this subsection we suppose that Assumption 6.C(η)ρ holds with η > 12 and we study the Level Shift Operator, denoted Γρ , of the triple K⊗K⊗Γ0s (Z⊕Z), Lfr , π(Vρ )−Jπ(Vρ )J . Deﬁne the following self-adjoint operators on K: (vρ∗ )k,k−p P(p − r)−1 vρk−p,k ∆R ρ := k∈sp(K) p∈Fk

=

(v ∗ )k,k−p (1 + ρ)P(p − h)−1 v k−p,k

k∈sp(K) p∈Fk

+

tr v k,k−p (v ∗ )k−p,k ρP(p + h)−1 ,

k∈sp(K) p∈Fk

∆Iρ : = −π

(vρ∗ )k,k−p δ(p − r)vρk−p,k

k∈sp(K) p∈Fk

= −π −π

k∈sp(K)

p∈Fk+ ∪{0}

k∈sp(K)

p∈Fk−

(v ∗ )k,k−p (1 + ρ)δ(p − h)v k−p,k

tr v k,k−p (v ∗ )k−p,k ρδ(p + h).

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Set

I ∆ρ := ∆R ρ + i∆ρ =

(vρ∗ )k,k−p (p + i0 − r)−1 vρk−p,k

k∈sp(K) p∈Fk

=

(v ∗ )k,k−p (1 + ρ)(p + i0 − h)−1 v k−p,k

k∈sp(K) p∈Fk

+

tr v k,k−p (v ∗ )k−p,k ρ(p + i0 + h)−1 .

k∈sp(K) p∈Fk

For B ∈ l2 (K) set also

Ξρ (B) := 2π

= 2π

(vρ∗ )k1 ,k1 −p B ⊗ δ(p − r) (τ vρ )k2 −p,k2

k1 ,k2 ∈sp(K) p∈Fk1 ∩Fk2

1 1 (v ∗ )k1 ,k1 −p B ⊗ δ(p − h)(1 + ρ) 2 ρ 2 v k2 −p,k2

k1 ,k2 ∈sp(K) p∈F + ∩F + ∪{0} k k

+ 2π

1

2

1

1

tr v k1 ,k1 −p B(v ∗ )k2 −p,k2 δ(p + h)(1 + ρ) 2 ρ 2 .

k1 ,k2 ∈sp(K) p∈F − ∩F − k1

k2

I The formulas for ∆R ρ , ∆ρ , ∆ρ and Ξρ are written in two equivalent forms. The ﬁrst forms involve the operators vρ ; the second involve v and ρ. Altough the second forms are more directly related to the basic physical quantities of interest, they are less compact and technically less convenient. Note in particular that in the formulas for ∆R ρ and ∆ρ , the terms with p = 0 need to be carefully interpreted. (The singularity of P(−r)−1 and (i0 − r)−1 is “cut” into two parts in these expressions. This problem is absent in the formulas involving vρ .) In the expression for Ξρ we used the operator τ vρ ∈ B K, K ⊗ (Z ⊕ Z) . Note that 1 1 1 1 1 1 τ vρ = ρ 2 v, (1 + ρ) 2 v = ρ 2 (1 + ρ)− 2 , ρ− 2 (1 + ρ) 2 vρ , (6.45)

τ vρ (p)

=

1

ρ 2 v(p),

p ≥ 0,

1 2

(1 + ρ) v (−p), p ≤ 0;

Theorem 6.11 Let B ∈ l2 (K). Then Γρ (B) = ∆ρ B − B∆∗ρ + iΞρ (B), R R ΓR ρ (B) = ∆ρ B − B∆ρ ,

ΓIρ (B) = ∆Iρ B + B∆Iρ + Ξρ (B).

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Proof. Using (5.32) we see that ˇ ρ∗ − 1K ⊗vρ∗ τ e1 ,e2 (e1 − e2 + i0 − r)−1 1K ⊗v ˇ ρ − 1K ⊗τ vρ e2 ,e1 , 1K ⊗v Γρ = e1 ,e2 ∈F

(6.46) where the superscripts e1 , e2 correspond to the decomposition of K ⊗ K into the eigenspaces of K ⊗ 1 − 1 ⊗ K. (Note that sp(K ⊗ 1 − 1 ⊗ K) = F .) Let us switch to superscripts in sp(K) and to the decomposition of K into spectral subspaces of K. For a ﬁxed p ∈ R we obtain ˇ ρ∗ e,e−p ˇ ρ∗ )k,k−p , = 1K ⊗(v 1K ⊗v e∈F k∈sp(K) e,e−p k−p,k 1K ⊗vρ∗ τ = 1K ⊗vρ∗ τ , e∈F k∈sp(K) ˇ ρ e−p,e ˇ ρk−p,k , 1K ⊗v = 1K ⊗v e∈F k∈sp(K) e−p,e 1K ⊗τ vρ = 1K ⊗τ vρ k,k−p . e∈F

k∈sp(K)

The terms on the right can be nonzero only if p ∈ Fk . Therefore, (6.46) becomes ˇ ρ∗ )k,k−p (p + i0 − r)−1 1K ⊗v ˇ ρk−p,k Γρ = 1K ⊗(v k∈sp(K) p∈Fk

+1K ⊗vρ∗ τ −

k1 ,k2 ∈sp(K) p∈Fk1 ∩Fk2

k−p,k

(p + i0 − r)−1 1K ⊗τ vρ k,k−p

ˇ ρ∗ )k2 ,k2 −p (p + i0 − r)−1 1K ⊗τ vρ k1 ,k1 −p 1K ⊗(v +1K ⊗vρ∗ τ

k1 −p,k1

ˇ ρk2 −p,k2 . (p + i0 − r)−1 1K ⊗v

Now let B ∈ l2 (K). We see that Γρ (B) consists of four types of terms: Type I. Using (4.28), we obtain ˇ ρ∗ )k,k−p (p + i0 − r)−1 1K ⊗v ˇ ρk−p,k B 1K ⊗(v = (vρ∗ )k,k−p (p + i0 − r)−1 vρk−p,k B. Summing up the above terms over k ∈ sp(K) and p ∈ Fk we obtain ∆ρ B. Type II. We switch the sign in p and rename k − p to k. Using ﬁrst (4.29) and then τ rτ = −r, we get −1 1K ⊗(τ vρ )k−p,k B 1K ⊗(vρ∗ τ )k,k−p (−p + i0 − r = B(vρ∗ τ )k,k−p (−p + i0 − r)−1 τ vρk−p,k = −B(vρ∗ )k,k−p (p − i0 − r)−1 vρk−p,k . Summing up the above terms over k ∈ sp(K) and p ∈ Fk we obtain −B∆∗ρ .

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Type III. We use (4.27) to obtain ˇ ρ∗ )k2 ,k2 −p )(p + i0 − r)−1 (1K ⊗ (τ vρ )k1 ,k1 −p )B (1K ⊗(v = (vρ∗ )k2 ,k2 −p B⊗(p + i0 − r)−1 (τ vρ )k1 −p,k1 . Type IV. We switch the sign in p as well as rename k1 − p and k2 − p to k1 and k2 . We use (4.26) and then τ rτ = −r: ˇ ρk2 ,k2 −p )B (1K ⊗(vρ∗ τ )k1 ,k1 −p )(−p + i0 − r)−1 (1K ⊗v = (vρ∗ )k2 ,k2 −p B⊗(−p + i0 − r)−1 τ vρk1 −p,k1 = (vρ∗ τ )k2 ,k2 −p B⊗(−p + i0 + r)−1 vρk1 −p,k1 = (vρ∗ )k2 ,k2 −p B⊗(−p + i0 + r)−1 (τ vρ )k1 −p,k1 . In the last step we used (6.45) and the fact that ρ commutes with h. The sum of type III and IV terms over k1 , k2 ∈ sp(K) and p ∈ Fk1 ∩ Fk2 equals iΞρ (B). Set p vρ p := vρk−p,k , τ vρ := (τ vρ )k−p,k , k∈sp(K)

vp :=

k∈sp(K)

v k−p,k ,

p

v :=

k∈sp(K)

(v )k−p,k .

k∈sp(K)

Here is another useful expression for ΓIρ : Theorem 6.12 Let B1 , B2 ∈ l2 (K). Then p p ∗ p −TrB1∗ ΓIρ (B2 ) = π Tr vρ B1 − B1 ⊗1 τ vρ δ(p − r) vρ p B2 − B2 ⊗1 τ vρ =π

p∈F

1

1

Tr (1 + ρ) 2 vp B1 − B1 ⊗1 ρ 2 vp

p∈F + ∪{0}

∗

δ(p − h)

1 1 (1 + ρ) 2 vp B2 − B2 ⊗1 ρ 2 vp ∗ 1 1 +π Tr (1 + ρ) 2 vp B2∗ − B2∗ ⊗1 ρ 2 vp δ(p − h) p∈F +

1 1 (1 + ρ) 2 vp B1∗ − B1∗ ⊗1 ρ 2 vp .

Proof. Recall that −∆Iρ = π

k,p

(vρ∗ )k,k−p δ(p − r)vρk−p,k .

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Hence −TrB1∗ ∆Iρ B2 = π

∗ Tr vρk−p,k B1 δ(p − r)vρk−p,k B2

k,p

=π

775

∗ Tr vρk1 −p,k1 B1 δ(p − r)(vρk2 −p,k2 B2 ).

k1 ,k2 ,p

There is an alternative formula for −∆Iρ , which follows from τ rτ = −r: −∆Iρ = π tr(vρ )k,k−p (vρ∗ )k−p,k δ(p − r) k,p

=π

tr(τ vρ )k,k−p (vρ∗ τ )k−p,k δ(p + r)

k,p

=π

tr(τ vρ )k−p,k (vρ∗ τ )k,k−p δ(p − r).

k,p

Hence −TrB1∗ B2 ∆Iρ = π

k−p,k ∗ Tr B1 (τ vρ ) δ(p − r)B2 (τ vρ )k−p,k

k,p

=π

∗ Tr B1 (τ vρ )k1 −p,k1 δ(p − r)B2 (τ vρ )k2 −p,k2 .

k1 ,k2 ,p

Recall that 1 2 Ξρ (B2 )

=π

(vρ∗ )k1 ,k1 −p B2 ⊗δ(p − r) (τ vρ )k2 −p,k2 .

k1 ,k2 ,p

(6.47)

Terms coming from Ξρ we split as 1 1 TrB1∗ Ξρ (B2 ) + TrB1∗ Ξρ (B2 ). 2 2

TrB1∗ Ξρ (B2 ) =

(6.48)

The ﬁrst term on the right of (6.48) we treat as follows: ∗ 1 ∗ Tr vρk1 −p,k1 B1 δ(p − r) B2 (τ vρ )k2 −p,k2 . 2 TrB1 Ξρ (B2 ) = π k1 ,k2 ,p

Then we transform the formula (6.47), using (6.45), (4.28) and then τ rτ = −r: 1 Ξρ (B2 ) = π (vρ∗ τ )k1 ,k1 −p B2 ⊗δ(p − r) vρk2 −p,k2 2 k1 ,k2 ,p =π tr(τ vρ )k1 ,k1 −p B2 (vρ∗ )k2 −p,k2 δ(p − r) k1 ,k2 ,p

=π

trvρk1 ,k1 −p B2 (vρ∗ τ )k2 −p,k2 δ(p + r)

k1 ,k2 ,p

=π

k1 ,k2 ,p

trvρk1 −p,k1 B2 (vρ∗ τ )k2 −p,k2 δ(p − r).

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Hence the second term in (6.48) also has the form ∗ 1 ∗ Tr B1 (τ vρ )k2 −p,k2 δ(p − r) B2 vρk1 −p,k1 . 2 TrB1 Ξρ (B2 ) = π k1 ,k2 ,p

This ends the proof of the ﬁrst identity of our theorem. Let us prove the second identity. We have p p ∗ p π Tr vρ B1 − B1 ⊗1 τ vρ δ(p − r) vρ p B2 − B2 ⊗1 τ vρ p∈F ∗ 1 1 Tr (1 + ρ) 2 vp B1 − B1 ⊗1 ρ 2 vp =π p∈F + ∪{0} 1 1 δ(p − h) (1 + ρ) 2 vp B2 − B2 ⊗1 ρ 2 vp 1 p p ∗ 1 Tr ρ 2 v B1 − B1 ⊗1 (1 + ρ) 2 v +π −p∈F + 1 p p 1 δ(p + h) ρ 2 v B2 − B2 ⊗1 (1 + ρ) 2 v . The second term on the right side can be transformed into ∗ 1 1 π Tr ρ 2 (vp ) B1 − B1 ⊗1 (1 + ρ) 2 (vp ) p∈F + 1 1 δ(p − h) ρ 2 (vp ) B2 − B2 ⊗1 (1 + ρ) 2 (vp ) ∗ 1 1 Tr B1∗ ⊗1 ρ 2 vp − (1 + ρ) 2 vp B1∗ =π p∈F + 1 1 δ(p − h) B2∗ ⊗1 ρ 2 v p − (1 + ρ) 2 v p B2∗ 1 1 =π Tr B1∗ ⊗1 ρ 2 vp − (1 + ρ) 2 vp B1∗ p∈F + ∗ 1 1 B2∗ ⊗1ρ 2 vp − (1 + ρ) 2 vp B2∗ δ(p − h) ∗ 1 1 Tr (1 + ρ) 2 vp B2∗ − B2∗ ⊗1 ρ 2 vp =π p∈F + 1 1 δ(p − h) (1 + ρ) 2 vp B1∗ − B1∗ ⊗1 ρ 2 vp . −p v p ) , then Proposition 4.6, Proposition 4.8 and In the ﬁrst step we used v = ( in the last step we used the cyclicity of trace. Since 1 (1 + ρ) 2 vp (p), p ≥ 0 p vρ (p) = 1 p ρ 2 v (−p), p ≤ 0; 1 p ρ 2 v (p), p≥0 p τ vρ (p) = p 1 (1 + ρ) 2 v (−p), p ≤ 0,

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the identity proven in Theorem 6.12 can be also written as ∗ p p Tr vρ (p) B1 − B1 ⊗1 τ vρ (p) −TrB1∗ ΓIρ (B2 ) = π p∈F

=π

p vρ p (p) B2 − B2 ⊗1 τ vρ (p)

∗ 1 1 Tr (1 + ρ) 2 vp (p) B1 − B1 ⊗1 ρ 2 vp (p)

p∈F + ∪{0}

1 1 × (1 + ρ) 2 vp (p)B2 − B2 ⊗1 ρ 2 vp (p) +π

(6.49)

∗ 1 1 Tr (1 + ρ) 2 vp (p) B2∗ − B2∗ ⊗1 ρ 2 vp (p)

p∈F +

1 1 × (1 + ρ) 2 vp (p) B1∗ − B1∗ ⊗1 ρ 2 vp (p) . We are now ready to state the main result of this subsection. Theorem 6.13 B ∈ KerΓIρ iﬀ the following two commutation relations hold: 1

1

(1 + ρ) 2 vp (p)B = B⊗1 ρ 2 vp (p), 1

p ∈ F + ∪ {0},

1

(1 + ρ) 2 vp (p)B ∗ = B ∗ ⊗1 ρ 2 vp (p),

p ∈ F +.

(6.50)

Proof. Note that B ∈ KerΓIρ iﬀ TrB ∗ ΓIρ (B) = 0. Hence B ∈ KerΓIρ iﬀ all the terms of (6.49) with B1 = B2 = B are zero, and this is precisely the condition (6.50).

6.8

Pauli-Fierz systems with several reservoirs

Let us describe our formalism in the case of a small system coupled to several independent reservoirs. Let Zi be Hilbert spaces and hi , ρi , positive commuting self-adjoint operators on Zi , i = 1, . . . , n. Zi and hi are the single particle space and energy operator of the ith reservoir and ρi is the corresponding radiation density. Let vi ∈ B(K, K⊗Zi ) be the form factor describing interaction of the small system with the ith reservoir. Set n n n Zi , h := hi , ρ := ρi . Z := i=1

i=1

i=1

If we impose Assumption 6.Fρ (which is equivalent to imposing Assumptions 6.Fρi for all i), then the corresponding composite Pauli-Fierz system is well deﬁned.

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If we impose Assumptions 6.C(η)ρi with η > 12 on vi for all i, then the Level Shift Operator of the composite system, Γρ , is well deﬁned and equal to Γρ =

n

Γi,ρi ,

(6.51)

i=1

where Γi,ρi is the Level Shift Operator of the ith subsystem.

7 Thermal Pauli-Fierz systems Given a Pauli-Fierz Hamiltonian we can deﬁne a family of Pauli-Fierz systems parameterized by β ∈]0, ∞] whose radiation density is given by the Planck law at the inverse temperature β. Such systems will be called thermal Pauli-Fierz systems. They are particularly important from the physical point of view and enjoy special mathematical properties.

7.1

Thermal Pauli-Fierz Liouvilleans

The setup of this section is very similar to the setup of the previous section. In particular, the operators K, h, v and H, as well as the spaces K and Z are such as those introduced in Subsection 6.1. Let 0 < β ≤ ∞. In this section we consider the family of densities ρβ := (eβh − 1)−1 , ρ∞ = 0. Note that 1 + ρβ = (1 − e−βh )−1 = eβh ρβ ,

1 + ρ∞ = 1.

(7.52)

We change slightly the notation by replacing the subscripts ρβ with β. For t semi , MAW instance we will write vβ , Lβ , Lsemi β β,l , Mβ and τβ instead of vρβ , Lρβ , Lρβ , t MAW ρβ ,l , Mρβ and τρβ . We warn the reader that the density ρ = 0 corresponds now to the inverse temperature β = ∞. Note that 1 vβ = |1 − e−βr |− 2 (v, v ),

Assumption 7.A

1

τ vβ

= |1 − eβr |− 2 (v , v),

τ vβ

= |1 − eβr |− 2 (v, v ) = e−βr/2 vβ .

1

(h−1/2 + h)v ∈ B(K, K ⊗ Z)

Proposition 7.1 Suppose Assumption 7.A holds. Then for any 0 < β ≤ ∞ Assumption 6.Fρ for ρ = ρβ holds.

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Proof. Clearly, for 0 < β ≤ ∞, (h−1/2 + h)−1 (1 + h)(1 − e−βh )−1/2 is bounded. Therefore Assumption 7.A implies the boundedness of (1 + h)(1 − e−βh )−1/2 v. The following theorem follows immediately from Proposition 7.1 and Theorem 6.10. Theorem 7.2 Suppose that Assumption 7.A holds. Then, for any 0 < β ≤ ∞, Lβ is essentially self-adjoint on D(Lfr ) ∩ D(π(Vβ )) ∩ D(Jπ(Vβ )J), the thermal Pauli-Fierz system (Mβ , τβ ) is well deﬁned and Lβ is its Liouvillean.

7.2

KMS states for thermal Pauli-Fierz systems

In this subsection we describe results concerning the existence of KMS states for Pauli-Fierz systems at positive temperatures. Since Pauli-Fierz W ∗ -algebras are factors, a thermal Pauli-Fierz system may have at most one KMS-state. For 0 < β < ∞, set √ (7.53) γβ := e−βK/2 / Tr e−βK . (γβ is the β-KMS vector of the small system). The free Pauli-Fierz system (Mβ , τfr ) has a unique β-KMS state and γβ ⊗ Ω is the corresponding β-KMS vector. Obviously, γβ ⊗ Ω ∈ KerLfr . The existence of KMS state for interacting Pauli-Fierz systems, which is a somewhat delicate problem because perturbation Vβ is not a bounded operator and Araki’s theory [Ar, BR2] cannot be applied directly, follows from the result of [DJP], reviewed in Section 2. Theorem 7.3 Suppose that Assumption 7.A holds and that 0 < β < ∞. Then the thermal Pauli-Fierz system (Mβ , τβ ) has a unique β-KMS state. Moreover, γβ ⊗ Ω ∈ D(e−β(L+λπ(Vβ ))/2 ) and the vector e−β(L+λπ(Vβ ))/2 γβ ⊗ Ω

(7.54)

is the β-KMS vector for (Mβ , τβ ). This vector belongs to KerLβ . Proof. By Theorem 2.4 (see also [DJP]), we need only to check that for all λ ∈ R, e−λβπ(Vβ )/2 γβ ⊗ Ω < ∞. To verify this, it suﬃces to show that there exists a constant c such that for all n, π(Vβ )n γβ ⊗Ω ≤ cn (n + 1)!. (7.55) Since

ˇ β (a∗ ) + 1K ⊗v ˇ β∗ (a) π(Vβ ) = 1K ⊗v

we can decompose π(Vβ )n into the sum of 2n -terms, each of which is a product of creation and annihilation operators. Applying the estimates (5.31) to each term we derive that (7.55) holds with c = 2vβ .

780

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Ann. Henri Poincar´e

Level Shift Operator for thermal Pauli-Fierz Liouvilleans

In this subsection we consider the Level Shift Operator in the context of thermal Liouvilleans. Let us formulate the following family of assumptions parameterized by η ≥ 0. Assumption 7.B(η)

sη |r|−1/2 r1/2 (v, v ) ∈ B(K, K ⊗ (Z ⊕ Z)).

Proposition 7.4 Suppose that Assumption 7.B(η) holds. Then sη vβ ∈ B(K, K ⊗ (Z ⊕ Z)) for all β ∈]0, ∞[. Proof. It is easy to see that the function 1

R p → |1 − e−βp |− 2 p−1/2 |p|1/2 ∈ R

(7.56)

is bounded with all bounded derivatives. Hence the operator 1

sη |1 − e−βr |− 2 r−1/2 |r|1/2 s−η is bounded for all η ≥ 0. Therefore, the boundedness of sη |r|−1/2 r1/2 (v, v ) 1 implies the boundedness of sη |1 − e−βr |− 2 (v, v ). The following proposition gives a condition which is easy to verify in practice and which implies Assumption 7.B(η): Proposition 7.5 Let n be a non-negative integer and assume that ∞ n −1/2 1/2 ∂p p p v(p)2 dp < ∞; 0 = (−1)j ∂pj p−1/2 v (p) , j = 0, . . . , n − 1. ∂pj p−1/2 v(p) p=0

p=0

Then Assumption 7.B(n) holds. Proof. See Proposition 7.15, which has a similar proof. Throughout this subsection we assume that Assumption 7.B(η) holds with η > 12 . We will describe the Level Shift Operator in the case ρ = ρβ , which, consistently with our notation, will be denoted Γβ . A special attention needs to be devoted to the infrared term in Γβ . Proposition 7.6 There exists vir := lim p↓0

0 Set v ir :=

pp p vir .

v(p) p

1 2

= lim p↓0

v (p) 1

p2

.

(7.57)

We have vir = (vir ) ,

0

0

( vir ) = v ir .

(7.58)

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Moreover, ΓIβ acts as follows:

∗ |eβp − 1|−1 Tr eβp/2 vp (p)B1 − B1 ⊗ 1 vp (p) + p∈F × eβp/2 v p (p)B2 − B2 ⊗ 1 vp (p) ∗ −βp |e − 1|−1 Tr vp (p)B2∗ − e−βp/2 B2∗ ⊗ 1 vp (p) +π + p∈F × v p (p)B1∗ − e−βp/2 B1∗ ⊗ 1 vp (p) ∗ 0 0 B − B ⊗ 1 v + πβ Tr v ir 1 1 ir 0 0 . × v ir B2 − B2 ⊗ 1 v ir (7.59)

−TrB1∗ ΓIβ (B2 ) = π

Proof. For p > 0,

1

vβ (p) = |1 − e−βp |− 2 v(p), 1

vβ (−p) = |1 − eβp |− 2 v (p). Since the function R p → vβ (p) is continuous, 1

1

vβ (0) = lim vβ (p) = β − 2 lim p− 2 v(p), p↓0

p↓0 1

1

vβ (0) = lim vβ (−p) = β − 2 lim p− 2 v (p). p↓0

p↓0

This implies the existence of the limits in (7.57) and the identities (7.58). The identity (7.59) follows from (6.49) if we take into account the identities (7.52) and (7.58). Let N := B ∈ B(K) : B⊗1 v p (p) = vp (p)B, p ∈ F + , B ∗ ⊗1 v p (p) = vp (p)B ∗ , 0 0 B⊗1 v ir = v ir B .

p ∈ F +,

Proposition 7.7 N is a ∗-subalgebra of B(K) containing C1K . Moreover, for any t ∈ C and B ∈ N we have eitK Be−itK ∈ N. Proof. It is easy to check that N is an algebra and obviously 1K ∈ N. To see that N is preserved by ∗, note that the ﬁrst two conditions are manifestly symmetric 0 0 wrt ∗. The relation ( vir ) = v ir and Proposition 4.6 imply that 0

0

0

0

∗ ∗ B⊗1 v ir = v ir B ⇒ B ⊗1 v ir = v ir B ,

and so N is a ∗-algebra.

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Note that eitK ⊗1 v p (p)e−itK = e−itp vp (p), 0

p ∈ F +,

0

−itK = v eitK ⊗1 v ir e ir .

This implies that N is invariant wrt eitK · e−itK .

Proposition 7.8 KerΓIβ consists of operators of the form e−βK/2 C with C ∈ N. Proof. Relation (7.59) can be rewritten as −TrB1∗ ΓIβ (B2 )

∗ |eβp − 1|−1 Tr vp (p)eβK/2 B1 − eβK/2 B1 ⊗ 1 vp (p) p∈F + ×e−βK ⊗ 1 vp (p)eβK/2 B2 − eβK/2 B2 ⊗ 1 vp (p) ∗ −βp |e − 1|−1 Tr v p (p)B2∗ eβK/2 − B2∗ eβK/2 ⊗ 1 vp (p) +π p∈F+ × vp (p)B1∗ eβK/2 − B1∗ eβK/2 ⊗ 1 v p (p) e−βK ∗ 0 βK/2 0 βK/2 + βπ Tr v e B − e B ⊗ 1 v ir 1 1 ir 0 βK/2 0 . ×e−βK ⊗ 1 v B2 − eβK/2 B2 ⊗ 1 v ir e ir =π

Hence, B ∈ KerΓIβ iﬀ vp (p)eβK/2 B − eβK/2 B⊗1 vp (p) = 0, p ∈ F + , vp (p)B ∗ eβK/2 − B ∗ eβK/2 ⊗1 vp (p) = 0,

p ∈ F +,

0 βK/2 0 vir e B − eβK/2 B⊗1 vir = 0.

Therefore, B ∈ KerΓIβ iﬀ B = e−βK/2 C for some C ∈ N. Our main eﬀective coupling assumption is:

Assumption 7.C N = C1K . Now Proposition 7.8 implies immediately Theorem 7.9 Assumption 7.C is satisﬁed iﬀ KerΓIβ is spanned by γβ .

7.4

Return to equilibrium for a fixed positive temperature

In this subsection we describe conditions which ensure that for any ﬁxed positive temperature the thermal Pauli-Fierz system has the property of return to equilibrium. The result will not be uniform in the temperature.

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Theorem 7.10 Let η > 2 and suppose that Assumptions 7.A, 7.B(η) and 7.C hold. Then for all β ∈]0, ∞[ there exists λ0 (β) > 0 such that for 0 < |λ| < λ0 (β), spp (Lβ ) = {0},

spsc (Lβ ) = ∅.

dim 10 (Lβ ) = 1,

In particular, under the above conditions the Pauli-Fierz system (Mβ , τβ ) has the property of return to equilibrium. Proof. It follows from Theorem 7.9 that dim KerΓIβ = 1 for β ∈]0, ∞[. Hence, by Theorem 5.5, there exists λ0 (β) such that for 0 < |λ| < λ0 (β) we have dim 1p (Lβ ) ≤ dim 10 (ΓIβ ) = 1,

spsc (Lβ ) = ∅.

By Theorem 7.3, dim 10 (Lβ ) ≥ 1, and the result follows.

7.5

Zero temperature Pauli-Fierz Liouvilleans

Recall that zero temperature corresponds to β = ∞. All the properties of the zero temperature Liouvillean, denoted L∞ , follow easily from the properties of the Pauli-Fierz Hamiltonian. They are described in this subsection. Note that after the identiﬁcation K⊗K ⊗Γs (Z ⊕Z) K⊗Γs (Z)⊗K ⊗ Γs (Z) the zero temperature Liouvillean becomes L∞ = H ⊗ 1 − 1 ⊗ H. Hence, the Level Shift Operator for L∞ , denoted Γ∞ , can be expressed in terms of the Level Shift Operator for H, denoted Γ, as follows. If Γ∞ is expressed in terms ∆∞ and Ξ∞ as in Theorem 6.11, then ∆∞ = Γ,

Ξ∞ = 0,

(7.60)

and so Γ∞ (B) = ΓB − BΓ∗ . The following theorem then follows immediately from Theorem 6.4: Theorem 7.11 Under assumptions of Theorem 6.4, there exists λ0 > 0 such that for 0 < |λ| < λ0 we have spp (L∞ ) ⊂ {0},

dim 10 (L∞ ) ≤ 1,

spsc (L∞ ) = ∅.

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Uniform in temperature estimate on the Level Shift Operator

In this subsection we describe conditions under which Γβ is uniformly dissipative on the orthogonal complement to γβ . The eﬀective coupling assumptions 6.D, 6.E, and 7.C will play a key role. Recall that γβ is deﬁned by (7.53) for 0 ≤ β < ∞. For β = ∞ we set γ∞ := 1k0 (K)/ Tr1k0 (K), where k0 := inf sp(K). Note that [0, ∞] β → γβ ∈ l2 (K) is a continuous function. Theorem 7.12 Let η > 12 and suppose that Assumption 7.B(η) holds. Then (1) The function ]0, ∞] β → ΓIβ ∈ B(l2 (K)) is continuous. (2) Assume in addition that 6.D, 6.E and 7.C hold. Let β0 > 0. Then there exists

> 0 such that for β ∈ [β0 , ∞] TrB ∗ ΓIβ (B) ≤ − (TrB ∗ B − |TrBγβ |2 ).

(7.61)

Proof. The continuity of ΓIβ in β ∈]0, ∞] follows from Relation (7.59). To prove (2), let us consider ﬁrst the case β = ∞. Assumption 6.D and (7.60) imply that there exists (∞) > 0 such that ∆I∞ ≤ − (∞)(1 − 1k0 (K)).

(7.62)

Since ΓI∞ (B) = ∆I∞ B + B∆I∞ , using (7.62) and Assumption 6.E we obtain TrB ∗ ΓI∞ (B) ≤ − (∞) TrB ∗ (1 − 1k0 (K))B + TrB ∗ B(1 − 1k0 (K)) ≤ − (∞) TrB ∗ B − TrB ∗ 1k0 (K)B1k0 (K) = − (∞)(TrB ∗ B − |TrBγ∞ |2 ). Consider now β < ∞. It follows from Assumption 7.C and Theorem 7.9 that for any β ∈]0, ∞[, there exists (β) > 0 such that TrB ∗ ΓIβ (B) ≤ − (β)(TrB ∗ B − |TrBγβ |2 ). The compactness of [β0 , ∞], the continuity of [β0 , ∞] β → ΓIβ and of [β0 , ∞] β → γβ imply that one can choose > 0 such that (7.61) holds.

7.7

Uniform in temperature return to equilibrium

In this subsection we describe the main result of this paper. We describe conditions under which for 0 < |λ| < λ0 and β ∈ [β0 , ∞[ the Liouvillean Lβ has purely absolutely continuous spectrum except for a simple eigenvalue at zero. This implies

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that for a small nonzero coupling constant the system (M, τβ ) has the property of return to equilibrium uniformly in the temperature. One of the ingredients of our proof is the uniform estimate on the Level Shift Operator Γβ obtained in the previous subsection. The second ingredient is an additional assumption on the regularity of the interaction, which we will formulate below. For any η ≥ 0 we introduce Assumption 7.D(η) sup1≤β≤∞ sη vβ < ∞. Proposition 7.13 Suppose that Assumption 7.D(η) holds. Then (1) Assumption 7.B(η) holds. (2) For any β0 > 0, supβ0 ≤β≤∞ sη vβ < ∞. Proof. By Assumption 7.D(η), sη |1 − e−r |−1/2 (v, v ) < ∞. Clearly, τ sη |1 − e−r |−1/2 (v, v ) = sη |1 − er |−1/2 (v, v ). Thus, using the boundedness of , we obtain (7.63) sη |1 − e−r |−1/2 + |1 − er |−1/2 (v, v ) < ∞. Using the arguments of the proof of Proposition 7.4 we see that −1 −η sη r1/2 |r|−1/2 |1 − e−r |−1/2 + |1 − er |−1/2 s

(7.64)

is bounded. Now (7.63) and (7.64) imply Assumption 7.B(η), which proves (1). For 0 < β0 ≤ 1, we have −1 −η sup sη |1 − e−βr |−1/2 |1 − e−r |−1/2 + |1 − er |−1/2 s < ∞.

β0 ≤β≤1

Hence (7.63) implies supβ0 ≤β≤1 sη vβ < ∞. This proves (2).

Theorem 7.14 Suppose Assumptions 7.A, 7.D(η) with η > 2, 6.D 6.E and 7.C are satisﬁed. Let 0 < β0 < ∞. Then there exists λ0 > 0 such that for 0 < |λ| < λ0 and β ∈ [β0 , ∞[ we have spp (Lβ ) = {0},

dim 10 (Lβ ) = 1,

spsc (Lβ ) = ∅.

Hence, under the same conditions, the Pauli-Fierz system (Mβ , τβ ) has the property of return to equilibrium. Moreover, for 0 < |λ| < λ0 , spp (L∞ ) ⊂ {0},

dim 10 (L∞ ) ≤ 1,

spsc (L∞ ) = ∅.

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Proof. By Theorem 7.2 the operator Lβ is essentially self-adjoint on D(Lfr ) ∩ D(π(Vβ ) − Jπ(Vβ )J) for all β ∈]0, ∞[ and λ ∈ R. By Theorem 6.3 the operator L∞ is self-adjoint on D(Lfr ) = D(Lfr ) ∩ D(π(V∞ ) − Jπ(V∞ )J) for all λ. By Theorem 7.12 there exists > 0 such that for all β ∈ [β0 , ∞], ΓIβ ≤ − (1 − 10 (ΓIβ )). By Assumption 7.D(η) and Proposition 7.13 (2), for all β ∈ [β0 , ∞] and η > 2, ˇ β − 1K ⊗ τ vβ ≤ 2c. sη 1K ⊗v Hence, all conditions of Theorem 5.5 are satisﬁed, and there exists λ0 > 0 such that for 0 < |λ| < λ0 and all β ∈ [β0 , ∞] we have dim 1p (Lβ ) ≤ dim ΓIβ = 1,

spsc (Lβ ) = ∅.

By Theorem 7.3 dim 10 (Lβ ) ≥ 1 for β ∈]0, ∞[, and the statement follows.

7.8

Simplified condition for return to equilibrium

In this subsection we describe conditions which are easy to verify in practice and which imply Assumption 7.D(η). Proposition 7.15 Suppose that n is a nonnegative integer and r1/2 sn |r|−1/2 (v, v ) < ∞; |r|1/2−n+j sj |r|−1/2 (v, v ) < ∞,

j = 0, . . . , n − 1.

(7.65) (7.66)

Then Assumption 7.D(n) holds. Proof. Set g(p) := |1 − e−p |−1/2 |p|1/2 . Note that g is smooth and |∂pj g(p)| ≤ cj p1/2−j .

(7.67)

Now sn |1 − e−βr |−1/2 (v, v )

= β −1/2 sn g(βr)|r|−1/2 (v, v ) n = j=0 (−i)n−j β −1/2+n−j g (n−j) (βr)sj |r|−1/2 (v, v ).

To estimate the term with j = n we use that for β ≥ 1 we have β −1/2 g(βr) ≤ β −1/2 βr1/2 ≤ r1/2 To estimate the terms with j = 0, . . . , n − 1 we use β −1/2+n−j g (n−j) (βr) ≤ hn−j (βr)|r|1/2−n+j , where, by (7.67), hn−j (p) = g (n−j) (p)|p|−1/2+n−j is a bounded function.

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Here are yet another set of conditions implying Assumption 7.D(n). Proposition 7.16 Suppose that ∞ p∂pn p−1/2 v(p)2 dp < ∞; 0 ∞ p1−2n+2j ∂pj p−1/2 v(p)2 dp < ∞, j = 0, . . . , n − 1; 0 = (−1)j ∂pj p−1/2 v (p) , j = 0, . . . , n − 1. ∂pj p−1/2 v(p) p=0

(7.68) (7.69) (7.70)

p=0

Then (7.65) and (7.66) hold, and hence Assumption 7.D(n) holds. Proof. Since K is ﬁnite-dimensional, the conjugation is a bounded linear map, and (7.68) and (7.69) hold with v instead of v. We will prove (7.65). A similar argument yields (7.66). Using (7.68) and (7.70) we see that n −1/2 1/2 n −1/2 ∂p p v(p), p ≥ 0, n 1/2 r s |r| (v, v ) (p) = (−i) p ∂pn |p|−1/2 v (|p|), p ≤ 0, and r1/2 sn |r|−1/2 (v, v )2

≤ =

∞

n −1/2 (v, v )(p)2 dp −∞ p∂p |p|

∞ n −1/2 p ∂p p v(p)2 + ∂pn p−1/2 v (p)2 dp. 0

7.9

Pauli-Fierz systems with several thermal reservoirs

In this subsection we prove that a generic Pauli-Fierz system with a small nonzero coupling constant has no normal invariant states. For shortness, we restrict ourselves to a result non-uniform in the temperature. We consider the same framework as in Subsection 6.8. Moreover, we assume that the energy density of the ith reservoir is given by ρβi = (eβi hi − 1)−1 , = (β1 , . . . , βN ) and, after replacing β with β, we adopt where βi ∈]0, ∞[. We set β the same notational convention as in Subsection 7.1. Theorem 7.17 Let η > 1. Assume that Assumptions 7.A and 7.B(η) hold for i = 1, . . . , N . Suppose also that βj = βk for some j, k ∈ {1, . . . , N }, and that

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>0 Assumption 7.C holds for the jth and kth reservoir. Then there exists λ0 (β) such that for 0 < |λ| < λ0 (β), spp (Lβ ) = ∅,

spsc (Lβ ) = ∅.

Consequently, under the above conditions the system (Mβ , τβ ) has no normal invariant states. Proof. The proof is very similar to the proof of Theorem 7.10. By Theorem 5.5 and the remark after it, it suﬃces to show that dim KerΓIβ = {0}. The relation (6.51) yields N KerΓIβ = KerΓIi,βi , i=1

where Γi,βi is the Level Shift Operator of the ith subsystem. By Assumption 7.C, KerΓIi,βi is spanned by γβi for i = j, k. Since βj = βk , KerΓIj,βj ∩ KerΓIk,βk = {0}.

8 Examples of gluing As we have already emphasized, the key ingredient of our method is the Jakˇsi´cPillet gluing condition. In this section we show that this condition is satisﬁed in a certain class of physically motivated models involving massless bosons. The gluing is accomplished by passing to the radial coordinates in the momentum representation.

8.1

Massless scalar bosons

In this subsection we consider the same model as in Section 3. Recall that Z = L2 (Rd ), where ξ ∈ Rd describes the momentum, and that h is the operator of multiplication by |ξ|. The gluing map is deﬁned as L2 (Rd ) ⊕ L2 (Rd ) (f+ , f − ) → f ∈ L2 (R) ⊗ L2 (S d−1 ), d−1 p 2 f+ (pω), p > 0, f (p, ω) := d−1 2 (−p) f − (−pω), p ≤ 0.

(8.71) (8.72)

Here, (p, ω) ∈ R×S d−1 . Moreover, the conjugation in L2 (Rd ) and L2 (R)⊗L2 (S d−1 ) is the standard complex conjugation. The map (8.71) is unitary. As in Section 3, we ﬁx a form-factor v : Rd → B(K). Recall that the corresponding Pauli-Fierz Hamiltonian is H := K ⊗ 1 + 1 ⊗ |ξ|a∗ (ξ)a(ξ)|ξ| dξ +λ v(ξ) ⊗ a∗ (ξ) + v ∗ (ξ) ⊗ a(ξ) dξ.

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We ﬁx the density Rd ξ → ρ(ξ) ∈ R+ and assume that Assumption 3.A holds. Recall that the semi-Liouvillean at density ρ is given by := K ⊗ 1 + 1 ⊗ |ξ|a∗l (ξ)al (ξ) − |ξ|a∗r (ξ)ar (ξ) dξ Lsemi ρ 1 1 +λ v(ξ) ⊗ (1 + ρ(ξ)) 2 a∗l (ξ) + ρ(ξ) 2 ar (ξ) dξ + hc. If we use the glued variables and introduce d−1 1 p 2 (1 + ρ(pω)) 2 v(pω), p>0 vρ (p, ω) := d−1 1 ∗ 2 2 (−p) ρ(−pω) v (−pω), p ≤ 0,

(8.73)

then the semi-Liouvillean can be written as Lsemi = K ⊗ 1 + 1 ⊗ pa∗ (p, ω)a(p, ω) dpdω ρ +λ vρ (p, ω) ⊗ a∗ (p, ω) + vρ∗ (p, ω) ⊗ a(p, ω) dpdω. Now, using (8.73), it is easy to give explicit conditions on v(ξ) needed for our results. For instance, Assumption 6.C(n)ρ is satisﬁed if ∂pn vρ (p, ω)2 dpdω < ∞. (8.74) vρ (p, ω)2 dpdω < ∞, Recall that in (6.35) we introduced the antilinear map κ on Z. In the context of scalar ﬁelds it is equal to κf (ξ) = f (−ξ). and satisﬁes κ2 = 1 (it is an internal conjugation). Assume that ρκ = κρ,

v = v.

In the context of scalar ﬁelds this means v ∗ (ξ) = v(−ξ),

Then vρ (p, ω) =

ρ(ξ) = ρ(−ξ).

(8.75)

1

d−1

p 2 (1 + ρ(pω)) 2 v(pω), p > 0 d−1 1 p ≤ 0. (−p) 2 ρ(pω) 2 v(pω),

(8.76)

Assume further that ρβ (ξ) := (eβ|ξ| − 1)−1 and set vβ := vρβ . Then vβ (p, ω) =

p 1 − e−βp

12

|p| 2 −1 v(pω). d

(8.77)

Now, if R p → |p| 2 −1 v(p ·) ∈ B(K, K ⊗ L2 (S d−1 )) is n times diﬀerentiable and d (8.78) ∂pj |p| 2 −1 p1/2 v(pω)2 dpdω < ∞, j = 0, n, d

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then (8.74) holds (the “gluing condition” (7.70) is automatically satisﬁed for ρ = ρβ d by the diﬀerentiability |p| 2 −1 v(p ·) at zero). Theorem 7.10 (the nonuniform result on return to equilibrium) applies whenever (8.74) holds with n = 3. The case d = 3 and v(ξ) ∼ |ξ|−1/2 as |ξ| ↓ 0, is sometimes called the “ohmic case” and is typical for the infrared regime of QED. Note that Theorem 7.10 covers the ohmic case. However, Theorem 7.14 (our uniform result) does not apply to the ohmic case.

8.2

Massless vector particles

In this section we brieﬂy explain how Jakˇsi´c-Pillet gluing works for transversal massless vector bosons (e.g., photons). Consider ﬁrst the Hilbert space of square integrable vector ﬁelds on Rd , that 2 is L (Rd ) ⊗ Cd L2 (Rd , Cd ) and the function Rd ξ → Ptr (ξ) := 1 − |ξ|−2 |ξ)(ξ| ∈ B(Cd ), where |ξ|−2 |ξ)(ξ| denotes the orthogonal projection onto ξ/|ξ|. We can view Ptr as an operator in L2 (Rd , Cd ). The Hilbert space of square integrable transversal vector ﬁelds on Rd is deﬁned as L2tr (Rd , Cd ) := RanPtr . We consider a Pauli-Fierz system with the 1-particle space L2tr (Rd , Cd ) and the 1-particle energy |ξ|. We assume that the interaction is of the form Rd ξ → −2 v(ξ) = 1−|ξ| |ξ)(ξ| v0 (ξ), for a certain function Rd ξ → v0 (ξ) ∈ B(K, K⊗Cd ). We can now repeat almost verbatim the constructions and statements of the previous subsection. Note that the formulas (8.73), (8.76) and (8.77) should be replaced by vρ (p, ω) = 1 − |ω)(ω|

vρ (p, ω) = 1 − |ω)(ω|

1

d−1

p 2 (1 + ρ(pω)) 2 v0 (pω), p>0 d−1 1 2 2 ρ(−pω) v0 (−pω), p ≤ 0; (−p)

(8.79)

1

d−1

p 2 (1 + ρ(pω)) 2 v0 (pω), p > 0 d−1 1 p ≤ 0; (−p) 2 ρ(pω) 2 v0 (pω),

vβ (p, ω) = 1 − |ω)(ω|

p 1 − e−βp

12

(8.80)

|p| 2 −1 v0 (pω). d

(8.81)

The condition (8.78) can be replaced by demanding that R p → |p| 2 −1 v0 (p ·) ∈ B(K, K ⊗ L2 (S d−1 )) is n times diﬀerentiable and d

∂pj |p| 2 −1 p1/2 v0 (pω)2 dpdω < ∞, d

j = 0, n.

(8.82)

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Acknowledgments. The research of the ﬁrst author was partly supported by the Postdoctoral Training Program HPRN-CT-2002-0277 and the grant SPUB127. A part of this work was done during a visit of the ﬁrst author to Aarhus University supported by MaPhySto funded by the Danish National Research Foundation, during his visit to University of Montreal and during a visit of both authors to Schr¨ odinger Institute in Vienna and Johns Hopkins University in Baltimore. The research of the second author was partly supported by NSERC.

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V. Jakˇsi´c, C.-A. Pillet, On a model for quantum friction II: Fermi’s golden rule and dynamics at positive temperature, Comm. Math. Phys. 176, 619 (1996).

[JP2]

V. Jakˇsi´c, C.-A. Pillet, On a model for quantum friction III: Ergodic properties of the spin-boson system, Comm. Math. Phys. 178, 627 (1996).

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V. Jakˇsi´c, C.-A. Pillet, Non-equilibrium steady states for ﬁnite quantum systems coupled to thermal reservoirs, Comm. Math. Phys. 226, 131 (2002).

[M]

M. Merkli, Positive commutators in non-equilibrium quantum statistical mechanics, Comm. Math. Phys. 223, 327 (2001).

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W. Pauli, M. Fierz, Nuovo Cimento 15, 167 (1938).

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T. Okamoto, K. Yajima, Complex scaling technique in non-relativistic qed, Ann. Inst. Henri Poincar´e 42, 391 (1981).

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D. Ruelle, Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98, 57 (2000).

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E. Skibsted, Spectral analysis of N -body systems coupled to a bosonic system, Rev. Math. Phys. 10, 989 (1998).

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[Sp1]

H. Spohn, An algebraic condition for the approach to equilibrium of an open N -level system, Lett. Math. Phys. 2, 33 (1977).

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H. Spohn, Ground states of the spin-boson Hamiltonian, Comm. Math. Phys. 123, 277 (1989).

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S. Stratila, Modular Theory in Operator Algebras, Abacus Press, Turnbridge Wells 1981.

Jan Derezi´ nski Department of Mathematical Methods in Physics Warsaw University Ho˙za 74 PL-00-682 Warszawa Poland email: [email protected] Vojkan Jakˇsi´c Department of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal, QC, H3A 2K6 Canada email: [email protected] Communicated by Gian Michele Graf submitted 22/09/02, accepted 06/05/03

Ann. Henri Poincar´e 4 (2003) 795 – 811 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/040795-17 DOI 10.1007/s00023-003-0147-3

Annales Henri Poincar´ e

Anderson Localization for 2D Discrete Schr¨ odinger Operators with Random Magnetic Fields Fr´ed´eric Klopp, Shu Nakamura, Fumihiko Nakano and Yuji Nomura Abstract. We prove Anderson localization near the bottom of the spectrum for twodimensional discrete Schr¨ odinger operators with random magnetic fields and no scalar potentials. We suppose the magnetic fluxes vanish in pairs, and the magnetic field strength is bounded from below by a positive constant. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, Anderson localization, i.e., pure point spectrum with exponentially decreasing eigenfunctions, is proved by the standard multiscale argument.

1 Introduction We consider a magnetic Schr¨odinger operator on Z2 deﬁned as follows. Let E = (x, y) x, y ∈ Z2 , |x − y| = 1 be the set of the directed edges on Z2 , and let A : E → T := R/(2πZ) be a vector potential such that A((x, y)) = −A((y, x))

for (x, y) ∈ E.

Then, our Hamiltonian is deﬁned by H(A)u(x) = u(x) − eiA((x,y)) u(y) ,

x ∈ Z2 ,

|x−y|=1

for u ∈ 2 (Z2 ). It is easy to show that H(A) is a bounded self-adjoint operator on 2 (Z2 ) and 0 ≤ H(A) ≤ 8 for any vector potential A. The magnetic ﬁeld induced by A is deﬁned as follows: let F = {x1 , x1 + 1} × {x2 , x2 + 1} ⊂ Z2 (x1 , x2 ) ∈ Z2 be the set of unit squares in Z2 . For f ∈ F, the boundary ∂f is deﬁned by ∂fx = (x, x + e1 ), (x + e1 , x + e1 + e2 ), (x + e1 + e2 , x + e2 ), (x + e2 , x) ⊂ E

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where fx = {x1 , x1 +1}×{x2, x2 +1}, x = (x1 , x2 ), and e1 = (1, 0), e2 = (0, 1) ∈ Z2 . The magnetic ﬁeld B = dA is then given by B(f ) = A(e), B : F → T. e∈∂f

It is well known that the spectral properties of H(A) depend only on B, and are independent of the choice of A such that B = dA. We suppose Assumption A. For any n, m ∈ Z, B ω (f(2n+1,m) ) = −B ω (f(2n,m) ), and B ω (f(2n,m) ) n, m ∈ Z are i.i.d. random variables on a probability space (Ω, B, µ). Let ν be the common distribution of B ω (f ). ν has a bounded density g(λ). Moreover, there exists c > 0 such that supp g ⊂ (T \ (−c, c)), ±c ∈ supp g, and g is Lipshitz continuous on T \ (−c, c). Under this assumption, σ(H(Aω )) = [4(1 − cos(c/4)), 4(1 + cos(c/4))] almost surely (cf. [12] Theorem 1 and Example 1). We set E0 = inf σ(H(Aω )) = 4(1 − cos(c/4)). If we set the vector potential Aω so that  ω  if e = ((2n + 1, m), (2n + 1, m + 1)), B (f(2n,m) ), ω A (e) = −B ω (f(2n,m) ), if e = ((2n + 1, m + 1), (2n + 1, m)),   0, otherwise, then, Aω generates the magnetic ﬁeld B ω . Thus we may consider our model to be a Schr¨ odinger operator with random vector potential, not random magnetic ﬁeld. Theorem 1.1. Suppose Assumption A holds. Then, Anderson localization holds near the bottom of the spectrum. Namely, there exist E > E0 such that H(Aω ) has dense pure point spectrum on [E0 , E ] almost surely, and each eigenfunction associated to an energy in this interval decays exponentially as |x| → ∞. Theorem 1.1 is proved by the standard multiscale argument (see, e.g., [6], [4], [15] and references therein), combined with the Lifshitz tail (Theorem 1.2) and the Wegner estimate (Theorem 1.3). In order to state these results explicitly, we introduce the integrated density of states (IDS). For L > 0, we set ΛL = [−L, L]2 ∩ Z2 ⊂ Z2

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be the ﬁnite lattice of size |ΛL | = (2L + 1)2 . Let HΛL (A) be the magnetic Schr¨ odinger operator restricted to ΛL (the precise deﬁnition is given in Section 2). For E ∈ R, the integrated density of states is deﬁned by k(E) = lim

L→∞

1 eigenvalues of HΛL (Aω ) ≤ E . |ΛL |

See Appendix C of [12] for the proof of the existence of k(E) for discrete magnetic Schr¨ odinger operators. Note that k(E) is a nonrandom quantity, i.e., it is almost surely independent of the choice of the sample ω. Theorem 1.2. Under the above assumptions, lim log(− log k(E))/ log(E − E0 ) ≤ −1.

E↓E0

(1.1)

This result implies, roughly speaking, k(E) e−(E−E0 )

−1

as E ↓ E0 ,

and hence, the IDS is very “thin” near the bottom of the spectrum. We note that, with an additional assumption on ν, (1.1) becomes an equality (cf. Theorem 2.3). On the other hand, the Wegner estimate implies that the distribution of the eigenvalues of HΛL (Aω ) admits a density in a low energy region. In particular, it follows from the Wegner estimate that the IDS is Lipschitz continuous. Theorem 1.3. Under the above assumptions, there exist E1 > E0 and C > 0 such that P dist(σ(HΛL (Aω )), E) < ε ≤ Cε|ΛL | for E ∈ [E0 , E1 ], L > 0 and ε > 0. Once Theorems 1.2 and 1.3 are proved, Theorem 1.1 follows by the standard multiscale argument, and we omit the detail. We note that the Combes-Thomas estimate and the decomposition of resolvents in the multiscale argument work for magnetic Schr¨ odinger operators with essentially no modiﬁcations. Whereas a large amount of work has been done on the spectral properties of Schr¨ odinger operators with random potentials (see, e.g., [2], [5], [15] and references therein), only a few results have been obtained on Schr¨ odinger operators with random magnetic ﬁelds. Ueki ([17]) proved the Lifshitz tail for a class of Gaussian random magnetic ﬁeld, and Nakamura ([12], [13]) proved it for the 2D discrete case, and the continuous case, respectively. Hislop and Klopp proved the Wegner estimate near the bottom edge of the spectrum for continuous case ([8]). They suggested Anderson localization (combined with the result of [13]), but it was not completely clear if there exists an interval in the spectrum that satisﬁes both conditions. In a recent paper [18], Ueki proved Anderson localization for Schr¨ odinger operator with a random potential and a correlated random magnetic ﬁeld.

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There have been active discussions about the spectrum of random magnetic Schr¨ odinger operators in the physics literature, most of them are mainly numerical computations (see, e.g., [7], [11], [9], [14]). There seems to be no agreement on the existence of continuous spectrum in the middle of the spectrum; however, it appears that the localization near the spectrum edges, which is the subject of this paper, is widely believed. We prove Theorem 1.2 under more general assumptions in Section 2, and Theorem 1.3 is proved in Section 3. In the appendix, we discuss another model of discrete magnetic Schr¨odinger operator, for which we prove Anderson localization.

2 The Lifshitz tail We ﬁrst deﬁne HΛL (Aω ) on 2 (ΛL ). We note 2 1 u(i(e)) − eiA(e) u(t(e)) u|Hu = 2 e∈E 2 1 u(i(e)) − eiA(e) u(t(e)) , = 2 f ∈F e∈∂f

where i(e) and t(e) denote the initial point and the terminal point of e, respectively. Namely, i(e) = x, t(e) = y for e = (x, y). Then, we set u|HΛL u =

2 1 u(i(e)) − eiA(e) u(t(e)) + |u(x)|2 , 2 f ∈F , e∈∂f f ⊂ΛL

x∈∂ΛL

where

∂ΛL = x ∈ ΛL |xi | = L for i = 1 or 2 . The boundary term ∂ΛL |u(x)|2 does not aﬀect the IDS since it is an operator of rank 8L << |ΛL |. One may consider our Hamiltonian HΛL an analogue to the Dirichlet Hamiltonian, (though they are slightly diﬀerent). In order to prove Theorem 1.2, we follow the argument of [12], but we need a more precise local energy estimate since inf σ(H(Aω )) > 0. We ﬁx √ 0 < α < 1 − 1/ 2, and deﬁne β(t) = min(1 − cos(t/4), α) for t ∈ T ∼ = [−π, π). We set WB (x) =

x∈f

Then, we have

β(B(f )),

x ∈ Z2 .

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Theorem 2.1. For u ∈ 2 (Z2 ),

u|Hu ≥ uWB u + γ |u|H0 |u|

where γ=

1 1 1 − √ − α > 0, 4 2

and H0 is the free discrete Schr¨ odinger operator on Z2 . Proof. We reﬁne the argument of Theorem 2 of [12]. We consider the Hamiltonian Hf on 2 (f ) ∼ = C4 deﬁned by

1 uf (i(e)) − eiA(e) uf (t(e))2 u f Hf u f = 2 e∈∂f

for uf ∈ 2 (f ). We may write f = y0 , y1 , y2 , y3 ,

ej = (yj , yj+1 ),

where y4 = y0 , and 3

1 uf (yj ) − eiA(ej ) uf (yj+1 )2 . u f Hf u f = 2 j=0

By a gauge transform, we may suppose A(ej ) = B/4 where B = B(f ). Indeed, there exists {gj }3j=0 with |gj | = 1 such that if we set u ˜f (yj ) = gj uf (yj ) then,

3 2

1 u ˜f (yj ) − eiB/4 u u f Hf u f = ˜f (yj+1 ) . 2 j=0

˜ f on 2 (f ) so that We deﬁne H

˜f u uf Hf uf = u˜f H ˜f . It is easy to show ˜ f ) = λj j = 0, 1, 2, 3 σ(Hf ) = σ(H where λj = 1 − cos((B + 2πj)/4), ˜ f are given by and the eigenvectors of H vj =

1 (1, eiπj/2 , e2iπj/2 , e3iπj/2 ), 2

j = 0, 1, 2, 3.

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Let Πj be the orthogonal projection to the eigenspace with the eigenvalue λj . Then, we have 3

uf |Hf uf =

λj Πj u ˜ f 2

j=0

≥ β(B) Π0 u˜f 2 +

3

λj Πj u ˜ f 2

j=1

= β(B) ˜ uf 2 +

(2.1)

3 (λj − β(B)) Πj u ˜ f 2 j=1

≥ β(B) uf 2 + 4γ (1 − Π0 )˜ uf 2 √ by the deﬁnitions of β(t) and γ, and the fact that λj ≥ 1 − 1/ 2 for j = 1, 2, 3. We then estimate the last term in the right-hand side of (2.1). For v ∈ 2 (f ), we have

(1 − Π0 )v 2 =

3 j=0

≥ where v¯ =

1 4

3

j=0

|v(yj ) − v¯|2 ≥

3

1 |v(yj ) − v(yj+1 )|2 4 j=0

3 2 1 |v(yj )| − |v(yj+1 )| 4 j=0

v(yj ) is the average of v. Hence, we have

uf 2 ≥

(1 − Π0 )˜

=

3 2 1 |˜ uf (yj )| − |˜ uf (yj+1 )| 4 j=0 3 2

1 1 |uf (yj )| − |uf (yj+1 )| = |uf |H0,f |uf | 4 j=0 2

where H0,f is the free Schr¨ odinger operator on 2 (f ). Combining these, we learn

uf |Hf uf ≥ β(B(f )) uf 2 + γ |uf |H0,f |uf | . If we set uf = uf and sum up this inequality over f ∈ F, then, we obtain u|Hu =

f ∈F

β(B(f )) uf 2 + γ

|uf |H0,f |uf | f ∈F

= uWB u + γ |u|H0 |u| .

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We can also prove the following estimate for HΛL in exactly the same way as above. Note that we use the boundary term in the deﬁnition of HΛL near the boundary. Now we denote FL = f ∈ F f ⊂ ΛL . Theorem 2.2. For u ∈ 2 (ΛL ),

u|HΛL u ≥ uWB,ΛL u + γ |u|H0,ΛL |u| ,

where WB,ΛL (x) =

β(B(f )),

x∈f ⊂ΛL

and H0,ΛL is the free Schr¨ odinger operator on 2 (ΛL ) defined by 1 u|H0,ΛL u = |u(i(e)) − u(t(e))|2 . 2 f ∈FL e∈∂f

Given this estimate, we can prove the following generalization of Theorem 1.2, using the argument of [12] and the large deviation argument of [10, 16]. We omit the details. Theorem 2.3. Suppose {B(f )|f ∈ F } are metrically transitive random variables with finite correlation length, i.e., there exists R > 0 such that {B(f )|f ∈ F1 } and {B(f )|f ∈ F2 } are independent if dist(F1 , F2 ) ≥ R. Let µ be the common distribution of B(f ), and suppose supp µ ⊂ [−b+ , −b− ] ∪ [b− , b+ ] with 0 < b− < b+ ≤ π. Then, lim log(− log k(E))/ log(E − E0 ) ≤ −1,

E↓E0

where E0 = 4(1 − cos(b− /4)). Moreover, if in addition, either µ([b− , b− + ε]) ≥ Cεa ,

or µ([−b− − ε, −b− ]) ≥ Cεa

for ε > 0,

with some C, a > 0, then lim log(− log k(E))/ log(E − E0 ) = −1.

E↓E0

3 The Wegner estimate In this section, we suppose Assumption A, and let Aω (e) be the vector potential deﬁned in Section 1. We decompose H as in the proof of Theorem 2.1, but we ﬁx the gauge as above. Namely, we write u|Hu = uf |Hf uf , u ∈ 2 (Z2 ), f ∈F

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where uf = u|f ∈ 2 (f ) ∼ = C4 . Hf is a self-adjoint matrix on 2 (f ). Let vj (j = 0, 1, 2, 3) be the normalized eigenvectors with the eigenvalues λj (B(f )) = 1 − cos((B(f ) + 2πj)/4),

j = 0, 1, 2, 3.

The eigenvectors vj are easily computed (cf. [12] Lemma 3). Lemma 3.1. Fix f ∈ F. Let uf ∈ 2 (f ) and denote αf = |v0 |uf |,

βf =

3

2

|vj |uf |

1/2 .

j=1

Then,

∂Hf |B(f )| 2 1 uf ≥ sin uf αf − c1 αf βf − c2 βf2 , ∂|B(f )| 4 4

where c1 = 8, c2 = 16 + 1/4. Proof. For simplicity, we denote b = |B(f )| in this proof. Since uf |Hf uf =

3

λj |uf |vj |2 ,

j=0

we have

3 ∂Hf ∂λj uf = |uf |vj |2 uf ∂b ∂b j=0 3 ∂vj ∂vj uf vj |uf + vj |uf uf + λj ∂b ∂b j=0 = I + II.

The ﬁrst term is estimated as 3 b b 1 1 1 ∂λj 2 2 |vj |uf | ≥ sin I = sin αf + α2f − βf2 . 4 4 ∂b 4 4 4 j=1 We note 0=

∂ vj |vk = ∂b

∂vj ∂vk v vj . + k ∂b ∂b

Hence, we have

3 3 ∂vj ∂vj ∂vk uf = vk vk |uf = − vj vk |uf . ∂b ∂b ∂b k=0

k=0

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Using this estimate we have 3

II =

j,k=0

∂vk vj vj |uf vk |uf . (λk − λj ) ∂b

Then, using the explicit form of vj , we learn |II| ≤ 4

|vj |uf | · |vk |uf |

j=k

≤ 4|v0 |uf |

3

3

|vj |uf | + 4

j=1

|vj |uf | · |vk |uf |

j,k=1

≤ 8αf βf + 16βf2 . The lemma follows from these estimates. Lemma 3.2. Suppose u ∈ 2 (ΛL ), u = 1, and HΛL u = Eu with E > E0 . Let uf = u|f , and let αf , βf be as in the previous lemma. Then,

βf2 ≤ c3 (E − E0 ),

f ∈FL

where c3 = 1 −

√1 2

α2f ≥ 4 − c4 (E − E0 ),

f ∈FL

−

E0 4

−1

, c4 = c3 + 3

√3 2

−2

−1

.

Proof. We note 1 = u 2 ≤

1 3

uf 2 + |u(x)|2 4 4 f

∂ΛL

1 2 1 2 3 = αf + βf + |u(x)|2 . 4 4 4 f

f

∂ΛL

On the other hand, by the deﬁnition of HΛL , we have E = u|HΛL u =

uf |Hf uf +

f

=

f

λ0 (B(f ))α2f +

|u(x)|2

∂ΛL 3 f

λj (B(f ))|vj |uf |2 +

j=1

E0 2 1 2 ≥ αf + 1 − √ βf + |u(x)|2 4 2 f f ∂Λ L

∂ΛL

|u(x)|2

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1 1 2 3 α2f + βf + |u(x)|2 4 4 4 f f ∂ΛL E0 3 1 βf2 + 1 − E0 |u(x)|2 + 1− √ − 4 4 2 f ∂ΛL E0 1 1 ≥ E0 + 1 − √ − βf2 + 1 − 3 1 − √ |u(x)|2 . 4 2 2 = E0

f

∂ΛL

The ﬁrst estimate of the lemma follows immediately from this estimate. Then, α2f ≥ 4 − βf2 − 3 |u(x)|2 f

f

∂ΛL

−1 1 ≥ 4 − c3 (E − E0 ) − 3 1 − 3 1 − √ (E − E0 ) 2 = 4 − c4 (E − E0 ).

Lemma 3.3. Let u ∈ 2 (ΛL ) as in the last lemma. Then, ∂H(A) u ≥ sin(c/4) − c5 (E − E0 ) − c6 E − E0 u ∂|B(f )|

(3.1)

f ∈FL

1/2

where c5 = c4 /4 + c2 c3 , c6 = 2c1 c3 . Proof. By Lemmas 3.1 and 3.2, we have ∂H(A) ∂Hf u = uf u uf ∂|B(f )| ∂|B(f )| f

f

1 α2f − c1 βf2 ≥ sin(c/4) αf βf − c2 4 f 1/2 1/2 c4 ≥ sin(c/4) 1 − (E − E0 ) − c1 − c2 βf2 α2f βf2 4 c4 1/2 ≥ sin(c/4) − E − E0 . + c2 c3 (E − E0 ) − 2c1 c3 4

Given Lemma 3.3, the Wegner estimate is proved by the standard argument. Theorem 3.4. Suppose E1 − E0 is so small that the right-hand side term of (3.1) be positive for E = E1 . Then, there exists C > 0 such that P dist(σ(HΛL (Aω )), E) < ε ≤ Cε|ΛL | for ε > 0, L > 0 and E ∈ [E0 , E1 ].

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Proof. The idea of the proof is similar to that in [3] and [8]. For simplicity, we suppose L is even. If L is odd, we have to take the contribution from the boundary area into account, but it is similarly handled and we omit it. We choose E1 and ε0 > 0 so that c7 = sin(c/4) − c5 (E1 − E0 − ε0 ) − c6 E1 − E0 − ε0 > 0. We ﬁx E ∈ [E0 , E1 ], and for 0 < ε < ε0 we set η ∈ C0∞ (R) so that 1, (|t − E| ≤ ε), η(t) = 0, (|t − E| ≥ 2ε), and 0 ≤ η(t) ≤ 1 for all t ∈ R. By Chebyshev’s inequality, we have P(dist(σ(HΛL (Aω )), E) < ε) ≤ E(Tr(η(HΛL (Aω )))).

(3.2)

Let {Ejω |j = 1, 2, . . . } be the eigenvalues of HΛL (Aω ), and let {ψjω } be the corresponding eigenfunctions. Then, by a standard computation of analytic perturbation theory, we have ∂HΛL (Aω ) ∂Ejω = ψj ψj . ∂|B ω (f )| ∂|B ω (f )|

Let ξ(t) =

t

∞

η(s)ds ∈ C ∞ (R).

Then, we learn ∂ ∂|B ω (f )|

Tr(ξ(HΛL (Aω ))) =

∂ ∂|B ω (f )|

ξ(Ejω ) = −

j

j

∂Ejω η(Ejω ), ∂|B ω (f )|

where the sum is taken over j such that Ejω ∈ [E − 2ε, E + 2ε]. Combining these with Lemma 3.3, we obtain ∂HΛ (Aω ) ∂ L ω Tr(ξ(HΛL (A ))) = ψj η(Ejω ) ψj ω (f )| ∂|B ω (f )| ∂|B f ∈FL f ∈FL j ≥ c7 η(Ejω ) ≥ c7 Tr(η(HΛL (Aω ))). (3.3) j

Next we compute the expectation of the left-hand side of (3.3). We denote ΛL = (2n + 1, m) ∈ ΛL n, m ∈ Z , and for y = (2n + 1, m) ∈ ΛL , we write e(y) = ((2n + 1, m), (2n + 1, m + 1)), f+ (y) = {2n + 1, 2n + 2} × {m, m + 1}, f− (y) = {2n, 2n + 1} × {m, m + 1}.

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The probability space for HΛL is ΩL = TΛL , and the probability measure is given by ν(Aω (e(y))) = g(Aω (e(y)))dAω (e(y)). y∈ΛL

y∈ΛL

Since ∂ ∂|Aω (e(y))|

=

∂ ∂|B ω (f

− (y))|

+

∂ ∂|B ω (f

+ (y))|

,

the left-hand side of (3.3) is f ∈FL

∂ ∂|B ω (f )|

Tr(ξ(HΛL (Aω ))) =

y∈ΛL

∂ ∂|Aω (e(y))|

Tr(ξ(HΛL (Aω ))).

Hence, its expectation is E

∂ ω Tr(ξ(H (A ))) ΛL ∂|Aω (e(y))| y∈ΛL ∂ ω Tr(ξ(H = (A ))) g(Aω (e(y )))dAω (e(y )). ··· Λ L ∂|Aω (e(y))| y∈ΛL

y ∈ΛL

We denote Kyt = HΛL (Aty ) with  ω  A (e), t Ay (e) = t,   −t,

(e = e(y), e(y)), (e = e(y)), (e = e(y)).

By an integration by parts, for y ∈ ΛL we have

∂ Tr(ξ(HΛL (Aω )))g(Aω (e(y)))dAω (e(y)) ∂|Aω (e(y))| ∂ Tr ξ(Kyt ) − ξ(Kyπ ) g(t)dt = ∂|t| π −c ∂ ∂ Tr ξ(Kyt ) − ξ(Kyπ ) g(t)dt − Tr ξ(Kyt ) − ξ(Kyπ ) g(t)dt = c ∂t −π ∂t c π = −g(c)Tr ξ(Ky ) − ξ(Ky ) − g(−c)Tr( ξ(Ky−c ) − ξ(Kyπ ) −c π t π Tr ξ(Ky ) − ξ(Ky ) g (t)dt + Tr ξ(Kyt ) − ξ(Kyπ ) g (t)dt − c

−π

≤ (2 sup |g| + 2π sup |g |) sup |Tr(ξ(Kyt ) − ξ(Kyπ ))|. t

(3.4)

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Since Kyt − Kyπ is an operator of rank 2, we have t π |Tr(ξ(Ky ) − ξ(Ky ))| = − ξ (s)|Ξ(s; Kyt , Kyπ ))|ds ≤ 2 η(s)ds ≤ 8ε,

807

(3.5)

where Ξ(s; A, B) denotes the spectral shift function for the pair of operators A and B. Note that Ξ(s; A, B) is uniformly bounded by the rank of A − B (see, e.g., [1]). Thus (3.4) and (3.5) imply ∂ ω E Tr(ξ(H (A ))) ΛL ∂|Aω (e(y))| y∈ΛL ≤ · · · c8 ε g(Aω (e(y )))dAω (e(y )) ≤ c8 ε|ΛL |, y∈ΛL

y =y

where c8 = 16 sup |g| + 16π sup |g |. From (3.2), (3.3) and this estimate, we learn P(dist(σ(HΛL (Aω )), E) < ε) ≤ Cε|ΛL | with C = c8 /c7 , and Theorem 3.4 is proved.

A

Appendix: Another model of magnetic Schr¨ odinger operator

Here we discuss another model of two-dimensional discrete magnetic Schr¨odinger operator for which we can prove the Wegner estimate, Lifshitz tails, and thus, Anderson localization. We suppose the vector potential A(e) has the following form. For 0 < b < π, we set b if x1 + x2 is even, b B0 (fx ) = −b if x1 + x2 is odd, and we ﬁx Ab0 so that dAb0 = B0b . For example, we set b/2 if x1 + x2 is even, b A0 ((x, x + e1 )) = −b/2 if x1 + x2 is odd, and Ab0 ((x, x + e2 )) = 0 for all x ∈ Z2 . Let aω (e) : E → [−1, 1] be i.i.d. random variables. Let λ > 0 and set Aω (e) = Ab0 (e) + λaω (e) ∈ T

for e ∈ E.

We denote the common distribution of aω (e) by ν. We note that σ(H(Ab0 )) = [4(1 − cos(b/4)), 4(1 + cos(b/4))]

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(see [12], Example 1). Moreover, if ν([−1, −1 + ε]) > 0,

ν([1 − ε, 1]) > 0 for ε > 0,

(A.1)

then, we also have σ(H(Aω )) = [4(1 − cos((b − 4λ)/4)), 4(1 + cos((b − 4λ)/4))] almost surely, provided b − 4λ ≥ 0. Assumption B. ν has a bounded density function g(t), and ν satisﬁes (A.1). Moreover, g(t) is Lipschitz continuous on [−1, 1]. Theorem A.1. Suppose Aω has the above form and suppose Assumption B. Then, there exists λ1 > 0 such that if 0 < λ < λ1 , then P dist(σ(HΛL (Aω )), E) < ε ≤ Cε|ΛL | for E ∈ [E0 , E2 ] and ε > 0, with some E2 > E0 and C > 0. This Wegner estimate is proved using the idea of Hislop and Klopp [8]. In particular, the next estimate is a key step of the proof. Lemma A.2. We denote F0 = inf σ(H(Ab0 )) = 4(1 − cos(b/4)). Let ψ be a normalized eigenfunction of HΛL (Aω ) with an eigenvalue E < F0 . Then, ∂HΛL (Aω ) 1 ω ψ ≥ F0 − E − 6λ2 , a (e) ψ ω (e) 2 ∂a e∈EL

where EL = e ∈ E e ⊂ ΛL . Proof. We decompose HΛL (Aω ) as follows: b HΛL (Aω )u(x) = u(x) − eiA0 ((x,y)) u(y) + χ∂ΛL (x)u(x) |x−y|=1

+

b ω 1 − eiλa ((x,y)) eiA0 ((x,y)) u(y)

|x−y|=1

= HΛL (Ab0 )u(x) + V ω u(x), with u ∈ 2 (ΛL ). By direct computations, we have 1 ∂V ω u uaω (e) ω 2 ∂a (e) e∈EL ω b i =− λaω (e)ei(λa (e)+A0 (e)) u(i(e))u(t(e)) 2 e ω b + λaω (e)e−i(λa (e)+A0 (e)) u(t(e))u(i(e)) e

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Anderson Localization for 2D Discrete Schr¨ odinger Operators

= −i

ω

λaω (e)ei(λa

(e)+Ab0 (e))

809

u(i(e))u(t(e))

e

=−

b ω eiλa (e) − 1 eiA0 (e) u(i(e))u(t(e)) + r(e)u(i(e))u(t(e)) e

e

= −u|V ω u + u|Ru where

(A.2)

b ω ω r(e) := (eiλa (e) − 1) − iλaω (e)eiλa (e) eiA0 (e) .

It is easy to observe by Taylor’s theorem that |r(e)| ≤ and hence,

3 3 2 ω λ |a (e)|2 ≤ λ2 , 2 2

R ≤ 4 sup |r(e)| ≤ 6λ2 . e

(A.3)

On the other hand, we have E = ψ|HΛL (Aω )ψ = ψ|HΛL (Ab0 )ψ + ψ|V ω ψ ≥ F0 + ψ|V ω ψ and hence,

ψ|V ω ψ ≤ −(F0 − E).

(A.4)

Combining (A.2) with (A.3) and (A.4), we obtain ω ∂HΛL (Aω ) 1 ψ a (e) ψ = −ψ|V ω ψ + ψ|Rψ ≥ (F0 − E) − 6λ2 . 2 ∂aω (e) e∈EL

Using this estimate, Theorem A.1 (the Wegner estimate) is proved as in [8]. Theorem 2.3 (the Lifshitz tail) applies to this model, and Anderson localization near the spectrum edges is proved by the standard method in the same way as Theorem 1.1. We omit the detail.

References [1] M.Sh. Birman, D.R. Yafaev, The spectral shift function. The papers of M.G. Krein and their further development, St. Petersburg Math. J. 4, 833–870 (1993). [2] R. Carmona, J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Birkh¨auser, Boston, MA, 1990. [3] J.M. Combes, P.D. Hislop, S. Nakamura, The Lp -theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators, Comm. Math. Phys. 218, 113–130 (2001).

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[4] H. von Dreifus, A. Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124, 285–299 (1989). [5] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Berlin, 1992. [6] J. Fr¨ ohlich, T. Spencer, Absence of diﬀusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88, 151–184 (1983). [7] B.I. Halperin, P.A. Lee, N. Read, Theory of half-ﬁlled Landau level, Phys. Rev. B 47, 7312–7343 (1993). [8] P.D. Hislop, F. Klopp, The integrated density of states for some random operators with nonsign deﬁnite potentials, Jour. Func. Anal. 195, 12–47 (2002). [9] T. Kawarabayashi, T. Otsuki, Diﬀusion of electrons in random magnetic ﬁelds, Phys. Rev. B 51, 10897–10904 (1995). [10] W. Kirsch, F. Martinelli, Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians, Comm. Math. Phys. 89, 27–40 (1983). [11] D.K. Lee, J.T. Chalker, Uniﬁed model of two localization problems: Electron states in spin-degenerate Landau levels and in a random magnetic ﬁeld, Phys. Rev. Lett. 72, 1510–1513 (1994). [12] S. Nakamura, Lifshitz tail for 2D discrete Schr¨ odinger operator with random magnetic ﬁeld, Ann. Henri Poincar´e 1, 823–835 (2000). [13] S. Nakamura, Lifshitz tail for Schr¨ odinger operator with random magnetic ﬁeld, Comm. Math. Phys. 214, 565–572 (2000). [14] D.N. Sheng, Z.Y. Weng, Delocalization of electrons in a random magnetic ﬁeld, Phys. Rev. Lett. 75, 2388–2391 (1995). [15] P. Stollmann, Caught by Disorder. Bound States in Random Media, Birkh¨ auser, Boston, MA, 2001. [16] B. Simon, Lifshitz tails for the Anderson model, Journal of Statistical Physics, 38, 65–76 (1985). [17] N. Ueki, Simple examples of Lifschitz tails in Gaussian random magnetic ﬁelds, Ann. Henri Poincar´e 1, 473–498 (2000). [18] N. Ueki, Seminar talk at RIMS, 2001.

Vol. 4, 2003

Anderson Localization for 2D Discrete Schr¨ odinger Operators

Fr´ed´eric Klopp1 L.A.G.A., Institut Galil´ee Universit´e Paris-Nord F-93430 Villetaneuse France email: [email protected] Shu Nakamura2 Graduate School of Mathematical Sciences University of Tokyo 7-3-1 Komaba Meguro Tokyo 153-8914 Japan email: [email protected] Fumihiko Nakano Mathematical Institute Tohoku University Aoba, Sendai 980-8578 Japan email: [email protected] Yuji Nomura Department of Mathematics 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 Gian Michele Graf submitted 01/10/02, accepted 16/04/03

1 F.K.’s 2 S.N.’s

research is partially supported by FNS 2000 “Programme Jeunes Chercheurs”. research is partially supported by JSPS grant Kiban C-13640155

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Annales Henri Poincar´ e

Scattering on Reissner-Nordstrøm Metric for Massive Charged Spin 1/2 Fields Fabrice Melnyk Abstract. We consider massive charged Dirac fields on the Reissner-Nordstrøm metric. We prove the existence and asymptotic completeness of wave operators, classical at horizon and modified at infinity.

1 Introduction In this paper, we consider the linear, charged, massive Dirac equation on the Reissner-Nordstrøm space-time. This space-time outside a charged, static blackhole is the globally hyperbolic manifold (M, g), M = Rt ×]r0 , +∞[r ×Sω2 , gµν dxµ dxν = F (r)dt2 − F −1 (r)dr2 − r2 dω 2 , 2

dω 2 = dθ2 + sin θdϕ2 , ω = (θ, ϕ) ∈ [0, π] × [0, 2π[, Q2 2M + 2, 0 < r0 = M + M 2 − Q2 < +∞, F (r) = 1 − r r

(1) (2)

0 ≤ |Q| < M,

(3)

where, Q, M and r0 are respectively the electric charge, the mass and the radius of the horizon of the black-hole. We introduce a radial coordinate r∗ , which straightens the radial null geodesics: r 1 1 2κ0 1 dr∗ r∗ = = F −1 , − ln(r − r0 ) − dx , κ0 = F (r0 ), 2κ0 x − r F (x) 2 dr 0 r0 (4) where κ0 is the surface gravity at the black hole horizon. This coordinate r∗ shifts the horizon to negative inﬁnity. Then, we can develop a time-dependent scattering theory in two separate asymptotic regions, constructing the wave operators, classical at the horizon and Dollard-modiﬁed at inﬁnity since the perturbations, when r∗ → +∞, are long-range. We prove the existence and the asymptotic completeness of these operators. This work is part of a research program concerning time-dependent scattering on black-hole-type space-times. For Schwarzschild’s metric (Q = 0 in (3)), we refer to several important papers. The existence and asymptotic completeness of classical wave operators for the wave equation has been studied by J. Dimock [Di]. The case of the Maxwell and Regge-Wheeler equations has been respectively investigated by A. Bachelot [Ba1] and A. Bachelot, Ag. Motet-Bachelot [BM-B].

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J. Dimock and B.S. Kay [DK] established the existence of modiﬁed wave operators for the Klein-Gordon equation. For the same equation, A. Bachelot [Ba2] proved the existence and asymptotic completeness of wave operators, classical at the horizon and Dollard-modiﬁed at inﬁnity. Recently, in the Kerr metric D. H¨ afner [Ha1][Ha2] developed a scattering theory for the Klein-Gordon ﬁeld. As regards the massless Dirac system on Schwarzschild’s space-time, J.-P Nicolas [N1] established the existence and asymptotic completeness of classical wave operators at the horizon and at inﬁnity. In the massive case, we need to modify the wave operators at inﬁnity because of the long-range interaction created by the mass. W.M. Jin [Ji] proved the existence of similar operators on Schwarzschild’s metric. In this paper, we construct the modiﬁed wave operators at inﬁnity as in [Me], using the ideas of B. Thaller [Th] and J. Dollard, G. Velo [DV] on Minkowski’s space-time. For scalar and matrix-valued long-range perturbations, the existence and asymptotic completeness of time-modiﬁed wave operators for the Dirac system in ﬂat space-time, have been proved by PL. Muthuramalingam, K.B. Sinha [MS] and V. Enss, B. Thaller [TE], B. Thaller [Th]. We can also refer to the paper by Y. Gˆ atel and D. Yafaev [GY], where the authors give a proof of asymptotic completeness of time-independent modiﬁed wave operators for Dirac in ﬂat spacetime perturbed by long-range electromagnetic potential. The main tools to obtain our result are the Enss and Thaller method [TE][Th] for Dirac’s equation with long-range potentials in ﬂat space-time, the Kato-Birman theorem and ﬁnally, the Mourre method to prove the absence of singular continuous spectrum.

2 The Dirac equation For particles with real charge q and mass m > 0, the Dirac equation on (M, g) takes the form (see [N2] and [Ba5]), F iγ 2 1 1 qQ 1 1 −2 0 1 2 + iF γ ∂t + i + iF γ ∂r + + ∂θ + cot θ r r 4F r 2

iγ 3 + ∂ϕ − m Ψ = 0, (5) r sin θ where the Dirac matrices γ k , satisfy µ, ν = 0, . . . , 3, η µν = Diag(1, −1, −1, −1). γ µ γ ν + γ ν γ µ = 2η µν I R4 , 0 σ0 0 σk 0 k , γ =i γ =i k = 1, 2, 3, −σ 0 0 σk 0 with the Pauli matrices, 1 0 1 0 , σ1 = , σ0 = 0 1 0 −1

σ2 =

0 1 1 0

,

σ3 = i

0 1

−1 0

(6) (7) . (8)

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With the new coordinate r∗ , (5) becomes, i∂t Ψ = IDΨ

F 1/2 (r) F (r) 1 1 qQ 1 + Γ ∂r∗ + + F (r) + Γ2 (∂θ + cot θ) ID = r r 4 r 2

(9) (10)

F 1/2 (r) ∂ϕ + Γ4 F 1/2 (r), r sin θ Γ1 = −iγ 0 γ 1 = −iDiag(−1, 1, 1, −1), Γ2 = −iγ 0 γ 2 , Γ3 = −iγ 0 γ 3 , Γ4 = mγ 0 . (11) + Γ3

We introduce the Hilbert space, H = L2 (Rr∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 .

(12)

The properties of the angular parts of the operator ID allow us to reduce (9) to a family of one-dimensional problems. To do this, we use spin-weighted harmonl l (see [GS],[Me]). For each spinorial weight s, 2s ∈ Z, the family {Ysn = ics Ysn −inϕ l 2 2 e usn ; l − |s| ∈ N, l − |n| ∈ N} forms a Hilbert basis of L (Sω , dω) where each ulsn satisﬁes the recurrence relations, −i (l ± n)(l ∓ n ± 1)ul s−n cos θ l l s,n∓1 (θ), ±l > −n, ∂θ usn (θ) ∓ sin θ usn (θ) = 0, l = ∓n (13) and −i (l ± s)(l ∓ s + 1)ul cos θ l s∓1,n (θ), ±l > −s. ∂θ ulsn (θ) ∓ n−s u (θ) = sn 0, l = ∓s. sin θ (14) We deﬁne ⊗4 , the operation between two vectors of C4 such that ∀v = (v1 , v2 , v3 , v4 ), u = (u1 , u2 , u3 , u4 ) ∈ C4 , v ⊗4 u = (u1 v1 , u2 v2 , u3 v3 , u4 v4 ). Since the families

1 Y 1l ,n ; (l, n) ∈ I , Y−l 1 ,n ; (l, n) ∈ I , I = (l, n)/ l − ∈ N, l − |n| ∈ N , 2 2 2 form a Hilbert basis of L2 (Sω2 ), we express H as a direct sum: Hln , Hln = L2w ⊗4 V ect[Yln ], L2w = L2 (Rr∗ , r2 F 1/2 (r)dr∗ )4 , H=

(15)

(l,n)∈I

Yln =

Y−l 1 ,n , Y 1l ,n , Y−l 1 ,n , Y 1l ,n . 2

2

2

2

Any Ψ(t, .) ∈ H, where Ψln (t, .) ∈ L2w , can be written in the following way: [Ψln ⊗4 Yln ](t, r∗ , ω). Ψ(t, r∗ , ω) = (l,n)∈I

(16)

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With (14) and s = ±1/2, we obtain ID = (l,n)∈I [IDln ⊗ 1 ω ] IDln = Γ1 ∂r∗ +

F (r) r

+ 14 F (r) +

qQ r

− iF

1/2

(r)

r

l + 12 Γ2 + F 1/2 (r)Γ4 .

(17)

Following Proposition 4.1 in [Me], we can show that ID is self-adjoint on H with the dense domain     2 Ψln ⊗4 Yln : Ψln ∈ D (IDln ) , IDln Ψln L2w < +∞ . D(ID) = Ψ =   (l,n)∈I

(l,n)∈I

(18) Hence, if Ψ0 ∈ H, equation (9) has a unique solution Ψ(t), such that Ψ(t) ∈ C 0 (Rt , H), Ψ(0) = Ψ0 and Ψ(t) = e−itID Ψ0 . (19)

3 Wave operators To investigate the behaviour of the ﬁeld near the horizon, we introduce the Hilbert space (20) H0 = L2 (Rr∗ × Sω2 ; dr∗ dω)4 . We consider H0 , the formal limit of H when r∗ → −∞, H0 = Γ1 ∂r∗ +

qQ , r0

(21)

and the associated Schr¨odinger equation i∂t Ψ = H0 Ψ.

(22)

H0 is self-adjoint on H0 with the dense domain D(H0 ) = H 1 [Rr∗ ; L2 (Sω2 )]4 . Hence, for all Ψ ∈ H0 , there exists an unique solution of (22) in C 0 (Rt , H0 ) given by e−itH0 Ψ0 , t ∈ R. The spectrum of H0 is purely absolutely continuous. Using the H0± , stable under H0 , of simple form of Γ1 , (see (11)), we deﬁne the subspaces + − incoming and outgoing waves, such that H0 = H0 H0 and H0− = Ψ = Ψln ; Ψln = t Φ1ln , Φ2ln , Φ3ln , Φ4ln (l,n)∈I

Yln ∈ Vln ; Φ1ln = Φ4ln = 0 , (23) 4

H0+

= Ψ=

(l,n)∈I

Ψln ; Ψln =

t

1 Φln , Φ2ln , Φ3ln , Φ4ln 4

Yln ∈ Vln ; Φ2ln = Φ3ln = 0 . (24)

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Then, we compare the solutions of (22) in H0± with the solution of (9) in H given by (19). For this purpose, we introduce the identifying operator between H0 and H such that, for all Ψ± ∈ H0± J0 : Ψ± (r∗ , ω) → J0 (Ψ± )(r∗ , ω) = χ0 (r∗ )r−1 F −1/4 (r)Ψ± (r∗ , ω),

(25)

where χ0 is a cut-oﬀ function:

∞

χ0 ∈ C (Rr∗ ),

∃ a, b ∈ R,

a b

(26)

Finally, the wave operator W0± is deﬁned for all Ψ± ∈ H0± by W0± Ψ± = lim eitID J0 e−itH0 Ψ± t→±∞

in H.

(27)

The Reissner-Nordstrøm metric is asymptotically ﬂat at inﬁnity. In Minkowski space-time the Dirac equation is expressed in Cartesian coordinates as: i∂t Ψ = H∞ Ψ, H∞ = α.pp + βm, H∞ is self-adjoint on

(28) 1

2

3

0

α = i(Γ , Γ , Γ ), β = γ , p = −i∇x .

(29)

H∞ = L2 (R3x , dx)4 ,

(30)

with the dense domain H 1 (R3x )4 . Moreover, σ(H∞ ) = (−∞, −m] ∪ [m, ∞) is purely absolutely continuous. If Ψ ∈ H∞ , there exists a unique solution of (28) in C 0 (Rt , H∞ ) given by e−itH∞ Ψ. With the spherical coordinates ((ρ = |x|, ω) ∈ R+ ∗ × [0, π] × [0, 2π[), we write the Dirac equation on ∞ = L2 (R+ × S 2 , ρ2 dρdω)4 H ρ ω

(31)

as ∞ Ψ, i∂t Ψ = H

Γ3 1 1 Γ2 1 (∂θ + cot θ) + ∂ϕ + Γ4 . (32) H∞ = Γ ∂ρ + + ρ ρ 2 ρ sin θ

∞ such that Now, we introduce the isometry Λ between H∞ and H

Λ:

ω) = T Ψ(x) Ψ(ρ, ∞ H∞ → H ϕ 1 2 1 2 π 2 3 θ π , , T = e 2 γ γ e− 4 γ γ e( 2 − 4 )γ γ = TΨ Ψ → Ψ x = (ρ sin θ cos ϕ, ρ sin θ sin ϕ, ρ cos θ) ∈H ∞ , ∀Ψ = Ψ Λ−1 Ψ H∞

H∞

,

(33)

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with the important property: ∞ Λ. H∞ = Λ−1 H

(34)

As we remark in [Me], the comparison of the solutions of (28) in H∞ with the solutions of (9) in H involves long-range perturbations. These perturbations are composed of the β-valued and scalar long-range potentials, respectively due to the mass m and the charges q and Q. Then, we construct the Dollard-modiﬁed propagator U (t). The form of the modiﬁcation of e−itH∞ is suggested by the work of B. Thaller [Th] and V. Enss, B. Thaller [TE], J. Dollard, G. Velo [DV]. For Ψ ∈ H∞ and |t| > 1, we deﬁne U (t) by U (t) = U + (t)P+0 + U − (t)P−0 ,

log(t) log(t) p) ± ∓itλ(p − ib U (t) = e exp ∓iam , u(pp)|λ(pp) u(pp)| |u |u

(35)

where P±0 are the projectors onto the positive and negative energy subspaces of H∞ and a = −mM, b = qQ, λ(pp ) = (|pp|2 + m2 )1/2 ,

Now, we deﬁne the bounded identifying operator J∞ J∞ [Ψ(x)](r∗ , ω) :=

J∞ : H∞ → H, χ∞ (ρ)r 0

−1

F

−1/4

log(t) = t|t|−1 ln |t|. (36) between H∞ and H:

u (pp) = p /λ(pp),

∀Ψ ∈ H∞ , (r)r∗ Λ[Ψ(x)](ρ, ω)

r∗ > 0 r∗ ≤ 0.

(37)

with ρ = |x| = r∗ ≥ 0 to make artiﬁcial long-range interactions disappear, and χ∞ ∈ C

∞

(R+ r∗ ),

1 < a b.

(39)

± for all Ψ ∈ H∞ : Hence, we deﬁne the wave operator W∞ ± Ψ = lim eitID J∞ U (t)Ψ in H. W∞ t→±∞

(40)

The main result of this work is the following theorem: ± ) on H0± (resp. H∞ ) exist and are Theorem 3.1 The operators W0± (resp. W∞ independent of χ0 (resp. χ∞ ) satisfying (25) (26) (resp. (37) (39)). Moreover for ± Ψ± 0 ∈ H0 (resp. Ψ∞ ∈ H∞ ), ± W0± Ψ± 0 H = Ψ0 H0

and

± (resp. W∞ Ψ∞ H = Ψ∞ H∞ )

± = H. Ran W0± ⊕ W∞

(41) (42)

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4 Proof of Theorem 3.1 In this section, we give the proof of the existence and asymptotic completeness of the wave operators (27) and (40). The demonstration consists in comparing the solutions of (9) on H with the solution of this same equation with MIT bag boundary condition (see [CJJ]) on the sphere {r∗ = 1}. This splitting allows us to study separately the problems at the horizon and at inﬁnity. In the neighbourhood of the horizon, the existence and asymptotic completeness follow from the BirmanKuroda theorem, since the perturbations are short-range as r∗ → −∞. At inﬁnity, using several intermediate dynamics, we reduce the problem to the construction of a complete scattering theory for Dirac’s equation on ﬂat space-time with bounded long-range matrix valued potentials. This is done in Part 4.1. Then, using the techniques introduced by E. Mourre[Mo], we conclude the proof by showing that the singular continuous spectrum of ID is empty (σsc (ID) = ∅).

4.1

Asymptotic completeness for perturbed Dirac fields on flat space-time

In this section, we establish the asymptotic completeness of the wave operators for the Dirac equation in ﬂat space-time perturbed by bounded, scalar and β-valued potentials. We consider a perturbation H of H∞ (29), such that: H = H∞ + V, ∞

V = aβVlm + bVlc ,

a, b ∈ R,

3

−1

Vlm , Vlc ∈ L (R ), with, |Vlm (x)| ≤ O(|x|

(43) −1

), |Vlc (x)| ≤ O(|x|

) |x| → ∞. (44)

Since V ∈ L∞ , by the Kato-Rellich theorem, H is self-adjoint with the dense domain D(H∞ ) = D(H) = H 1 (R3 )4 in H∞ . With the Theorem XI-20 in [RS], (H∞ −z)−1 −(Hsp −z)−1 = (Hsp −z)−1 V (H∞ −z)−1 is compact, for all z ∈ ρ(H∞ ). Hence, by the Weyl theorem: σess (H) = σ(H∞ ) = R\] − m, m[.

(45)

Now, we deﬁne the spaces, cont H∞ = P cont H∞ ,

cont H∞± = P±cont H∞ ,

(46)

where P±cont are the projectors on positive and negative continuous subspaces of H, such that P+cont + P−cont = P cont . We recall the deﬁnition of u (pp): −1 0 u (pp) = p λ−1 (pp ) = ∇p λ(pp ) = ±ppH∞ P± .

(47)

The Hypotheses (43) and (44), allow us to apply the theory of [TE]. By a study of quantum asymptotic observables, the authors analyse precisely the long-time behaviour and the propagation properties of e−itH in phase space. These properties expressed by (54), (57), (58) and (59) in the proof of the following theorem. The asymptotic completeness of modiﬁed wave operators has been established in

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F. Melnyk

Ann. Henri Poincar´e

[TE] for scalar Coulombian-type perturbations of H∞ and in [Th] for β-valued Coulombian-type perturbations. Now, we give a theorem which states the asymptotic completeness for the sum of scalar and β-valued bounded long-range perturbations: Theorem 4.1 If there exists R > 0 such that |11 (|x| > R)|x|−1 − Vlm (x)| ≤ O(|x|−2 ), |x| → ∞, −1

−2

), |x| → ∞,

(49)

W ± (H∞ , H) := s − lim U (−t)e−itH , in H∞ ,

(50)

|11 (|x| > R)|x|

− Vlc (x)| ≤ O(|x|

(48)

then, the wave operator t→±∞

cont cont with U (t) deﬁned by (35) exists on H∞ and is an isometry from H∞ onto H∞ . cont the Proof. To prove the existence of (50), we write for |t|, |τ | > 1 and Ψ ∈ H∞ Cauchy criterion: I = U (−t − τ )e−iH(t+τ ) − U (−τ )e−iHτ Ψ

=

−iH(t+τ ) e − U (t + τ )U (−τ )e−iHτ Ψ

=

−iHt e − U (t + τ, τ ) e−iHτ Ψ ,

(51)

with U (t + τ, τ ) = U (t + τ )U (−τ ).

(52)

To obtain the result, we must establish that: lim

sup I = 0,

|τ |→+∞ |t|≥1

cont for Ψ ∈ H∞ .

(53)

Since cont cont D± = Ψ ∈ H∞± , / ∃ e1 , e2 ≷ ±m, Ψ = 1 (e1 < H < e2 )Ψ ∈ H∞±

(54)

cont is a dense subspace of H∞± , we only need to prove (53) for Ψ ∈ D± . We do this for Ψ ∈ D+ , the proof is similar for D− . With U ± (t + τ, τ ) = U ± (t + τ )U ± (−τ ) and Ψ ∈ D+ , we have −iHt e − U (t + τ, τ ) e−iHτ Ψ − e−iHt − U + (t + τ, τ ) P+cont e−iHτ Ψ

≤ e−iHt − U + (t + τ, τ ) P+0 − P+cont e−iHτ Ψ + e−iHt − U − (t + τ, τ ) P−0 e−iHτ Ψ .

(55)

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The two last norms vanish, when |τ | → ∞, because the evolution-groups are bounded and also thanks to part b) of Corollary 1.4 in [TE]. So, it is suﬃcient to establish: lim sup e−iHt − U + (t + τ, τ ) e−iHτ Ψ = 0, pour Ψ ∈ D+ . (56) |τ |→+∞ |t|≥1

In the proof of Theorem 4-1 in [TE], the authors of the paper have shown, for Ψ ∈ D+ and g ∈ C0∞ (R+ ) satisfying suppg ⊂ [4u2 , 1],

u = 4−1 ue1 ,

1/2 ue1 := (1 − m2 e−2 , 1 )

g|[u2

2 e1 ,ue2 ]

≡ 1,

|g|∞ ≤ 1

1/2 ue2 := (1 − m2 e−2 , 2 )

(57) (58)

that: lim φ(t) − e−iHt Ψ = 0,

|t|→∞

u |2 (pp))11 (|x| < δ|t|)eiλ(pp )t e−iHt Ψ, φ(t) = e−iλ(pp )t P+0 g(|u

δ > 0. (59)

Therefore −iHt − U + (t + τ, τ ) e−iH(τ ) Ψ ≤ 2 φ(τ ) − e−iHτ Ψ e + e−iHt − U + (t + τ, τ ) φ(τ ) (60) = I1 + I2 . Thanks to (59), it is now suﬃcient to estimate I2 . We apply Cook’s method with positive time, since the negative time case is treated in the same way. Hence, for t, τ > 1, we have, t

b am iHs + − I2 ≤ ds e V (x) − U (s + τ, τ )φ(τ ) , u (pp)(s + τ )|λ(pp ) |u u (pp)(s + τ )| |u 0 (61) therefore, t

1 iHs + ds e Vlc (x) − U (s + τ, τ )φ(τ ) I2 ≤ |b| u (pp)(s + τ )| |u 0 t iHs +|a| ds e βVlm (x) − 0

=

m u (pp)(s + τ )|λ(pp ) |u

U + (s + τ, τ )φ(τ )

I21 + I22 .

(62) We ﬁrst show that limτ →+∞ supt≥1 I22 = 0. Given |x| 0 r < 1/2 VR (x) = J , (63) |x|−1 , R > 0, J ∈ C ∞ (R+ ), J(r) = 1 r>1 R

822

F. Melnyk

Ann. Henri Poincar´e

with the property, Vt.s =

1 Vt (x/s). s

(64)

As, βVlm (x) −

m 1 = β Vlm (x) − u(pp)(s + τ )|λ(pp ) u (pp)(s + τ )| |u |u 1 m , (65) + β− u (pp)(s + τ )| λ(pp) |u

and using

−1 (β − mλ−1 (pp ))P+0 = GP+0 , G = β − mH∞ ,

we obtain, I22

≤ |a|

0

∞

u (pp)(s + τ )|−1 U + (s + τ, τ )φ(τ ) ds Vlm (x) − |u t iHs −1 + u (pp)(s + τ )| U (s + τ, τ )φ(τ ) ds e G|u + |a| 0

= I221 + I222 .

(66) We can estimate I221 as follows ∞ u(pp)(s + τ )|−1 U + (s + τ, τ )φ(τ ) I221 ≤ |a| ds VR (x) − |u 0 ∞ ds {VR (x) − Vlm (x)} U + (s + τ, τ )φ(τ ) + |a| = I2211 + I2212 .

(67)

0

We set R = (s + τ )u, δ = 2−1 u. To investigate I2212 , we establish the Lemma 4.1 There exists Q > 1 such that, for all N > 0, Q < t, τ and u > 0 ﬁxed, we have, u|2 (pp))11 (|x| < 2−1 τ u) 11(|x| < (t + τ )u)U + (t + τ, τ )e−iτ λ(pp ) P+0 g(|u ≤ CN (1 + t + τ )−N . u |2 (pp))| decays Proof. We only need to prove that |U + (t + τ, τ )e−iτ λ(pp ) P+0 (pp)g(|u rapidly in t+τ uniformly on |x| ≤ (t+τ )u. With the help of the Fourier transform, we write, p) 0 + −iτ λ(p 2 u| (pp)) = u |2 (ξ)), (68) P+ (pp)g(|u dξ ei(t+τ )κt,τ (ξ) P+0 (ξ)g(|u U (t + τ, τ )e R3

t+τ

κt,τ (ξ) =

log τ x.ξ − λ(ξ) − t+τ t+τ

ib iam + u(ξ)| |u u(ξ)|λ(ξ) |u

.

(69)

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log t+τ ib iam x τ ∇ξ κt,τ (ξ) = − u (ξ) − ∇ξ + , u(ξ)| |u u(ξ)|λ(ξ) t+τ t+τ |u

(70)

and by assumption |x| < (t + τ )u,

u (ξ)| > 2u > 0, 1 > |u

∀ξ.

(71)

Therefore

log t+τ ib iam τ ∇ξ . + |∇ξ κt,τ (ξ)| ≥ 2u − u − u (ξ)| |u u (ξ)|λ(ξ) t+τ |u

(72)

The last term vanishes when t, τ → ∞. Hence, there exists Q > 1 such that for t, τ > Q we have |∇ξ κt,τ (ξ)| > 0. Moreover, for α ∈ N3 with α1 + α2 + α3 ≥ 2 we have supt,τ,x |∂ξα κt,τ (ξ)| < ∞. Hence the non-stationary phase Theorem XI-14 in [RS] entails the result. Now, ∞ ds {VR (x) − Vlm (x)} 1 (|x| > (s + τ )u) U + (s + τ, τ )φ(τ ) I2212 ≤ |a| ∞0 ds VR (x) − Vlm (x)∞ 1 (|x| < (s + τ )u)U + (s + τ, τ )φ(τ ) . (73) + |a| 0

Hence, the ﬁrst integral tends to zero when τ → ∞, because, by assumption |V (x)−VR (x)| ≤ O(|x|−2 ), as |x| → ∞, R > 0 and U + (s+τ, τ )φ(τ ) is uniformly bounded in s and τ . The same conclusion can be obtained for the second integral with the help of Lemma 4.1. Finally, we get limτ →∞ I2212 = 0. Besides, since on the u (pp)(s + τ )|−1 = VR (u u (pp)(s + τ )), limτ →∞ I2211 = 0 is a consequence support of g, |u of Lemma 4.2 With the deﬁnitions (64), (59) and (52) we have: ∞ u (pp)(s + τ ))} U + (s + τ, τ )φ(τ ) = 0. lim ds {VR (x) − VR (u |τ |→∞

0

Proof. With the Formula ∞ 1 ds J= s + τ 0

(64) and R = (s + τ )u, we write

x + Vu u (pp)) U (s + τ, τ )φ(τ ) − Vu (u . s+τ

The above norm is bounded by q.x

d3 q Fx (Vu )(q) ei s+τ − eiq.uu(pp) U + (s + τ, τ )φ(τ ) . 3 R "# $ ! K

(74)

(75)

824

F. Melnyk

Ann. Henri Poincar´e

Since, q.x

q.x

q

eiλ(pp)t ei t e−iλ(pp )t = ei t e−it[λ(pp)−λ(pp + t )] q.x q.x q e−iq.uu(pp ) ei t = ei t e−iq.uu (pp+ t ) ,

(76)

then K

q.x

= eiλ(pp)(s+τ ) ei s+τ − eiq.uu (pp) U + (s + τ, τ )φ(τ ) q.x

q = ei s+τ e−i(s+τ )[λ(pp)−λ(pp+ s+τ )] eiλ(pp )(s+τ ) − eiq.uu (pp) eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ )

q.x q = e−iq.uu(pp ) ei s+τ e−i(s+τ )[λ(pp)−λ(pp + s+τ )] eiλ(pp )(s+τ ) − eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ ) q.x

q q = ei (s+τ ) e−iq.uu(pp+ s+τ ) e−i(s+τ )[λ(pp)−λ(pp+ s+τ )] − 1 eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ ) % q.x

q.x

& q q ≤ ei (s+τ ) e−iq.uu(pp + s+τ ) e−i(s+τ )[λ(pp)−λ(pp+ s+τ )] − 1 − ei s+τ − 1 eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ ) q.x

+ ei s+τ − 1 eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ ) = J1 + J2 .

(77) The Sobolev embeddings give: |q|2 Fx Vu ∈ L1 (R3 ) and |q|Fx Vu ∈ L1 (R3 ), u > 0. Therefore, (75) is bounded by,

1 d q Fx (Vu )(q)|q| × 2 J1 + |q| R3 3

2

R3

d3 q Fx (Vu )(q)|q| ×

1 J2 . |q|

(78)

u∞ < ∞, then, The function, eiy , y ∈ R is Lipschitz, u (x) = ∇x λ(x) and ∇u q q iq.uu (ξ+ s+τ ) − e−i(s+τ )[λ(ξ)−λ(ξ+ s+τ )] e q q u(ξ + s+τ ≤ | − (s + τ )[λ(ξ) − λ(ξ + s+τ )] − q.u )| 1 sq q ≤ ds q. u ξ + −u ξ + s+τ s+τ 0 1 s uq 1 ds du q.∇ u ξ + .q ≤ |s+τ | s+τ 0 1 2

|q| u ≤ C |s+τ | ∇u ∞ ,

C > 0. (79)

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825

With the help of the Fourier transform and Parseval’s theorem, we get, 2 q q 1 2 iq.uu(ξ+ s+τ ) −i(s+τ )[λ(ξ)−λ(ξ+ s+τ )] J ≤ − e | e q4 1

(80)

Fx (eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ ))(ξ)|2 d3 ξ ≤C

1 u2∞ Ψ2 . ∇u |s + τ |2

(81)

Furthermore, 1 |q| J2

q.x i s+τ

≤

1 |q|

≤

1 s+τ

0

1

dl e

q.x i s+τ l iλ(p p )(s+τ )

e

U (s + τ, τ )φ(τ ) +

x eiλ(pp )(s+τ ) U + (s + τ, τ )φ(τ )

(82)

and writing U + (s + τ, τ )φ(τ ) explicitely, we obtain,

ib 1 iam s+τ 1 x exp J2 ≤ − log u (pp)| |u u(pp)|λ(pp ) |q| s+τ |u τ

u |2 (pp))11 (|x| < δτ )eiλ(pp )τ e−iHτ Ψ P0+ g(|u

≤

=

log s+τ τ {x(|u u (pp)|λ(pp ))−1 }P0+ g(|u u |2 (pp))11 |a|m s+τ (|x| < δτ )eiλ(pp )τ e−iHτ Ψ log s+τ τ u |2 (pp))11 (|x| < δτ )eiλ(pp )τ e−iHτ Ψ u(pp)|−1 }P0+ g(|u +|b| {x|u s+τ 1 + u |2 (pp))11 (|x| < δτ )eiλ(pp )τ e−iHτ Ψ + xP0 g(|u s+τ J21 + J22 + J23 . (83)

With the Fourier transform in the variable x and since all derivatives of u and λ are bounded, we show that the norm J21 and J22 are uniformly bounded in τ . Moreover, J23 ≤

1 u (pp))]11 (|x| < δτ )eiλ(pp )τ e−iHτ Ψ [x, P0+ g(u s+τ 1 + x11(|x| < δτ )eiλ(pp )τ e−iHτ Ψ . (84) s+τ

Hence as above with the Fourier transform and since g ∈ C0∞ (R+ ), we show that the ﬁrst norm is bounded uniformly in τ . The second term is bounded by δτ (s + τ )−1 . Then, for τ suﬃciently large we have, ∞ δτ + 1 + log s+τ τ J ≤C ds ≤ 2δ, δ small, C > 0. (85) 2 (s + τ ) 0

826

F. Melnyk

Ann. Henri Poincar´e

Hence, limτ →+∞ I221 = 0. Now, we investigate I222 . We write e−iHs U − (s + τ, τ ), as the derivative of an integral on [0, s]. With P±0 G = GP∓0 and integrating by part, we get t iHs + −1 u (pp)(s + τ )| φ(τ ) I222 = |a| ds e U (s + τ, τ )G|u 0 s ∞ 1 iHσ + −1 u (pp)| φ(τ ) ≤ |a| ds dσ e U (σ + τ, τ )G|u 2 (s + τ ) 0 0 t |a| u (pp)|−1 φ(τ ) + dσ eiHσ U + (σ + τ, τ )G|u (t + τ ) 0 = I2221 + I2222 . (86) We calculate

L(t, τ ) :=

t

dσ eiHσ U + (σ + τ, τ )

(87)

0

writing eiλ(pp )σ as the derivative of an integral between 0 and σ, integrating by part, ' t log σ+τ log σ+τ p) iHσ −iσλ(p τ τ − iam dσ e e exp −ib L(t, τ ) = u(pp)| u(pp)|λ(pp ) |u |u 0 σ+τ ' σ+τ t log τ log τ e−iλ(pp)σ d − iam eiHσ exp −ib dσ = u(pp)| u(pp)|λ(pp ) dσ |u |u iλ(pp) 0 1 . (88) − eiHt U + (t + τ, τ ) − 1 iλ(pp) But, d dσ

e

' log σ+τ log σ+τ τ τ − iam exp −ib u (pp)| u(pp)|λ(pp ) |u |u iam ib iHσ − −iλ(pp) + iV − =e u (pp)|(σ + τ ) |u u (pp)|λ(pp)(σ + τ ) |u ' σ+τ log σ+τ log τ τ − iam , (89) exp −ib u(pp )| u(pp)|λ(pp ) |u |u

iHσ

hence, L(t, τ ) = −

t

dσ e 0

iHσ

+ U (σ + τ, τ ) b am + −V u(pp)|(σ + τ ) |u u (pp)|λ(pp )(σ + τ ) |u 2λ(pp) iHt + 1 + e U (t + τ, τ ) − 1 . (90) 2iλ(pp)

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Then, for I2222 , we deduce the following estimate: I2222

≤

|a| λ−1 (pp)G|u u (pp)|−1 φ(τ ) t+τ t |a| u (pp)|−1 U + (σ + τ, τ )φ(τ ) + dσ V (x)λ−1 (pp)G|u t+τ 0 + |a|m + |b| t 1 U (σ + τ, τ )φ(τ ) . + dσ t+τ (σ + τ ) 0

(91)

The ﬁrst and third term on the right-hand side, tend to zero when t, τ → ∞, whereas their norms are uniformly bounded in σ and τ . For the second term, we have the following estimate: t |a| u(pp)|−1 U + (σ + τ, τ )φ(τ ) dσ V (x)11 (|x| > u(σ + τ )) |u t+τ 0 t |a| u (pp)|−1 U + (σ + τ, τ )φ(τ ) . + dσ V ∞ 1 (|x| < u(σ + τ ))λ−1 (pp)G|u t+τ 0 (92) u(pp)|−1 U + (σ + τ, τ )φ(τ ) is bounded uniformly in Since V (x) = O(|x|−1 ) and |u σ and τ , the ﬁrst integral tends to zero when t, τ → ∞. Now, we treat the second integral using the following lemma: Lemma 4.3 There exists Q > 1 such that, for all N > 0, Q < t, τ and u > 0 ﬁxed, we have, u 2 (pp))λ−1 (pp)G(pp )|u u (pp)|−1 11(|x| < (t + τ )u)U + (t + τ, τ )e−iτ λ(pp ) P−0 g(u 1 (|x| < 2−1 τ u) ≤ CN (1 + t + τ )−N . Proof. As in the proof of Lemma 4.1, we write, u |2 (pp))λ−1 (pp)G(pp )|u u (pp)|−1 U + (t + τ, τ )e−iτ λ(pp) P−0 (pp)g(|u u |2 (ξ))λ−1 (ξ)G(ξ)|u u (ξ)|−1 , (93) = dξ ei(t+τ )κt,τ (ξ) P−0 (ξ)g(|u R3

log t+τ x.ξ ib iam τ κt,τ (ξ) = − λ(ξ) − + u (ξ)| |u u (ξ)|λ(ξ) t+τ t+τ |u and we conclude with the same arguments.

(94)

Furthermore, with the same calculation as for I2222 , we have limτ →∞ I2221 = 0. Then, limt→+∞ supt≥1 I222 = 0 and consequently limτ →∞ supt≥1 I22 = 0. I21 is treated like I221 to obtain limτ →+∞ supt≥1 I21 = 0 and ﬁnally limτ →∞ supt≥1 I2 = 0. We have proved limτ →∞ supt≥1 I = 0, therefore also limit (50). To establish that the range of W ± (H∞ , H) is equal to H∞ , we prove this last lemma:

828

F. Melnyk

Ann. Henri Poincar´e

Lemma 4.4 With U (t) deﬁned by (35), we have W ± (H, H∞ ) = s − lim eitH U (t), in H∞ t→±∞

(95)

exists on H∞ . Proof. To demonstrate the existence of the limit (95), for Ψ ∈ H∞ and |t|, |τ | > 1, we write the Cauchy criterion:

iH(t+τ ) U (t + τ ) − eiHτ U (τ ) Ψ = U (t + τ ) − e−iHt U (τ ) Ψ e = U (t + τ )U (−τ ) − e−iHt U (τ )Ψ . (96) To obtain the result, we prove for all Ψ in D± = Ψ ∈ P±0 H∞ / ∃ e1 , e2 ≷ ±m, Ψ = 1 (e1 < H∞ < e2 )Ψ ∈ P±0 H∞

(97)

which are dense subspaces of P±0 H∞ , that lim sup U ± (t + τ )U ± (−τ ) − e−iHt U ± (τ )Ψ = 0.

(98)

|τ |→+∞ |t|≥1

Concentrating on the positive energies of H∞ and under Hypotheses (57) and (58), Formula (59) remains true when H is replaced by H∞ . Hence, for Ψ ∈ D+ we must show that log(τ ) log(τ ) lim sup U + (t + τ )U + (−τ ) − e−iHt e{−iam |uu(pp)|λ(pp) +ib |uu(pp)| } φ(τ ) = 0. |τ |→+∞ |t|≥1

(99) To prove (99), we use the same estimates as for I2 . However, we remark that the exponent term in (99), leads us to modify the statements of Lemmas 3.1, 3.2 and 3.3. Apart from this modiﬁcation, the proof of Lemma 3.2 remains the same. For Lemmas 3.1 and 3.3 in Formulas (69), (70), (72) and (94), the term log(t + τ ) is put in place of log((t + τ )τ −1 ). The proof of these lemmas remains similar because limt,τ →+∞ (t + τ )−1 log(t + τ ) = 0. Hence, W ± (H, H∞ ) exists on H∞ . cont and W ± (H, H∞ ) on H∞ by the above lemma, W ± (H∞ , H) exists on H∞ ± cont onto H∞ . This concludes the therefore W (H∞ , H) is an isometry from H∞ proof of the theorem.

4.2

Spectral properties of ID

We show that the spectrum of ID is purely absolutely continuous. In particular, we prove that the singular continuous spectrum of this operator is empty by Mourre’s method. This requires to use an auxiliary self-adjoint operator. First time, we prove that the auxiliary operators that we have choosen (A and Ac ), are actually

Vol. 4, 2003

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self-adjoint. Then, using a technical compactness lemmas, we check Mourre’s hypothesis as expressed in the paper of E. Mourre [Mo] completed by simpliﬁcations suggested by V. Georgescu and C. G´erard [GG]. We can also use the hypothesis given by W.O. Amrein, A.M. Boutet de Monvel-Berthier and V. Georgescu in [ABG1]. For r∗ ∈ R, let J+ , J− ∈ C ∞ (Rr∗ ) such that, J− (r∗ ) + J+ (r∗ ) = 1,

|J− (r∗ )| ≤ 1, |J+ (r∗ )| ≤ 1, ∀r∗ ∈ R

J− (r∗ ) = 1 if r∗ ≤ −3 and J− (r∗ ) = 0 if r∗ ≥ −2.

(100) (101)

We put: i A = − (r∗ ∂r∗ + ∂r∗ r∗ ) , 2 B = A + cr∗ J− (r∗ ), c ∈ R, 0 1

Ac = A + cγ γ r∗ J− (r∗ ), c =

(102) (103) qQr0−1

∈ R.

(104)

Lemma 4.5 The operators A and B (resp. Ac ) are essentially self-adjoint on C0∞ (Rr∗ ) (resp. C0∞ (Rr∗ )4 ). Proof. The proof consists in applying Theorem X.37 in [RS]. Thanks to Theorem X.28 in [RS] the auxiliary operator N = −∂r2∗ + r∗2 + 1 ≥ 1 is essentially selfadjoint on C0∞ (Rr∗ ). We still denote by N its closure with the dense domain D(N ) = {Ψ ∈ L2 (R); N Ψ ∈ L2 (R)}. Moreover, in the sense quadratic forms on C0∞ (Rr∗ ), we have: N2

=

(r∗2 − ∂r2∗ )2 + 2(r∗2 − ∂r2∗ ) + 1

≥

r∗4 + ∂r4∗ − r∗2 ∂r2∗ − ∂r2∗ r∗2 + 1

=

r∗4 + ∂r4∗ − 2r∗ ∂r2∗ r∗ − [r∗ , [r∗ , −∂r2∗ ]] + 1

=

r∗4 + ∂r4∗ − 2r∗ ∂r2∗ r∗ − 1

≥

r∗4 + ∂r4∗ − 1,

(105)

hence, for Ψ ∈ C0∞ (Rr∗ ), we obtain r∗2 Ψ2 ≤ N Ψ2 + Ψ2 ∂r2∗ Ψ2

2

2

≤ N Ψ + Ψ .

(106) (107)

Then, D(N ) = H 2 (Rr∗ ) ∩ D(r∗2 ). Since C0∞ (Rr∗ ) is dense in D(N ) then, C0∞ (Rr∗ ) is a core for N . As, for all Ψ ∈ C0∞ (Rr∗ ), N 2 ≥ (r∗2 − ∂r2∗ )2 + 1 ≥ −2r∗ ∂r2∗ r∗ − 1 = 2A2 − 1/2, then, with (106), (107), we get AΨ ≤ CN Ψ C > 0.

(108)

830

F. Melnyk

Ann. Henri Poincar´e

BΨ2 ≤ CN Ψ2 + |c| r∗ J− (r∗ )Ψ2

(109)

Moreover, thanks to (106) and (108)

2

≤ C1 N Ψ ,

C1 > 0.

(110)

On the other hand, for all Ψ ∈ C0∞ (Rr∗ ) |(AΨ, N Ψ) − (N Ψ, AΨ)| = 2|(Ψ, {r∗2 − ∂r2∗ }Ψ)| ≤ 2N 1/2 Ψ2 .

(111)

Finally, with the following inequality − ∂r2∗ + [(r∗ J− (r∗ )) ]2 ∓ i (r∗ J− (r∗ )) ∂r∗ + ∂r∗ (r∗ J− (r∗ ))

= (i∂r∗ ∓ (r∗ J− (r∗ )) )2 ≥ 0, (112) and (111), we have, |(BΨ, N Ψ) − (N Ψ, BΨ)|

≤ 2N 1/2 Ψ2 + |c| (r∗ J− (r∗ )Ψ, −∂r2∗ Ψ) − (−∂r2∗ Ψ, r∗ J− (r∗ )Ψ)

≤ 2N 1/2 Ψ2 + |c| Ψ, (r∗ J− (r∗ )) ∂r∗ + ∂r∗ (r∗ J− (r∗ )) Ψ %

& 2 ≤ 2N 1/2 Ψ2 + |c| Ψ, (r∗ J− (r∗ )) − ∂r2∗ Ψ ≤ C2 N 1/2 Ψ2 ,

C2 > 0.

(113)

The Formulas (108), (110), (111) and (113) enable us to apply Theorem X.37 of [RS] which concludes the proof. Then, with the closure of A and Ac still denoted A and Ac , we deﬁne, for all t ∈ R, the unitary groups eitA and eitAc , respectively on L2 (Rr∗ ) and L2 (Rr∗ )4 . We introduce two self-adjoint operators h and h0 on L2 (Rr∗ )4 with dense domain D(h) = H 1 (Rr∗ )4 , such that: h = −iγ 0 γ 1 ∂r∗ + Vrn , Vrn (r∗ ) = Vq (r∗ ) + Vl (r∗ ) + Vm (r∗ ),

(114)

1/2

F (r) qQ , Vl (r∗ ) = −i(l + 1/2)Γ2 , Vm (r∗ ) = mγ 0 F 1/2 (r) r r h0 = −iγ 0 γ 1 ∂r∗ + mγ 0 . Vq (r∗ ) =

(115) (116)

Moreover, Vl and Vm are exponentially decreasing as r∗ → −∞. Now, we give the compactness lemmas that will be useful to check the Mourre hypothesis. Lemma 4.6 If χ ∈ C0∞ (] − m, m[), m > 0 and J+ is deﬁned by (100) and (101), then J+ χ(h) is compact on L2 (R).

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Proof. First, we remark that σ(h0 ) = σac (h0 ) = R\] − m, m[. Hence χ(h)J+ = {χ(h) − χ(h0 )}J+ . Thanks to Proposition C.2.1 in [DG] and the spectral theorem, there exists χ ∈ C0∞ (C) with χ |R = χ and for all N ∈ N |∂z¯χ (z)| ≤ CN |Imz|N , such that i ∂z¯χ (z) (h − z)−1 − (h0 − z)−1 J+ dz ∧ d¯ z . (117) {χ(h) − χ(h0 )}J+ = 2π C To prove the compactness of (117), it suﬃces to check that: (z − h)−1 (h − h0 )(z − h0 )−1 J+ ≤ C|Imz|−2 , z ∈ K ⊂⊂ C

(118)

(z − h)−1 (h − h0 )(z − h0 )−1 J+ is compact for z ∈ C \ (σ(h) ∪ σ(h0 )).

(119)

and

We establish easily (118), because (z − h)−1 ≤ |Imz|−1 , (z − h0 )−1 ≤ |Imz|−1 and h − h0 = Vrn − mγ 0 , J + are bounded. We estimate (z − h)−1 (h − h0 )(z − h0 )−1 J+ = (z − h)−1 {Vrn − mγ 0 }J+ (z − h0 )−1 + (z − h)−1 {Vrn − mγ 0 }[(z − h0 )−1 , J+ ]. (120) On the one hand, since lim|x|→∞ {Vrn − mγ 0 }J+ = 0, {Vrn − mγ 0 }J+ (z − h0 )−1 is compact thanks to Theorem XI-20 in [RS]. On the other hand, [(z − h0 )−1 , J+ ] = −(z − h0 )−1 [h0 , J+ ](z − h0 )−1 = i(z − h0 )−1 γ 0 γ 1 J+ (z − h0 )−1 ,

(121)

∈ C0∞ (R). Then (121) is compact thanks to Theorem XI-20 in [RS] and with J+ (119) is satisﬁed.

Lemma 4.7 If O is a self-adjoint operator with dense domain H 1 (Rx )4 in L2 (Rx )4 and f, g ∈ C 0 (Rx ; C) such that lim|x|→+∞ (|f (x)| + |g(x)|) = 0, then f (x)g(O) is compact on L2 (R). Proof. Given χn the cut-oﬀ function such that χn ∈ C0∞ (Rx ), χ ≡ 1 on [−n, n], 0 on R \ [−n − 1, n + 1] and 0 ≤ χn ≤ 1. Therefore, putting fn = χn f and gn = χn g fn gn (O) − f g(O) ≤ fn − f ∞ g∞ + gn − g∞ f ∞ → 0, n → +∞. (122) Yet, for n ﬁxed, χn f is compactly supported and gn (O) is an operator from L2 (Rx ) into H 1 (Rx ). Thanks to Sobolev embeddings, gn (O) is compact from L2 (Rx ) into L2 (Rx ). Then, we conclude that f (x)g(O) is compact. Theorem 4.2 The spectrum of h and ID is purely absolutely continuous on R.

832

F. Melnyk

Ann. Henri Poincar´e

Proof. Since, (see (15), (17)) ID = ⊕(l,n)∈I [IDln ⊗4 1 ω ] , on H = ⊕(l,n)∈I Hln , with IDln = r−1 F −1/4 hrF 1/4 on L2w

(123)

it suﬃces to establish the result for h on L2 (Rr∗ )4 , for all (l, n) ∈ I. – From [Ba5] (Lemma VI.1), we already know that h has no eigenvalue. – To prove that the singular continuous spectrum of h is empty, we use the Mourre method according to the papers of E. Mourre [Mo] and V. Georgescu, C. G´erard [GG] or W.O. Amrein, A.M. Boutet de Monvel-Berthier and V. Georgescu [ABG1]. We must check that the hypotheses are satisﬁed: i) i[h, D] with D = A or Ac is bounded from D(h) = H 1 (Rr∗ )4 into L2 (Rr∗ )4 . Indeed on C0∞ (Rr∗ )4 we get:

|(Ψ, i[h, A]Φ)| = |(Ψ, {−iγ 0γ 1 ∂r∗ −r∗ Vrn (r)}Φ)| ≤ ΨL2 (Rr∗ )4 ΦH 1 (Rr∗ )4 (124)

since r∗ Vrn (r) is bounded. Moreover i[h, Ac ] = i[h, A] + i[h, qQr0−1 γ 0 γ 1 r∗ J− (r∗ )],

(125)

with

i[h, qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] = qQr0−1 r∗ J− (r∗ ) + qQr0−1 J− (r∗ ) + i[Vl , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] + i[Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )].

(126)

We note that (126) is bounded on R with the last two commutators vanishing at negative and positiv inﬁnity. Thanks to (125), (124) and (126), |(Ψ, i[h, Ac ]Φ)| ≤ CΨL2(Rr∗ )4 ΦH 1 (Rr∗ )4 ,

C > 0.

(127)

Hence, i[h, A] and i[h, Ac ] extend to bounded operators from H 1 (Rr∗ )4 into L2 (Rr∗ )4 . These extensions are still denoted by i[h, A] and i[h, Ac ]. ii) eitA D(h) ⊂ D(h) and eitAc D(h) ⊂ D(h), t ∈ R. For Ψ ∈ C0∞ (Rr∗ )4 , eitAc Ψ satisﬁes a Schr¨odinger type equation, hence we have: ih∂t e−itAc Ψ = hAc e−itAc Ψ ⇔ i∂t he−itAc Ψ = Ac he−itAc Ψ + [h, Ac ]e−itAc Ψ. (128) With (127), we get, −itA c he Ψ ≤ hΨ +

t 0

[h, Ac ]e−iτ Ac Ψ dτ ≤ ΨH 1 (R )4 r∗ t −iτ A c e +C ΨH 1 (Rr 0

∗)

4

dτ.

(129)

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Then, there exists C > 0 such that −itA c e ΨH 1 (R

r∗

)4

≤ C ΨH 1 (Rr∗ )4 + C

0

t

−iτ A c e ΨH 1 (R

r∗ )

4

dτ.

(130)

Using the Gronwall lemma and a density argument, we conclude the proof for Ac . To obtain the result for A, in the above demonstration we take qQr0−1 = 0. iii) [[h, A], A] and [[h, Ac ], Ac ] are bounded from D(h) = H 1 (Rr∗ )4 into D∗ (h) = H −1 (Rr∗ )4 . Indeed, on C0∞ (Rr∗ )4 we have:

|(Ψ, [[h, A], A]Φ)| = |(Ψ, {−iγ 0 γ 1 ∂r∗ + r∗ Vrn (r) + r∗2 Vrn (r)}Φ)| ≤ ΨH 1 (Rr∗ )4 ΦH 1 (Rr∗ )4 , (131)

since r∗ Vrn (r) and r∗2 Vrn (r) are bounded. Also we obtain: |(Ψ, [[h, Ac ], Ac ]Φ)| ≤ ΨH 1 (Rr∗ )4 ΦH 1 (Rr∗ )4 ,

(132)

thanks to (131) and because the following double commutators are bounded: [[h, qQr0−1 γ 0 γ 1 r∗ J− (r∗ )], qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] = [[Vl , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )], qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] + [[Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )], qQr0−1 γ 0 γ 1 r∗ J− (r∗ )]. [[h, qQr0−1 γ 0 γ 1 r∗ J− (r∗ )], A] = r∗ i[h, qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] ,

(133) (134)

[[h, A], qQr0−1 γ 0 γ 1 r∗ J− (r∗ )] = −qQr0−1 r∗ J− (r∗ ) − qQr0−1 J− (r∗ )

+ i[r∗ Vrn , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )].

(135)

Therefore [[h, A], A] and [[h, Ac ], Ac ] extend to bounded operators from H 1 (Rr∗ )4 into H −1 (Rr∗ )4 . iv) The last assumption consists in checking the Mourre inequality: for I, a bounded borelian subset of R with I ∩ {0, −m, m} = ∅, there exists ε > 0 and a compact operator K, such that 1 I (h)i[h, D]11 I (h) ≥ 11I (h)+11I (h)K11I (h), (D = A or Ac depending on I). (136) Because the form of Vrn , to establish the inequality (136), we must examine the problem according to the parameters qQr0−1 and m. With (124), (125), (126), (127) and the conclusion of part i), we estimate the commutators in the sense of quadratic forms on H 1 (Rr∗ )4 . If χ ∈ C0∞ (R \ {0, −m, m}) and 0 ≤ qQr0−1 , m > 0 with

834

F. Melnyk

Ann. Henri Poincar´e

– suppχ ⊂]m, +∞[: From calculations (124), (125) and (126), with the deﬁnition of J+ and J− (see (100, 101)), we get χ(h)i[h, Ac ]χ(h) = χ(h)hχ(h) − χ(h)Vrn χ(h) + χ(h)qQr0−1 J− χ(h)

(r∗ ) + qQr0−1 r∗ J− + i[Vl + Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )])χ(h) + χ(h)(−r∗ Vrn

≥ (ε + m)χ2 (h) − χ(h)Vm χ(h) − χ(h)Vq J+ χ(h) − χ(h)Vq J− χ(h) + χ(h)qQr0−1 J− χ(h) − χ(h)Vl χ(h) + k1 ≥ εχ2 (h) + k,

(137)

with ε > 0. By Lemma 4.7, k1 is compact operator because

lim (|r∗ Vrn (r∗ )| + |qQr0−1 r∗ J− ||) = 0

|r∗ |→∞

and

lim |[Vl + Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )]| = 0.

|r∗ |→∞

Finally, since lim (|Vq J+ | + |Vl |) = 0,

|r∗ |→∞

k = k1 + χ(h)(Vq J+ + Vl )χ(h) is also compact. – suppχ ⊂]0, m[: As above, from calculations (124), (125) and (126), with the deﬁnition of J+ and J− , we have χ(h)i[h, Ac ]χ(h) = χ(h)hχ(h) − χ(h)Vrn χ(h) + χ(h)qQr0−1 J− χ(h)

+ χ(h)(r∗ Vrn (r∗ ) + qQr0−1 r∗ J− + i[Vl + Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )])χ(h)

≥ εχ2 (h) − χ(h)Vrn J+ χ(h) − χ(h)Vrn J− χ(h) + χ(h)qQr0−1 J− χ(h) + k1 = εχ2 (h) − χ(h)Vq J− χ(h) + χ(h)qQr0−1 J− χ(h) − χ(h)Vrn J+ χ(h) − χ(h)(Vl + Vm )J− χ(h) + k1 = εχ2 (h) + k,

(138)

with ε > 0. By Lemma 4.7, k1 is compact operator because

lim (|r∗ Vrn (r∗ )| + |qQr0−1 r∗ J− ||) = 0

|r∗ |→∞

and

lim |[Vl + Vm , qQr0−1 γ 0 γ 1 r∗ J− (r∗ )]| = 0.

|r∗ |→∞

χ(h)Vrn J+ χ(h) is also a compact operator thanks to Lemma 4.6 and ﬁnally, since lim (|Vl J− | + |Vm J− |) = 0, |r∗ |→∞

k = k1 − χ(h)Vrn J+ χ(h) + χ(h)(Vl + Vm )J− χ(h) is compact.

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– suppχ ⊂] − ∞, −m[: Because Vq ≥ 0, we have χ(h)i[h, −A]χ(h) = −χ(h)hχ(h) + χ(h)r∗ Vrn (r∗ )χ(h) + χ(h)Vrn χ(h)

≥ (m + ε)χ2 (h) + k + χ(h)Vm χ(h) = εχ2 (h) + k, with ε > 0. By Lemma 4.7, k is compact since

(139) (r∗ )| + |Vl |) = 0. lim (|r∗ Vrn

|r∗ |→∞

– suppχ ⊂] − m, 0[: From the deﬁnition of J+ and J− (100,101) and because Vq J− (r∗ ) ≥ 0, we have χ(h)i[h, −A]χ(h) = −χ(h)hχ(h) + χ(h)r∗ Vrn (r∗ )χ(h) + χ(h)Vrn χ(h) (r∗ ) ≥ εχ2 (h) + χ(h)Vq J− χ(h) + χ(h)(Vrn J+ − r∗ Vrn + Vl J− + Vm J− )χ(h)

≥ εχ2 (h) + k,

(140)

(r∗ )| + with ε > 0. By Lemma 4.7, k is compact, because lim|r∗ |→∞ (|r∗ Vrn |Vl J− | + |Vm J− |) = 0 and thanks to Lemma 4.6 for the compactness of J+ χ(h).

Finally, for qQr0−1 < 0, we obtain the estimates in a symmetric way. From Proposition 4 in [ABG1] or the theorem in [Mo], we obtain that σsc (h) = ∅ on R.

4.3

Proof of asymptotic completeness

We begin by establishing some technical lemmas. Lemma 4.8 Given φ ∈ {Ψ ∈ H∞ : Fx (Ψ) ∈ C0∞ (R3x \ {0})4 } and f ∈ Lp (R3x , dx) with p ∈ [2, +∞], then, for all |t| > 1 and > 0 small, we get 3 f U (t)φH∞ ≤ O |t|− p + . Proof. As P+0 φ+ P−0 φ = φ, we give the proof for positive energies, since it is similar for negative energies. According to the H¨older inequality (1/p + 1/q = 1/2), f U (t)P+0 φ ≤ f Lp(R3 ) U (t)P+0 φLq (R3 )4 . (141) H∞ The estimate

U (t)P+0 φ q 3 4 ≤ O |t|− p3 + L (R )

is established as in Lemma 6.1 in [Me].

(142)

836

F. Melnyk

Ann. Henri Poincar´e

We consider the operators O[1,+∞[ , O]−∞,1] expressed in spherical coordinates (ρ, ω) as O[1,+∞[ , O]−∞,1] = Γ1 ∂ρ + Γ2 A(ρ)(∂θ +

A(ρ) 1 cot θ) + Γ3 ∂ϕ + Γ4 B(ρ) + C(ρ), 2 sin θ (143)

(A, B, C ∈ L∞ (Rρ , dρ), A > 0), or (A ≡ B ≡ 0, C(ρ) = c ∈ R).

(144)

Their domains are: D O[1,+∞[ = Ψ ∈ L2 ([1, +∞[ρ ×Sω2 , dρdω)4 ; O[1,+∞[ Ψ ∈ L2 ([1, +∞[ρ ×Sω2 , dρdω)4 ,

− γ 1 Ψ(1, .) = iΨ(1, .) , (145)

D O]−∞,1] = Ψ ∈ L2 (] − ∞, 1]ρ × Sω2 , dρdω)4 ; O[−∞,1[ Ψ ∈ L2 (] − ∞, 1]ρ × Sω2 , dρdω)4 , γ 1 Ψ(1, .) = iΨ(1, .) . (146) Note that the MIT bag boundary conditions in (145) and (146) are well deﬁned. Indeed, if A is strictly positive, O[1,+∞[ −Γ4 B−C, (resp. O]−∞,1] −Γ4 B−C) is elliptic. 1 1 Therefore, Ψ ∈ Hloc ([1, +∞[, L2 (Sω2 ))4 , (resp. Ψ ∈ Hloc (]−∞, 1], L2 (Sω2 ))4 ). In the same way, when A ≡ 0, B ≡ 0 and C ≡ c ∈ R we have Ψ ∈ H 1 ([1, +∞[, L2 (Sω2 ))4 , (resp. Ψ ∈ H 1 (] − ∞, 1], L2 (Sω2 ))4 ). Lemma 4.9 The operators O[1,+∞[ and O[−∞,1] are self-adjoint with dense domains, D(O[1,+∞[ ) and D(O]−∞,1] ) deﬁned by (145) and (146). Proof. We give the proof for O[1,+∞[ , since it is similar for O[−∞,1] . As for the operator ID (see (15, 17)), we decompose O[1,+∞[ with the help of spin-weighted spherical harmonics: % & ln O[1,+∞[ = ⊕(l,n)∈I O[1,+∞[ ⊗4 1 ω , on ⊕(l,n)∈I Lln , Lln = L2 (]1, +∞[ρ , dρ)4 ⊗4 V ect[Yln ], (147) ln O[1,+∞[ = Γ1 ∂ρ − iΓ2 A(ρ)(l + 1/2) + Γ4 B(ρ) + C(ρ).

(148)

ln Since the functions A, B, C are bounded, by the Kato-Rellich theorem, O[1,+∞[ is self-adjoint with the dense domain:

ln D O[1,+∞[ = Ψln ∈ L2 ([1, +∞[ρ , dρ)4 ;

ln O[1,+∞[ Ψln ∈ L2 ([1, +∞[, dρ)4 , −γ 1 Ψln (1) = iΨln (1) , (149)

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ln equiped with graph norm for O[1,+∞[ . Then, O[1,+∞[ is self-adjoint with the dense domain:   ln D(O[1,+∞[ ) = Ψ = Ψln ⊗4 Yln : Ψln ∈ D(O[1,+∞[ ),  (l,n)∈I   2 ln < +∞ . O[1,+∞[ Ψln 2  L ([1,+∞[,dρ)4 (l,n)∈I

According to the above lemma, the operator e−itO]−∞,1] is well deﬁned on L (]− ∞, 1]ρ × Sω2 , dρdω)4 and is given in the explicit form by the following lemma: 2

Lemma 4.10 If B ≡ A ≡ 0, c ∈ R and Ψ0 =t (Ψ10 , Ψ20 , Ψ30 , Ψ40 ) ∈ C 1 (] − ∞, 1]ρ × Sω2 )4 then,  −ict t 3 e Ψ0 (2 − ρ − t, ω), Ψ20 (ρ − t, ω), Ψ30 (ρ − t, ω),      −Ψ20 (2 − ρ − t, ω) , ρ + t ≥ 1,      e−ict t Ψ1 (ρ + t, ω), Ψ2 (ρ − t, ω), Ψ3 (ρ − t, ω) , 0 0 −itO]−∞,1] 0 e Ψ0 =  Ψ40 (ρ + t, ω) , ρ + t ≤ 1, ρ − t ≤ 1,       e−ict t Ψ10 (ρ + t, ω), −Ψ40 (2 − ρ + t, ω), Ψ10 (2 − ρ + t, ω) ,    Ψ40 (ρ + t, ω) , ρ − t ≥ 1. Proof. For Ψ ∈ C 1 (] − ∞, 1]ρ × Sω2 )4 and thanks to (11), we obtain:  ∂t Ψ1 = ∂ρ Ψ1 − icΨ1 ,    1 Ψ3|ρ=1 = Ψ1|ρ=1 , i∂t Ψ = Γ ∂ρ Ψ + cΨ, ∂t Ψ2 = −∂ρ Ψ2 − icΨ2 , ⇔ with 1 3 3 3 γ Ψ|ρ=1 = iΨ|ρ=1 , ∂t Ψ = −∂ρ Ψ − icΨ , Ψ2|ρ=1 = −Ψ4|ρ=1 .    4 4 4 ∂t Ψ = ∂ρ Ψ − icΨ , (150) We conclude by studying the characteristics of each equation of the above system. For two self-adjoint operators O1 on HO1 and O2 on HO2 , we deﬁne formally the wave operators W ± (O1 , O2 , J ) = s − lim eitO1 J e−itO2 Pac (O2 ), t→±∞

(151)

where Pac (O2 ) is the projector on the absolutely continuous subspace of O2 and J the bounded identifying operator between HO2 and HO1 . When HO1 = HO2 and J = Id, we denote W ± (O1 , O2 , Id) simply by W ± (O1 , O2 ).

838

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Ann. Henri Poincar´e

Lemma 4.11 We consider two operators O1 and O2 (resp. O3 and O4 ) deﬁned by (143), (144) and (145) (resp. (143), (144) and (146)) with the functions A1 , A2 , B1 , B2 ∈ L∞ ([1, +∞[) (resp. A3 , A4 , B3 , B4 ∈ L∞ (] − ∞, 1])) satisfying A1 − A2 , B1 − B2 , C1 − C2 ∈ L1 ([1, +∞[, dρ) 1

(resp. A1 − A2 , B1 − B2 , C1 − C2 ∈ L (] − ∞, 1], dρ)). Then

W ± (O1 , O2 )

(resp. W ± (O3 , O4 ) )

(152) (153) (154)

exists and is complete from L2 ([1, +∞[ρ ×Sω2 , dρdω)4

onto

Pac (O1 )L2 ([1, +∞[ρ ×Sω2 , dρdω)4

(resp. from L2 (] − ∞, 1]ρ × Sω2 , dρdω)4 onto Pac (O3 )L2 (] − ∞, 1]ρ × Sω2 , dρdω)4 ). Proof. We give the proof for W ± (O1 , O2 ), since it is similar for W ± (O3 , O4 ). Thanks to the properties of the angular parts of O1 and O2 , we can decompose O1 and O2 using spin-weighted harmonics. We have (O1 ± i)−1 − (O2 ± i)−1 = ⊕(l,n)∈I [Rln ⊗4 1 ω ], on ⊕(l,n)∈I Lln , Lln = L2 ([1, +∞[ρ , dρ)4 ⊗4 V ect[Yln ], Rln = (Γ1 ∂ρ + R1 ± i)−1 [Γ2 (l + 1/2)E1 + Γ2 E2 + E3 ](Γ1 ∂ρ + R2 ± i)−1 Rk = Γ2 Ak (ρ)(l + 1/2) + Γ4 Bk (ρ) + Ck (ρ), k = 1, 2, E1 = A2 − A1 , E2 = B2 − B1 , E3 = C2 − C1 . Now, we introduce the following Hamiltonians, T0 ,T1 ,T2 = Γ1 ∂ρ ,D(T0 ) = H 1 (R)4 , D(T1 ) = Ψ ∈ L2 ([1,+∞[ρ ,dρ)4 ;T1 Ψ ∈ L2 ([1,+∞[ρ ,dρ)4 ,−γ 1 Ψ(1) = iΨ(1) , D(T2 ) = Ψ ∈ L2 (R\]1,+∞[ρ ,dρ)4 ;T2 Ψ ∈ L2 (R\]1,+∞[ρ ,dρ)4 ,γ 1 Ψ(1) = iΨ(1) . On L2 (R)4 , for each j = 1, 2, 3, we have |Ej |1/2 (T1 ± i)−1 ⊕ 0 = (|Ej |1/2 ⊕ 0)(T0 ± i)−1 + (|Ej |1/2 ⊕ 0)[(T1 ⊕ T2 ± i)−1 − (T0 ± i)−1 ]. (155) (T1 ⊕ T2 ± i)−1 − (T0 ± i)−1 is of ﬁnite rank. From (152) and Theorem XI.20 in [RS], (|Ej |1/2 ⊕ 0)(T0 ± i)−1 are Hilbert-Schmidt operators. Then |Ej |1/2 (T1 ± i)−1 has the same property. Besides, since (|Ej |1/2 (T1 ± i)−1 )∗ = (T1 ∓ i)−1 |Ej |1/2 are Hilbert-Schmidt, (T1 ± i)−1 |Ej |1/2 |Ej |1/2 (T1 ± i)−1 are trace class. Now, we write: Rln = (Γ1 ∂ρ + R1 ± i)−1 (T1 ± i) (T1 ± i)−1 [Γ2 (l + 1/2)E1 + Γ2 E2 + E3 ] (T1 ± i)−1 (T1 ± i)(Γ1 ∂ρ + R2 ± i)−1 . (156)

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By the closed graph theorem, we show that the two operators between the brackets are bounded. We have already seen that the operator of the center is trace class. Therefore, Rln is trace class for each (l, n) ∈ I. According to the KurodaBirman theorem (see Theorem XI-9 in [RS]), W ± (O1 |Lln , O2 |Lln ) exists and is complete on Lln , which leads to the same conclusion, on L2 ([1, +∞[ρ ×Sω2 , dρdω)4 , for W ± (O1 , O2 ). Proof of Theorem 3.1. – We begin with a ﬁrst splitting to separate the problems at the horizon and at inﬁnity. We introduce the self-adjoint operator H − ⊕ H + on H with the dense − domain D (H ) D (H + ), such that: H −, H + =

F 1/2 (r) qQ F (r) 1 1 + Γ1 ∂r∗ + + F (r) + Γ2 (∂θ + cot θ) r r 4 r 2 + Γ3

F 1/2 (r) ∂ϕ + Γ4 F 1/2 (r), r sin θ

D H − = Ψ ∈ L2 (] − ∞, 1]r∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 ;

H − Ψ ∈ L2 (] − ∞, 1]r∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 , γ 1 Ψ(1, .) = iΨ(1, .) ,

D H + = Ψ ∈ L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 (r)dr∗ dω)4 ;

H + Ψ ∈ L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 (r)dr∗ dω)4 , −γ 1 Ψ(1, .) = iΨ(1, .) .

Using the properties of the angular parts of ID, H − and H + , on Hln we show that, (ID|Hln ± i)−1 − (H − ⊕ H|+H

ln

± i)−1

(157)

is of ﬁnite rank, hence trace class for all (l, n) ∈ I. The Birman-Kuroda theorem gives the existence on Hln of W ± ID|Hln , H − ⊕ H|+H (158) ln

with

Ran W ± (ID|Hln , H − ⊕ H|+H ) = Pac (ID|Hln )Hln . ln

Hence, the wave operator

W ± ID, H − ⊕ H +

exists on H, and by Theorem 4.2: Ran W ± (ID, H − ⊕ H + ) = H.

(159)

(160)

(161)

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+ – Now, on L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 (r)dr∗ dω)4 , we compare H + with Hp+ the + self-adjoint operator with the dense domain D Hp+ , such that: Γ3 1 1 Γ2 M qQ + Hp+ ∂ϕ +Γ4 1 − , (162) = Γ1 ∂r∗ + + (∂θ + cot θ)+ + r∗ r∗ 2 r∗ sin θ r r

and + D Hp+ = Ψ ∈ L2 ([1, +∞[r∗ ×Sω2 , r∗2 dr∗ dω)4 ; + Ψ ∈ L2 ([1, +∞[r∗ ×Sω2 , r∗2 dr∗ dω)4 , Hp+

− γ 1 Ψ(1, .) = iΨ(1, .) . (163)

+ Hp+ is the perturbed Dirac operator in spherical coordinates on ﬂat space-time deﬁned outside the unit sphere; we have chosen

ρ = |x| = r∗ ≥ 0

(164)

to avoid the creation of artiﬁcial long-range interactions when we compare the dynamics. Moreover, we introduce Jr∗ and Jr deﬁned by Jr∗ : Ψ(r∗ , ω) → Jr∗ (Ψ)(r∗ , ω) = r∗ Ψ(r∗ , ω) Jr : Ψ(r∗ , ω) → Jr (Ψ)(r∗ , ω) = r

−1

F

−1/4

(165)

(r)Ψ(r∗ , ω).

(166)

+ It suﬃces to apply the result of Lemma 4.11 to W ± (Jr−1 H + Jr , Jr∗ Hp+ Jr−1 ), to ∗ obtain the existence of the following wave operator + W ± H + , Hp+ (167) , Jr Jr∗

on L2 ([1, +∞[r∗ ×Sω2 , r∗2 dr∗ dω)4 , with + Ran W ± (H + , Hp+ , Jr Jr∗ ) = Pac (H + )L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 (r)dr∗ dω)4 . (168) + , the operator on L2 ([0, 1]r∗ ×Sω2 , r∗2 dr∗ dω)4 with the dense – Now, we consider Hp− + domain D Hp− , such that: Γ3 1 1 Γ2 M qQ + 1 4 Hp− = Γ ∂r∗ + ∂ϕ + Γ 1 − , (∂θ + cot θ) + + + r∗ r∗ 2 r∗ sin θ r r (169) + 2 2 2 4 (170) D Hp− = Ψ ∈ L ([0, 1]r∗ × Sω , r∗ dr∗ dω) ; + Ψ ∈ L2 ([0, 1]r∗ × Sω2 , r∗2 dr∗ dω)4 , Hp−

γ 1 Ψ(1, .) = iΨ(1, .) . (171)

+ Hp− is self-adjoint, since unitarily equivalent to the Dirac Hamiltonian in ﬂat space-time deﬁned on the unit ball, perturbed by bounded potentials, with conservative boundary conditions on the unit sphere. Also, we introduce the self-adjoint operator H on H∞ with the dense domain H 1 (R3x )4 :

H = H∞ + V, V = qQr−1 − mM γ 0 r−1 .

(172)

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H∞ is deﬁned by (29) and V satisﬁes the assumptions of Theorem 4.1. Therefore, + + we compare the dynamics induced by H on H∞ and Hp+ ⊕ Hp− on L2 (R+ r∗ × 2 2 4 Sω , r∗ dr∗ dω) . We put, JΛ : Ψ(x) → JΛ (Ψ)(r∗ , ω) = Λ(Ψ(x))(r∗ = |x|, ω).

(173)

+ + Thanks to the properties of the angular parts of Hp+ , Hp− and H, we obtain:

−1 −1 + + ±i − JΛ HJΛ−1 ± i = ⊕(l,n)∈I [Rpln ⊗4 1 ω ] on ⊕(l,n)∈I Lpln , Hp− ⊕ Hp+ (174) 2 2 4 Lpln = L2 (R+ r∗ × Sω , r∗ dr∗ dω) ⊗4 V ect[Yln ], + + ⊕ Hp− ± i)−1 − (JΛ HJΛ−1 | p Rpln = (Hp+ | p L L ln

± i)−1 .

(175)

ln

Rpln is of ﬁnite rank for all (l, n) ∈ I. Therefore, by the Kuroda-Birman theorem + and since the spectrum of Hp− is purely discrete, we deduce, on Lpln , the existence and the completeness of + ⊕ 0|Lp , JΛ HJΛ−1 | W ± (Hp+ ln

p L ln

).

(176)

Hence, on H∞ + + + W ± (Hp+ ⊕ 0, H, JΛ ) = W ± (Hp+ ⊕ Hp− , H, JΛ )

(177)

exists and + + Ran W ± (Hp+ , H, Jl ) = Pac (Hp+ )L2 ([1, +∞[r∗ ×Sω2 , r∗2 dr∗ dω)4 .

(178)

Finally, we put d J∞ = Jr Jr∗ JΛ ,

(179)

and from Theorem 4.1 and the completeness of (167) and (177), we conclude that + + d ) = W ± (H + , Hp+ , Jr Jr∗ )W ± (Hp+ , H, JΛ )W ± (H, H∞ ), W ± (H + , H∞ , J∞ (180) exists on H∞ , with d Ran W ± (H + , H∞ , J∞ ) = Pac (H + )L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 dr∗ dω)4 . (181)

– At the neighbourhood of the horizon, we compare H − with H0− the self-adjoint operator on L2 (] − ∞, 1]r∗ × Sω2 )4 with the dense domain D(H0− ), given by qQ H0− = Γ1 ∂r∗ + , r0 D(H0− ) = Ψ ∈ L2 (] − ∞, 1]r∗ × Sω2 , dr∗ dω)4 ; H0− Ψ

2

∈ L (] − ∞, 1]r∗ ×

Sω2 ,

(182) (183) 4

1

dr∗ dω) , γ Ψ(1, .) = iΨ(1, .) . (184)

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Since the perturbation due to the comparison is exponentially decreasing and |r − r0 | = O e2κ0 r∗ as r∗ → −∞, Lemma 4.11 applied to W ± (Jr−1 H − Jr , H0− ) assures that (185) W ± H − , H0− , Jr exists on L2 (] − ∞, 1]r∗ × Sω2 , dr∗ dω)4 with Ran W ± (H − , H0− , Jr ) = Pac (H − )L2 (]− ∞, 1]r∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 . (186) – We now study the following wave operator: W ± (H0− , H0 , J−∗ ) : H0± ⊂ H0 → L2 (] − ∞, 1]r∗ × Sω2 , dr∗ dω)4 where J−∗ is the adjoint of χ1 Ψ J− : Ψ → J− Ψ = 0 χ1 ∈ C ∞ (Rr∗ ),

r∗ ≤ 1 r∗ ≥ 1,

∃ a, b ∈ R,

(187)

(188)

a b

(189)

± ± ∞ 2 4 2 If we take Ψ± 0 ∈ H0 ∩ C0 (Rr∗ × Sω ) such that supp(Ψ0 ) ⊂ [−R + 1, R + 1] × Sω for a given R > 0, then for all t ∈ R± and ±T ≥ R we have −

itH0 J−∗ e−i(T +t)H0 Ψ± J−∗ e−iT H0 Ψ± 0 = e 0.

(190) −1

itqQr0 Ψ± Moreover, since supp(χ21 − 1) ⊂ [a, +∞[ and e−itH0 Ψ± 0 (r∗ , ω) = e 0 (r∗ ± ± ± ∞ 2 4 t, ω) for all Ψ0 ∈ H0 ∩ C0 (Rr∗ × Sω ) : lim J− J−∗ − 1 e−itH0 Ψ± (191) 0 H0 = 0. t→±∞

Then, by a density argument, we conclude that W ± (H0− , H0 , J−∗ ) is well deﬁned and is an isometry on H0± . For Ψ0 ∈ C0∞ (] − ∞, 1]r∗ × Sω2 )4 , thanks to Lemma 4.10 and since ∂r∗ χ1 is compactly supported and supp(χ21 − 1) ⊂ [a, +∞[, we obtain, − H0 J− − J− H0− e−itH0 Ψ0 H0 −itH0− = (∂r∗ χ1 ) e Ψ0

2 , dr dω)4 L2 (]−∞,1]r∗ ×Sω ∗

− lim J−∗ J− − 1 e−itH0 Ψ0

t→±∞

∈ L1 (Rt ),

2 , dr dω)4 L2 (]−∞,1]r∗ ×Sω ∗

= 0.

(192) (193)

A density argument assures that the wave operator W ± (H0 , H0− , J− ) is well deﬁned and is an isometry on L2 (]−∞, 1]r∗ ×Sω2 , dr∗ dω)4 . Hence, on H0± , the operator (187) exists and (194) Ran W ± (H0− , H0 , J−∗ ) = Pac (H0− )L2 (] − ∞, 1]r∗ × Sω2 , dr∗ dω)4 .

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If we put

843

J0d = Jr J−∗ ,

(195)

then, we deduce from the completeness of (187) and (185) that the wave operator W ± (H − , H0 , J0d ) = W ± (H − , H0− , Jr )W ± (H0− , H0 , J−∗ )

(196)

exists on H0± , with Ran W ± (H − , H0 , J0d ) = Pac (H − )L2 (] − ∞, 1]r∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 . (197) – Then, we deﬁne the following operators: Jd∞ : L2 ([1, +∞[r∗ ×Sω2 , r2 F 1/2 (r)dr∗ dω)4 → H;

Ψ → Jd∞ Ψ =

Jd0 : L2 (] − ∞, 1]r∗ × Sω2 , r2 F 1/2 (r)dr∗ dω)4 → H;

Ψ → Jd0 Ψ =

Ψ 0

r∗ ≥ 1 , r∗ ≤ 1 (198)

Ψ r∗ ≤ 1 . 0 r∗ ≥ 1 (199)

Therefore, with the Conclusions (160), (161), (180), (181) and (196), (197), we obtain, on H0± ⊕ H∞ , the existence of d W ± ID, H0 , Jd0 J0d ⊕ W ± ID, H∞ , Jd∞ J∞ (200) with

d Ran W ± (ID, H0 , Jd0 J0d ) ⊕ W ± (ID, H∞ , Jd∞ J∞ ) = H.

(201)

– To ﬁnish this demonstration, there remains to show that d ± W ± (ID, H0 , Jd0 J0d ) ⊕ W ± (ID, H∞ , Jd∞ J∞ ) = W0± ⊕ W∞

in H.

(202)

In other words, we must check that ± ∀Ψ± 0 ∈ H0 ,

with

lim A = 0 and ∀Ψ∞ ∈ H∞ ,

t→±∞

lim B = 0,

t→±∞

A = r−1 F −1/4 (r)χ0 − Jd0 Jr J−∗ e−itH0 Ψ± 0 H, B = r−1 F −1/4 (r)r∗ χ∞ Λ − Jd∞ Jr Jr∗ JΛ U (t)Ψ∞ H .

(203)

(204)

± To begin with, we treat the ﬁrst limit of (203). As above, taking Ψ± 0 ∈ H0 ∩ −1 itqQr0 Ψ± C0∞ (Rr∗ × Sω2 )4 we have e−itH0 Ψ± 0 (r∗ , ω) = e 0 (r∗ ± t, ω). Thanks to the properties of the identifying operators (166), (188) ,(189), (199), we obtain A = {χ0 − χ1 } e−itH0 Ψ± 2 (205) 2 4 . 0

L (]−∞,1]r∗ ×Sω ,dr∗ dω)

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Hence (205) is zero for t large enough, since χ0 − χ1 is compactly supported. By density, the result follows on H0± . For the second limit, from the properties of the identifying operators (165), (166), (173), (198) and Λ, B satisﬁes B = {χ∞ − 1} ΛU (t)Ψ∞ L2 ([1,+∞[r∗ ×S 2 ,r2 dr∗ dω)4 ω ∗ ≤ Λ−1 1 [1,+∞[ (χ∞ − 1) ΛU (t)Ψ∞ H∞ .

(206)

As Λ−1 1 [1,+∞[ (χ∞ − 1) Λ is compactly supported, we apply Lemma 4.8 for Ψ∞ ∈ {Ψ ∈ H∞ : Fx (Ψ) ∈ C0∞ (R3x \ {0})4 }. By density of this last subspace in H∞ , we get the result on H∞ .

5 Conclusion In this paper, we have developed a complete scattering theory for the linear massive charged Dirac system on the Reissner-Nordstrøm metric. This result is a suitable generalization of the work of J.-P. Nicolas [N1], since, if we take Q = 0 and m = 0 in our deﬁnition of the wave operators, we ﬁnd again J.-P. Nicolas’s wave operators. With our scattering result and the techniques developed by A. Bachelot in the important papers about quantum vacuum polarization respectively for the Klein-Gordon [Ba3] and Dirac equations [Ba5], and the Hawking eﬀect [Ba4] in the case of Klein-Gordon ﬁeld, it must be possible to investigate the Hawking eﬀect for Dirac ﬁelds.

Acknowledgments I would like to thank warmly Alain Bachelot for his advices and comments. I am also grateful to Jean-Philippe Nicolas for his remarks concerning this work.

References [ABG1] W.O. Amrein, A.M. Boutet de Monvel-Berthier, V. Georgescu, On Mourre’s approach to spectral theory, Helvetica Physica Acta 62, 1–20 (1989). [Ba1] A. Bachelot, Gravitational scattering of electromagnetic ﬁeld by Schwarzschild black-hole, Ann. Inst. H. Poincar´e Phys. Th´eor. 54, 261–320 (1991). [Ba2] A. Bachelot, Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric, Ann. Inst. H. Poincar´e Phys. Th´eor. 61, 411– 441(1994). [Ba3] A. Bachelot, Quantum Vacuum Polarization at the Black-Hole Horizon, Ann. Inst. H. Poincar´e Phys. Th´eor. 67-2, 181–222 (1997).

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[Ba4] A. Bachelot, The Hawking eﬀect, Ann. Inst. H. Poincar´e Phys. Th´eor. 70-1, 41–99 (1994). [Ba5] A. Bachelot, Creation of Fermions at the Charged Black-Hole Horizon, Ann. Henri Poincar´e, 1, 1043–1095 (2000). [BM-B] A. Bachelot, Ag. Motet-Bachelot, Les r´esonnances d’un trou noir de Schwarzschild, Ann. Inst. H. Poincar´e Phys. Th´eor. 59, 3–68 (1993). [BG]

A. Boutet de Monvel-Berthier, V. Georgescu, Spectral and scattering theory by the congugate operator method, St. Petersburg Math. J. 4, no. 3, 469– 501 (1993).

[BMG] A. Boutet de Monvel, D. Manda, R. Purice, Limit absorption principle for the Dirac operator, Ann. Inst. H. Poincar´e Phys. Th´eor. 58, no. 4, 413–431 (1993). [CJJ] A. Chodos, R.L. Jaﬀe, K. Johnson, C.B. Thorn, V.F. Weisskopf, New extended model of hadrons, Phys. Rev. D (3) 9, no. 12, 3471–3495 (1974). [DG] J. Derezi´ nski, C. G´erard, Scattering theory of classical and quantum nparticle systems, (1997) Springer. [Di]

J. Dimock, Scattering for the Wave Equation on the Schwarzschild Metric, Gen. Rel. Grav. 17, 353–69 (1985).

[DK] J. Dimock, B.S. Kay, Scattering for massive scalar ﬁelds on Coulomb potentiels and Schwarzschild metric, Class. Quantum Grav. 3, 71–90 (1986). [DV]

J. Dollard, G. Velo, Asymptotic behaviour of a Dirac particle in Coulomb ﬁeld, Nuovo Cimento, (1966), no. 45, p801-812.

[GY] Y. Gˆatel, D. Yafaev, Scattering theory for the Dirac operator with a longrange electromagnetic potential, Prepublication de l’IRMAR, Universit´e de Rennes I. [GS]

M. Gel’Fand, Z. Ya. Sapiro, Representations of the group of rotations of 3-dimensional space and their applications, Amer. Math. Soc. Transl. 11 2, 207–316 (1956).

[GG] V. Georgescu, C. G´erard, On the Virial Theorem in Quantum Mechanics, Commun. Math. Phys. 208, 275–281 (1999). [Ha1] D. H¨ afner, Th´eorie de la diﬀusion en relativit´e g´en´erale: ´equation des ondes dans des espaces-temps stationnaires asymptotiquement plats et ´equation de Klein-Gordon dans l’espace-temps de Kerr, Th`ese de doctorat de l’Ecole Polytechnique.

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[Ha2] D. H¨ afner, Compl´etude asymptotique dans une classe d’espace temps stationnaire et asymptotiquement plat, Ann. Inst. Fourier 51 no. 3, 779–833 (2001). [Ji]

W.M. Jin, Scattering of massive Dirac ﬁelds on the Schwarzschild black hole spacetime, Class. Quantum Grav. 15, 3163–3175 (1998).

[Me]

F. Melnyk, Wave operator for the massive charged linear Dirac ﬁeld on the Reissner-Nordstrøm metric, Class. Quantum Grav. 17, 2281–2296 (2000).

[Mo]

E. Mourre, Absence of Singular Continuous Spectrum for Certain Selfadjoint Operators, Commun. Math. Phys. 78, 391–408 (1981).

[MS]

P.L. Muthuramalingam, K.B. Sinha, Existence and completeness of wave operators for the Dirac operator in an electro-magnetic ﬁeld with long-range potentials, J. Indian Math. Soc. 50 1–4, 103–130 (1986).

[N1]

J.-P. Nicolas, Scattering of linear Dirac ﬁelds by a spherically symmetric Black-Hole, Ann. Inst. H. Poincar´e Phys. Th´eor. 62, no. 2, 145–179 (1995).

[N2]

J.-P. Nicolas, Global exterior Cauchy problem for spin 3/2 zero rest-mass ﬁelds in the Schwarzschild space-time, Comm. Partial Diﬀerential Equations 22, no. 3–4, 465–502 (1997).

[RS]

M. Reed, B. Simon, Methods of modern mathematical physics , Vol. I, II, III, IV, (1972-75-78-79), Academic Press.

[Th]

B. Thaller, Relativistic Scattering Theory for Long-Range Potential of the Nonelectrostatic Type, Lett. Math. Phys. 12, no. 1, 15–19 (1986).

[TE]

B. Thaller, V. Enss, Asymptotic observables and Coulomb scattering for the Dirac equation, Ann. Inst. H. Poincar´e Phys. Th´eor. 45 , no. 2, 147–171 (1986).

Fabrice Melnyk Math´ematiques Appliqu´ees de Bordeaux UMR CNRS 5466 Universit´e Bordeaux I 351, Cours de la Lib´eration F-33405 Talence Cedex France email: [email protected] Communicated by Piotr Chrusciel submitted 24/05/02, accepted 14/02/03

Ann. Henri Poincar´e 4 (2003) 847 – 896 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/050847-50 DOI 10.1007/s00023-003-0149-1

Annales Henri Poincar´ e

Surface Tension and Wulﬀ Shape for a Lattice Model without Spin Flip Symmetry T. Bodineau and E. Presutti

Abstract. We propose a new deﬁnition of surface tension and check it in a spin model of the Pirogov-Sinai class without symmetry. We study the model at low temperatures on the phase transitions line and prove: (i) existence of the surface tension in the thermodynamic limit, for any orientation of the surface and in all dimensions d ≥ 2; (ii) the Wulﬀ shape constructed with such a surface tension coincides with the equilibrium shape of the cluster which appears when ﬁxing the total spin magnetization (Wulﬀ problem).

1 Introduction During the past decade, progress was made in the understanding of the phase segregation starting from microscopic models. To summarize, two approaches prevail to derive the Wulﬀ construction for Ising type models. The ﬁrst one enables to describe the phenomenon of phase coexistence in two dimensions with an extremely high accuracy, in particular it provides a sharp control of the phase boundaries w.r.t. the Hausdorﬀ distance (see, e.g., [DKS, I1, I2, ISc, Pf, PV2]). The second strategy is much less precise and gives only L1 estimates; however it can be also implemented in higher dimensions (see, e.g., [ABCP, BCP, BBBP, BBP, Ce, B1, CePi1, BIV1, CePi2]). Phase segregation occurs in a wide range of physical systems, but the two strategies mentioned above have been mainly implemented in models with symmetry among phases and in some cases, the speciﬁc microscopic structure of the interactions has been at the heart of the proofs (duality, FK representation, ferromagnetic inequalities . . . ). The goal of this paper is to extend the L1 -approach to a class of systems without symmetry, which can be studied by the Pirogov-Sinai Theory. The L1 -theory is at ﬁrst sight not model dependent, it is based on a coarse grained description of the system and provides a general framework to relate the surface tension to L1 -estimates. Nevertheless, its concrete implementation has been restricted to a speciﬁc class of models: Bernoulli percolation [Ce], Ising model [B1, BIV1, CePi1] and Potts model [CePi2]. These three instances have a common structure which arises in the FK representation. The coarse graining developed by Pisztora [Pi] played a key role in the derivation of the L1 -approach for the three models above. This hinders the generalization to a broader class of models, since parts of the

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proof relied on Pisztora’s coarse graining and thus on the FK representation. Notice also that the proofs were based on the symmetry of the model and on the ferromagnetic inequalities. In particular the analysis of the surface tension was completely model dependent. There are some works which deal with surface tension in non-symmetric models [BKL, HKZ1, HKZ2, MMRS], but a general theory of surface tension (including the thermodynamic limit for all slopes) seems still to be missing. In this paper we propose a new deﬁnition of surface tension. The advantage is that its existence in the thermodynamic limit for arbitrary slopes of the surface and in all dimensions d 2, does not rely on the symmetry of the pure phases nor on ferromagnetic inequalities, at least when the Pirogov-Sinai theory can be applied. The validity of the deﬁnition is then conﬁrmed by the proof that the Wulﬀ construction using this surface tension actually determines the equilibrium shape of a droplet in the system. The surface tension is characterized by two speciﬁc features, a cutoﬀ on the interface ﬂuctuations and the notion of perfect walls. The precise deﬁnition and heuristics are postponed to Section 3. The thermodynamic limit of the surface tension is derived by a recursive procedure. The rest of the L1 -approach (including the coarse graining) is presented in Section 6 following the usual scheme. In the present paper, we focus on a particular model in order to stress the main ideas in the most simple context. We actually believe that the proof holds for a general class of two phase models in the Pirogov-Sinai Theory (see the last paragraph of Subsection 2.1). The liquid/vapour phase coexistence is also the subject of current investigations and it seems possible to generalize our strategy for particles in the continuum with Kac potentials as considered by Lebowitz, Mazel, Presutti [LMP]. The main ideas in this work have been developed in collaboration with Dima Ioﬀe.

2 Model and main theorem 2.1

The model

We consider a lattice model on Zd , d 2, which is made of interacting spins σx taking values {−1, 1}. The interaction depends on a 2d -body potential deﬁned so that its ground states are the conﬁgurations with all spins equal to +1 and all spins equal to −1. However the interaction is not invariant under spin ﬂip and the analysis of the Gibbs measures at positive temperatures relies on the Pirogov-Sinai theory and phase coexistence occurs at non zero values of the magnetic ﬁeld. We call cell and denote it by c a cube in Zd of side 2 (meaning that it contains d 2 lattice sites); denoting by σc the restriction of σ to c, we deﬁne the cell potential V (σc ) as equal to 0 if σc ≡ 1 and σc ≡ −1, otherwise V (σc ) is equal to the number

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of +1 spins present in σc . The Hamiltonian in the ﬁnite set Λ with b.c. σΛc is then V (σc ) − hσx . HhσΛc (σΛ ) = c∩Λ=∅

x∈Λ

¯ we will also write Hhσ¯ (σΛ ). If σΛc is the restriction to Λc of a conﬁguration σ The Gibbs measure associated to the spin system with boundary conditions σ ¯ is ¯ (σΛ ) = µσβ,h,Λ

1 σ ¯ Zβ,h,Λ

exp −βHhσ¯ (σΛ ) ,

σ ¯ where β is the inverse of the temperature and Zβ,h,Λ is the partition function. If σ ¯ is uniformly equal to 1 (resp −1), the Gibbs measure will be denoted by µ+ β,h,Λ (resp µ− ). β,h,Λ

Classical Pirogov-Sinai theory ensures that for any β large enough, there exists a value of the magnetic ﬁeld h(β) such that a ﬁrst order phase transition is located on the curve (β, h(β)). In particular on the phase coexistence curve, one can deﬁne (see Theorem 4.2 below) two distinct Gibbs measures µ+ β,h(β) and Z µ− β,h(β) which are measures on the space {±1} . They are obtained by taking the + − thermodynamic limit of µβ,h(β),Λ and µβ,h(β),Λ . Each of these measures represents a pure state. The averaged magnetization in each phase is denoted by d

+ m+ β = µβ,h(β) (σ0 )

− and m− β = µβ,h(β) (σ0 ) .

(2.1)

Observe that if we replace cells by bonds we recover (modulo an additive constant) the energy of the nearest neighbor Ising model. Our system is in our intentions the simplest modiﬁcation of the nearest neighbor Ising model where the spin ﬂip symmetry is broken but the ground states are kept unchanged. This choice has been to give up any attempt of generality and instead to focus on a particular model, where the main ideas are not obscured by too many technicalities. Nevertheless, we believe our analysis extends to ﬁnite range, many body Hamiltonians of the form VX (σX ) X⊂Λ

provided they are into the Pirogov-Sinai class and under the assumptions that the potentials VX are symmetric and translational invariant, with ground states the constant conﬁgurations. The symmetry assumption means (2.2) for all X, VRX (Rσ)RX = VX (σX ) , where R denotes the symmetry w.r.t. the origin and (Rσ)j = σ(R)−1 (j) . We will pursue the discussion on possible extensions and open questions in Subsection 3.4.

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Ann. Henri Poincar´e

Phase coexistence

The phenomenon of phase segregation will be described in the framework of the L1 -approach. Let us ﬁrst recall the functional setting. On the macroscopic level, the = [0, 1]d of Rd and a macroscopic conﬁguration system is conﬁned in the torus T + where the pure phases coexist is described by a function v taking values {m− β , mβ }. The function v should be interpreted as a signed indicator representing the local order parameter: if vr = m+ β for some r ∈ T, then the system should be locally at r in equilibrium in the + phase. To deﬁne the macroscopic interfaces, i.e., the boundary of the set {v = m− β }, − + a convenient functional setting is the space BV(T, {m , m }) of functions of β

β

bounded variation with values m± β in T (see [EG] for a review). For any v ∈ − + BV(T, {mβ , mβ }), there exists a generalized notion of the boundary of the set − ∗ {v = m− β } called reduced boundary and denoted by ∂ v. If {v = mβ } is a regular ∗ set, then ∂ v coincides with the usual boundary ∂v. The interfacial energy associated to a domain is obtained by integrating the surface tension along the boundary of the domain. The surface tension is a function τβ : S(d−1) → R+ on the set of unit vectors S(d−1) , which in our model has the as expression speciﬁed in Section 3. The Wulﬀ functional Wβ is deﬁned in L1 (T) follows {m− , m+ }), τ (nx ) dHx , if v ∈ BV(T, β β ∂∗v β Wβ (v) = (2.3) ∞, otherwise.

we associate the function 1IA = m+ 1Ac + m− 1A To any measurable subset A of T, β β and simply write Wβ (A) = Wβ (1IA ). + Fix an interval [m1 , m2 ] included in (m− β , mβ ). The equilibrium crystal shapes are the solutions of the Wulﬀ variational problem, i.e., they are the minimizers of the functional Wβ under a volume constraint

{m− , m+ }), min Wβ (v) v ∈ BV(T, β β

m1

T

vx dx m2 .

(2.4)

Let D(m1 , m2 ) be the set of minimizers of (2.4).

2.3

Local magnetization

The correspondence between the microscopic quantities and the functional setting described above can be obtained only after some averaging procedure, as the one we are going to describe. We ﬁrst need a few extra notations. Let B (K) , K = 2k , k ∈ N, be the partition of Zd into cubes BK : the seed of the partition is

BK (0) = x ∈ Zd : 0 xi < K, i = 1, . . . , d

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and the other cubes of B (K) are obtained by translations by integer multiples of K in all coordinate directions. The sequence B (K) , k ∈ N, is then a compatible sequence of partitions of Rd , namely each cube BN ∈ B (N ) is union of cubes BK in B (K) , if K = 2k N = 2n . Given K = 2k , we denote by BK (x) the box in B (K) which contains the point x ∈ Zd . The local averaged magnetization is deﬁned by MK (x) =

1 |BK (x)|

σy .

(2.5)

y∈BK (x)

By abuse of notation, MK (·) can be viewed also as a piecewise constant function on Rd . For simplicity the microscopic region Λ is chosen as BN (0) and, imposing periodic b.c. it becomes the torus TN . We call ψN the map from TN onto T, obtained by shrinking by a factor 1/N . We then deﬁne the local magnetization −1 (r) , MN,K (r) = MK ψN

r∈T

(2.6)

piecewise constant on the boxes ψN (BK ), BK ⊂ TN . which is a function on T The local order parameter MN,K characterizes the local equilibrium. The total magnetization in TN is simply denoted by MN . We can now state a result on phase coexistence. Theorem 2.1. There exists β0 > 0 such that for any β > β0 and [m1 , m2 ] ⊂ + (m− β , mβ ) (with m1 < m2 ), the following holds: for every δ > 0 there is a scale K0 = K0 (β, δ) such that for any K K0 lim µβ,h(β),N

N →∞

inf

v∈D(m1 ,m2 )

MN,K

− v1 δ m1 MN m2 = 1 ,

where D(m1 , m2 ) denotes the set of the equilibrium crystal shapes (2.4) (where the surface tension is the one deﬁned in Section 3) and µβ,h(β),N is the Gibbs measure on TN with periodic boundary conditions.

3 Surface tension For any given unit vector n = (n1 , . . . , nd ), we are going to deﬁne the surface tension τβ (n) in the direction orthogonal to n. Contrary to the Ising model, the lack of symmetry between the two pure phases requires a more complex deﬁnition of surface tension which relies on two new features: a cutoﬀ of the interface ﬂuctuations and the introduction of perfect walls.

852

3.1

T. Bodineau and E. Presutti

Ann. Henri Poincar´e

Interface ﬂuctuations cutoﬀ

We associate to any unit vector n = (n1 , . . . , nd ) a coordinate direction j ∈ {1, . . . , d} in such a way that ni nj for all i while ni < nj , for any i > j. For notational simplicity suppose j = d, the other cases are treated similarly. We set

∀i < d, − xi ; −m (x · n) m . (3.1) Λ,m (n) = x ∈ Zd , As n is ﬁxed throughout this section, we will drop it from the notation. The surface tension τβ (n) will be the thermodynamic limit of ratios of partition functions deﬁned on subsets of the slab ΛL, 11ε . The limit will be taken 10 L for appropriate sequences of the parameters (L, ε), in particular we require L and (ε/10)L to be in {2n , n ∈ N}. We will ﬁrst introduce the partition function with mixed boundary conditions. We want to impose + and − boundary conditions on top and bottom of our domains; it will be convenient to leave some freedom on their exact location and with this in mind we introduce the notion of barriers. A barrier in a slab Λ,m is a connected set of cells in Λ,m which connects the faces of Λ,m parallel to ed and it is such that its complement in Λ,m is made of two distinct components ε which are not -connected. Let then C + and C − be two barriers in ΛL, 10 ed L + εL + − ε 11ε and ΛL, 10 − εL e . The subset of Λ lying between C and C is denoted by L d L, 10 L ¯ ± outside Λ(C + , C − ) are deﬁned as Λ(C + , C − ). The mixed boundary conditions σ follows +1, if (x · n) 0 , + − ± σ ¯x = ∀x ∈ Λ(C , C ), −1, if (x · n) < 0 . We denote by S + (resp S − ) the set of spin conﬁgurations for which there is a εL ε barrier included in ΛL, 10ε L + εL ed (resp ΛL, 10 ed ) where all spins are equal L− 2 2 to 1 (resp −1). Finally, we introduce the following constrained partition function on Λ(C + , C − ) with mixed boundary conditions (see Figure 1) ± C + ,C − ZL,ε (S + , S − ) = 1{σ∈S + ∩S − } exp −βHhσ¯ (σ) . (3.2) σ∈{±1}Λ(C+ ,C− )

The barriers S + , S − act as a cutoﬀ of the interface ﬂuctuations: they decouple the interface from the boundary conditions outside Λ(C + , C − ). In the following, we will explain the role of this screening.

3.2

Perfect walls

A perfect wall is such that its contribution to the ﬁnite volume corrections to the pressure is inﬁnitesimal w.r.t. the area of its surface, best examples are walls deﬁned by periodic boundary conditions. Under suitable assumptions on the interaction it is in fact well known that with periodic boundary conditions the corrections

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to the pressure decay exponentially with the size of the box. Periodic boundary conditions are however not useful in our context, because we want to impose one of the two phases on some of the walls; but, as we are going to see, it is possible to deﬁne some sort of periodic conditions on single walls of the container. We start by deﬁning a symmetric partition of Zd by the hyperplane Σ orthogonal to n and containing 0. Let us ﬁrst suppose that the orientation n is such that Σ ∩ Zd = {0}. We then set d−1 ni d d d xd − xi \ {0}, ∪ {0}) . Zd− = Zd \ (Z+ Z+ = x ∈ Z n d i=1 Then Zd = Zd+ ∪ Zd− ∪ {0},

Zd− = R(Zd+ )

(3.3)

where R is the symmetry w.r.t. 0. If there are sites in Zd ∩Σ besides 0, we split them between Zd+ and Zd− in such a way that (3.3) is preserved. Notice ﬁrst that if x ∈ Σ ∩ Zd , also Rx ∈ Σ ∩ Zd . Then if x = (x1 , . . . , xd ) = 0, we call i the ﬁrst integer so that xi = 0 (i.e., x1 = · · · = xi−1 = 0) and we put x ∈ Zd+ if xi > 0 and x ∈ Zd− otherwise. Thus   d d−1  nj  Zd+ = xj (3.4) x ∈ Zd ∀i < k, xi = 0, xk > 0, xd −   n j=1 d k=1   d−1  nj  , ∪ x ∈ Zd ∀i < k, xi = 0, xk < 0, xd > − xj  nd  j=1

A drawback of the deﬁnition is that for n oriented along one of the axis of coordinates the bottom of Zd+ is not ﬂat. This could be avoided at the price of considering a more complicated mapping than the simple symmetry w.r.t. 0. We now proceed in deﬁning the reﬂected Hamiltonian in Zd . The idea is to use R in order to glue together diﬀerent regions touching the surface Σ so that if, for instance, x ∈ Zd+ interacts across Σ with y ∈ Zd− then x will now interact with R(y) ∈ Zd+ . As the energy is deﬁned in terms of cells, this can be easily achieved by introducing a new set of cells {c}R . Cells which are entirely contained either in Zd+ , or in Zd− or in B = {−1, 0, 1}d are unchanged. Instead any cell c containing sites both in Zd+ and in Zd− is replaced by c+ = (c \ Zd− ) ∪ R(c ∩ Zd− ) c→ − . (3.5) c = (c \ Zd+ ) ∪ R(c ∩ Zd+ ) Notice that both cells c and R(c) generate the same pair c± , so that the “total number” of old and new cells is the same.

854

T. Bodineau and E. Presutti

Ann. Henri Poincar´e

Extending the deﬁnition of V (σc ) to the new set of cells, the reﬂected Hamiltonian is then R,σΛc Hh,Λ (σ) = V (σc ) − h σx . (3.6) c∈{c}R c∩Λ=∅

x∈Λ

We will always consider regions which do not contain B, so that the spins in B will act as boundary conditions: thus the structure of cells entirely contained in B is unimportant. In preparation to the deﬁnition of the surface tension and using the notation of Subsection 3.1, we deﬁne the upper half of Λ(C + , C − ) by Λ+ (C + ) = Λ(C + , C − ) ∩ Zd+ \ B . The partition function with reﬂection and + boundary conditions outside Λ+ (C + ) (+,R) C + ,R ZL,ε = exp −βHΛ+ (C + ) (σ) , (3.7) σ∈{±1}Λ+ (C+ )

where the Hamiltonian on the right-hand side is deﬁned in (3.6) with Λ replaced by Λ+ (C + ). Notice that the boundary conditions outside Λ+ (C + ) are imposed also C − ,R around the center of reﬂection on B = {−1, 0, 1}d. The partition function ZL,ε is deﬁned similarly on the lower half, Λ− (C − ) of Λ(C + , C − ) and with − reﬂected boundary conditions on the top (see Figures 1 and 2). Let ΣL be the bottom face of Λ+ (C + ), i.e., the face with the reﬂected interactions (the side-length of ΣL is L). As we will see in Lemma 4.7, away from 0 and from its boundaries, ΣL behaves as a wall with periodic boundary conditions; indeed, the overall contribution of ΣL to the ﬁnite volume corrections to the pressure will be proportional to Ld−2 which is therefore a “perfect wall” in the sense speciﬁed at the beginning of this subsection. Finally notice that one could also consider a mapping diﬀerent from the symmetry w.r.t. 0 provided that it respects the topological structure of Zd and that most of the points are far apart from their images. This will be made clear in Section 4.

3.3

Deﬁnition of the surface tension

We can ﬁnally introduce Deﬁnition 3.1. The surface tension in the direction n, is deﬁned by +

−

C ,C ZL,ε (S + , S − ) (n · ed ) log , τβ (n) = lim inf lim inf +inf − − + C ,R C − ,R ε→0 L→∞ C ,C βLd−1 ZL,ε ZL,ε

(3.8)

ε ed . where the inﬁmum is taken over the barriers (C + , C − ) in the slabs ΛL, 10 L ± εL

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855

Λ(C + )

Λ(C + )

n εL

0

Λ(C − )

S+ S− C−

Λ(C − )

Figure 1. On the left, the domain ΛL,εL is depicted with mixed boundary conditions in the direction n and with the interface cutoﬀ. The action of the perfect walls boils down to fold Λ(C + ) and Λ(C − ) around the point 0 (see right picture and also Figure 2).

There are two important points in this deﬁnition, one is that the perfect walls should give negligible surface corrections to the pressure. Moreover, due to decay of correlations, the inf over C + , C − should not matter because of cancellations among numerator and denominator: the barriers S + and S − screen the eﬀect of the boundary conditions. The main step towards the derivation of phase coexistence (Theorem 2.1) will be to prove the convergence of the thermodynamic limit for the surface tension: Theorem 3.1. For any β large enough (such that the model is in the Pirogov-Sinai regime, see Section 4), the following holds +

τβ (n)

−

C ,C ZL,ε (S + , S − ) (n · ed ) = lim lim sup − log , + − ε→0 L→∞ C + ,C − βLd−1 Z C ,R Z C ,R L,ε

(3.9)

L,ε

ε where the supremum is taken over the barriers (C + , C − ) in the slabs ΛL, 10 ed . L ±εL In (3.9), the supremum can also be replaced by an inﬁmum.

The derivation of Theorem 3.1 and of the properties of the surface tension is postponed to Section 5.

3.4

Heuristics on the surface tension

We are going to discuss heuristically the representation of the surface tension and explain the choice of the perfect walls and of the cutoﬀ. We believe that the ultimate justiﬁcation for Deﬁnition 3.1 is to be the surface tension for which Theorem 2.1 is valid.

856

T. Bodineau and E. Presutti

Ann. Henri Poincar´e

Let us start by a rough expansion of log ZL+,− , which denotes the partition function on the cube ∆L = {−L, . . . , L}d with mixed boundary conditions in the direction n. log ZL+,− =

Ld + Ld−1 + − (P + P − ) + τβ (n) + (τbd + τbd ) d Ld−1 + O(Ld−2 ) . 2 (n · ed ) (3.10)

The ﬁrst term is of volume order and corresponds to the pressures of the diﬀerent pure phases P + and P − (which are equal on the curve of phase coexistence, see Lemma 4.2). The surface tension τβ (n) arises at the next order, but there are as well other terms of order Ld−1 which can be interpreted as surface energies due to the boundary conditions. The lack of symmetry of our model implies that the + produced by the + boundary conditions diﬀers from the surface surface energy τbd − energy τbd produced by the − boundary conditions. In order to extract the surface tension factor, one has to compensate not only + − and τbd . In a symmetric case (e.g., the bulk term, but also the surface energies τbd + − the Ising model) τbd = τbd therefore the partition function in ∆L with + boundary conditions is the appropriate normalization factor. As this is no longer the case for non-symmetric models, the following alternative deﬁnition seems to be the most natural τβ (n) = lim − L→∞

(n · ed ) ZL+,− log . Ld−1 Z+ Z− L

(3.11)

L

Notice that this representation of the surface tension would also require an assumption on the potential similar to (2.2) in order to produce exact cancellations between the numerator and the denominator. The representation (3.11) of the surface tension is the most commonly used, nevertheless, to our best knowledge, the existence of its thermodynamic limit is not known in general. The surface tension can be studied for diﬀerent types of models, in particular, let us mention the Ashkin-Teller model [Ve], the Blume Capel model [HK], the Potts model at the critical temperature [MMRS, LMR] and general 3D lattice models [HKZ1, HKZ2]. Depending on the dimension, the results are of diﬀerent nature. In 2 dimensions, the interface has a uni-dimensional structure and a very accurate control can be obtained by using renewal theory. In particular it should be possible to derive in a general context a complete expansion of the right-hand side of (3.11) which would include the Ornstein-Zernike corrections1. Such results would also provide a description of the ﬂuctuations of the interface. We refer the reader to the paper by Hryniv and Kotecky [HK] for an implementation of these methods in the case of Blume-Capel model (see also [Al, CIV]). 1 Private

communication by D. Ioﬀe.

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In dimension 3 or higher, if n coincides with one of the axis, the interface generated by the Dobrushin conditions is rigid and an extremely accurate description of the non-translation invariant Gibbs states can be obtained. As a byproduct of this description, (3.11) can be derived for a broad class of models (see Holicky, Kotecky, Zaradhnik [HKZ1, HKZ2]; Messager, Miracle-Sol´e, Ruiz, Shlosman [MMRS]). However a derivation of (3.11) in dimensions larger or equal to 3 for general slopes n seems still to be missing. In general, the ground states of tilted interfaces are degenerated, this complicates seriously the implementation of a perturbative approach of the thermodynamic limit (3.11). The representation (3.8) of the surface tension was motivated by the Wulﬀ construction and it has been designed primarily to prove the phase coexistence (Theorem 2.1). The ﬁrst step to evaluate the surface energy of a droplet is to decompose the interface of the droplet and to estimate locally the surface tension. As the system is random, one is lead to consider partition functions with mixed boundary conditions on arbitrary domains of the type Λ(C + , C − ) and not only on regular sets like ∆L . Locally, the occurrence of an interface means a term like the numerator of (3.8) can be factorized from the global partition function. At this point, the local surface tension factor is extracted from the global partition function by removing the numerator of (3.8) and replacing it instead by the denominator of (3.8). In (3.11), the cancellation of the terms corresponding to the boundary surface tension imposes to choose symmetric domains in the denominator. This constraint is too stringent to apply the procedure previously described to arbitrary domains. The perfect walls provide an alternative way to control the surface order corrections without using symmetry. The second important feature of Deﬁnition 3.1 is the interface ﬂuctuation cutoﬀ. The Pirogov-Sinai theory describes accurately the bulk phenomena in a low temperature regime, nevertheless it cannot be applied directly to study Gibbs measures with mixed boundary conditions. The cutoﬀ decouples the interface from the boundary conditions and therefore enables us to control the dependence between the surface tension and the domain shapes. In fact, the problem in the domain between C + and S + (resp S − and C − ) is set in the regime associated to the pure phase with + (resp. −) boundary conditions where again cluster expansion applies. The derivation of the thermodynamic limit (Theorem 3.1) relies on a recursive procedure which is reminiscent of the proof of the Wulﬀ construction. The basic idea is to approximate the interface on large scales by using the Deﬁnition 3.1 on smaller scales. Concretely, the energy in the small regions along the interface is evaluated by pasting the a priori estimates provided by Deﬁnition 3.1. The iteration is possible thanks to the very loose structure of the deﬁnition of the surface tension. The limit w.r.t. ε has no impact on the value of the surface tension, the main motivation is technical: it is useful in the iteration procedure and afterwards in the completion of Theorem 2.1. We are going now to compare the representations τβ (n) and τβ (n) of the surface tension. According to Theorem 3.1 the convergence (3.9) is uniform over

858

T. Bodineau and E. Presutti

Ann. Henri Poincar´e

the domains Λ(C + , C − ) and thus it is enough to deﬁne the surface tension on regular domains of the type ∆L . Furthermore, the perfect walls are such that

lim

L→∞

1 Ld−1

log

ZL+ ZL− ZL+,R ZL−,R

= 0.

It remains only to analyze the role of the cutoﬀ of interface ﬂuctuations. Deﬁnition 3.1 would coincide with (3.11) if the following holds lim

1

L→∞ Ld−1

log

ZL+,− (S + , S − ) 1 + − = lim d−1 log µ+,− β,∆L (S , S ) = 0 . L→∞ L ZL+,−

(3.12)

This statement boils down to prove a very weak form of localization of the interface. In fact, a much stronger √ localization is expected since the ﬂuctuation√of the interface are of the order L in 2D and believed to be at most of the order log L in 3D. For the ferromagnetic ﬁnite range Ising model and the Kac-Ising models, (3.12) holds and Deﬁnition 3.1 of the surface tension coincide with the usual one (3.11). Since the ingredients used in the proof of Theorem 3.1 are the typical ones of cluster expansion, the extension to more general Pirogov-Sinai models, as those described at the end of Subsection 2.1, seems possible. For more general models several questions remain. In particular, Deﬁnition 3.1 does not seem appropriate to deal with periodic ground states. For multi-phase models, the solution of the variational problem is not known and thus a macroscopic description of phase coexistence is a mathematical challenge. The probabilistic point of view is slightly diﬀerent since one is interested to derive the macroscopic variational problem (without solving it) from the microscopic system. In this case, the difﬁculties are of two distinct natures: geometric and probabilistic. For a thorough study of the geometric problems we refer the reader to Cerf, Pisztora [CePi2]. For the issues related to the coarse graining and the surface tension, we hope that our approach can provide a step towards the derivation of phase coexistence for multiphase models. Nevertheless, it should be stressed that the interesting phenomena, as boundary layers, occurring in multi-phase models cannot be capture in the L1 framework. A more reﬁned analysis of the microscopic structure of the interface is necessary to describe these subtle mechanisms (see, e.g., [HK, MMRS, HKZ2]).

4 Peierls estimates, cluster expansion In this section we will see that notion and procedures of the Pirogov-Sinai theory can be modiﬁed to apply when reﬂecting walls are present. In particular we will derive formulas for the ﬁnite volume corrections to the pressure which show that the contribution of the reﬂecting walls is negligible.

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We need to generalize the context considered in the previous section because in the next ones we will have simultaneously several reﬂecting surfaces {Σi } (introduced to decouple diﬀerent regions of the whole domain). An example is depicted in Figure 2. These reﬂecting surfaces are separated in such a way that there will be no interference among them and we can consider each one separately. Let us then call Σ one of them (dropping momentarily the label i) and describe its structure. Σ is the intersection of a hyperplane H and a coordinate cylinder π with cubic cross section of integer side. The axis of the cylinder is the coordinate direction associated to the normal to Σ, in the sense of Subsection 3.1, and its intersection with Σ, called the center of Σ, is supposed to be in Zd . We then introduce the set of boundary sites associated to Σ, i.e., the sites close to the border of Σ and to the center of Σ. Deﬁning Zd± as the spaces above and below H, in the sense of Subsection 3.2, we call B the “boundary of Σ” as the sites x of Zd± which are ∗ connected to Zd∓ as well as ∗ connected to π c , if in π, and to π, if in π c . B is deﬁned as the union of B with the center of Σ and the sites ∗ connected to it. We then call {c }R the set of all new cells c determined by the reﬂection through the hyperplane H which are in π, are not contained in B and diﬀer from original cells. This refers to the generic surface Σi with Bi and {c }R i , we are now resuming the notation with the subscript i. The union of all Bi will be called B while R {c }R is the union of all {c }R i . We then deﬁne the set of new cells {c} , as the R collection of {c } and of all cells which have not been modiﬁed by reﬂections through any of the surfaces Σi . Thus {c}R are the new cells and {c} the old ones. The new Hamiltonian is given by the same expression (3.6) but with {c}R the above collection of cells. Finally, we set Zd,R = Zd \ B and ﬁx hereafter the spins in B. In the sequel Λ will denote regions in Zd,R and the spins in B will always act as boundary conditions. The collection {c}R deﬁnes a new topology, where the nearest neighbor sites of x ∈ Zd is the union of all cells c ∈ {c}R which contain x. Without reﬂection, this reduces to the usual notion in Zd , where the n.n. sites of x are those ∗ connected to x. It is convenient to add a metric structure, deﬁning the “ball of radius n ∈ N and center x ∈ Zd ”, denoted by K(x, n) for the old and, respectively, by K R (x, n) for the new cells, by setting K(x, 0) = K R (x, 0) = {x} and K(x, n) = y ∈ Zd : y ∈ c, c ∩ K(x, n − 1) = ∅, c ∈ {c} (4.1) K R (x, n) = y ∈ Zd : y ∈ c, c ∩ K R (x, n − 1) = ∅, c ∈ {c}R The external boundary of Λ in the old and new topology are δ(Λ) = y ∈ Λc : y ∈ c, c ∩ Λ = ∅, c ∈ {c} y ∈ Λc : y ∈ c, c ∩ Λ = ∅, c ∈ {c}R δ R (Λ) = where Λ ⊂ Zd,R (we recall that B belongs to Λc ).

(4.2)

(4.3)

860

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Ann. Henri Poincar´e

K(x, n) B

K R (x, n)

B

Figure 2. The two examples above represent the diﬀerent types of reﬂecting surfaces which will be used in this paper. The gray rectangles stand for the location of the boundary conditions B. On the left, a domain with two reﬂecting surfaces on its bottom face; a reﬂected contour is also depicted. This type of domain will be used in the analysis of surface tension (Section 5). The domain on the right contains several reﬂecting surfaces where the structure of the cells is modiﬁed (see Subsections 6.3 and 6.4).

The whole analysis in this section is based on a simple geometric property of the collection {c}R , which is a consequence of the way reﬂections on a single surface have been deﬁned and the fact that the reﬂecting surfaces are separated from each other. Given x ∈ Zd,R , call n(x) the smallest integer n such that K R (x, n) ∩ B = ∅ and n (x) the smallest integer n such that K R (x, n) reaches two distinct reﬂecting surfaces Σi and Σj , i.e., contains sites on either side of Σi and on either side of Σj . Theorem 4.1. Suppose that for all x ∈ Zd,R , n(x) < n (x), then, for any n n(x), there is a bijective map T from K(x, n) onto K R (x, n) which transforms bijectively all cells of {c} in K(x, n) onto the cells of {c}R in K R (x, n). Consequently, for any ∆ ⊂ K R (x, n) with also δ R (∆) ⊂ K R (x, n) σ

HhR,σ∆ (σ∆ ) = Hh T c

−1 (∆c )

(σT −1 (∆) ),

σ

−1

R,σ∆ T (∆ ) Zβ,h,∆ = Zβ,h,T −1 (∆) c

c

(4.4)

Proof. Since n < n (x), it is enough to consider a reﬂection w.r.t. a single surface and modulo a change of variables to work in the framework of Subsection 3.2. Suppose x is in the upper part, x ∈ Zd+ , then, by induction on k n it is easy to see that K R (x, k) = T (K(x, k)), where T is equal to the identity on K(x, n) ∩ Zd+ and to R on K(x, n) ∩ Zd− . We next check that T is one-to-one. If it was not the case, there would be two distinct sites y, z ∈ K(x, n) such that T (y) = T (z). This would mean that z = R(y) and, since K(x, n) is a convex set, then 0 would be in K(x, n), which is excluded because n n(x). Since T maps the cells of {c} in K(x, n) bijectively in those {c}R in K R (x, n), (4.4) follows. The theorem is proved.

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The previous theorem implies that away from the set B, the reﬂections have no impact on the energy. This will be useful to evaluate the corrections to the pressure in presence of reﬂected boundaries. The particular structure of the reﬂecting surfaces will not matter in the sequel, the analysis only relying on the following assumption: Assumption: For all x ∈ Zd,R , n(x) < n (x).

(4.5)

Having deﬁned the setup, we can now start the analysis which begins by recalling the fundamental notion of contours, adapted to the case of reﬂecting surfaces.

4.1

Contours

We will refer explicitly to the case of reﬂections, as underlined by the superscript R; without R the expressions refer to the case without reﬂections for which the classical proofs apply directly and which can anyway be recovered from our analysis by replacing {c}R by {c}. We deﬁne the phase indicator at x, ηxR (σ), as equal to 1 (resp. −1) if σ is identically 1 (resp. −1) on all c x, c ∈ {c}R ; otherwise ηx (σ) = 0. Calling R-connected two sites x and y if they both belong to a same cell in {c}R , the spatial supports sp(Γ) of the R contours Γ of σ are the maximal Rconnected components of the set {ηxR = 0}. We will tacitly suppose in the sequel that they are all bounded sets. Let ¯= Γ

K R (x, 2)

(4.6)

x∈sp(Γ)

¯ σΓ¯ ), with σΓ¯ the restriction of σ Then the R contours Γ of σ are the pairs Γ = (Γ, ¯ to Γ. ¯ sp(Γ), σx is identically Notice that in each R connected component of Γ\ ¯ equal either to 1 or to −1, while the values outside Γ are not determined by Γ and therefore can be arbitrary. Let ¯ \ sp(Γ) D := Γ

(4.7)

and call D0 and Di± the maximal R connected components of D. D0 is the one ¯ c , D+ (resp. D− ) are which is R connected to the unbounded component of Γ i i the components where σx (as speciﬁed by Γ) is equal to 1 (resp. −1). We also ± ¯c call int± i (Γ) the component of Γ which is R connected to Di . Finally Γ is a ± contour, if σ = ±1 on D0 . The R contours in a bounded domain Λ ⊂ Zd,R with + [−] boundary conditions are deﬁned as the contours of the conﬁguration (σΛ , 1Λc ) [resp. and of (σΛ , −1Λc )].

862

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The weight wR,+ (Γ) of a + R contour is w

R,+

R,− R n− e−βHh (σΓ¯ ) Zβ,h,int− i (Γ) (Γ) = . ¯ R,+ eβh|Γ| Z i=1 β,h,int− (Γ)

(4.8)

i

The superscript R recalls that all quantities are deﬁned using the collection of cells ¯ {c}R . The term eβh|Γ| in the denominator is the Gibbs factor of the conﬁguration ¯ ¯ e−βHh (1Γ¯ ) = eβh|Γ| . 1Γ¯ identically equal to 1 in Γ, R,− The weight w (Γ) of a − R contour is deﬁned symmetrically with the role of + and − interchanged. With these deﬁnitions, we have the identity R,± Zβ,h,Λ = e±βh|Λ| wR,± (Γi ) , (4.9) ± {Γi }± Λ {Γi }Λ

− where {Γi }+ Λ [{Γi }Λ ] is a compatible collection of + [−] R contours in Λ. Two contours are compatible iﬀ their spatial supports are not R-connected. For the case without reﬂections we can apply directly the classical PirogovSinai theory:

Theorem 4.2. There is c > 0 and, for any β large enough, h(β) ∈ (0, ce−β/2 ) so that the thermodynamic limits of µ± β,h(β),Λ deﬁne distinct DLR measures. Moreover, for any contour Γ, the weight without reﬂection satisfy 0 < w± (Γ) e−βNΓ /2 ,

(4.10)

where NΓ is the number of distinct cells which cover sp(Γ). In the following the bound (4.10) will be referred as a Peierls estimate since it leads −βNΓ /2 (4.11) µ± β,h(β),Λ (Γ) e The bound (4.10) is actually the crucial point of the theorem, the small weight of the contours is in fact responsible for the memory of the boundary conditions to survive the thermodynamic limit, thus yielding the phase transition. Moreover, as we will see, if β is large, (and the weight small), by cluster expansion, it is possible to exponentiate the right-hand side of [the analogue without reﬂections] of (4.9) and thus to compute the ﬁnite volume corrections to the pressure. This is on the other hand also the key point in the proof of (4.10), which at ﬁrst sight makes all the above to look circular. The main goal in this section is to prove the bound (4.10) in case of reﬂections.

4.2

Restricted ensembles

Following Zahradnik, we construct a much simpler, ﬁctitious model which, as a miracle, in the end, turns out to coincide with the real one. In the whole sequel β is large enough and h = h(β), see Theorem 4.2, will often drop from the notation.

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Inspired by (4.9), we set for any bounded region Λ ⊂ Zd,R , ±βh|Λ| ΞR,± w ˆR,± (Γi ) β,Λ = e {Γi }± Λ

w ˆ

R,+

(Γ) = min e

−βNΓR /2

(4.12)

{Γi }± Λ

R,− R n− e−βHh (σΓ¯ ) Ξβ,int− i (Γ) , ; ¯ R,+ eβh|Γ| i=1 Ξβ,int− (Γ) i

R,+ R n+ −βHh (σΓ ¯) Ξ β,int+ R,− −βNΓR /2 e i (Γ) , w ˆ (Γ) = min e ; −βh|Γ| ¯ R,− e i=1 Ξβ,int+ (Γ)

Γ a + R contour (4.13) Γ a − R contour

i

(4.14) where NΓR is the number of R cells in sp(Γ). In this way the weights automatically satisfy the crucial bound (4.10), but ﬁrst let us check that (4.12)–(4.14) do ˆR,± (Γ). Indeed, really deﬁne the “partition functions” ΞR,± β,Λ and the “weights” w the triple (4.12)–(4.14) should be regarded as an equation in the unknowns ΞR,± β,Λ and w ˆR,± (Γ). Existence and uniqueness are proved by induction on |Λ|. If |Λ| = 1, any contour in Λ has no interior, hence (4.13)–(4.14) specify its weight and conseR,± quently (4.12) determines ΞR,± β,Λ for such a Λ. If on the other hand we know Ξβ,Λ for all Λ ⊂ Zd,R , |Λ| n, we can use (4.13)–(4.14) to determine the weights of all Γ if all their interior parts have volume n; since regions Λ with |Λ| = n + 1 cannot contain contours whose interior part has volume > n, we can use (4.12) to determine ΞR,± β,Λ for such a Λ, and the induction step is proved. For β large enough the weights w ˆR,± (Γ) become so small, that the general theory of cluster expansion can be applied, see for instance [KP], so that ω ˆ R,± (π) , (4.15) log ΞR,± β,Λ = ±βh|Λ| + π∈ΠR,± Λ

where ΠR,+ [ΠR,− Λ Λ ] is the collection of all + [−] polymers π contained in Λ and R,± ω ˆ (·) their weights, all such notions being deﬁned next. Analogous expressions are valid in the absence of reﬂections. (the deﬁnition of − polymers is similar A + R polymer π = [Γεi i ] in ΠR,+ Λ and omitted), is an unordered, ﬁnite collection of distinct + R contours Γi taken with positive integer multiplicity εi , and such that, setting X(π) = sp(Γi ), π = [Γεi i ] (4.16) i

X(π) is a R connected subset of Λ. The weights ω ˆ R,± (π), π = [Γεi i ], are given in terms of the weights of contours, R,± w ˆ (Γ): ω ˆ R,± (π) = r(π) w ˆR,± (Γi )εi , (4.17) i

864

T. Bodineau and E. Presutti

where r(π) =

(εi !)−1

Ann. Henri Poincar´e

(−1)|G | ,

G ⊂G(π)

i

with G(π) the (abstract) graph of π, which consists of vertices, labelled by the i εi contours in π, and of edges, which join any two vertices labelled by contours with intersecting supports. By deﬁnition G(π) is connected and the sum in (4.18) is over all the connected subgraphs G of G(π) which contain all the i εi vertices; |G | denotes the number of edges in G . The number of connections of each site is not increased by the reﬂection procedure. Thus, for β large enough, [KP], the series on the right-hand side of (4.15) is absolutely convergent and, given any ﬁnite sequence Γ1 , . . . , Γn of contours,

|ˆ ω R,± (π)|

n

e−NΓi (β/2−2

d

α)

,

(4.18)

i=1

π∈ΠR,± ,πΓi ,i=1,...,n

where ΠR,± denotes the collection of all + [−] polymers in the whole space Zd,R and α > 0 is large enough, in particular we will also require that 22|D| e−α|D| < 1 , (4.19) D0

where the sum is over all R connected sets D in Zd,R which contain the origin (supposing 0 ∈ Zd,R ). D represents the spatial support of a contour and 2|D| bounds the number of contours with same spatial support D. The extra 2 in 22|D| is for convenience. The factor 2d in the last term of (4.18) enters via the relation (2d )ND ≥ |D|, ND the number of cells needed to cover D. Since by Theorem 4.2, the weights w± (Γ) satisfy the same bounds as the w ˆR,± (Γ), we have, analogously to (4.15), ± log Zβ,Λ = ±βh|Λ| +

ω ±(π) ,

(4.20)

π∈ΠR,± Λ

ˆR,± (Γ). As in (4.18), with ω ± (π) deﬁned by (4.17) having w± (Γ) in the place of w π∈Π,πΓi ,i=1,...,n

|ω ± (π)|

n

e−NΓi (β/2−2

d

α)

.

(4.21)

i=1

We will often use the following corollary of (4.19)–(4.21): Lemma 4.1. For any β large enough and any x ∈ Zd,R d+1 |ˆ ω R,± (π)| e−β/2+2 α X(π)x

(4.22)

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and, for any x and n,

|ˆ ω R,± (π)| e−(β/2−2

d+1

α)n

(4.23)

X(π)x,X(π)∩K R (x,n)c =∅

Both (4.22) and (4.23) remain valid in the case without reﬂections. Proof. By (4.18),

|ˆ ω R,± (π)|

e−NΓ (β/2−2

d

α)

Γ:sp(Γ)0

X(π)0

e−(β/2−2

d+1

α)

e−α|sp(Γ)| ,

Γ:sp(Γ)0

where we used that |sp(Γ)| = 2d NΓ . Applying (4.19), we obtain (4.22). To prove (4.23), we denote by {Γ1 , . . . , Γk } any sequence of contours such that sp(Γ1 ) x, sp(Γk )∩K c = ∅, K ≡ K R (x, n), and sp(Γi ) ∼ sp(Γi+1 ), i = 1, . . . , k−1, (where A ∼ B shorthands that A is R connected to B). Then the left-hand side of (4.23) is bounded by

|ˆ ω

R,±

(π)|

k,{Γ1 ,...,Γk } π:Γi ∈π,i=1,...,k

k

e−NΓi (β/2−2

d

α)

k,{Γ1 ,...,Γk } i=1

e−(β/2−2

d+1

α)n

k

k

D1 x,Dj ∼Dj+1 ,j=1,...,k−1

i=1

2|Di | e−α|Di |

which proves (4.23) because, as we are going to see, the sum over k, that we denote by S(x), is bounded by 1. Calling SN (x) the sum with k ≤ N , since S(x) is the limit as N → ∞ of SN (x), it suﬃces to prove that for all y and N , SN (y) 1. The proof is by induction on N . S1 (y) < 1 by (4.19). Suppose SN −1 (x) 1 for all x, then 1 + SN −1 (y) 2|D1 | e−α|D1 | 2|D1 | 2|D1 | e−α|D1 | SN (x) D1 x

y∈D1

D1 x

the second factor 2|D1 | coming from the induction hypothesis. Then, by (4.19), SN (x) 1 for any x and (4.23) is proved. The lemma is proved. By the analogue of (4.22) we conclude convergence of the series on the righthand side of ω ± (π) 1 (4.24) P± := ±h + β |X(π)| π∈Π(±),X(π)0

To study the weights of the polymers obtained by reﬂection we will use the following three lemmas, where Λ is tacitly supposed to be a bounded region in Zd .

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They refer to the case without reﬂections and since the magnetic ﬁeld h is equal to h(β) they are part of the classical Pirogov-Sinai Theory. For convenience, we give an explicit proof, consequence of Theorem 4.2. Lemma 4.2. For β large enough, P+ = P− = P

(4.25)

where P is the thermodynamic pressure at inverse temperature β and magnetic ﬁeld h = h(β). Moreover, ± log Zβ,Λ

=

±βh|Λ| +

x∈Λ x∈X(π)⊂Λ

=

β|Λ|P −

X(π)∩Λc =∅

ω ± (π) |X(π)|

|X(π) ∩ Λ| ± ω (π) |X(π)|

(4.26) (4.27)

Proof. (4.26) is just a rewriting of (4.20); (4.25) follows from (4.26) by taking the thermodynamic limit and using Lemma 4.1. (4.27) is also a rewriting of (4.26). The lemma is proved. Lemma 4.3. For β large enough, and calling δ(Λ) the union of all sites in Λc which are ∗-connected to Λ, d+1 ± − β|Λ|P e−β/2+2 α |δ(Λ)| (4.28) log Zβ,Λ Proof. By (4.27) ± − β|Λ|P log Zβ,Λ

|ω ± (π)|

x∈δ(Λ) X(π)x

which, by (4.22), yields (4.28). The lemma is proved. The ﬁnal lemma proves that the bound (4.10) was too conservative. Lemma 4.4. There is a constant c so that, for β large enough, d+1 + − − log Zβ,Λ log Zβ,Λ 2e−β/2+2 α |δ(Λ)| w± (Γ) ≤ exp

− βNΓ 1 − ce−β/2

(4.29) (4.30)

Proof. (4.29) follows directly from (4.28). By the analogue of (4.8) without reﬂections, ¯ + 2e−β/2+2d+1 α |δ(int− w± (Γ) ≤ exp − βNΓ + 2β|h||Γ| i (Γ))| i

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Notice also that the constraint on h(β) can be easily recovered. By equating the two right-hand side of (4.24) and then using (4.22) in the version without reﬂections, we get d+1 β|h| e−β/2+2 α (4.31) Moreover, if x ∈ δ(int− i (Γ)) there is a cell c such that c ∩ K(x, 2) = ∅, c ∩ sp(Γ) = ∅, so that to each x ∈ ∪i δ(int− i (Γ)) we can associate a cell contributing to NΓ , in such a way that the same cell is counted at most |K(0, 3)| times. Thus d+1 ¯ + 2e−β/2+2d+1 α |K(0, 3)|NΓ (4.32) w± (Γ) ≤ exp − βNΓ + 2e−β/2+2 α |Γ| The inequality ¯ ≤ |sp(Γ)||K(2, 0)| ≤ NΓ 2d |K(2, 0)| |Γ| concludes the proof of the lemma.

We turn now back to the main goal of the section, namely to prove that the bound (4.10) holds also for the weights with reﬂections. The proof is obtained in two steps. Theorem 4.3. For any β large enough the following holds. Let x ∈ Zd,R and R,± n n(x); then if Λ ∪ ∂ R (Λ) ⊂ K R (x, n), ΞR,± β,Λ = Zβ,Λ and if Γ is a ±, R ¯ ⊂ K R (x, n), then w contour with Γ ˆ R,± (Γ) = wR,± (Γ) < e−βNΓ /2 . Proof. Under the assumption that n n(x), Theorem 4.1 applies and therefore the proof will follow from the previous results on the weights without reﬂection and from the one-to-one correspondence between K(x, n) and K R (x, n). In particular (4.4) implies that for domains strictly contained in K R (x, n) R,± ± Zβ,T −1 (Λ) = Zβ,Λ

(4.33)

In the case |Λ| = 1, any contour in Λ has no interior and (4.12)–(4.13) allow to compute w ˆR,± (Γ), getting, as in the proof of Lemma 4.4, ¯ w ˆR,± (Γ) exp − βNΓ + 2β|h||Γ| hence, for β large enough, w ˆR,± (Γ) = w± (Γ) < e−βNΓ /2 . Suppose by induction R,± ± that for any |Λ| k (Λ as in the text of the theorem), ΞR,± β,Λ = Zβ,T −1 (Λ) = Zβ,Λ . Then if Γ is as in the text of the theorem and moreover all its interior parts have volume k, then the second term on the right-hand side of (4.12)–(4.13) is equal to w± (T −1 Γ), with the obvious meaning of the notation, which by Lemma 4.4 is, for β large enough, < e−βNΓ /2 . Then the second term on the right-hand side of (4.12)–(4.13) is smaller than the ﬁrst one, hence w ˆ R,± (Γ) = wR,± (Γ). Since all contours inside Λ have interior parts with volume k, (4.10) shows ±,R ± that ΞR,± β,Λ = Zβ,Λ = Zβ,T −1 (Λ) , thus proving the induction step. The theorem is proved.

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Before extending the result to general Λ, we state and prove the following lemma. Lemma 4.5. For β large enough, log ΞR,± β,Λ = β|Λ|P +

X(π)∩K R (x;n(x))c =∅;x∈X(π)⊂Λ

x∈Λ

−

X(π)∩K(0;n(x))c =∅;0∈X(π)

ω ˆ R,± (π) |X(π)|

ω ± (π) . (4.34) |X(π)|

Proof. We write log ΞR,± β,Λ

= ±βh|Λ| +

x∈X(π)⊂K R (x;n(x))

x∈Λ

+

X(π)∩K R (x;n(x))c =∅;x∈X(π)⊂Λ

β|Λ|P

=

±βh|Λ| +

i∈Λ

ω R,± (π) |X(π)|

X(π)⊂K(x;n(x));x∈X(π)

+

ω R,± (π) |X(π)|

X(π)∩K(x;n(x))c =∅;x∈X(π)

ω ± (π) |X(π)|

ω ± (π) |X(π)|

Then log ΞR,± β,Λ − β|Λ|P is equal to the diﬀerence of the right-hand side of the last two equations. The ﬁrst terms in the sum over x cancel with each other, see the proof of Theorem 4.1, and (4.34) follows after recalling that the weights without reﬂections are translational invariant. The theorem is proved. R,± Theorem 4.4. For any β large enough, for any bounded Λ ⊂ Zd,R , ΞR,± β,Λ = Zβ,Λ and for any bounded, ±, R contour Γ, w ˆ R,± (Γ) = wR,± (Γ) < e−βNΓ /2 .

Proof. By (4.34) and (4.22), denoting by n(x, y) the maximal integer such that y∈ / K R (x; n(x, y)), d+1 R,± e−[β/2−2 α] n(x) log Ξβ,Λ − β|Λ|P 2 x∈Λ

2 2

x∈Λ

y∈δ R (Λ)

y∈δ R (Λ)

n≥1

e−[β/2−2

d+1

e−(β/2−2

α] n(x,y)

d+1

α)n

(2n + 1)d

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which yields

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d+1 R,± log Ξβ,Λ − β|Λ|P cα e−β/2+2 α δ R (Λ) ,

with cα = 2eα

e−αn (2n + 1)d .

(4.35) (4.36)

n≥1

An inductive argument as in the proof of Theorem 4.3 completes then the proof of the theorem. Before ending this section, we collect some estimates used in the next sections. Lemma 4.6. Given any positive integer n, ± − {±βh|Λ| + log Zβ,Λ x∈Λ

x∈X(π)⊂K(x;n)∩Λ

R,± log Zβ,Λ − {±βh(β)|Λ| + x∈Λ

d ω ± (π) } |Λ|e−(β/2−2 α)n (4.37) |X(π)|

x∈X(π)⊂K R (x,n)∩Λ

d ω R,± (π) } |Λ|e−(β/2−2 α)n |X(π)|

(4.38) Proof. (4.37) and (4.38) follow from (4.23) and its analogue without reﬂections. This lemma will enable us to estimate the corrections to the pressure. Let us also examine two other consequences which will be crucial in the rest of the paper. The ﬁrst consequence justiﬁes the notion of perfect walls introduced in Subsection 3.2. We consider the slab ΛL,ε and the reﬂection w.r.t. to the hyperplane Σ = {x ∈ Rd , (n · x) = 0} which splits ΛL,ε into two non-interacting domains. Lemma 4.7. There exists c > 0 such that d R,+ + − log ZL,ε log ZL,ε ce−(β/2−2 α) Ld−2 ,

(4.39)

+ denotes the partition function on ΛL,εL with + boundary conditions where ZL,ε R,+ and ZL,ε is the partition function obtained by reﬂection (see Subsection 3.2). The same statement holds with − boundary conditions.

Proof. Let B = {−1, 0, 1}d. For any x in ΛL,ε , we set n ¯ (x) = min{n, Then +,R + − log ZL,ε log ZL,ε

K(x, n) ∩ B = ∅, K(x, n) ∩ ΛcL,ε = ∅} x∈ΛL,εL

X(π)∩K R (x;¯ n(x))c =∅;x∈X(π)⊂ΛL,εL

+

X(π)∩K(x;¯ n(x))c =∅;x∈X(π)

ω R,+ (π) |X(π)| ω + (π) . |X(π)|

870

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The contribution of the polymers with X(π) in K(x, n) and the reﬂected ones in X(π) in K R (x, n) with n n ¯ (x) cancel with each other by Theorem 4.1. Since the weights of the polymers are exponentially small (see Lemma 4.1), the result follows. The second consequence will be used in Section 6. Let TN be the torus {−N, . . . , N }d and we consider a collection of reﬂections inside TN for which the assumption (4.5) is satisﬁed. Let B denote the boundary conditions imposed by the reﬂections, i.e., the centers and the boundaries of each reﬂecting surfaces. We have d R − log Zβ,N cα e−(β/2−2 α) |B| , (4.40) log Zβ,N R where Zβ,N (resp. Zβ,N ) denotes the partition functions in TN with periodic boundary conditions and with (resp. without) reﬂection.

5 Properties of the surface tension In the following, β is ﬁxed large enough such that the results of Section 4 are satisﬁed and h refers to h(β). We ﬁrst derive the existence of the thermodynamic limit for the surface tension and then its convexity and positivity.

5.1

Proof of Theorem 3.1

The proof can be split into three steps. First, we are going to prove that the choice of the barriers (C + , C − ) has almost no contribution on the ratio of the partition functions. Then, an inductive procedure enables us to improve (3.8) and to derive the convergence (3.9). Step 1: The ﬁrst step is to prove that +

τβ (n) =

−

C ,C ZL,ε (S + , S − ) (n · ed ) lim inf lim inf sup − log . + − ε→0 L→∞ C + ,C − βLd−1 Z C ,R Z C ,R L,ε

(5.1)

L,ε

This boils down to check that there are constants (C1 , C2 ) such that for any (L, ε) and for any (C + , C − ) and (C˜+ , C˜− ) C˜+ ,C˜− C + ,C − ZL,ε ZL,ε (S + , S − ) (S + , S − ) log C1 Ld exp(−C2 εL) . − log (5.2) C + ,R C − ,R C˜+ ,R C˜− ,R ZL,ε ZL,ε ZL,ε ZL,ε The events S + , S − decouple the interface from the boundary eﬀects thus (5.2) can be derived by using only estimates in a pure phase.

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It is enough to consider C˜− = C − . In this case, (5.2) becomes C + ,C − C + ,R ZL,ε ZL,ε (S + , S − ) log − log ˜+ C1 Ld exp(−C2 εL) . + − ˜ C ,C C ,R ZL,ε (S + , S − ) ZL,ε

(5.3)

For any spin conﬁguration in S + , let us denote by s+ the support of the + barrier in ΛL, 10ε L + εL ed which is the closest to the hyperplane Σ = {x; (x · n) = 2 0}. This particular choice of s+ will be stressed by the notation s+ S + . The constrained partition function can be decomposed as follows + C + ,C − C + ,s+ s+ ,C − + (S + , S − ) = eβh|s | ZL,ε ZL,ε (s S + , S − ) , ZL,ε s+

with the ﬁrst partition function free of constraints so that cluster expansion applies and the second partition function which takes into account the constraint that εL ε ed below s+ . there is no + barrier in ΛL, 10 L + 2 We ﬁrst write +

−

C ,C ZL,ε (S + , S − ) ˜+

−

C ,C ZL,ε (S + , S − )

+

=

˜+

+

C ,s ZL,ε ˜+

e

+

C ,s ZL,ε

s+

βh|s+ |

+

+

−

˜+

−

C ,s s ,C ZL,ε ZL,ε (s+ S + , S − ) C ,C ZL,ε (S + , S − )

.

(5.4)

Let N = εL/10, (suppose, for notational simplicity, N an integer), then +

d −(β/2−2α)N

exp{−4L e

}

+

C ,s ZL,ε

C˜+ ,s+ ZL,ε

˜+

C ,R ZL,ε C + ,R ZL,ε

exp{4Lde−(β/2−2α)N }

(5.5)

follows from crossed cancellations among the terms in the numerator and denominator. We are going to apply the expansion of the partition function derived in Lemma 4.6 with n = N . The factor 4 is because there are 4 partition functions involved. With reference to (4.37) and (4.38), the contribution of x such that the C + ,s+ C + ,R and ZL,ε cancel with each scalar product (x · n) 8εL/10 coming from ZL,ε ˜+

+

C ,s other, as well as those from ZL,ε

˜+

C ,R and ZL,ε . Symmetrically, the contribution +

+

˜+

+

+

−

C ,s C ,s and ZL,ε cancel with each of x such that (x · n) < 8εL/10 arising from ZL,ε ˜+

+

C ,R C ,R other, as well as those from ZL,ε and ZL,ε .

Finally, by applying (5.5), we get from (5.4): +

−

˜+

−

C ,C ZL,ε (S + , S − ) C ,C ZL,ε (S + , S − ) ˜+

+

C ,R ZL,ε ˜+

C ,R ZL,ε

e

4Ld e−(β/2−2α)N

s+

+

C ,R ZL,ε

C˜+ ,R ZL,ε

e4L

d −(β/2−2α)N

e

+

+

C ,s s ,C ZL,ε eβh|s | ZL,ε (s+ S + , S − ) ˜+

−

C ,C ZL,ε (S + , S − )

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Ann. Henri Poincar´e

In the same way we get +

−

C ,C (S + , S − ) ZL,ε C˜+ ,C − ZL,ε (S + , S − )

+

C ,R ZL,ε C˜+ ,R ZL,ε

e−4L

d −(β/2−2α)N

e

.

Recalling that N = εL/10, we have thus completed the proof of (5.2). Step 2: The goal is to derive a lower bound for +

+

−

φ(L, ε, C , C ) =

−

C ,C ZL,ε (S) +

−

C ,R C ,R ZL,ε ZL,ε

in terms of τβ (n). For simplicity S = (S + , S − ). The previous step (see (5.1)) implies that there exists a sequence (εk , Lk )k 0 such that (n · e ) 1 d + − (5.6) inf log φ(L , ε , C , C ) + τ ( n ) . k k β + − βLd−1 k C ,C k We ﬁx (εk , Lk ) and consider a pair (ε, L) such that εk Lk εL and Lk L. In order to derive a lower bound on φ(L, ε, C + , C − ), we are going to localize the interface in the slab ΛL,εk Lk . We set Λ0k = ΛLk ,εk Lk , the upper-script 0 is to distinguish it from its translates (which will be introduced below). We call Ck0 = (Ck+,0 , Ck−,0 ) and Sk0 = (Sk+,0 , Sk−,0 ) the set of all spin conﬁgurations which have ± barriers as required from the deﬁnition of the surface tension. The “maximal barriers” are denoted by ±,0 is the ﬁrst barrier coming from the top [resp. the bottom] c±,0 k , meaning that ck ±,0 0 of Λk . We also write ck Ck±,0 for the event where c±,0 are the maximal barriers k in Ck±,0 . We ﬁnally call Uk±,0 the union of all sites outside Λ0k and at distance 1 from its faces parallel to n; The ± labels distinguish those where the b.c. in the deﬁnition of the surface tension are set equal to ±1. Let (Λik ) be those translates of Λ0k which are contained in ΛL,εL , where ∀i = (i1 , . . . , id );

 n j (Lk + 2) ij + ξi  , Λik = Λ0k + (Lk + 2)i1 , . . . , (Lk + 2)id−1 , − n d j=1 

d−1

with ξi ∈ [0, 1) chosen such that Λik ⊂ Zd . The same translation which carries Λ0k onto Λik is used to deﬁne Cki = (Ck+,i , Ck−,i ), Ski = (Sk+,i , Sk−,i ), c±,i Ck±,i , k ±,i Uk as translates of the corresponding quantities with i = 0. Notice that the

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distance between two distinct Λik and Λjk is always larger than the range of the interaction and indeed two distinct Uk±,i have at most their external surfaces in common. We denote by Uk+ the union of all Uk+,i with the addition of the regions Λik ∩ ΛL,εL ∩ {(x · n) 0}, when i ranges over all values such that Λik is not contained in ΛL,εL . Uk− is deﬁned analogously and Uk = Uk+ ∪ Uk− . The volume of Uk is bounded (for L so large that (Lk + 2)2 < L) by |Uk | {(Lk + 2)d−2 2}εk Lk

Ld−1 + Ld−2 (Lk + 2)εk Lk 4εk Ld−1 . (5.7) (Lk + 2)d−1

The ﬁrst term bounds the contribution of all i where Λik ⊂ ΛL,εL , the second term the remaining ones; the ﬁnal estimate uses that (Lk + 2)2 < L. ∆({c+,i k })

C+ εk L k

Uk+,0

c+,i k

Uk−,0

c−,i k ∆({c−,i k })

C−

Uk = Uk+ ∪ Uk− Figure 3. Decomposition at the scale Lk of the domain Λ(C + , C − ) by means of the subsets (Λik )i (depicted by dashed boxes).

Let Qk be the intersection of the events C i = (Ck+,i , Ck−,i ), Ski = (Sk+,i , Sk−,i ) over all i such that Λik ⊂ ΛL,εL . Call c±,i the maximal barriers realizing the event k Ck±,i (maximal in the sense described previously). In order to decouple the events in the diﬀerent regions (Λik ), we ﬁx the spin conﬁgurations in Uk as equal to 1± Uk , ± where the latter is the conﬁguration where the spins are equal to ±1 on Uk , we −,i i call Q k such a further constraint. On Qk we set Λ(c+,i k , ck ) as the region in Λk +,i which goes from the maximal top barrier ck down to the maximal bottom barrier c−,i (both included), and set k ∆({c±,i k })

+

−

= Λ(C , C )

\

"

−,i Λ(c+,i k , ck )

# Uk

,

i

∆+ ({c+,i k }) =

∆− ({c−,i k }) =

∆({c±,i n) 0}, k }) ∩ {x; (x ·

∆({c±,i n) < 0} . k }) ∩ {x; (x ·

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Ann. Henri Poincar´e

Imposing the constraint Qk , Q k , and decomposing the partition function with respect to (Ck+,i , Ck−,i ), we get $ $ C + ,C − C + ,C − S ZL,ε S Qk (5.8) ZL,ε Q k ± + − −βHh (1U ) C +,i C − −,i k Z S + , c+,i Z∆ e , ck Ck−,i = − ({c−,i }) S k Ck ∆+ ({c+,i }) k

−,i (c+,i k ,ck )

k

+,i −,i βh(|c+,i |−|c−,i |) ck ,ck i ZLk ,εk (Sk ) . × e i

By introducing the partition functions in each Λik with reﬂected boundary conditions at the scale Lk , we will recover an approximation of the surface tension. c+,i ,c−,i

For each factor ZLkk ,εk k (Sk ) in the last product, we write (see (5.6)) c+,i ,c−,i ZLkk ,εk k (Sk )

c+,i ,R(k) c−,i ,R(k) ZLkk ,εk ZLkk ,εk

"

# β Ld−1 k exp − , τβ (n) + 1/k (n · ed )

we are using the notation of Subsection 3.2 with R(k) instead of R to underline c±,i ,R(k)

take into account the multiple reﬂections at that the partition functions ZLkk ,εk the scale Lk (see Figure 2). By taking the product over all i, we get c+,i ,c−,i c+,i β Ld−1 ,R(k) c−,i ,R(k) k k τβ (n) + 1/k ZLk ,εk (Sk ) exp − ZLkk ,εk ZLkk ,εk . (n · ed ) i i We are going to plug the previous inequality in (5.8) in order to reconstruct two partition functions on the domains ± ±,i ∆± ({c±,i . ∆± = , R(k)) ∪ c±,i k }k ) ∪ Λ (c k i

Notice that the sets ∆± are slightly diﬀerent from Λ± (C ± ) since they are built according to the rules of the reﬂection at the scale Lk . We ﬁnally obtain +

C ,C ZL,ε

−

$ $ S Qk Q k

C + ,R(k) +

Z∆ +

C − ,R(k) − −,i −βHh (1± ) Uk S , Ck+,i Z∆− S , Ck e β Ld−1 τβ (n) + 1/k exp − , (n · ed )

C + ,R(k) + S , Ck+,i denotes the partition function on ∆+ with a perfect wall where Z∆+ made of multiple reﬂections on the scale Lk and taking into account the occurrence of the barriers S + and {Ck+,i }i . By (5.7) d−1 |Hh (1± Uk )| c|Uk | c4εk L

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so that, it only remains to check that lim lim

inf

k→∞ L→∞ C + ,C −

C + ,R(k) +

1 Ld−1

log

C − ,R(k) − −,i S , Ck+,i Z∆− S , Ck

Z∆ +

+

−

C ,R C ,R ZL,ε ZL,ε

0,

(5.9)

because, if we suppose that the previous inequality holds, then lim inf L→∞

(n · ed ) inf log φ(L, ε, C + , C − ) − τβ (n) , βLd−1 C + ,C −

which completes the Theorem 3.1. Step 3: The ﬁnal step is devoted to the derivation of (5.9). This amounts to prove that the corrections to the pressure for the diﬀerent types of reﬂected boundary conditions are negligible. C + ,R(k) + S , Ck+,i is First, we check that the constrained partition function Z∆+ C + ,R(k)

asymptotically equivalent to the non-constrained partition function Z∆+ +,R(k) be the corresponding Gibbs measure. Then the following holds µ∆+

. Let

+,R(k) +

µ∆+

N S , Ck+,i 1 − Ld−1 exp(−cεL) 1 − Ld−1 exp(−cεk Lk ) k . (5.10) k

This can be derived as follows. The occurrence of a barrier with blocks uniformly labelled by 1 in the slab Λl,m implies that there is no connected set of blocks la+,R(k) belled by −1 joining the two faces of Λl,m orthogonal to n. Under µ∆+ , a Peierls estimate similar to (4.11) (see Theorem 4.4). A Peierls type argument implies then that a connected set of − blocks with length at least m has a probability smaller than exp(− β2 m). Applying recursively the Peierls argument, we derive (5.10). By hypothesis on the sequence (εk , Lk ), for k large enough (5.10) implies d−1 +,R(k) + S , Ck+,i 2−1−L exp(−cεk Lk ) . µ∆+ Therefore lim lim

k→∞ L→∞

1 Ld−1

S , (Ck+,i )

C + ,R(k) +

inf log

Z∆ +

C+

C + ,R(k)

ZL,ε

0.

(5.11)

= 0.

(5.12)

This reduces the proof of (5.9) to lim lim

k→∞ L→∞

1 Ld−1

C + ,R(k)

inf

C + ,C −

log

Z∆ +

+

C − ,R(k)

Z∆ − −

C ,R C ,R ZL,ε ZL,ε

Again this estimate will follow from cross cancellations between the 4 partition functions.

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Ann. Henri Poincar´e

Following the strategy of step 1, the bulk contribution and the correction to the pressure from the boundary terms C + , C − can be estimated by Lemma 4.6; they are of the order Ld exp 2Lde−(β/2−2α)εL/10

.

Thus it is enough to check that the contribution of the perfect walls involved in each partition function will be negligible w.r.t. to the surface order. We ﬁrst C + ,R C − ,R , ZL,ε . By an analogous argument of the consider the partition functions ZL,ε one used to derive (4.39), we see that the corrections to the pressure induced by the perfect wall Σ are of the order Ld−2 . We consider now the partition functions with multiple reﬂections. The perfect wall associated to the box Λik is denoted Σi and to each reﬂection corresponds a particular set Bi of boundary conditions. The set Bi comprises the sites around the center of reﬂection in Σi as well as the sites outside Λik which are connected to Σi . The union of the Bi is denoted by B (see Figure 2). In order to use the estimate of Section 4, we should ﬁrst check that the assumption (4.5) holds for the multiple reﬂections at the scale Lk . Suppose that for some x, n (x) n(x). Following the proof of Theorem 4.1 there exists a bijective map T such that K R (x, n) = T (K(x, n)) for any n < n (x). Thus K R (x, n (x)− 1) contains only sites in K(x, n (x) − 1) or in the reﬂection of K(x, n (x) − 1) w.r.t. one perfect wall. By construction K R (x, n (x)) is obtained by adding all the cells connected to K R (x, n (x) − 1), so that it is impossible that K R (x, n (x)) contains sites in two distinct perfect walls Σi and Σj without intersecting the boundaries of Σi and Σj which are included in B. This shows that n(x) < n (x) and that assumption (4.5) is satisﬁed. d−1 C + ,R(k) C − ,R(k) or Z∆− there are LLk reﬂections In each partition function Z∆+ at the scale Lk . Each reﬂection leads to corrections of the order Ld−2 and overall k d−1 we get an eﬀect of the order LLk . As k diverges this leads to vanishingly small contributions w.r.t. the surface order Ld−1 . Combining the previous estimates, we conclude (5.12).

5.2

Properties

We are going to establish some basic properties of the surface tension Proposition 5.1. For any β large enough such that the model is in the Pirogov Sinai regime inf τβ (n) > 0.

n∈Sd−1

The positivity of the surface tension deﬁned in (3.11) was already derived in [BKL] (nevertheless the existence of the thermodynamic limit was an assumption in [BKL]).

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The homogeneous extension on Rd of the surface tension is deﬁned by x d ∀x ∈ R , τβ (x) = x2 τβ , τβ (0) = 0 . x2 Proposition 5.2. The surface tension τβ is convex on Rd . As a consequence [Am], the functional Wβ is lower semi-continuous. The deﬁnition (3.8) of the surface tension in the direction n relies on the arbitrary choice of the orientation of the slab along one of the axis (see Section 3). Nevertheless, since τβ is convex, it is also continuous and therefore the value of the surface tension is independent of the arbitrary choices in the deﬁnition. Proof of Proposition 5.1. According to Theorem 3.1, it is enough to prove that there is cβ > 0 such that uniformly over n the following holds +

1 ∀L > 0, ∀ε > , L

inf log

C + ,C −

−

C ,C ZL,ε (S + , S − ) C + ,R ZL,ε

C − ,R ZL,ε

− cβ Ld−1 .

(5.13)

At this stage the constraint (S + , S − ) plays no role and can be dropped. Furthermore, it is enough to select the most simple barriers C + , C − and to derive ∀L > 0, ∀ε >

1 , L

log

+,− ZL,ε

+,R −,R ZL,ε ZL,ε

− cβ Ld−1 ,

(5.14)

+,− denotes the partition function with mixed boundary conditions in the where ZL,ε √ domain ΛL,εL . For simplicity we suppose that nd = (n · ed ) 1/ d. As explained after the heuristic expansion (3.10), the precise derivation of the + surface tension requires to compensate precisely the boundary surface tensions τbd − and τbd appearing in the numerator and the denominator. For (5.13), only a crude + − bound on τbd and τbd is necessary. More precisely, by (4.38), there is C1 > 0 such that C1 d−1 −β/2 +,R −,R L e . (5.15) log ZL,ε + log ZL,ε − βP|ΛL,εL | nd +,− have Due to the mixed ± b.c. the spin conﬁgurations which contribute to ZL,ε necessarily an “open” contour Γ whose spatial support, sp(Γ), ∗-disconnects the ¯ see Subsection 4.1 for deﬁnitop and bottom faces of ΛL,εL . The complement of Γ, tions, is made by a ﬁnite number of regions, say ∆1 , . . . , ∆n , with their boundaries, δ∆i (i.e., all cells in ∆ci , ∗-connected to ∆i ) where the spins have a constant sign, denoted by ξi . Then n +,− ξi = e−βHh (σΓ¯ ) Z∆ . ZL,ε i Γ

i=1

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Ann. Henri Poincar´e

By (4.28), we get +,− ZL,ε eβ|ΛL,εL |P

¯

e−βHh (σΓ¯ )+β|Γ|P

Γ

n

−β/2+2α

ee

Nδ∆i

.

i=1

In the last product we use the inequality n

Nδ∆i 3d NΓ

i=1

(as each cell in δ∆i is ∗-connected to a cell of sp(Γ) and the correspondence is at most 3d to 1). Moreover, by the deﬁnition of contours and using the fact that h d+1 belongs to (0, e−β/2+2 α ) (see (4.31)) ¯ − βNΓ − β[1 − e−β/2+2 −βHh (σΓ¯ ) β|h||Γ|

d+1

α d

6 ]NΓ .

The previous estimate implies that for β large enough, +,− ZL,ε eβ|ΛL,εL |P Γ

d+1 d+1 d+1 exp − β[1 − e−β/2+2 α 6d ] − 6d 2e−β/2+2 α − 3d e−β/2+2 α NΓ d e−β/2ND 23 |D| , eβ|ΛL,εL |P

Dx∗ ,ND 2−d Ld−1

where the sum is over all connected sets D of cells (D standing for sp(Γ)) which contain x∗ a point of ΛL,εL ∗-connected to the surface which separates the + and d − boundary conditions; 23 |D| counts the number of contours with given spatial support. This leads to −d

+,− eβ|ΛL,εL |P e−(β/2−α)2 ZL,ε

Ld−1

.

(5.16)

Inequalities (5.15) and (5.16) imply +,− ZL,ε

exp

C1 −β/2 β . − Ld−1 2−d ( − α) − e 2 nd

+,R −,R ZL,ε ZL,ε √ Since nd 1/ d, for β large enough (5.14) holds.

Proof of Proposition 5.2. The convexity is equivalent to the pyramidal inequality (see, e.g., [MMR]). To any collection of unit vectors (n1 , . . . , nd+1 ), one associates a pyramid ∆(n1 , . . . , nd+1 ) with faces (Fi )i orthogonal to (ni )i . Let |Fi | be the area of Fi . Then the pyramidal inequality means that |F1 | τβ (n1 )

d+1 i=2

|Fi | τβ (ni ) .

(5.17)

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The derivation of the pyramidal inequality follows closely the approximation scheme explained in the second step of the proof of Theorem 3.1. For a given (L, ε), instead of approximating the surface tension in the slab ΛL,εL (n1 ) by localizing the interface in the smaller slabs ΛLk ,εk Lk (n1 ), the interface is constrained to follow a more complicated periodic pattern. More precisely, the hyperplan orthogonal to n1 and going through 0, is paved () by unit (d − 1)-dimensional cubes denoted by (C () ) . For any , let F1 be a translate of F1 rescaled appropriately to ﬁt in the cube C () . The corresponding pyramid is denoted by ∆() . In this way, a periodic structure is created Q=

() C () ∪ ∆() \ F1 .

The interface will be forced to cross ΛL,εL (n1 ) by following the periodic pattern N Q, where N = ε2 L. This is done by decomposing each ﬂat region of N Q orthogonal to ni into slabs ΛLk ,εk Lk (ni ), with Lk N . The interface is allowed to ﬂuctuate inside each slab, thus an approximation of the surface tension in each direction ni can be recovered. Since the portion of the interface outside the slabs is small w.r.t. the surface order, its contribution is negligible and we obtain Ld−1 τβ (n1 ) nd

d+1 () () |Fi | |C () \ F1 | τβ (n1 ) + τβ (ni ) . nd nd i=2

(5.18)

Thus inequality (5.18) follows.

6 Wulﬀ construction In this section, β is ﬁxed large enough such that the results of Section 4 on the phase transition regime hold. The Gibbs measure with magnetic ﬁeld h(β) and periodic boundary conditions on TN is denoted by µβ,N .

6.1

Coarse graining

A key step in the analysis of the equilibrium crystal shapes is to extract a precise information from the L1 -estimates by means of a coarse graining. For this purpose, we adapt in our context a coarse graining which was introduced in [B2]. The typical spin conﬁgurations are deﬁned at the mesoscopic scale K = 2k . Let ∂BK = BK+K α \ BK be the enlarged external boundary of the box BK , where α is in (0, 1). The parameter ζ > 0 will control the accuracy of the coarse graining. Let x be in TN and denote by BK (x) the corresponding B (K) -measurable box. For any ε = ±1, the box BK (x) is ε-good if the spin conﬁguration inside the enlarged box BK+K α (x) is typical, i.e.,

880

T. Bodineau and E. Presutti

Ann. Henri Poincar´e

(P1) The box BK (x) is surrounded by at least a connected surface of cells in ∂BK (x) with η-labels uniformly equal to ε. (P2) The average magnetization inside BK (x) is close to the equilibrium value mεβ of the corresponding pure phase MK (x) − mεβ ζ

and MK (x) =

1 (2K + 1)d

σi .

i∈BK (x)

See Figure 4.

K |MK − m+ β| ζ

d2

+ Kα Figure 4. Coarse grained conﬁguration with overlapping + good blocks.

On the mesoscopic level, each B (K) -measurable box BK (x) is labelled by a mesoscopic phase label if BK (x) is ε-good , mεβ , ζ uK (x) = ∀x ∈ TN , 0, otherwise. For large mesoscopic boxes, the typical spin conﬁgurations occur with overwhelming probability. Theorem 6.1. Then for any ζ > 0, the following holds uniformly over N ∀{x1 , . . . , x }, µβ,N uζK (x1 ) = 0, . . . , uζK (x ) = 0 ρζK ,

(6.1)

where the parameter ρζK vanishes as K goes to inﬁnity. Despite the fact that the mesoscopic phase labels are not independent, the theorem above ensures that the occurrence of the bad-blocks is dominated by a Bernoulli measure. For the sake of completeness, the proof of Theorem 6.1 is recalled in the Appendix.

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As in (2.6), the macroscopic counterpart of the phase labels is deﬁned by −1 . uζN,K (x) = uζK ψN (x) , x∈T N,K (x). The images of B (K) boxes by ψN are denoted by B 1 Any discrepancy in the L -norm between the coarse graining and the local order parameter can be neglected with superexponential probability. By construc either |MN,K (x) − uζ (x)| is smaller than ζ or the block tion, for any x ∈ T N,K N,K (x) has label uζ (x) = 0. Using the domination by Bernoulli percolation, B N,K the following holds. Given any δ > 0, one can choose the accuracy ζ of the coarse graining and a scale K0 (δ, β) such that for any mesoscopic K K0 lim

N →∞

1 N d−1

log µβ,N MN,K − uζN,K 1 > δ = − ∞ .

(6.2)

This estimate will enable us to rephrase statements on the local parameter in terms of the phase labels uζN,K which are much easier to handle.

6.2

Equilibrium crystal shapes

The concentration in L1 of MN,K to the solutions of the variational problem requires the derivation of precise logarithmic asymptotic in terms of the surface tension. {m− , m+ }), then one can choose δ0 = δ0 (v), Proposition 6.1. Let v be in BV(T, β β such that uniformly in δ < δ0 lim inf N →∞

1 log µβ,N MN,K − v1 δ − Wβ (v) − o(δ) , N d−1

where the function o(·) depends only on β and v and vanishes as δ goes to 0. {m− , m+ }) such that Wβ (v) is ﬁnite, one can Proposition 6.2. For all v in BV(T, β β choose δ0 = δ0 (v), such that uniformly in δ < δ0 lim sup N →∞

1 log µβ,N MN,K − v1 δ − Wβ (v) + o(δ) , N d−1

where the function o(·) depends only on β and v and vanishes as δ goes to 0.

6.3

Upper bound

The proof of Proposition 6.2 follows the general scheme of the L1 Theory. First the boundary ∂ ∗ v is approximated; this enables us to reduce the proof to local computations in small regions. Then in each region the interface is localized on the mesoscopic level by using the minimal section argument. In the last step, the representation of the surface tension (see Deﬁnition 3.1) enables us to conclude.

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Ann. Henri Poincar´e

Step 1: Approximation procedure. We approximate ∂ ∗ v with a ﬁnite number of parallelepipeds. Theorem 6.2. For any δ positive, there exists s positive such that there are disjoint included in T with basis B 1, . . . , B of size length s and 1 , . . . , R parallelepipeds R i divides R i in 2 parallelepipeds R i,+ and R i,− and the height δs. The basis B i is denoted by ni . Furthermore, the parallelepipeds satisfy the following normal to B properties

i R

i ) |XR i (x) − v(x)| dx δ vol(R

and

i=1

i B

τβ (ni ) dHx(d−1) − Wβ (v) δ,

− d i i where XR i = m+ i,+ + mβ 1R i,− and the volume of R is vol(R ) = δs . β 1R

The proof follows from standard arguments of geometric measure theory (see for example [Ce, B1]). Theorem 6.2 enables us to decompose the boundary into regular sets (see Figure 5) so that it will be enough to consider events of the type MN,K ∈

$

i

i

V(R , δvol(R ))

,

i=1

i , ε) is the ε-neighborhood of X i where V(R R i , ε) = V(R

v ∈ L1 T

i R

|v (x) − XR i (x)| dx ε .

h i,+ R

i B ni

1 δh 2

{v = 1} i,− R

{v = −1}

Figure 5. Approximation by parallelepipeds.

According to (6.2), the local averaged magnetization can be replaced by the mesoscopic phase labels. Therefore Proposition 6.2 is equivalent to the following

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statement: for any δ positive, there exists K0 = K0 (δ, h), ζ0 = ζ0 (δ, h) such that uniformly in K K0 , ζ ζ0 lim sup N →∞

"

1 N d−1

log µβ,N

uζN,K

∈

$

# i i V(R , δvol(R ))

i=1

− Wβ (v) + C(β, v)δ.

(6.3)

The previous inequality localizes the L1 -estimates into regular macroscopic i which are the counterparts of the domains ΛsN,δsN (ni ) introduced domains RN in Section 3. To use the deﬁnition of the surface tension, one has ﬁrst to establish i the existence of 4 barriers in RN which will play the roles of C + , C − and S + , S − . The derivation of this boils down to transfer the macroscopic L1 -bounds into a i microscopic statement on the localization of an interface inside each RN . This is 1 a key step in the L -approach and the coarse graining will play a major role. Step 2: Minimal section argument. i,± in TN are denoted Ri,± and we set Ri = The microscopic images of R N N i,+ i,− RN ∪RN . For simplicity, we will only prove the existence of a + barrier C i,+ lying i in the upper part of RN and refer to [B2] for a complete derivation. We consider i,+ top i i ∂ RN the face of RN orthogonal to the vector ni and contiguous to RN . Let i,top i,+ δs top i RN be the set of sites in RN at distance smaller than 10 N of ∂ RN . At a given mesoscopic scale K, we associate to any spin conﬁguration the set of bad i,top with uζK labels equal to 0 or −1. boxes which are the boxes BK intersecting RN i,j i For any integer j, we set BN = BN + j cd K ni and deﬁne i,j i,top i,j = y ∈ RN | ∃x ∈ BN , BN

y − x 10 .

i The sections Bji of the parallelepiped RN are deﬁned as the smallest connected set i,j of B (K) -measurable boxes BK intersecting BN . The parameter cd is chosen such that the Bji are disjoint surfaces of boxes. For j positive, let n+ i (j) be the number i of bad boxes in Bj and deﬁne

= min n+ n+ i i (j) :

δs N 9δs N <j< 10cd K cd K

.

Call j + the smallest location where the minimum is achieved and deﬁne the mini,top imal section in RN as Bji + (see Figure 6). % i , δvol(R i )), For any spin conﬁguration such that uζN,K belongs to i=1 V(R the number of bad boxes in a minimal section is bounded by i n+ i δvol(R )

10cd δs

N K

d−1 10cdδsd−1

N K

d−1 .

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bad blocks

Ri,− N

Bij

+

Bij

−

{v = −1}

{v = 1} Ri,+ N

bad blocks Figure 6. Minimal sections.

i | = sd−1 can be controlled in terms of the perimeter of ∂ ∗ v, the total As i=1 |B number of bad boxes is bounded by d−1 N n+ δ C(v) . (6.4) i K i=1 From the very construction of the coarse graining, the + spin surfaces associated to overlapping boxes with uζN,K labels equal to 1 are connected. As each minimal section contains mainly + good blocks, there exist almost a + barrier in each minimal section. By modifying the spin conﬁgurations σ on the bad boxes, we will complete these + barriers. More precisely, we associate to any conﬁguration σ the conﬁguration σ ¯ with spins equal to + on the boundary of each bad box in the minimal section Bji + and equal to σ otherwise. The cost of this surgical procedure can be estimated as follows. # " $ i , δvol(R i )) ∈ V(R µβ,N uζ N,K

i=1

+ µβ,N {n+ 1 , . . . , nk } . (6.5)

+ (i1 ,...,ik ) (j1 ,...,jk ) (n+ 1 ,...,nk )

The right-hand side takes into account the fact that in the domains Ri1 , . . . , Rik , + + δsN the minimal sections are at heights j1 , . . . , jk ∈ [ 9δsN 10K , K ] and contain n1 , . . . , nk

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bad boxes such that (6.4) holds. Once the location of the bad boxes is ﬁxed, the N d−1 d−1 K . By connumber of spin ﬂips to modify σ into σ ¯ is at most C(v) K struction σ ¯ belongs to the set A1 of spin conﬁgurations which contain a + barrier i in the upper part of each domain RN k + (sN/K)d−1 + µβ,N {n1 , . . . , nk } exp δ C2 (v, β)N d−1 µβ,N A1 , n+ α α=1

where (sN/K)d−1 refers to the total number of blocks in each minimal section. Summing over all the conﬁgurations and using (6.4) again, we obtain + d−1 µβ,N A1 . (6.6) µβ,N {n+ 1 , . . . , nk } exp o(δ) C3 (v, β)N + (n+ 1 ,...,nk )

Finally replacing (6.6) in (6.5), we get " µβ,N

uζN,K

∈

$

# i

i

V(R , δvol(R ))

i=1

2

N K

exp o(δ) C3 (v, β)N d−1 µβ,N A1 . (6.7)

Repeating the same argument, we can consider instead of A1 an event A i which contains at least 4 barriers in each RN . For any spin conﬁguration in A, i,+ i,+ as the support of the + barrier in RN which is the we deﬁne the set of sites C top i i,− is the location of the − barrier in the closest to ∂ RN . In the same way, C i i c lower part of RN which is the closest to (RN ) . By analogy with the notation of Section 3, the set of spin conﬁgurations which contain a + and a − barrier in the domain Λ(C i,+ , C i,− ) is denoted by S i = (S i,+ , S i,− ). Step 3: Surface tension estimates. As a consequence of the previous step, for any spin conﬁguration in A, there i exists a microscopic interface localized in each cube RN . Thus we are now in a good shape to check that lim sup N →∞

1 N d−1

log µβ,N (A) −

i=1

i B

τβ (ni ) dHx + C(β, v, δ) ,

(6.8)

where C(β, v, δ) vanishes as δ tends to 0. Combining the previous inequality with (6.7), we deduce (6.3). We now proceed in deriving (6.8). i by imposing that the boxes We ﬁrst pin the interfaces on the sides of each RN i,+ i,− on the boundary of each RN (resp RN ) parallel to ni have η labels equal to 1 i (resp −1). Since the height of RN is δs, this procedure requires to modify at most

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δsd−1 N d−1 spins. Therefore this has no further impact on the evaluation of the statistical weights of the conﬁgurations because the cost of ﬂipping these spins is bounded by exp(δC(v) N d−1 ). In this way, the domain TN is partitioned into the domains Λ(C i,+ , C i,− ) and a remainder which will be denoted by ∆. µβ,N (A) =

1 Zβ,N

ω Z∆

(C i,+ ,C i,− )

C ,C ZN,δN i,+

i,−

(S i ) ,

i=1

where the boundary conditions ω are imposed by the values of the spins outside ∪i Λ(C i,+ , C i,− ) . Introducing by force the partition functions with the perfect walls we get µβ,N (A) =

1 Zβ,N

ω Z∆

(C i,+ ,C i,− )

C ,C ZN,δN i,+

C i,+ ,R C i,− ,R ZN,δN ZN,δN

C i,+ ,R

i=1

i=1

i,−

(S i )

C ,R ZN,δN ZN,δN i,−

.

(6.9)

By Deﬁnition 3.1 of the surface tension, the last term in the right-hand side is bounded by " '# &

C i,+ ,C i,− ZN,δN (S i ) d−1 exp −N τβ (ni )dHx + |Bi |c(β, N, δ) , C i,+ ,R C i,− ,R i=1 ZN,δN ZN,δN i=1 Bi (6.10) where the remainder c(β, N, δ) satisﬁes lim sup lim sup c(β, N, δ) = 0 . δ→0

N →0

In order to complete the derivation of (6.8), it remains to check that  lim

N →∞

1 N d−1

1 log  Zβ,N

(C i,+ ,C i,− )

ω Z∆

 C ,R C ZN,δN ZN,δN,R  i,+

i,−

i=1

= lim

N →∞

1 N d−1

log

R Zβ,N = 0, Zβ,N

R denotes the partition function in TN where the interactions have been where Zβ,N i reﬂected in the middle of each RN . The previous statement follows readily from (4.40) where the contribution of the reﬂected boundary conditions to the pressure are proven to be of order N d−2 . Nevertheless in order to apply (4.40), we have ﬁrst to check that the assumption (4.5) holds for the particular topology imposed by the reﬂections. If assumption (4.5) fails, it is easy to see that one can decompose i,k i each parallelepiped RN into smaller parallelepipeds {RN }k of side-length h h

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for which Theorem 6.2 still holds (see the proof in [B1]). If h is smaller than i the mutual distance between the parallelepipeds {RN }i , a set K R (x, n) cannot j,k j ,k intersect two regions RN and RN with j = j without touching the boundary conditions B. Following the argument detailled in the third step of Subsection 5.1, i,k we can then exclude multiple reﬂections between cubes {RN }k . Thus assumption (4.5) is also valid in this setup.

6.4

Lower bound

In order to derive Proposition 6.1, it is enough to consider the typical spin conﬁgurations which contain a microscopic contour in a neighborhood of the boundary of ∂ ∗ v. At this stage, Theorem 3.1 becomes necessary. Step 1: Approximation procedure. We ﬁrst start by approximating the boundary ∂ ∗ v by a regular surface ∂ V . A polyhedral set has a boundary included in the union of a ﬁnite number of hyperplanes. The surface ∂ ∗ v can be approximated as follows (see Figure 7) Theorem 6.3. For any δ positive, there exists a polyhedral set V such that Wβ (V ) − Wβ (v) δ. and 1IV − v1 δ 1 , . . . , R with basis For any s small enough there are disjoint parallelepipeds R 1 ,...,B included in ∂ V of side-length s and height δs. Furthermore, the sets B 1, . . . , B cover ∂ V up to a set of measure less than δ denoted by U δ = ∂ V \ B ( i B and they satisfy i=1

i=1

i B

τβ (ni ) dHx(d−1) − Wβ (v) δ,

i is denoted by ni . where the normal to B The proof is a direct application of Reshtnyak’s Theorem and can be found in the paper of Alberti, Bellettini [AlBe]. Using Theorem 6.3, we can reduce the proof of Proposition 6.1 to the computation of the probability of {MN,K − 1IV 1 δ}. According to (6.2) the estimates can be restated in terms of the mesoscopic phase labels. It will be enough to show that: for any δ > 0, there exists ζ = ζ(δ) and K0 (δ) such that for all K K0 lim inf N →∞

1 N d−1

log µβ,N uζN,K − 1IV 1 δ − Wβ (V ) − o(δ),

where the function o(δ) vanishes as δ goes to 0.

(6.11)

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j B

ni i R

{v = −1}

Figure 7. Polyhedral approximation.

Step 2: Localization of the interface. i and U δ in TN will be denoted by VN , Ri and U δ . We The images of V , R N N i,− i,+ i i and split RN into RN and RN which are the microscopic counterparts of V ∩ R i \ V . R We will enforce the occurrence of a microscopic interface along the boundary i ∂ V . As in the derivation of the upper bound, the domains RN are the counterparts i,+ i,+ and Ai,− of ΛN,δN (ni ). Let A be the event that there are two + barriers in RN i,− i,± the analogous event with two − barriers in RN . The ± barrier in RN which is % i c the closest from (RN ) is denoted by C i,± . We set A = i=1 Ai,+ ∩ Ai,− . Let us also deﬁne Di,+ (resp Di,− ) the set of spin conﬁgurations such that the η-labels i,+ i,− are equal to 1 (resp −1) on the sides of RN (resp RN ) parallel to ni . In order to construct a closed contour of spins surrounding VN , we deﬁne D as the set δ of conﬁgurations in Di,+ and Di,− such that the blocks on one side of UN have η-labels − and + in the other side. Any spin conﬁguration in A ∩ D contains a microscopic interface which decouples VN from its complement. One has µβ,N uζN,K − 1IV 1 δ µβ,N uζN,K − 1IV 1 δ ∩ A ∩ D . (6.12) The spin conﬁgurations inside VN (resp VNc ) are surrounded by − (resp +) boundary conditions, so that they are in equilibrium in the − (resp +) pure phase. Bulk estimate imply that one can choose s small enough, ζ = ζ (δ) and K0 = K0 (δ) such that

δ or |uζN,K (x) − m+ lim µβ,N β | dx N →∞ 2 c V

δ ζ − |uN,K (x) − mβ | dx A ∩ D = 0 . 2 V (This limit can be obtained by using a proof similar to the one of Theorem 6.4.)

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So that (6.12) can be rewritten for N large enough as 1 µβ,N uζN,K − 1IV 1 δ µβ,N (A ∩ D) . 2

(6.13)

Step 3: Surface tension. Let Λ be the union of the sets Λi = Λ(C i,+ , C i,− ). The conﬁgurations in the event A ∩ D contain two closed surfaces with + and − blocks which partition the domain TN into 3 regions. TN = Λ ∪ ∆+ ∪ ∆− , where ∆± represents the location of the ± pure phases and Λ is concentrated along the interface. We proceed now to evaluate the right-hand side of (6.13) C i,+ ,C i,− 1 + − µβ,N (A ∩ D) Z∆ ZΛi (Si ) , + Z∆ − ZN i,+ i,− i C

,C

where we used analogous notation to Section 3 for the partition function with mixed boundary conditions. Introducing the partition functions with reﬂected boundary conditions we get 1 µβ,N (A ∩ D) ZN

ZΛC i

i,+

C i,+ ,C i,−

C i,+ ,R + − Z∆ + Z∆− ZΛi

i,− ZΛC i ,R

i

,C i,−

i,+ ZΛC + ,R i

(Si )

i,− ZΛC − ,R i

, (6.14)

Λ± i

refers to the sets Λ± (C i,± ) which were introduced in Subsection 3.2. The where last term in the right-hand side is an approximation of the surface tension in each domain Λi , therefore Theorem 3.1 implies inf

C i,+ ,C i,−

1 N d−1

i

ZΛC i

i,+

log

,C i,−

i,+ ZΛC + ,R i

(Si )

i,− ZΛC − ,R i

−

i

i B

τβ (ni ) dHx(d−1) − P (v)c(δ, N ) ,

where limδ→0 limN →∞ c(δ, N ) = 0 and P (v) is the perimeter of v. It remains to check that   i,+ i,− 1 1 lim log  Z ++ Z −− Z C + ,R ZΛC − ,R  = 0 . N →∞ N d−1 ZN i,+ i,− ∆ ∆ Λi i C

(6.15)

(6.16)

,C

Combining inequalities (6.15) and (6.16) we see that

1 τβ (ni ) dHx(d−1) − o(δ) . lim inf d−1 log µβ,N (A ∩ D) − N →∞ N i B i=1 Using Theorem 6.3 and letting δ vanish, we conclude the proof of Proposition 6.1.

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We turn now to the derivation of (6.16). Since the reﬂected boundary conditions decouple the system, the numerator should be understood as the prod¯ + = ∆+ ∪i Λ+ (C i,+ ) and uct of two partition functions associated to the sets ∆ − − − i,− i,± ¯ ∆ = ∆ ∪i Λ (C ), where Λ(C ) denotes the part of Λi . It is important to ¯ ± are independent of the choice of the surfaces note that contrary to ∆± , the sets ∆ i,± C . In particular, following the notation of Section 3, C i,+

C + Z∆ + Z + Λ

i,+

,R

i

R i,+ = Z∆ ), ¯ + (C

¯ + under the conwhere the right-hand side denotes the partition function on ∆ i,+ straint that in each RN there is a + barrier. Applying the same strategy as for the derivation of (5.11), we can check that 1

lim

N →∞

N d−1

R i,+ Z∆ ) ¯ + (C = 0. R Z∆ ¯+

log

This implies that (6.16) is equivalent to lim

1

N →∞

N d−1

log

R R Z∆ ¯ + Z∆ ¯− = 0. ZN

(6.17)

The partition functions in the numerator take also into account the conδ and on the sides of straints imposed by the set D on the spins along the set UN i RN parallel to ni . These constraints can be released up to a small cost w.r.t. the surface order. This comes from the fact that the event D is supported by at most c(d, δ)N d−1 edges where c(d, δ) vanishes as δ goes to 0. Therefore the probability of D is negligible with respect to a surface order and we get R R Z∆ ¯ + Z∆ ¯− log c(d, δ)N d−1 , R ZN

(6.18)

R where ZN is the unconstrained partition function on TN for which the interactions i have been modiﬁed and replaced by perfect walls. Again in the middle of each RN by the same considerations as in the last argument of the proof of the upper bound (see Subsection 6.3), one check that one can ﬁnd a polyhedral approximation for which assumption (4.5) is satisﬁed. The corrections to the pressure induced by the reﬂection are negligible w.r.t. the surface order (see (4.40)) so that

lim

1

N →∞ N d−1

log

R ZN = 0. ZN

This, combined with (6.18) implies the validity of (6.16).

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Exponential tightness

The purpose of this subsection is to prove that phase coexistence cannot occur by creation of many small droplets. Rephrased in a mathematical way, this means that with an overwhelming probability, the conﬁgurations will concentrate close to the compact set

{m− , m+ }) | P ({v = m− }) a , Ka = v ∈ BV(T, (6.19) β β β where P denotes the perimeter and a will be chosen large enough. Proposition 6.3. There exists a constant C(β) > 0 such that for all δ positive one can ﬁnd K0 (δ) such that for K K0 ∀a > 0,

lim sup N →∞

1 N d−1

log µβ,N (MN,K ∈ V(Ka , δ)) − C(β) a,

where V(Ka , δ) is the δ-neighborhood of Ka in L1 (T). The estimate (6.2) allows us to shift our attention from the local averaged magnetization to the mesoscopic phase labels. In particular Proposition 6.3 follows from Theorem 6.4. Fix ζ > 0. For every a > 0 and δ > 0 there exists a ﬁnite scale K0 (δ), such that for all K K0 1 lim sup d−1 log µβ,N uζN,K ∈ V(Ka , 2δ) − c(β, K)a , (6.20) N →∞ N where c(β, K) is a positive constant. The core of the proof relies on the control of the phase of small contours by means of an entropy/energy argument. The argument is standard and depends only on the structure of the coarse graining. We refer the reader to [BIV1] (Theorem 2.2.1), where Proposition 6.4 was derived in a complete generality. Finally, notice that similar arguments can easily be adapted to multi-phase models (see Remark 3.4 in [BIV2]). Theorem 2.1 can be obtained by combining Propositions 6.3, 6.1, 6.2. Since Ka is compact with respect to the L1 topology (see [EG]), the exponential tightness property 6.3 enables us to focus only on a ﬁnite number of conﬁgurations close to Ka . The precise asymptotic of these conﬁgurations is then estimated by Propositions 6.1, 6.2 (see [B1] for details).

A

Proof of Theorem 6.1

The magnetic ﬁeld is equal to h(β) and omitted from the notation throughout the proof. The proof follows the argument developed in [B2].

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Step 1. Let us start with a single box. If BK (x) is not a good box then either there is a contour of length at least K α crossing the enlarged boundary or conditionally on the event that the box BK (x) is surrounded by a surface of η-block spins of sign εx , the magnetization MK (x) is atypical. These two occurrences can be estimated separately. Applying the Peierls estimate (4.11), we get µβ,N there is a contour crossing ∂BK (x) K d−1 exp(−cβ K α ) . (A.1) Conditionally on the occurrence of a connected surface S of η-block spins of sign εx surrounding the box BK (x), the conﬁgurations inside BK (x) are decoupled from the exterior. We ﬁrst use Tchebyshev inequality   1 2 εx ε ε x  σi − mβx  , µβ,N {|MK (x) − mβ | ζ} S 2 2d µβ,int(S) ζ K i∈BK (x)

where int(S) is the region surrounded by S. As S has been chosen as the closest surface to (BK+K α )c , the magnetization inside the box BK (x) is measurable after the conditioning. Classical Pirogov-Sinai theory ensures also that under the assumptions of Theorem 4.2, the correlations decay exponentially in the εx -pure phase, so that we obtain 1 x {|MK (x) − mεβx | ζ} ≤ 2 d χ , (A.2) µεβ,h,int(S) ζ K where the susceptibility χ = i∈Zd µ+ β (σ0 ; σi ) is ﬁnite. Step 2. In order to evaluate the probability of the event uζK (x1 ) = 0, . . . , uζK (x ) = 0 (K)

the partition B (K) is sub-divised into cd sub-partitions (Bi )i cd such that two (K) cubes of size K +K α centered on two sites of Bi are disjoint. By applying H¨older inequality, the estimate (6.1) is reduced to cubes which are not nearest neighbors. cd (K) µβ,N ∀xj ∈ Di , µβ,N uζK (x1 ) = 0, . . . , uζK (x ) = 0

uζK (xj ) = 0

c1

d

.

i=1

Step 3. The event uζK (x1 ) = 0, . . . , uζK (x ) = 0 can be decomposed into 2 terms:

on boxes the density is atypical, whereas there are contours crossing the − enlarged boundaries of the remaining boxes. For a given collection of j boxes, we deﬁne Aj

=

{The j boxes are surrounded by ± surfaces, but their averaged magnetizations are non-typical}

Bj

=

{There are contours crossing the j enlarged boundaries of the boxes} .

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The probabilities of both events can be evaluated as follows. As the j boxes are disjoint and the surfaces of blocks decouple the conﬁgurations inside each box j j µβ,N (Aj ) µβ,N (A1 ) αK , where the constant αK =

µβ,N (Bj ) =

j

χ ζ2Kd

was introduced in (A.2).

µβ,N {∃ i contours crossing the j enlarged boundaries} .

i=1

We choose i blocks as starting points of these contours. Then we have to evaluate |Γ1 |+···+|Γi |

µβ,N (Γ1 , . . . , Γi ) , jK α

where the contours (Γ1 , . . . , Γi ) have also to cross each boundaries of the j cubes. Let nr be the number of boundaries crossed by the contour r

µβ,N (Γ1 , . . . , Γi )

µβ,N (Γ1 , . . . , Γi ) .

n1 +···+ni =j (Γr ,nr )

|Γ1 |+···+|Γi | jK α

If a contour crosses nr boundaries then it has a length at least nr K α + (nr − 1)K because the distance between the boxes is at least K. Thus µβ,N (Γ1 , . . . , Γi ) |Γ1 |+···+|Γi | jK α

i

exp(−cβ nr K α − cβ (nr − 1)K)

n1 +···+ni =j r=1

"

exp(−cβ jK ) α

∞

#i exp(−cβ (n − 1)K)

n=1 α

C i exp(−cβ jK ) . Finally µβ,N (Bj )

j j K (d−1)i C i exp(−cβ jK α ) i i=1

exp(−cβ jK α )(1 + CK d−1 )j = (α K )j , where the constant α K vanishes as K goes to inﬁnity.

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Combining both estimates, we obtain µβ,N uζK (x1 ) = 0, . . . , uζK (x ) = 0 µβ,N (A )1/2 µβ,N (B− )1/2 αK + α K . =1

This completes the proof.

Acknowledgment We are indebted to R. Kotecky, S. Shlosman and Y. Velenik for many helpful discussions. The ﬁrst part of this work was done at the IHP where both authors participated to the semester on “hydrodynamic limits”. T.B. acknowledges kind hospitality at Roma Tor Vergata where this paper has been completed.

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[BIV2] T. Bodineau, D. Ioﬀe, Y. Velenik, Winterbottom construction for ﬁnite range ferromagnetic models: a L1 -approach, J. Stat. Phys. 105, No 1/2, 93–131 (2001). [BKL] J. Bricmont, K. Kuroda, J. Lebowitz, Surface tension and phase coexistence for general lattice systems, J. Stat. Phys. 33, 59–75 (1983). [Ce]

R. Cerf, Large deviations for three-dimensional supercritical percolation, Ast´erisque 267 (2000).

[CePi1] R. Cerf, A. Pisztora, On the Wulﬀ crystal in the Ising model, Ann. Probab. 28, No. 3, 947–1017 (2000). [CePi2] R. Cerf, A. Pisztora, Phase coexistence in Ising, Potts and percolation models, Ann. Inst. H. Poincar´e Probab. Statist. 37, No. 6, 643–724 (2001). [CIV]

M. Campanino, D. Ioﬀe, Y. Velenik, Ornstein-Zernike Theory for the ﬁnite range Ising models above Tc , Probab. Theory Related Fields 125, No. 3, 305–349 (2003).

[DKS] R.L. Dobrushin, R. Koteck´ y, S. Shlosman, Wulﬀ construction: a global shape from local interaction, AMS translations series, vol 104, Providence R.I. (1992). [EG]

L. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, London (1992).

[F]

I. Fonseca, The Wulﬀ theorem revisited, Proc. Royal London Soc. Sect. A, 432, 125–145 (1991).

[FM]

I. Fonseca, S. Mueller, A uniqueness proof of the Wulﬀ Theorem, Proc. Roy. Soc. Edinburgh; Sect A, 119, 125–136 (1991).

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O. Hryniv, R. Kotecky, Surface tension and the Ornstein-Zernike behaviour for the 2D Blume-Capel model, Jour. Stat. Phys., 106, No. 3–4, 431–476 (2002).

[HKZ1] P. Holicky, R. Kotecky, M. Zahradnik, Rigid interfaces for lattice models at low temperatures., Jour. Stat. Phys. 50, No. 3–4, 755–812, (1988). [HKZ2] P. Holicky, R. Kotecky, M. Zahradnik, Phase Diagram of Horizontally Invariant Gibbs States for Lattice Models, Ann. Henri Poincar´e Phys. Theo., 3, no. 2, 203–267 (2002). [KP]

R. Kotecky and D. Preiss, Cluster Expansion for Abstract Polymer Models, Commun. Math. Phys., 103, 491–498 (1986).

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D. Ioﬀe, Large deviations for the 2D Ising model: a lower bound without cluster expansions, J. Stat. Phys. 74, 411–432 (1994).

[I2]

D. Ioﬀe, Exact deviation bounds up to Tc for the Ising model in two dimensions, Prob. Th. Rel. Fields 102, 313–330 (1995).

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D. Ioﬀe, R. Schonmann, Dobrushin-Koteck´ y-Shlosman theory up to the critical temperature, Comm. Math. Phys. 199, 117–167 (1998).

[LMR] L. Laanait, A. Messager, J. Ruiz, Phases coexistence and surface tensions for the Potts model., Comm. Math. Phys. 105, No. 4, 527–545 (1986). [LMP] J. Lebowitz, Mazel, Presutti, Liquid-vapor phase transitions for systems with ﬁnite-range interactions., J. Stat. Phys. 94, No.5-6, 955–1025 (1999). [MMR] A. Messager, S. Miracle-Sol´e, J. Ruiz, Surface tension, step free energy and facets in the equilibrium crystal, J. Stat. Phys. 67, No. 3–4, 449–470 (1992). [MMRS] A. Messager, S. Miracle-Sol´e, J. Ruiz, S. Shlosman, Interfaces in the Potts model. II. Antonov’s rule and rigidity of the order disorder interface, Comm. Math. Phys. 140, No.2, 275–290 (1991). [Pf]

C.E. Pﬁster, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta 64, 953–1054 (1991).

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C.E. Pﬁster, Y. Velenik, Large deviations and continuum limit in the 2D Ising model, Prob. Th. Rel. Fields 109, 435–506 (1997).

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Y. Velenik, Phase separation as a large deviations problem: a microscopic derivation of surface thermodynamics for some 2D spin systems, Th`ese 1712 EPF-L, (1997).

T. Bodineau† Universit´e Paris 7 and Laboratoire de Probabilit´es et Mod`eles Al´eatoires C.N.R.S. UMR 7599 U.F.R. Math´ematiques, Case 7012 2, Place Jussieu F-75251 Paris, France email: [email protected] E. Presutti Errico Presutti Dipartimento di Matematica Universit` a di Roma Tor Vergata I-00133 Roma, Italy email: [email protected] Communicated by Vincent Rivasseau submitted 24/01/03, accepted 12/04/03 † Research

partially supported by MURST and NATO Grant PST.CLG.976552

Ann. Henri Poincar´e 4 (2003) 897 – 945 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/050897-49 DOI 10.1007/s00023-003-0150-8

Annales Henri Poincar´ e

Dissipative Transport: Thermal Contacts and Tunnelling Junctions J¨ urg Fr¨ ohlich, Marco Merkli and Daniel Ueltschi Abstract. The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production rate, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable smallness and regularity assumptions on the interaction between the reservoirs.

1 Introduction 1.1

Description of the problems

Simple transport processes, such as those observed near a spatially localized thermal contact or tunnelling junction between two macroscopically extended metals at diﬀerent temperatures and chemical potentials, have been studied experimentally and theoretically for a long time; see, e.g., [Ma]. One is interested, for example, in measuring or predicting energy and charge transport through a contact between two metals, as well as the rate of entropy production. The natural theoretical description of such processes is provided by quantum theory, more precisely by non-equilibrium quantum statistical mechanics. The results of experiments or theoretical calculations can, however, often be expressed in the language of thermodynamics. In this paper we attempt to study such transport processes in a mathematically precise way, extending or complementing results in [DFG, EPR, JP1, JP2, Ru1, Ru2]. In recent years, interest in transport processes has been driven by various experimental and theoretical developments in mesoscopic physics and the discovery of rather unexpected phenomena. Among them we mention dissipation-free transport in incompressible Hall ﬂuids [La, TKNN, BES, ASS, FST], or in ballistic quantum wires [vW, Bee, ACF, FP1]. In such systems, “transport in thermal equilibrium” and the quantization of conductances are observed. Further interesting transport processes are electron tunnelling into an edge of a Hall ﬂuid [CPW, CWCPW, LS, LSH], and tunnelling processes between two diﬀerent quantum Hall

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edges through a constriction leading to measurements of fractional electric charges of quasi-particles (see, e.g., [SGJE], and [FP2] for theoretical considerations). At present, these processes are only partially understood theoretically. Other examples are Josephson junctions and Andreev scattering [SR1, SR2], or energy transport in chains (see, e.g., [Af] and references given there). In this paper, the main emphasis is put, however, on conceptual aspects of the theory of simple dissipative transport processes between two quantummechanical reservoirs and on an illustration of the general theory in a simple example, namely transport of energy and charge between two metals, described as non-interacting electron liquids, at diﬀerent temperatures and chemical potentials. Of particular interest to us are connections between theoretical descriptions based on non-equilibrium quantum statistical mechanics, on the one hand, and on thermodynamics, on the other hand. Our quantum-mechanical description involves equilibrium and non-equilibrium states of macroscopic reservoirs with many degrees of freedom. We show that, on intermediate time scales, tunnelling processes can be described in terms of non-equilibrium stationary states (NESS), examples of which have recently been studied in [DFG, EPR, JP1, JP2, Ru1, Ru2, BLR]. Our construction of non-equilibrium stationary states is based on methods of algebraic scattering theory and is inspired by ideas in [He, Rob, BR, BM, Ha]. Links between quantum statistical mechanics and thermodynamics are constructed by providing precise deﬁnitions of energy and particle currents and of entropy production and by deriving a suitable form of the ﬁrst and second law of thermodynamics. The general theory developed in Section 2 reviews ideas and results scattered over numerous articles and books and represents an attempt to provide a somewhat novel and, we believe, rather clear synthesis. A more complete version, including the treatment of systems with time-dependent Hamiltonians, appears in [FMSU] and in a forthcoming paper. It is illustrated on the example of two reservoirs of non-interacting electrons coupled through local many-body interactions (Sections 3 through 5). Examples of non-equilibrium stationary states supporting particle and/or energy currents are constructed with the help of a convergent perturbation (Dyson) series in the many-body interaction terms. The currents and the entropy production rate are calculated quite explicitly to leading order. This enables us to show that, under natural hypotheses, they are strictly positive. Onsager reciprocity relations are established to lowest non-trivial order in the many-body interaction terms. Positivity of the entropy production rate has also been established recently in [AP, MO] for XY chains, and in [CNP] for wave turbulence.

1.2

Contents of paper

In Section 2.1, quantum-mechanical reservoirs are introduced, whose time evolution is given in terms of a one-parameter group of ∗automorphisms of a kinematical C ∗ -algebra of operators. Conservation laws of reservoirs are described by commuting conserved charges. The equilibrium states of such reservoirs are introduced and parameterized by temperature and chemical potentials. Two general assumptions,

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(A1) and (A2), are formulated. They state that the thermodynamic limits of the time evolution and of the gauge transformations of operators in the kinematical algebra of an inﬁnite system and of the equilibrium states exist. In Section 2.2 we study two interacting reservoirs at diﬀerent temperatures and chemical potentials. Each reservoir is required to satisfy the assumptions formulated in Section 2.1. A class of many-body interactions coupling the two reservoirs is introduced. It is assumed that the thermodynamic limit of the interactions and of the corresponding time evolutions exist. Energy and charge currents for ﬁnite and inﬁnite systems are then deﬁned. Connections between quantum statistical mechanics and thermodynamics are elucidated in Section 2.3. The entropy production rate is deﬁned and expressed in terms of the currents and of thermodynamic parameters. An inequality expressing the positivity of relative entropy is shown to imply that the total entropy production is non-negative (see also [Ru2, JP1]). In Section 2.4, non-equilibrium stationary states for coupled reservoirs are introduced. They can be constructed with the help of scattering (Møller) endomorphisms of the kinematical algebra of the inﬁnite coupled system. A precise assumption concerning the existence of scattering endomorphisms is formulated. Our approach has its roots in Hepp’s work on the Kondo problem [He] and Robinson’s analysis of return to equilibrium in the XY spin chain [Rob]. Robinson’s ideas have been put into a general context in [BR, Ha]. The scattering approach is the starting point for numerous heuristic studies of thermal contacts and tunnelling junctions (see, e.g., [Ma]). The ﬁrst mathematically rigorous implementation of this approach in a study of energy transport in the XY spin chain and of tunnelling between free-fermion reservoirs appeared in [DFG]. In Section 2.5, long-time stability properties of equilibrium and non-equilibrium stationary states against perturbations of the initial state of the coupled system are studied, and conditions for the existence of temperature or density proﬁles in non-equilibrium stationary states are identiﬁed. The general theory of Section 2 is illustrated in Sections 3, 4 and 5 on the example of two coupled free-electron reservoirs. In Section 3, the quantum theory of ﬁnite and inﬁnite reservoirs of free electrons is brieﬂy recalled, and a class of local many-body interactions between two such reservoirs satisfying the general assumptions formulated in Section 2 is introduced. Our main technical result, the existence of scattering (Møller) endomorphisms, deﬁned on an appropriate kinematical C ∗ -algebra describing two inﬁnite free-electron reservoirs, is established in Section 4. A similar result has previously been proven in [BM]. We show that, under appropriate smallness and regularity assumptions on the many-body interaction terms, the scattering endomorphisms are given by a (norm-) convergent Dyson series. As a consequence, non-equilibrium stationary states can be constructed with the help of a convergent perturbation expansion.

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The results of Section 4 are used in Section 5 to derive explicit expressions for the energy and charged-particle currents to leading order in the many-body interaction terms. These expressions, along with the convergence of the Dyson series, prove that, for small coupling constants, the entropy production rate is strictly positive, Ohm’s law holds to leading order in the voltage drop between the reservoirs, with a resistance whose temperature dependence can be determined, and the Onsager reciprocity relations hold to leading order. We conclude this introduction with explicit formulae for the leading contributions to the particle current, J , and to the energy current, P, between two reservoirs, I and II, of free electrons coupled to each other by a quadratic local interaction term with a form factor w((−k, II), (l, I)) and a brief discussion of the qualitative implications of these formulae. These currents are given by 2 dk dl δ(|k|2 − |l|2 ) |w((−k, II), (l, I))| (ρII (k) − ρI (k)) , J 2π R6 2 dk dl |k|2 δ(|k|2 − |l|2 ) |w((−k, II), (l, I))| (ρII (k) − ρI (k)) , (∗) P 2π R6

where ρr (k) is the Fermi distribution of the free electron gas (r = I, II labels the reservoirs), and w((−k, II), (l, I)) is the interaction kernel describing scattering of a particle in an initial state with energy |l|2 localized in reservoir I to a ﬁnal state with energy |k|2 localized in reservoir II. If both reservoirs have the same chemical potential and the temperatures satisfy T I < T II then J and P are positive; particles and energy ﬂow from the hotter to the colder reservoir. Similarly, if the reservoirs have the same temperature, then particles and energy ﬂow from the reservoir with the higher chemical potential to the other one. Formulae (∗) prove that the leading contribution to the entropy production rate is strictly positive, unless both reservoirs are at the same temperature and at the same chemical potential. Another consequence of (∗) is that, to leading order in the interaction, and for a small voltage drop, ∆µ = µII − µI (at a ﬁxed temperature, T , for both reservoirs), Ohm’s law is valid, i.e., the voltage drop is proportional to the current, ∆µ R(µI , T )J . Our calculations show that the resistance R(µI , T ) grows linearly in T , for large T , it has a positive value at T = 0 and may increase or decrease in T , at small temperatures, depending on properties of the interaction kernel w modeling the junction between the two reservoirs. This paper is dedicated to Klaus Hepp and David Ruelle on the occasion of their retirement from active duty, but not from scientiﬁc activity. Some of their work plays a signiﬁcant rˆ ole in the analysis presented in this paper. J. F. is deeply grateful to them for their generous support and for everything they have contributed to making his professional life at ETH and at I.H.E.S. so pleasant.

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2 Elements of a general theory of junctions and of non-equilibrium stationary states 2.1

Quantum theory of reservoirs

We start our general analysis by describing “quantum-mechanical reservoirs”. A reservoir is a quantum system with very many degrees of freedom, e.g., an electron liquid in a normal metal, a superconductor, a gas of atoms, or a large array of coupled, localized spins, but with a small number of observable physical quantities. It is conﬁned to a macroscopically large, but compact subset, Λ, of physical space R3 . Its pure states correspond, as usual, to unit rays in a separable Hilbert space, HΛ , and its dynamics is generated by a selfadjoint Hamiltonian, H Λ , acting on the space HΛ . The kinematics of the reservoir is encoded into an algebra, AΛ , of operators contained in (or equal to) the algebra of all bounded operators on HΛ 1 . The time evolution of an operator a on HΛ is given, in the Heisenberg picture, by itH αΛ t (a) := e

Λ

/

a e−itH

Λ

/

,

(2.1)

Λ Λ and it is assumed that αΛ t (a) ∈ A , for every a ∈ A . There may exist a certain number of linearly independent, commuting conΛ Λ servation laws, which are represented by selfadjoint operators QΛ 1 , . . . , QM on H commuting with the dynamics of the reservoir, i.e., Λ Λ Λ H , Qj = 0, QΛ = 0, QΛ (2.2) i , Qj i , a = 0,

for all i, j, = 1, . . . , M , and for all “observables” a ∈ AΛ . More precisely, one asM sumes that all operators exp itH Λ /, exp i sj QΛ j j=1 commute with one another, for arbitrary real values of t, s1 , . . . , sM . A typical example of a conservation law is the particle number operator, N Λ , of a reservoir consisting of a gas of non-relativistic atoms. On the algebra, B(HΛ ), of all bounded operators on the Hilbert space HΛ , we deﬁne “gauge transformations of the ﬁrst kind” by setting Λ

Λ

is·Q a e−i s·Q , ϕΛ s (a) := e

for a ∈ B(HΛ ), where s · QΛ :=

M

sj Q Λ j .

(2.3)

(2.4)

j=1

s ∈ RM is an M -parameter Abelian group of ∗ automorphisms (see Then ϕΛ s (2.16), below) of the algebra B(HΛ ), and Λ

Λ

Λ αΛ (2.5) t ϕs (a) = ϕs αt (a) , 1 The algebra AΛ is sometimes called algebra of “observables”, a commonly used, but unfortunate expression.

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for all a ∈ B(HΛ ), by (2.2). It is natural to deﬁne the “observable algebra” AΛ as the algebra of all those operators a ∈ B(HΛ ) for which M ϕΛ s (a) = a , for all s ∈ R .

(2.6)

To every conservation law QΛ j there corresponds a conjugate thermodynamic parameter, µj , commonly called a chemical potential. Thermal equilibrium of the reservoir at inverse temperature β and chemical potentials µ = (µ1 , . . . , µM ) is described by a mixed state, or density matrix, given by Λ −1 (2.7) exp −β H Λ − µ · QΛ , Ξβ,µ where

Λ Λ ΞΛ β,µ := tr exp −β H − µ · Q

(2.8)

is the grand-canonical partition function. Of course, it is assumed that the operators exp −β H Λ − µ · QΛ are trace-class, for all chemical potentials µ in a region M ⊆ RM , for all β > 0, and for arbitrary compact subsets Λ of physical space R3 . It is also commonly assumed that reservoirs are thermodynamically stable, in the sense that the thermodynamic potential, G, given by

(2.9) βG (β, µ, V ) := − ln ΞΛ β,µ is extensive, i.e., proportional to the volume, V , of the set Λ, up to boundary corrections, for arbitrary β > 0, µ ∈ M. The expectation value of an operator a ∈ B(HΛ ) in the equilibrium state corresponding to the density matrix (2.7) is given by

−1 −β [H Λ −µ·QΛ ] Λ (a) := ΞΛ tr e ωβ,µ a . (2.10) β,µ Λ has some remarkable properties to be discussed next. The state ωβ,µ It is time-translation invariant, i.e., Λ

Λ Λ ωβ,µ αt (a) = ωβ,µ (a),

(2.11)

for arbitrary a ∈ B(HΛ ). It obeys the celebrated KMS condition Λ Λ

Λ Λ ωβ,µ αt (a) b = ωβ,µ b αt+iβ ϕΛ , −iβµ (a)

(2.12)

for arbitrary a and b in B(HΛ ). In particular, if a ∈ AΛ then

Λ Λ Λ Λ ωβ,µ αt (a) b = ωβ,µ b αt+iβ (a) ,

(2.13)

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for arbitrary b ∈ B(HΛ ); see (2.6). Equation (2.12) is an easy consequence of Equations (2.10) and (2.3) and of the cyclicity of the trace, i.e., tr (a b) = tr (b a). For further details concerning these standard facts of quantum statistical mechanics we refer the reader to [Ru 3, BR]. Next, we recall some conventional wisdom concerning the thermodynamic limit of a reservoir. We are interested in understanding asymptotics of physical quantities, as the region Λ to which the reservoir is conﬁned increases to all of R3 , or to an inﬁnite half-space (2.14) R3± := (x, y, z) ∈ R3 x ≷ 0 . We use the notation “Λ ∞” to mean that Λ R3 (or Λ R3± ), in the sense of Fisher (meaning, in essence, that the ratio between the surface and the volume goes to zero); see [Ru 3]. We introduce an operator algebra F r , called the “ﬁeld algebra”, convenient for the description of the thermodynamic limit of a reservoir: B(HΛ ), (2.15) F r := Λ∞

where Λ∞ B(HΛ ) is the algebra generated by all the operators in the increasing sequence of algebras

· · · ⊆ B HΛ ⊆ B HΛ ⊆ · · · , Λ ⊆ Λ , and (·) denotes the closure in the operator norm. Technically speaking, F r is a C∗ -algebra, [BR]. The superscript r stands for “reservoir”. Below, we consider two interacting reservoirs, labelled by r = I, II. A group {τt t ∈ Rn } of homomorphisms, of a C∗ -algebra F is an n-parameter ∗ automorphism group of F iﬀ τt=0 (a) = a, τt (τt (a)) = τt+t (a), and

τt (a)∗ = τt (a∗ ),

(2.16)

for all a ∈ F and arbitrary t, t ∈ R . For the purposes of this paper we shall require the following two assumptions concerning the existence of the thermodynamic limit. n

(A1) Existence of the thermodynamic limit of the dynamics and the gauge transformations For every operator a ∈ F r , the limits in operator norm n − lim αΛ t (a) =: αt (a) Λ∞

(2.17)

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and

Ann. Henri Poincar´e

n − lim ϕΛ s (a) =: ϕs (a)

(2.18)

Λ∞

exist, for all t ∈ R, s ∈ RM , and deﬁne ∗ automorphism groups of the ﬁeld algebra F r . The convergence in (2.17) and (2.18) is assumed to be uniform for t in any compact interval of R and for s in any compact subset of RM , respectively. The ∗ automorphism groups αt and ϕs may be assumed to be norm-continuous in t and s, respectively; (but “weak∗ continuity” will usually be suﬃcient). We deﬁne the “kinematical algebra”, Ar , to be the largest subalgebra of F r pointwise invariant under {ϕs }s∈RM , i.e., Ar := a ∈ F r ϕs (a) = a, for all s ∈ RM . (2.19) Since, by (2.5), (2.17) and (2.18), αt and ϕs commute, αt (a) ∈ Ar , for every a ∈ Ar . (A2) Existence of the thermodynamic limit of the equilibrium state For every a ∈ F r , Λ (a) =: ωβ,µ (a) lim ωβ,µ

(2.20)

Λ∞

exists and is time-translation invariant, i.e., ωβ,µ (αt (a)) = ωβ,µ (a),

(2.21)

for all a ∈ F r , t ∈ R . ◦

We assume that F r contains a norm-dense subalgebra F r with the property that the operator αt (ϕs (a)) extends to an entire function of t ∈ C, s ∈ CM , for every ◦

a ∈ F r . If αt and τs are norm-continuous in t and s, respectively, the existence of ◦

an algebra F r ⊂ F r with these properties is an easy theorem. From the KMS condition (2.12), and from (2.17), (2.18), (2.20), it follows that the inﬁnite-volume state ωβ,µ on F r obeys the KMS condition ωβ,µ (αt (a) b) = ωβ,µ (b αt+iβ (ϕ−iβµ (a))), ◦

(2.22)

◦

for all a ∈ F r , b ∈ F r . If a ∈ Ar ∩ F r then Equation (2.22) simpliﬁes to ωβ,µ (αt (a) b) = ωβ,µ (b αt+iβ (a)).

2.2

(2.23)

Thermal contacts and tunnelling junctions between macroscopic reservoirs

We consider two reservoirs, I and II, with all the properties described in Section 2.1. These reservoirs may or may not have the same physical properties. For

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example, they may be ordinary metals located in two complementary half-spaces of R3 ; or I may be a metal and II a superconductor, etc. Later, we shall consider the example where I and II are ordinary metals, i.e., non-interacting electron liquids. In the following, “I” will be a shorthand notation for (I, ΛI ), and “II” for (II, ΛII ), where ΛI and ΛII are arbitrary compact subsets of R3 . Realistically ΛI and ΛII should not intersect; but we shall ignore this constraint. The Hilbert space of the system obtained by composing the two reservoirs is given by H = HI ⊗ HII (2.24) and the dynamics, before the reservoirs are brought into contact, is generated by the Hamiltonian H 0 := H I ⊗ 1l + 1l ⊗ H II . (2.25) The natural algebra of operators of the coupled system is given by B(HI )⊗B(HII ) and, in the thermodynamic limit (ΛI R3− and ΛII R3+ , or ΛI R3 and ΛII R3 ), by (2.26) F := F I ⊗ F II , where F I and F II are the ﬁeld algebras of the two reservoirs (see (2.15)), and the closure is taken in the operator norm. A contact or tunnelling junction between the two reservoirs is described in terms of a perturbed Hamiltonian, H, of the coupled system. The operator H has the form H = H 0 + W (ΛI , ΛII ) (2.27) where W (ΛI , ΛII ) is a bounded, selfadjoint operator on H for each choice of ΛI and ΛII . We shall always require the following assumption. (A3) Existence of the thermodynamic limit of the contact interaction

n − lim W ΛI , ΛII =: W

(2.28)

ΛI ∞ ΛII ∞

exists (as a selfadjoint operator in F ). Let αΛ and αΛ be the time evolutions of the reservoirs before they are t t brought into contact; see Equation (2.1). The time evolution of operators in I ΛII and is generated by the HamilB(HI ) ⊗ B(HII ) is then given by αΛ t ⊗ αt tonian H 0 introduced in (2.25). After the interaction W (ΛI , ΛII ) has been turned on the time evolution of operators in the Heisenberg picture is given by I

II

αI∪II (a) = ei(tH/) a e−i(tH/) , t with H as in (2.27), for a ∈ B(HI ) ⊗ B(HII ).

(2.29)

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It follows from Assumptions (A1) and (A3), Equations (2.17) and (2.28), that the thermodynamic limit of the time evolution of the coupled reservoirs exists: For arbitrary a ∈ F the limits Λ α0t (a) = n − lim αΛ t ⊗ αt (a)

(2.30)

(a) αt (a) = n − lim αI∪II t

(2.31)

I

II

Λ ∞ ΛII ∞ I

and ΛI ∞ ΛII ∞

exist, and the convergence is uniform on arbitrary compact intervals of the time axis. Given (2.30), (2.31) follows by using the Lie-Schwinger series for αt (α0−t (a)). We distinguish between diﬀerent types of contacts or junctions between the two reservoirs, according to symmetry properties of the contact interactions W (ΛI , ΛII ). (J1) Thermal contacts. The interaction W (ΛI , ΛII ) commutes with all the I MI ΛII M II conservation laws {QΛ j ⊗ 1l}j=1 and {1l ⊗ Qj }j=1 of the reservoirs, i.e., I II

I II ϕΛ = ϕΛ W ΛI , ΛII = W ΛI , ΛII , s W Λ ,Λ s

(2.32)

Λ Λ for arbitrary ΛI and ΛII , where ϕΛ s is a shorthand notation for ϕs ⊗ id, and ϕs II stands for id ⊗ ϕΛ s ; (see Equation (2.3)). It follows from (2.27) and (2.2) that I ΛII the operators {QΛ j ⊗ 1l} and {1l ⊗ Qi } are conservation laws of the perturbed dynamics. By Assumption (A1), the limits I

I

r n − lim ϕΛ s (a) =: ϕs (a) r

Λr ∞

II

(2.33)

exist for r = I or II and for all a ∈ F. By Assumption (A3) and (2.32) it follows that (2.34) ϕIs (W ) = ϕII s (W ) = W, in the thermodynamic limit. Energy appears to be the only thermodynamic quantity that can be exchanged through a thermal contact. (J2) Tunnelling junctions. There are m ≤ min (M I , M II ) linear combinaI ΛI ΛII ΛII Λ tions, Q 1 , . . . , Qm and Q1 , . . . , Qm , of conservation laws of the two reservoirs with the property that the operators ΛI ⊗ 1l + 1l ⊗ Q ΛII , := Q QI∪II j j j

(2.35)

j = 1, . . . , m, are conservation laws of the perturbed dynamics generated by the Hamiltonian H of Equation (2.27). Without loss of generality, we may assume that

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ΛI = QΛI and Q ΛII = QΛII , for j = 1, . . . , m. Of course, there may be further Q j j j j

Λ conservation laws of the reservoirs, QΛ i ⊗ 1l, 1l ⊗ Qj , for some i > m and/or some j > m, which are conservation laws of the perturbed dynamics. “Leaky junctions” are contacts where the interaction W (ΛI , ΛII ) violates some of the conservation I ΛII laws QΛ i ⊗ 1l and/or 1l ⊗ Qj , i, j > m. For convenience, we shall sometimes assume that the operators QI∪II , j = 1, . . . , m, are the only conservation laws of j I the perturbed dynamics, and M = M II = m. Let s = (s1 , . . . , sm , 0, . . . , 0). We deﬁne I ΛII (a) := ϕΛ (2.36) ϕI∪II s ⊗ ϕs (a), s I

II

for a ∈ B(HI ) ⊗ B(HII ), and (a), ϕs (a) := n − lim ϕI∪II s

(2.37)

ΛI ∞ ΛII ∞

for a ∈ F; see Assumptions (A1), Equation (2.18). Tunnelling junctions can then be characterized by the requirement that I II

W Λ ,Λ = W ΛI , ΛII , (2.38) ϕI∪II s for arbitrary ΛI , ΛII , and hence, using (2.18) and (2.28), we ﬁnd that, in the thermodynamic limit, (2.39) ϕs (W ) = W. As an initial state of a tunnelling junction we shall usually choose a state ω close to I II a tensor product state, ωβΛI ,µI ⊗ωβΛII ,µII , of two equilibrium states of the uncoupled reservoirs, where β I , µI and β II , µII are arbitrary, (with µI ∈ MI , µII ∈ MII ). Two reservoirs joined by a tunnelling junction can exchange energy and I ΛII “charge” (as measured by the conservation laws QΛ j ⊗ 1l, 1l ⊗ Qj , j = 1, . . . , m), Λ or leak some “charge” corresponding to QΛ j ⊗ 1l, or to 1l ⊗ Qj , for some j > m. I

II

Energy current. The operator corresponding to a measurement of the gain of internal energy per second of reservoir r, with r = I or II, at time t is conveniently deﬁned in the Heisenberg picture by P r (t)

:= =

d I∪II r α (H ) dt t i I∪II α ([H, H r ]), t

(2.40)

is as in (2.29), and H r = H I ⊗ 1l or = 1l ⊗ H II , for r = I or II, where αI∪II t respectively. By (2.25) and (2.27), P r (t) =

i I∪II αt W (ΛI , ΛII ), H r .

(2.41)

908

J. Fr¨ ohlich, M. Merkli and D. Ueltschi

Ann. Henri Poincar´e

By (2.17), (2.28) and (2.31), the operator corresponding to the energy gain per second of reservoir r has a thermodynamic limit given by P r (t) = −

d αt αrs (W ) s=0 , ds

(2.42)

where αrs is the time evolution of reservoir r in the thermodynamic limit, in the absence of any contacts. It follows from (2.41) and (2.42) that P I (t) + P II (t)

i I∪II α W (ΛI , ΛII ), H 0 t

i I∪II α W (ΛI , ΛII ), H = t

d I∪II α W (ΛI , ΛII ) , = − dt t =

(2.43)

where H 0 and H are as in Equations (2.25), (2.27), and, in the thermodynamic limit, d αt (W ), (2.44) P I (t) + P II (t) = − dt with W ∈ F. We observe that if ω is an arbitrary time-translation invariant state of the coupled system, we have that

(2.45) ω P I (t) + P II (t) = 0, for all times. Charge current. The operator corresponding to a measurement of the gain of charge Qrj per second at time t, in reservoir r, is conveniently deﬁned by Ijr (t)

= =

d I∪II r α (Qj ) dt t i I∪II α ([H, Qrj ]), t

(2.46)

II Λ for j = 1, . . . , m, r = I, II, with QIj := QΛ j ⊗ 1l, and Qj := 1l ⊗ Qj . Since H is given by

H = H I ⊗ 1l + 1l ⊗ H II + W ΛI , ΛII , I

II

and since H I ⊗ 1l and 1l ⊗ H II commute with Qri , it follows that Ijr (t) =

i I∪II αt W (ΛI , ΛII ), Qrj .

(2.47)

In the thermodynamic limit, Ijr (t) = −

1 ∂ αt ϕrs (W ) s=0 ∂ sj

(2.48)

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with ϕrs as in (2.33); r = I, II, j = 1, . . . , M r . Recall that, for j = 1, . . . , m, the I ΛII = QΛ are conservation laws of the perturbed operators QI∪II j ⊗ 1l + 1l ⊗ Qj j dynamics, see (2.38), and therefore IjI (t) + IjII (t) =

i I∪II I II I∪II

α W Λ , Λ , Qj = 0, t

(2.49)

for j = 1, . . . , m; this can be transferred to the thermodynamic limit. Apparently, charge lost by one reservoir is gained by the other one. For j > m, (2.49) does not hold in general and the operators Ijr (t), r = I, II, describe the leakage of charge Qrj at the junction.

2.3

Connections with thermodynamics

We start by recalling the 1st and 2nd law of thermodynamics. For the reservoir r, the ﬁrst and second law of thermodynamics can be summarized in the equation dU Λ

r

= T r dS Λ + µr · dq Λ − pr dV r , r

r

(2.50)

where U Λ is the expectation value of the Hamiltonian H r in a state of reservoir r r close to, or in thermal equilibrium, i.e., U Λ is the internal energy of the reservoir r r r; T r the temperature; S Λ the entropy; qjΛ is the expectation value of the charge Qrj , j = 1, . . . , M r , in the state describing the reservoir; pr is the pressure, and V r = vol(Λr ) the volume. The diﬀerential “d” indicates that we consider the r r variation of U Λ , S Λ , etc. under small, reversible changes of the state of reservoir r (which may include small changes of the region Λr ). We shall be interested in studying small, slow changes in time of the state of reservoirs I and II, at approximately ﬁxed values of the thermodynamic parameters T r , µr and pr , brought about by opening a contact or junction between the r r two reservoirs. Then U Λ , S Λ , . . . are time-dependent, and (2.50) becomes r

r r r U˙ Λ = T r S˙ Λ + µr · q˙ Λ − pr V˙ r ,

(2.51)

where the “dots” indicate time derivatives. By (2.40), the energy gain per second, r U˙ Λ , of the reservoir r is given by r U˙ Λ (t) = ω I∪II (P r (t)) ,

(2.52)

and the gain in the jth charge per second by

r q˙jΛ (t) = ω I∪II Ijr (t) ,

(2.53)

see (2.46), where ω I∪II is the state of the system consisting of the two reservoirs. By (2.51), the change in entropy per second of reservoir r is given by (β r := 1/T r )

r r r (2.54) S˙ Λ = β r U˙ Λ − µr · q˙ Λ + pr V˙ r .

910

J. Fr¨ ohlich, M. Merkli and D. Ueltschi

Ann. Henri Poincar´e

We deﬁne the entropy production rate, E I∪II , by I II E I∪II := S˙ Λ + S˙ Λ .

(2.55)

The main property of E I∪II is its sign: thermodynamic systems should exhibit positive entropy production, (2.56) E I∪II ≥ 0, in the limit where ΛI ∞ and ΛII ∞. In [Ru2], Ruelle has proven (2.56) for the special case of thermal contacts between inﬁnitely large reservoirs. Below, we shall derive (2.56), under more general conditions, from the positivity of “relative entropy”; see also [JP1]. We consider a situation in which the state of the system consisting of reservoirs I and II, before a contact or junction is opened, is given by the tensor product of two equilibrium states ω I∪II (a) = ωβΛI ,µI ⊗ ωβΛII ,µII (a), I

II

(2.57)

for any operator a ∈ B(HΛ ) ⊗ B(HΛ ), where the equilibrium states ωβΛr ,µr have been deﬁned in (2.10). The state ω I∪II is invariant under the unperturbed time r evolutions αΛ t , r = I, II, of the reservoirs. At some time t0 , the contact between the reservoirs is opened, and we are interested in the evolution of the state ω I∪II under the perturbed time evolu, introduced in (2.27), (2.29). In particular, we are interested in caltion, αI∪II t r culating the rate of energy gain, or loss, U˙ Λ (t), the gain or loss of charge j, Λr r q˙j (t), j = 1, . . . , M , per second and the entropy production rate E I∪II (t), under the perturbed time evolution, in the state ω I∪II . By Equations (2.40) and (2.41), I

r U˙ Λ (t)

II

= =

r

ω I∪II (P r (t))

i I∪II I∪II ω αt W (ΛI , ΛII ), H r

(2.58)

and, by (2.46), q˙jΛ (t) r

= ω I∪II Ijr (t)

i I∪II I∪II ω = αt W (ΛI , ΛII ), Qrj .

(2.59)

By Assumption (A2), see (2.20), the states ω I∪II have a thermodynamic limit ω 0 (a) =

lim

ΛI ∞ ΛII ∞

ω I∪II (a),

(2.60)

for a ∈ F = F I ⊗ F II . It follows from this property, from Assumption (A3), and r r from equations (2.31), (2.42), and (2.48), that the quantities U˙ Λ (t) and q˙jΛ (t)

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have thermodynamic limits P r (t) :=

lim

Λ ∞ ΛII ∞ I

−

=

r U˙ Λ (t)

d 0 r ω αt αs (W ) s=0 ds

(2.61)

and Jjr (t)

:=

lim

Λ ∞ ΛII ∞ I

−

=

q˙jΛ (t) r

1 ∂ ω 0 αt ϕrs (W ) s=0 . ∂sj

(2.62)

These limits are uniform on compact intervals of the time axis. By (2.43) and (2.49),

d P I (t) + P II (t) = − ω 0 αt (W ) (2.63) dt and JjI (t) + JjII (t) = 0, (2.64) for j = 1, . . . , m. Next, we study the entropy production rate for ﬁnite reservoirs. Let

r −1 r (2.65) ρr := ΞΛ exp − β r H r − µr · QΛ β r ,µr be the density matrix corresponding to the equilibrium state, ωβΛr ,µr , for the reservoir r; see (2.10). Then r (2.66) − ln ρr = β r H r − µr · QΛ − β r G (β r , µr , V r ) · 1l , r

where the thermodynamic potential G is as in (2.9). If the conﬁnement region Λr is kept constant in time, so that V˙ r = 0, then it follows from (2.51), (2.52), (2.53) and (2.66) that

r r r S˙ Λ = β r U˙ Λ − µr · q˙ Λ = =

d I∪II I∪II ω αt (ln ρr ) dt

d (ln ρr ) . − tr ρI ⊗ ρII αI∪II t dt −

(2.67)

Thus, by (2.55), S˙ I∪II (t) := =

E I∪II (t)

d ln ρI ⊗ ρII . − tr ρI ⊗ ρII αI∪II t dt

(2.68)

912

J. Fr¨ ohlich, M. Merkli and D. Ueltschi

Ann. Henri Poincar´e

By (2.61) and (2.62) this quantity has a thermodynamic limit β r [P r (t) − µr · J r (t)] . E(t) =

(2.69)

r=I,II

Integrating (2.68) in time, we ﬁnd that

S I∪II (t) − S I∪II (0) = −tr ρI ⊗ ρII αI∪II ln ρI ⊗ ρII − ln ρI ⊗ ρII . (2.70) t This equation shows that S I∪II (t) − S I∪II (0) is nothing but the relative entropy of the density matrix αI∪II (ρI ⊗ ρII ) with respect to the density matrix ρI ⊗ ρII ; t see, e.g., [BR, vol II] for a deﬁnition of relative entropy, which diﬀers from ours by the sign, and [JP1] for similar, independent considerations. If A is a non-negative matrix and B is a strictly positive matrix then −tr (A ln B − A ln A) ≥ tr (A − B),

(2.71)

(ρI ⊗ ρII ), see Lemma 6.2.21 of [BR, vol II]. Setting A = ρI ⊗ ρII and B = αI∪II t we ﬁnd that

S I∪II (t) − S I∪II (0) ≥ tr ρI ⊗ ρII − αI∪II (ρI ⊗ ρII ) = 0, (2.72) t by the unitarity of time evolution and the cyclicity of the trace. It follows that T

1 1 I∪II S E I∪II (t) dt = (T ) − S I∪II (0) ≥ 0, (2.73) T 0 T and this inequality remains obviously valid in the thermodynamic limit: T 1 E(t) dt ≥ 0 . T 0

(2.74)

Thus, if the limit lim E(t) =: E

t→∞

exists then E = lim

T →∞

1 T

0

T

E(t) dt ≥ 0 ,

(2.75)

(2.76)

i.e., the entropy production rate E, in the thermodynamic limit, is non-negative, as time t tends to ∞; see [Ru2]. In Section 5, we shall study examples where E is strictly positive. Let us assume that the entropy production rate E(t) converges as t → ∞. It follows from (2.74) and (2.76) that it is non-negative. Let us set (2.77) P := P I = −P II and, for j = 1, . . . , m, Jj := JjI = −JjII .

(2.78)

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In the case where each reservoir has precisely m conservation laws {Qrj }m j=1 , r = m I, II (that is, M I = M II = m and {QIj ⊗ 1l + 1l ⊗ QII } are conservation laws of j j=1 the coupled system), non-negativity of the entropy production rate implies that

(2.79) E = β I − β II P − β I µI − β II µII · J ≥ 0 . The currents Jjr vanish for thermal contacts, and (2.79) shows that energy is transferred from the hotter to the colder reservoir – as expected. The thermoelectric situation corresponds to M I = M II = m = 1, and r Q1 = N r is the particle number operator. For identical temperatures but different chemical potentials, (2.79) shows that particles are transferred from the reservoir with the higher chemical potential to the reservoir with the lower chemical potential. Notice that energy may ﬂow from the colder reservoir to the hotter one when the chemical potentials are diﬀerent (consider, e.g., β I µI β II µII but β I > β II ). Also interesting is the case of adiabatic thermal contacts between two reservoirs, i.e., contacts without heat exchange. A general discussion of systems with time-dependent interactions conﬁned to time-dependent regions is given in [FMSU], and will be elaborated upon in a forthcoming paper.

2.4

Existence of stationary states in the thermodynamic limit

The above considerations, and in particular (2.61), (2.62), and (2.75), suggest to study the question whether the inﬁnite-volume states ωt (a) := ω 0 (αt (a)) , a ∈ F ,

(2.80)

have a limit, as t → ∞. The state ω 0 , deﬁned in (2.60), is obviously invariant under the unperturbed time evolution α0t deﬁned in (2.30). Thus

ωt (a) = ω 0 α0−t (αt (a)) , a ∈ F. (2.81) A suﬃcient condition for the existence of a stationary (i.e., time-translation invariant) limiting state, ωstat (a) = lim ωt (a), a ∈ F , t→∞

(2.82)

is given in (A4) Existence of a scattering endomorphism The limits σ± (a) = n − lim α0−t (αt (a)) t→±∞

(2.83)

exist, for all a ∈ F, and deﬁne ∗ endomorphisms of F , i.e., σ± are endomorphisms of the C ∗ -algebra F with the property that σ± (a)∗ = σ± (a∗ ), for all a ∈ F.

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J. Fr¨ ohlich, M. Merkli and D. Ueltschi

Ann. Henri Poincar´e

The usefulness of these so-called scattering (or Møller) endomorphisms has ﬁrst been recognized in [He, Rob]; interesting examples have been constructed in [BM]. In the context of thermal contacts and tunnelling junctions, they have ﬁrst been used in [DFG]; see also [Ma]. It is important to note that scattering endomorphisms do not exist in ﬁnite volume, because the free and the perturbed time evolutions of the two reservoirs are generated by Hamiltonians H0

= H I ⊗ 1l + 1l ⊗ H II ,

H

= H 0 + W (ΛI , ΛII ),

see (2.25) and (2.27), with pure-point spectra when ΛI and ΛII are compact. It is thus natural to wonder about the meaning of scattering endomorphisms for large but ﬁnite reservoirs. Let us sketch the answer to this question. We ﬁx an arbitrarily small, but positive number ε. For every operator a ∈ F, there exist I II compact regions Λr (ε, a), r = I, II, and an operator aε ∈ B(HΛ ) ⊗ B(HΛ ), with Λr = Λr (ε, a), r = I, II, such that a − aε <

ε . 4

Then, by (2.82) and (2.83), there is some T (ε, a) < ∞ such that ωstat (a) − ωt (aε ) < ε , 2 for all t > T (ε, a). Assumption (A3), Equation (2.31), tells us that, for an arbitrary T < ∞, there are compact sets Λr (ε, a, T ) ⊇ Λr (ε, a) such that if Λr ⊃ Λr (ε, a, T ), r = I, II, then ε , αt (aε ) − αI∪II (aε ) < t 4 for all t ∈ [0, T ]. Finally, by Assumption (A2), one can choose Λr (ε, a, T ) so large that I∪II I∪II

ε ω , αt (aε ) − ω 0 αI∪II (aε ) < t 4 provided Λr ⊃ Λr (ε, a, T ), r = I, II, for all times t ∈ [0, T ]. It follows that, for any T , with 0 < T (ε, a) < T < ∞ , and for Λr ⊃ Λr (ε, a, T ), r = I, II,

ωstat (a) − ω I∪II αI∪II (2.84) (aε ) < ε , t for all times t, with T (ε, a) < t < T . These simple considerations, combined with (2.44) and (2.48), show that r r the energy-gain rates U˙ Λ (t) and the currents q˙jΛ (t) of two very large, but ﬁnite reservoirs, r = I, II, are well approximated by the energy-gain rates P r := lim P r (t) = − t→∞

d ωstat (αrs (W )) s=0 ds

(2.85)

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and the currents Jjr := lim Jjr (t) = − t→∞

1 ∂ ωstat (ϕrs (W )) s=0 , ∂sj

(2.86)

respectively, for a large range of suﬃciently large, but not exceedingly large times t; (see (2.61), (2.62)). Remark. It is usually much easier to prove that the limits

(a) := n − lim α−t α0t (a) σ± t→±∞

(2.87)

exist and are operators in F , for arbitrary a ∈ F, rather than to establish the existence of the scattering endomorphisms σ± in (2.83). If the unperturbed dynamics of the reservoirs is dispersive, (as for non-interacting, non-relativistic electrons), one may hope to prove (2.87) by using a simple Cook argument; see, e.g., [He, Rob, CFKS]. If both limits (2.83) and (2.87) exist then

(a) = σ± (σ± (a)) = a , (2.88) σ± σ± is a left and right inverse of σ± , and hence σ± is a ∗ automorphism of F . i.e., σ± This will turn out to hold in the examples discussed in subsequent sections.

2.5

Uniqueness and stability properties of stationary states

We ﬁrst describe the property of return to equilibrium for a single reservoir. Let ω be a state on the ﬁeld algebra of a single reservoir, F r , i.e., ω is a positive, linear functional on F r normalized such that ω(1l) = 1. From F r and ω one can construct a Hilbert space Hω , a representation πω of F r on Hω , and a unit vector Ω ∈ Hω (unique up to a phase) such that Hω = πω (a)Ω a ∈ F r , (2.89) where the closure is taken in the norm on Hω , i.e., Ω is “cyclic” for πω (F r ), and ω(a) = Ω, πω (a) Ω , (2.90) where ·, · is the scalar product on Hω . This is the content of the Gel’fandNaimark-Segal (GNS) construction. If the state ω is time-translation invariant then there exists a one-parameter unitary group Uω (t) t ∈ R on Hω such that πω (αt (a)) = Uω (t) πω (a) Uω (t)∗ , and Uω (t) Ω = Ω .

(2.91)

Under standard continuity assumptions on Uω (t), we can summon Stone’s theorem to conclude that (2.92) Uω (t) = eit Lω / , where the generator Lω is a selfadjoint operator on Hω with Lω Ω = 0.

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Ann. Henri Poincar´e

A state ρ on F r is called normal relative to ω iﬀ there exists a density matrix, P , on Hω such that ρ (a) = trHω (P πω (a)) , (2.93) for all a ∈ F r . Let ω := ωβ,µ be an inﬁnite-volume equilibrium state on F r obeying the KMS condition (2.22). We assume that the Hilbert space Hω obtained from the GNS construction is separable and that the cyclic vector Ω ∈ Hω is the only eigenvector (up to phases) of the operator Lω of (2.92), which, in this context, is called the Liouvillian or thermal Hamiltonian. In other words, the spectrum of Lω is purely continuous, except for a simple eigenvalue at 0. This assumption implies the property of “return to equilibrium”: If ρ is an arbitrary state normal relative to ω = ωβ,µ then 1 T →∞ T

lim

0

T

dt ρ (αt (a)) = ω(a),

(2.94)

for all a ∈ F r . If the spectrum of Lω is absolutely continuous, except for a simple eigenvalue at 0, then (2.95) lim ρ (αt (a)) = ω(a), t→∞

for all a ∈ F . Equations (2.94) and (2.95) follow from our assumptions on the spectrum of Lω and the KMS condition (2.22); see, e.g., [BFS]. Assuming the existence of the endomorphism σ+ , see (A4), we now address the question of uniqueness and dynamical stability of the stationary state ωstat in (2.82). We suppose that the property of return to equilibrium, (2.95), holds for each reservoir separately; (this can be shown for reservoirs consisting of free fermions as considered in Sections 3 and 4). Recalling that the reference state ω 0 is a product of two KMS states (see (2.60), (2.57)), it is not diﬃcult to extend the arguments in Section III, D of [BFS] to show that if ρ is an arbitrary state on F normal relative to the state ω 0 in (2.60) then

lim ρ α0t (a) = ω 0 (a), a ∈ F , (2.96) r

t→∞

where α0t is the time evolution of the reservoirs before they are coupled; see (2.30). Equation (2.96) and the existence of a scattering endomorphism σ+ , see Equation (2.83), now imply that lim ρt (a) =

=

lim ρ(αt (a))

lim ρ α0t σ+ (a) t→∞

ω 0 σ+ (a)

=

ωstat (a) ,

t→∞

=

t→∞

(2.97)

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for a ∈ F. Thus, if the initial state ρ is an arbitrary state normal relative to ω 0 then the states ρt tend to the stationary state ωstat , as t → ∞, (uniqueness). Next, let ρ be an arbitrary state normal relative to ωstat . We claim that if Equation (2.95) holds for each reservoir then lim ρt (a) = ωstat (a), a ∈ F.

t→∞

(2.98)

Equation (2.98) is the property of “return to the stationary state” (stability), for states ρ normal relative to ωstat . To prove Equation (2.98), we follow the arguments in Section III, D of [BFS]: Since ρ is normal relative to ωstat , there exist non∞ negative numbers, pn , n = 1, 2, 3, . . ., with n=1 pn = 1, and nets of operators α {uα n }α∈In , n = 1, 2, 3, . . ., with un ∈ F, for all α and all n, such that ρ(a) =

∞ n=1

α pn lim π uα n Ω, π(a) π un Ω α,α

where π is the GNS representation and Ω the cyclic vector corresponding to (ωstat , F ); see (2.89), (2.90), and (2.93). Then ρt (a) =

∞ n=1

Let

α

pn lim π uα n Ω, π αt (a) π un Ω . α,α

α b1 := uα n , b2 := un ,

for some ﬁxed n, α, α . Then

π(b1 )Ω, π αt (a) π(b2 )Ω = ωstat b∗1 αt (a)b2 .

(2.99)

Since ωstat (a) = ω 0 σ+ (a) , and since, by Equation (2.83),

σ+ αt (a) = α0t σ+ (a) ,

(2.100)

for arbitrary a ∈ F, the right side of (2.99) is given by

ωstat b∗1 αt (a)b2 = ω 0 σ+ (b∗1 ) α0t σ+ (a) σ+ (b2 ) . It follows from (2.96) by polarization that

lim ω 0 σ+ (b∗1 ) α0t σ+ (a) σ+ (b2 ) = ω 0 σ+ (b∗1 ) σ+ (b2 ) ω 0 σ+ (a) t→∞

= ωstat b∗1 b2 ωstat (a) = π(b1 )Ω, π(b2 )Ω ωstat (a) . Our contention, Equation (2.98), follows from this.

(2.101)

918

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Cluster properties and profiles in d>2 dimensional systems

The last question we wish to address, in this summary of the general theory, concerns cluster properties of the stationary state ωstat , which will show that ωstat cannot be an equilibrium (KMS) state for the dynamics, αt , of the coupled reservoirs and is, in general, not normal relative to the product state, ω 0 , of the uncoupled reservoirs. We consider two increasing families of reservoirs conﬁned to regions Λr ⊂ R3 , with (2.102) Λr R3 , r = I, II , joined together by a thermal contact or a tunnelling junction localized near the origin, x = 0, of physical space. The more realistic situation where the reservoirs are conﬁned to two complementary half-spaces, R3+ and R3− , respectively, with a junction localized near the origin, has been considered in [DFG]; see also [Ru1, Ru2]. It will be studied in more detail elsewhere. In order to describe spatial properties of the system, we make the following assumption. (A5) Existence of space translations For each reservoir r = I, II, there exists a ∗ automorphism (semi-) group τxr x ∈ R3(±) (2.103) of the ﬁeld algebra F r , representing space translations of R3 (R3± , respectively) on F r . For the system of two coupled reservoirs in R3 , τx := τxI ⊗ τxII , x ∈ R3 ,

(2.104)

deﬁnes a representation of space translations as a 3-parameter group of ∗ automorphisms on the ﬁeld algebra F = F I ⊗ F II . It is plausible that space translations satisfy the following assumption. (A6) Asymptotic abelianness of space translations, and homogeneity of reservoirs The action of τx on F is norm-continuous in x ∈ R3 and for all operators a and b in F , (2.105) lim [τx (a), b] = 0 . |x|→∞

Furthermore, the dynamics and the equilibrium states of the uncoupled reservoirs are homogeneous, in the sense that

(2.106) α0t τx (a) = τx α0t (a) and for all a ∈ F.

ω 0 τx (a) = ω 0 (a) ,

(2.107)

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The local nature of the perturbation, W , of the dynamics of the system due to the contact or junction, see Assumption (A3), Equations (2.27) and (2.28), and Assumption (A6), then imply that, for all a ∈ F,

(2.108) lim αt τx (a) − α0t τx (a) = 0 , |x|→∞

for all times t. A proof of (2.108) follows from the Lie-Schwinger series for α−t (α0t (a)) and use of (2.28) and (2.105). Relation (2.108) shows that observables localized far from the junction evolve according to the non-interacting dynamics. It is tempting, and can be justiﬁed in examples, to strengthen Assumption (A4) (existence of scattering endomorphism) as follows. (A7) Cluster properties of the scattering endomorphism The limits

n − lim α0−t αt τx (a) = σ± τx (a) t→±∞

(2.109)

are uniform in x ∈ R3 , for every a ∈ F. Equations (2.108) and (2.109) imply that

lim σ± τx (a) − τx (a) = 0 , |x|→∞

(2.110)

for every a ∈ F. From this property we conclude that

lim ω 0 σ+ τx (a) lim ωstat τx (a) = |x|→∞ |x|→∞

= lim ω 0 τx (a) |x|→∞

=

ω 0 (a), a ∈ F ,

(2.111)

i.e., very far from the junction, the stationary state ωstat resembles the product state ω 0 of the uncoupled reservoirs. Remark. It is not hard to understand that if the two reservoirs occupy complementary half-spaces, R3+ and R3− , then (2.111) is replaced by

lim ωstat τ(x,0,0) a ⊗ 1l = ωβ I ,µI (a) , x→−∞

for a ∈ F I , and lim

x→+∞

ωstat τ(x,0,0) 1l ⊗ b = ωβ II ,µII (b) ,

for b ∈ F II . This may prove the presence of a proﬁle of temperature or density, in the stationary state, ωstat , of the system.

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In the examples studied in subsequent sections, which concern junctions between ordinary three-dimensional metals, Assumptions (A6) and (A7) can be veriﬁed. Instead of three-dimensional reservoirs, we could consider one-dimensional (wires), or two-dimensional (layers) reservoirs joined by a thermal contact or a tunnelling junction. Our analysis in Section 4 will show that, in dimensions d = 1, 2, Assumption (A7) will fail, in general. In fact, the example of the one-dimensional XY spin chain treated in [DFG, AP] and the example of quantum wires studied in [ACF] show that, in one dimension, ωstat may well be space-translation invariant, i.e., it does not exhibit any proﬁle. In the example of quantum wires, ωstat is actually a homogeneous thermal equilibrium state. Thus, we observe that the validity of Assumption (A7) critically depends on the dimension of the reservoirs. In dimension d > 2, this assumption can be expected to hold, while it usually fails in dimension d = 1, 2. For people familiar with elementary facts of scattering theory this will not come as a surprise. In a subsequent paper, we will show that, for a large class of reservoirs, one can construct “observable at inﬁnity”, see, e.g., [BR, Vol. II], corresponding to the operators P r (t) and Ijr (t) deﬁned in Equations (2.42), (2.48), respectively. Clearly, the expectation values of these operators vanish in the product state ω 0 of the uncoupled reservoirs and are given by P r and Jjr in the stationary state, ωstat , of the coupled reservoirs. If we can show that P r = 0, or that Jjr = 0, for r = I or II and some j, then it follows that ωstat is not normal relative to ω 0 . In the examples studied in Sections 4 and 5, we shall encounter instances where P r and J r do not vanish.

3 Reservoirs of non-interacting fermions This section serves to introduce a class of simple, but physically important examples of reservoirs to which the general theory outlined in Section 2 can and will be applied. Our examples describe a quantum liquid of non-interacting, nonrelativistic electrons in a normal metal or a semi-conductor, possibly subject to an external magnetic ﬁeld, or an ideal quantum gas of fermionic atoms or molecules. In a subsequent paper, we shall also consider examples describing chiral Luttinger liquids, which arise in connection with the quantum Hall eﬀect. We start by considering a system consisting of a single, non-relativistic quantum-mechanical particle conﬁned to a region Λ of physical space Rd , d = 1, 2, 3. The Hilbert space of pure state vectors of this system is given by the space

L2 Λ, dd x

(3.1)

of square-integrable wave functions with support in Λ. If the particle has spin and/or if there are several species of such particles then L2 (Λ, dd x) must be replaced by the space

(3.2) hΛ := L2 Λ, dd x ⊗ Ck ,

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where k = lα=1 (2Sα + 1), Sα is the spin of species α, and l is the number of species. The one-particle dynamics is generated by the following selfadjoint operator, tΛ , acting on hΛ , 2 ∆ ⊗ 1l , (3.3) tΛ = − 2M where M is the mass of the particle and ∆ is the Laplace operator on L2 (Λ, dd x) with selfadjoint boundary conditions (e.g., Dirichlet, Neumann, or periodic) imposed at the boundary, ∂Λ, of Λ. In the following, we choose units in which = 1 and M = 12 . Other operators are physically interesting. Electrons in semi-conductors would involve a potential operator that is diagonal in the space representation. A magnetic ﬁeld could also be considered; the Laplacian should be replaced by the covariant Laplacian, and a coupling between the spin of the particle and the magnetic ﬁeld should be introduced. In this paper, we restrict our attention to the situation (3.3). Next, we consider a system consisting of n identical particles of the kind just considered, all conﬁned to the region Λ. Its state space is given by a subspace of the n-fold tensor product of hΛ of ﬁxed symmetry type, Λ ⊗n hΛ , hΛ n := P (h ) 0 := C ,

(3.4)

where P is the orthogonal projection onto the subspace of wave functions of the selected symmetry type under permutations of the n particle variables. If the particles are bosons then P ≡ P+ projects onto completely symmetric n-particle wave functions; while, for fermions, P ≡ P− projects onto totally anti-symmetric wave functions. In this paper, we focus our attention on fermions. If the particles do not interact with each other the Hamiltonian, TnΛ of the n-particle system is given by TnΛ

:=

n

1l ⊗ · · · ⊗ tΛ j ⊗ · · · ⊗ 1l ,

(3.5)

j=1 th where tΛ factor in the n-fold tensor product in (3.4). j acts on the j If the number of particles can ﬂuctuate (e.g., because the system is coupled to a particle reservoir such as a battery) then it is convenient to use the formalism of “second quantization”, which we brieﬂy recall. The Fock space is deﬁned by

HΛ :=

∞

hΛ n .

(3.6)

n=0

The free dynamics on HΛ is generated by the Hamiltonian H Λ :=

∞ n=0

TnΛ ,

(3.7)

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with TnΛ as in (3.5). The particle number operator, N Λ , is deﬁned by N Λ :=

∞

n · 1lhΛ .

n=0

(3.8)

n

Let κ be a symmetric k × k matrix acting on Ck . We set KnΛ :=

n

1l ⊗ · · · ⊗ 1l ⊗ κj ⊗ · · · ⊗ 1l ,

j=1

where 1l⊗ κj acts on the j th factor, L2 (Λ, dd x)⊗ Ck , in the n-fold tensor product Λ Λ Λ deﬁning hΛ n . For t as in (3.3), a typical charge operator, Q ≡ Q (κ), is of the form ∞ QΛ := KnΛ . (3.9) n=0

The operators H Λ , N Λ and QΛ (κ) are unbounded, selfadjoint operators on Λ H ; see, e.g., [RS]. Next, we describe the structure of HΛ in some more detail and introduce creation and annihilation operators. Let x, y, . . . denote points in physical space Rd , and let s = 1, . . . , k label an orthonormal basis in Ck . Vectors fn in the n-particle space hΛ n can be represented as square-integrable wave functions, fn (x1 , s1 , . . . , xn , sn ) , with support in Λ ⊂ R , which, for fermions, are totally anti-symmetric under permutations of their n arguments. Vectors ψ, φ, . . . in Fock space correspond to sequences, ∞ (3.10) ψ = (fn )∞ n=0 , φ = (gn )n=0 , . . . dn

dn

Λ of n-particle wave functions in hΛ n . The scalar product on H is deﬁned by ∞ n ψ, φ := dxj fn (x1 , s1 , . . . , xn , sn ) gn (x1 , s1 , . . . , xn , sn ) . n=0 s1 ,...,sn

Λn j=1

(3.11) The vector represented by the sequence (fn )∞ n=0 , with f0 = 1, fn ≡ 0, for n ≥ 1, is denoted by Ω and is called the vacuum (vector). Λ Let D = DΛ be the linear domain of vectors ψ = (fn )∞ n=0 in H with the property that all but ﬁnitely many fn ’s vanish. Clearly, D is dense in HΛ . For f ∈ hΛ , we deﬁne an annihilation operator, a(f ), by

a(f )ψ n x1 , s1 , . . . , xn , sn := k √

n+1 dx f (x, s) fn+1 x, s, x1 , s1 , . . . , xn , sn , (3.12) s=1

for arbitrary ψ =

(fn )∞ n=1

Λ

∈ D, and a(f )Ω := 0 .

(3.13)

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The creation operator, a∗ (f ), is deﬁned to be the adjoint of a(f ) on HΛ and is easily seen to be well deﬁned on D. It is well known, see, e.g., [RS, BR], that, for fermions, the following “canonical anti-commutation relations” (CAR) hold: # a (f ), a# (g) = 0 , (3.14) for arbitrary f, g in hΛ where a# = a or a∗ , and {A, B} := AB + BA is the anti-commutator of two operators A and B; {a(f ), a∗ (g)} = (f, g) · 1l ,

(3.15)

where (f, g) := s Λ dx f (x, s) g(x, s) is the scalar product on hΛ . For bosons, (3.14) and (3.15) hold if anti-commutators are replaced by commutators (CCR). Formally, dx f (x, s) a(x, s), and a∗ (f ) = dx a∗ (x, s) f (x, s) , a(f ) = s

with

Λ

s

Λ

{a(x, s), a∗ (x , s )} = δss δ (d) (x − x ) .

(3.16)

A remarkable consequence of the CAR is that the operators a(f ) and a∗ (f ) are bounded in norm by a(f ) = a∗ (f ) = f := (f, f ) . (3.17) To see this, we choose an arbitrary ψ ∈ D and note that a(f )ψ2 + a∗ (f )ψ2 = a(f )ψ, a(f )ψ + a∗ (f )ψ, a∗ (f )ψ = ψ, {a(f ), a∗ (f )} ψ = (f, f ) ψ, ψ , so that

a# (f )ψ ≤ f · ψ .

(3.18)

Equality in (3.18) is seen from examples. Equation (3.17) is false for bosons, a(f ) and a∗ (f ) being unbounded operators. For fermions, polynomials in a(f ), a∗ (f ), f ∈ hΛ , form a ∗ algebra of operators Λ on H which is weakly dense in B(HΛ ). The “observable algebra” AΛ is the norm closure of the algebra of these polynomials in a(f ), a∗ (f ), f ∈ hΛ , which commute with the number operator N Λ and, possibly, with further charge operators QΛ (κ), for certain choices of κ. Every monomial in a and a∗ belonging to AΛ has equally many factors of a and a∗ , since it must conserve the total particle number. A general monomial in a and a∗ is Wick-ordered if all a∗ ’s are to the left of all a’s.

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In terms of creation and annihilation operators, the operators H Λ , N Λ and Q can be expressed as follows. dx a∗ (x, s)(tΛ a)(x, s) , (3.19) HΛ = Λ

NΛ = and Λ

Q (κ) =

s

Λ

s

Λ

s,s

Λ

dx a∗ (x, s) a(x, s) ,

(3.20)

dx a∗ (x, s) κss a(x, s ) .

(3.21)

In the examples discussed below and in Sections 4 and 5, we usually regard N Λ = QΛ (κ = 1l) to be the only conservation law, besides H Λ , relevant for the description of the reservoirs. In a general discussion, we consider M conservation Λ Λ laws, QΛ j = Q (κj ), j = 1, . . . , M , and choose t as in (3.3). The main result of this section is the following theorem. Λ Theorem 3.1 For tΛ as in (3.3), and QΛ j = Qj (κj ), j = 1, . . . , M , with κ1 , . . . , κM Λ arbitrary, commuting symmetric k × k matrices, the equilibrium states ωβ,µ introM duced in Equation (2.10) exist, for arbitrary β > 0 and µ ∈ R . Assumptions (A1), (A2), (A5) and (A6) of Section 2, concerning the existence of the thermodynamic limit, Λ Rd , hold.

The proof of Theorem 3.1 is standard. A careful exposition can be found in [BR], Section 5.2. In Section 4 we shall consider a system consisting of two identical reservoirs, I and II, both composed of non-interacting, non-relativistic fermions conﬁned to some region Λ = ΛI = ΛII of Rd . A convenient notation for creation and annihilation operators for the two reservoirs is the following one. a# (x, s, I) := a# (x, s)HΛI ⊗ 1lHΛII , a# (x, s, II) := 1lHΛI ⊗ a# (x, s)HΛII . (3.22) We note that all the operators a# (f, I) = a# (f )⊗1l commute with all the operators a# (f, II) = 1l ⊗ a# (f ). If, for convenience, we prefer that they anti-commute we can accomplish this feature by a standard Klein-Jordan-Wigner transformation: a# (f, I) a# (f, II)

→ a# (f, I) , → a# (f, II) eiπ(N

ΛI

⊗1l)

.

(3.23) #

The operators on the right side of (3.23) will again be denoted by a (f, r), r = I, II. We introduce the following notation. X := (x, s, r) ∈ Rd × {1, . . . , k} × {I, II}, X (N ) := (X1 , . . . , XN ) ,

(3.24) (3.25)

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x(N ) := (x1 , . . . , xn ) ∈ RdN , s

(N )

(3.26)

:= (s1 , . . . , sN ) ∈ {1, . . . , k} , N

(N )

:= (r1 , . . . , rN ) ∈ {I, II} , k dX := dx, r

N

Λ

r=I,II s=1

dX (N ) := ΛN

N

j=1

a# (X (N ) ) :=

N

Λ

925

(3.27) (3.28) (3.29)

Λ

dXj ,

(3.30)

a# (Xj ) ,

(3.31)

j=1

with a# = a∗ or a. We are now prepared to describe the interactions, W (ΛI , ΛII ), (see Equation (2.27)), corresponding to thermal contacts or tunnelling junctions between the two reservoirs. We shall always assume that the total particle number of the system consisting of the two reservoirs is conserved. Thus the interaction Hamiltonian W (ΛI , ΛII ) must commute with the operator N I∪II := N Λ ⊗ 1l + 1l ⊗ N Λ I

II

,

(3.32)

(see Equation (2.35)). It follows that W (ΛI , ΛII ) must have the form W (ΛI , ΛII ) =

∞

WN (ΛI , ΛII ) ,

(3.33)

N =1

where

I

II

WN (Λ , Λ ) =

dX ΛN

(N )

ΛN

Λ dY (N ) a∗ (X (N ) ) wN

I

,ΛII

(X (N ) , Y (N ) ) a(Y (N ) ) ,

, and, for each choice of s1 , r1 , . . ., sN , rN , s1 , r1 , . . ., sN , rN ΛI ,ΛII (N ) (N ) (N ) (N ) (N ) (N )

wN ,r x ,s ,r , y ,s

(3.34)

is a smooth function of x(N ) ∈ ΛN and y (N ) ∈ ΛN vanishing if x(N ) ∈ ΛN or y (N ) ∈ ΛN . In Section 4, we introduce weighted Sobolev spaces, WN equipped with norms || · ||N with the property that Λ WN (ΛI , ΛII ) ≤ ||wN

I

,ΛII

||N ,

(3.35)

for all N = 1, 2, 3, . . .. We shall assume that, for each N , there is a function wN ∈ WN such that ∞ g(w) := ||wN ||N < ∞ , (3.36) N =1

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where w = (wN )∞ N =1 , and lim

∞

Λ ∞ N =1 ΛII ∞ I

Λ ||wN

I

,ΛII

− wN ||N = 0 .

(3.37)

It then follows that n − lim W (ΛI , ΛII ) =: W

(3.38)

W ≤ g(w) .

(3.39)

ΛI ∞ ΛII ∞

exists, and Thus, Assumption (A4), Equation (2.29), of Section 2 follows from (3.34), (3.35), (3.36) and (3.37). If we want to describe thermal contacts we shall require that [N Λ ⊗ 1l, W (ΛI , ΛII )] = [1l ⊗ N Λ , W (ΛI , ΛII )] = 0 , I

II

(3.40)

while, for tunnelling junctions, only [N I∪II , W (ΛI , ΛII )] = 0

(3.41)

is required, for arbitrary ΛI = ΛII = Λ ⊂ Rd . To conclude this section, we remark that a system of two reservoirs of noninteracting fermions, with a one-particle Hamiltonian tΛ as in Equation (3.3), and with interactions W (ΛI , ΛII ) as in (3.34)–(3.38), satisﬁes Assumptions (A1)–(A3) and (A5), (A6) of Section 2. Assumptions (A4) and (A7) are established in the next section for d ≥ 3 and small g(w).

4 Existence of Møller endomorphisms The goal of this section is to illustrate the general theory of Section 2 by providing a complete mathematical description of a concrete system, namely two coupled reservoirs of free fermions in dimension d ≥ 3. An illustration can be found in Fig. 1. The reservoirs are inﬁnite and without boundary, and the coupling W is localized near the origin, in the the sense that W , or W , is ﬁnite (see (4.3), (4.4), or (4.25)). The Hamiltonian for each reservoir has been introduced in Section 3, see Equation (3.7). The coupling between reservoirs is represented by the interaction in Equations (3.33) and (3.34). As stated in Theorem 3.1, there exist time evolution automorphisms α0t and αt in the thermodynamic limit. The former corresponds to the free dynamics and the latter to the dynamics for interacting reservoirs. In this section, we establish the existence of the Møller endomorphisms σ± deﬁned in (2.83). We start with the Dyson series for α0−t αt , namely ∞ 0 m α−t αt (a) = i dt1 . . . dtm [W (tm ), . . . , [W (t1 ), a] . . . ], (4.1) m=0

t>tm >···>t1 >0

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I W

II Figure 1: Two-dimensional reservoirs coupled by a local interaction W . We actually consider the three-dimensional analogue of this situation. where we set W (t) := α0−t (W ). Convergence of this series for ﬁnite t is clear since W < ∞, but we need to consider the limit t → ∞. Let us deﬁne an operator Dm (t) on the ﬁeld algebra F by Dm (t)a := dt1 . . . dtm [W (tm ), . . . , [W (t1 ), a] . . . ], (4.2) t>tm >···>t1 >0

for arbitrary a ∈ F. It is understood that D0 a = a. We deﬁne the norm of an interaction W by setting W =

N 2(d+2)N

k

k

s1 ,...,sN =1 s1 ,...,sN =1

N ≥1

(N ) (N )

wN (·, s(N ) , r(N ) ), (·, s , r ) 2dN ; (4.3)

=I,II r1 ,...,rN =I,II r1 ,...,rN

the function wN is viewed above as a function on R2dN , and the norm · M is deﬁned by f M =

1 23M/2

dx(M) f (x(M) ) RM

1/2 M 3 d2 − 2 + x2k + 1 f (x(M) ) . dxk

(4.4)

k=1 2

d 2 Note that the operators − dx 2 + xk + 1 are bounded below by 2, which implies the k inequality (3.35), namely f L2 (RM ) ≤ f M ; this inequality is saturated when f −x2k /2 is a product of Gaussians centered at the origin, f (x(M) ) = M . k=1 e

Theorem 4.1 For d ≥ 3 we have the bound ∞ 1 8πd m m [W (t), Dm−1 (t)a]dt ≤ a (W ) . m d − 2 0

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A similar statement holds for negative times. That is, one can rewrite the Dyson series in (4.1) and the operator Dm (t) in (4.2) for t < 0 by integrating over negative times 0 > t1 > · · · > tm > t. Then Theorem 4.1 holds with an integral from −∞ to 0. The proof for negative times is identical to the one for positive times. Before proving Theorem 4.1, let us work out its main consequence, the existence of Møller endomorphisms. Corollary 4.2 If

8πd d−2 W

< 1, there exist σ± such that lim α0−t αt (a) − σ± (a) = 0

t→±∞

for all a with a < ∞. Corollary 4.2 implies the existence of a scattering automorphism, see Assumption (A4). We comment below that the norm (4.4) can be replaced by an object that is translation invariant, see (4.25). Therefore the scattering automorphism is given by a limit of inﬁnite times, and this limit exists in norm, uniformly with respect to space translations (for both the interaction and the operator a ∈ F). Hence Assumption (A7) holds. Proof of Corollary 4.2. Observe that α0−t αt (a) = a +

im

m≥1

0

t

[W (s), Dm−1 (s)a]ds.

(4.5)

Then, for t < t , α0−t αt (a)

−

α0−t αt (a)

≤

m≥1

t

[W (s), Dm−1 (s)a]ds.

t

By Theorem 4.1 and the dominated convergence theorem, the right side vanishes as t, t → ∞. This implies the norm-convergence of α0−t αt (a). We will make use of Hermite functions in the proof of Theorem 4.1; so we collect a few useful facts on them. The Hermite functions are denoted by {φq }q∈N , where (−1)q 1 x2 d q −x2 2 e , φq (x) = √ 1 e dx 2q q!π 4 with x ∈ R. These functions satisfy the equation

d2 − 2 + x2 φq (x) = (2q + 1)φq (x). dx Lemma 4.3 (i) φq 1 ≤ 4π(q + 1). (ii) |(eit∆ φp , φq )| ≤

φp 1 φq 1

√

4π|t|

.

(4.6)

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2

d Proof. By Cauchy-Schwarz and since the operator − dx 2 is positive deﬁnite, we have that ∞ 1 dx φq 1 = x2 + 1 |φq (x)| √ 2 x +1 −∞

d2 1/2 ∞ dx 1/2 ≤ φq , − 2 + x2 + 1 φq . (4.7) 2 dx −∞ x + 1 √ The ﬁrst factor is equal to 2(q + 1) and the second one equals 2π, which proves (i). Claim (ii) immediately follows from ∞ i ∞

2 it∆ dx dye−i(x−y) /4t φp (y)φq (x). e φp , φq = 4πt −∞ −∞

Hermite functions form an orthonormal basis of L2 (R). We use them to express the interaction as a polynomial of creation and annihilation operators of fermions in states described by Hermite functions. The free time evolution of the interaction can be described as an evolution of these functions, and their decorrelation in time can be controlled using Lemma 4.3 (ii). Finally, Hermite functions will be removed at the expense of introducing diﬀerential operators in the deﬁnition of the norm of the interaction; see (4.4). We use from now on the following notation: for q = (q1 , . . . , qd ) ∈ Nd and x = (x1 , . . . , xd ) ∈ Rd , d φqi (xi ). (4.8) φq (x) = i=1

Proof of Theorem 4.1. We start by rewriting the interaction in the basis of Hermite functions. Let Q(N ) := (Q1 , . . . , QN ), with Qj = (q j , sj , rj ) ∈ Nd × {1, . . . , k} × {I, II}.

(4.9)

We set w ˜N (Q

(N )

,Q

(N )

)=

(N )

dx RdN

RdN

dy (N ) wN (X (N ) , Y (N ) )

N

φqj (xj )φqj (y j ).

j=1

(4.10) Here, X (N ) is determined by Q(N ) and x(N ) , namely Xj = (xj , sj , rj ) if Qj = (q j , sj , rj ). The interaction (3.33), (3.34) is given by a sum over (N )

W =

Q

WQ

Q := (N, Q(N ) , Q ), (N ) (N ) = a∗ (Q(N ) )w ˜N (Q(N ) , Q )a(Q ), Q

(4.11)

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J. Fr¨ ohlich, M. Merkli and D. Ueltschi

where a∗ (Q(N ) ) =

N j=1

Ann. Henri Poincar´e

a∗ (Qj ), and a∗ (Qj ) is the creation operator for a fermion

in the state φq j , of spin sj , in the reservoir rj . The annihilation operators a(Q (N ) ) are deﬁned similarly. The operator a also has a Hermite expansion (N ) (N ) aQ = a∗ (Q(N ) )˜ aN (Q(N ) , Q )a(Q ). (4.12) a= Q

Q

With this notation, we have that 0

∞

[W (t), Dm−1 (t)a]dt ≤

Q0 ,...,Qm

∞>tm >···>t1 >0

dt1 . . . dtm

[WQm (tm ), . . . , [WQ1 (t1 ), aQ0 ] . . . ]. (4.13) The multiple commutator above involves operators W and a, which in turn involve creation and annihilation operators of particles in both reservoirs. The latter satisfy anticommutation relations for particles in the same reservoir, or commutation relations for particles in diﬀerent reservoirs. This introduces a complication when estimating the multicommutator above. This complication can be avoided by using the Klein-Jordan-Wigner transformation explained in (3.23). For simplicity, we keep the same notation, but we assume from now on that all creation and annihilation operators satisfy anticommutation relations. Because α0−t is a ∗-automorphism, its action on the interaction W simply amounts to replacing operators a# (Q) = a# (φq , s, r) by α0−t (a# (φq , s, r)) = a# (eit∆ φq , s, r) := a# (Q, t).

(4.14)

We note that Lemma 4.3 yields the bound 0 α−t (a# (Q)), α0−t (a# (Q )) = |(eit∆ φq , eit ∆ φq )|δss δrr ≤ 1∧

d/2 4π |t−t |

d 1 1 (qi + 1) 2 (qi + 1) 2 .

(4.15)

i=1

The multicommutator in (4.13) can be written as [WQm (tm ), . . . , [WQ1 (t1 ), aQ0 ] . . . ] = m (N ) (N ) (N ) (N ) (N ) (Nm ) a ˜N0 (Q0 0 , Q0 0 ) w ˜Nj (Qj j , Qj j ) a∗ (Qm , tm )a(Qm m , tm ), . . . j=1

(N ) (N ) (N ) (N ) . . . , a∗ (Q1 1 , t1 )a(Q1 1 , t1 ), a∗ (Q0 0 )a(Q0 0 ) . . . . (4.16) Here, we set (Nj )

a# (Qj

, tj ) =

Nj =1

(Nj )

where Qj, is the -th element of Qj

.

α0−tj (a# (Qj, )),

(4.17)

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A commutator of products of operators can be expanded according to contraction schemes. The following equation holds when k is even: [a1 . . . ak , b1 . . . b ] = (−1)i+j+1 a1 . . . ai−1 b1 . . . bj−1 1≤i≤k 1≤j≤

{ai , bj }bj+1 . . . b ai+1 . . . ak . (4.18)

The multicommutator of (4.16) can thus be expanded in contraction schemes for operators at diﬀerent times. An operator at time t1 contracts necessarily with an operator at time t0 = 0; an operator at time t2 contracts with an operator at time tr2 with r2 = 0, 1; . . . ; an operator at time tm contracts with an operator at time trm with rm = 0, . . . , m − 1. See Fig. 2 for an illustration. To a set of contrac-

m rm = m−2

m−1 rm−1 = 1

m−2

...

rm−2 = 3

3

r3 = 1

2

1

0

r2 = 0

Figure 2: Illustration for the numbers rm , . . . , r2 that occur in the choice of contractions. We see that they deﬁne a tree. tions corresponds a monomial of creation and annihilation operators, multiplied by anticommutators of contracted operators. The monomial of creation and annihilation operators is bounded in operator norm by 1. Contracted operators are estimated using (4.15). This yields a factor involving times, namely m

d/2 4π 1 ∧ tj −t . r j

j=1

Second, one obtains a factor involving indices of Hermite functions for the contracted operators. An upper bound on this factor is obtained by writing a product over all indices, namely Nj d m

1

1

(qjki + 1) 2 (qjki + 1) 2 .

j=0 k=1 i=1

Here, q jk , q jk ∈ Nd are indices for Hermite functions determined by the k-th element of Qj . It remains to estimate the number of contraction schemes, given r2 , . . . , rm . We deﬁne (4.19) ej = {k : rk = j} + 1 − δj0 , 0 ≤ j ≤ m. Notice that 1 ≤ e0 ≤ m and 1 ≤ ej ≤ m − j + 1 if j = 0. ej is the number of operators at time tj that belong to a contraction and it is necessarily smaller than 2Nj .

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Since there are 2Nj operators at time tj , the number of possible contractions is m

(2Nj )! . (2N j − ej )! j=0 The above estimates could be improved by observing that many contraction schemes yield zero; namely, in the case where both operators are creation or annihilation operators; or if the spins are diﬀerent; or if they belong to diﬀerent reservoirs. It is not easy to take advantage of this, however. We now gather the above estimates to obtain the bound

∞

0

[W (t), Dm−1 (t)a]dt ≤

(N0 )

|˜ aN0 (Q0

, Q0

(N0 )

)|

Q0 ,...,Qm

×

m

Nj d m

m−1

, Qj

(Nj )

)|

j=1

(qjki + 1)

1 2

(qjki

+ 1)

1 2

∞>tm >···>t1 >0

j=0 k=1 i=1

×

(Nj )

|w ˜Nj (Qj

m−2

···

rm =0 rm−1 =0

1 m 1∧ r2 =0 j=1

4π tj −trj

dt1 . . . dtm

m

d/2 χ[ej ≤ 2Nj ](2Nj )! . (4.20) (2Nj − ej )! j=0

A sequence of numbers r2 , . . . , rm can be represented by a graph with set of vertices {0, . . . , m}, and an edge between i and j whenever rj = i. This graph is a tree: there are m edges, and each vertex j = 0 is directly connected to a vertex i < j, hence each vertex is eventually connected to 0. The numbers ej deﬁned in (4.19) are then the incidence numbers of the tree – ej is the number of edges containing the vertex j. This is illustrated in Fig. 2. We can symmetrize the bound by summing over all trees T with m + 1 vertices; this step will allow to deal with the time integrals. Reorganizing, we obtain

∞

0

[W (t), Dm−1 (t)a]dt ≤

×

×

Q0

k=1 i=1

∞>tm >···>t1 >0

m

j=1

Qj

N0 d 1 1 (q0ki + 1) 2 (q0ki + 1) 2

(N ) (N ) |˜ aN0 (Q0 0 , Q0 0 )|

(Nj )

|w ˜Nj (Qj

dt1 . . . dtm 1∧

, Qj

T (Nj )

Nj

)|

(i,j)∈T d

1

d/2 χ[e0 ≤ 2N0 ](2N0 )! 4π |ti −tj | (2N0 − e0 )! 1

(qjki +1) 2 (qjki +1) 2

k=1 i=1

! χ[ej ≤ 2Nj ](2Nj )! . (2Nj − ej )! (4.21)

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Let us focus on the time integrals. The integrand is a symmetric function of t1 , . . . , tm , because of the sum over arbitrary trees. We can therefore extract a factor 1/m!, at the cost of integrating over all positive times t1 , . . . , tm without the ordering condition. Since there is no integral over t0 = 0, we have ∞ ∞ m

d/2 ∞

4π d/2 dt1 . . . dtm ≤ dt , (4.22) 1 ∧ |ti4π 1 ∧ −tj | |t| 0

0

−∞

(i,j)∈T

for any tree T . The last integral is equal to 8πd/(d − 2). The number of trees with m + 1 vertices and incidence numbers e0 , . . . , em is equal to

m−1 e0 − 1, e1 − 1, . . . , em − 1

=

(m − 1)! ; (e0 − 1)! . . . (em − 1)!

see for instance [Ber], Th´eor`eme 2 p. 86. We sum over incidence numbers, using 2N 2N N e=0 ( e ) = 4 , and we get 0

∞

[W (t), Dm−1 (t)a]dt ≤

1 8πd m (N ) (N ) N0 4N0 |˜ aN0 (Q0 0 , Q0 0 )| m d−2 Q0

×

N0 d

1

1

(q0ki + 1) 2 (q0ki + 1) 2

k=1 i=1

×

N 4N |w ˜N (Q(N ) , Q

Q

(N )

)|

N d

1

1

(qki + 1) 2 (qki + 1) 2

!m . (4.23)

k=1 i=1

The last step consists in removing the Hermite functions. We ﬁx N , s(N ) , (N ) (N ) s , r(N ) , r , and perform the summation over q (N ) and q . Using CauchySchwarz, we obtain (N )

d N 1 1 (N ) (N ) w ˜ (Q , Q ) (qki + 1) 2 (qki + 1) 2 N

q(N ) ,q(N )

≤

"

k=1 i=1

q(N ) ,q (N )

#1/2 d N 2 (N ) (N ) 3 3 ) (qki + 1) (qki + 1) ˜N (Q , Q w k=1 i=1

×

" q≥0

1 (q + 1)2

#dN . (4.24)

The last factor on the right side is bounded by 2dN . The ﬁrst factor on the right side can be viewed as an expectation value of a certain operator expressed in the

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basis of Hermite functions. Rewriting it in the x-space representation, we ﬁnd that it is given by the square root of the following expression

1

(N )

dx

26dN

RdN

RdN

×

N d k=1 i=1

dy (N ) wN (X (N ) , Y (N ) )

−

3 d2 3 d2 2 − 2 + yki + x2ki + 1 + 1 wN (X (N ) , Y (N ) ). 2 dxki dyki

This motivates the use of the norm (4.4) and concludes the proof of Theorem 4.1. We end this section by remarking that an estimate can be obtained that is invariant under space translations. Such an estimate follows by repeating the steps above with translates of Hermite functions. Recall that w ˜N was deﬁned in (4.10) by integrating 2dN Hermite functions centered at the origin. We can choose z ∈ R2dN and translate the j-th function by zj . Lemma 4.3 still holds with translates of Hermite functions, so that the proof goes through without a change until (4.24). Since a Hermite function translated by z ∈ R satisﬁes the diﬀerential equation (4.6) with (x − z)2 instead of x2 , one gets a bound where the diﬀerential operators in the norm (4.4) are translated by z ∈ RM . This holds for all z; let us introduce · M by f M = 1 23M/2

inf

y∈RM

(M)

dx RM

f (x(M) )

M k=1

1/2 3 d2 2 (M) − 2 + (xk − yk ) + 1 f (x ) . dxk (4.25)

This object is translation invariant but it is not a norm. We have || · ||M ≤ || · ||M . Theorem 4.1 holds when A and W are replaced by A and W , whose deﬁnition is like (4.3) with · M instead of · M .

5 Explicit perturbative calculation of particle and energy currents In this section, we consider two reservoirs of non-relativistic non-interacting free spinless fermions. Such systems are a special case of the ones introduced in Section 3. For explicit calculations, it is convenient to represent the system in Fourier space, see Subsection 5.1, since the one-particle energy operator t = −∆ is diagonal in this representation. Subsection 5.2 is devoted to the calculation of the particle and energy currents for tunnelling junctions, in the lowest non-vanishing order in W . This establishes the relation between the particle current and chemical potentials of the reservoirs, the current voltage characteristics. If the diﬀerence of

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chemical potentials ∆µ = µI − µII is small, then the particle current is proportional to the voltage drop ∆µ. This linear relation is known as Ohm’s law. We calculate the (inverse of the) proportionality factor, which is called the resistance of the junction. Moreover, we explicitly verify that the entropy production rate is strictly positive, provided the two reservoirs are at either diﬀerent temperatures or chemical potentials. Let us recall that the single particle Hilbert space (in the thermodynamic limit) is h = L2 (Rd , dx), with x ∈ Rd , d ≥ 3 (see equation (3.2)). The dynamics is determined by t = −∆, see (3.3). For each reservoir, we take the particle number to be the only conservation law. Recall that for tunnelling junctions, the interaction W commutes with the total particle number operator, N ⊗ 1l + 1l ⊗ N , while for thermal junctions, W commutes separately with N ⊗ 1l and 1l ⊗ N ; see equations (3.41) and (3.40). To quantify the interaction, we introduce two coupling constants, g and ξ, and set ∞ ξ N WN . (5.1) W =g N =1

Let Jk,l , Pk,l denote the term of order g k ξ l of the particle current J (see (2.86), (2.78) (2.82)) and the energy current P (see (2.85), (2.77), (2.82)). Accordingly, we deﬁne Ek,l , where E is the entropy production rate in (2.69) and (2.75). We give now explicit expressions for some lower order terms of the currents and the entropy production rate. The calculations are presented in Subsection 5.2. Tunnelling junctions. The lowest order terms of the particle current are given by J1,1

=

J2,2

=

J1,2 = 0, (5.2) 2 2π dk dl δ(|k|2 − |l|2 ) |w 1 ((−k, II), (l, I))| (ρII (k) − ρI (k)) , R2d

(5.3) is the Fourier transform of w, and the function ρr (k) is deﬁned where k, l ∈ Rd , w as 1 ρr (k) = β r (|k|2 −µr ) . e +1 We obtain for the energy current the expressions P1,1

=

P2,2

=

P1,2 = 0, (5.4) 2π dk dl |k|2 δ(|k|2 − |l|2 ) |w 1 ((−k, II), (l, I))|2 (ρII (k) − ρI (k)) . R2d

(5.5) Assuming that w 1 is not identically zero, the above formulas show the following qualitative behaviour of the ﬂows.

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Ann. Henri Poincar´e

– If (β I , µI ) = (β II , µII ) then J2,2 = P2,2 = 0. The ﬂows vanish if both reservoirs are at the same temperature and chemical potential. – If µI = µII and β I > β II then ρII (k) − ρI (k) > 0, for all k. Consequently, J2,2 , P2,2 > 0. At constant chemical potential, there is a particle and energy ﬂow from the hotter to the colder reservoir. – If µI > µII and β I = β II , then ρII (k) − ρI (k) < 0, for all k. Consequently, J2,2 , P2,2 < 0. At constant temperature, there is a particle and energy ﬂow from the reservoir with higher chemical potential to the reservoir with lower chemical potential. Ohm’s law and the resistance of the junction. Suppose that β I = β II = β and µI = µ, µII = µ + ∆µ, with ∆µ small. Retaining only the leading order in ∆µ in the expression of the particle ﬂow yields J2,2 ≈

∆µ , R(µ, β)

(5.6)

where the resistance R(µ, β) is determined by −1

R(µ, β)

|w 1 ((−k, II), (l, I))|2 eβ(|k| = 2πβ dk dl δ(|k| − |l| ) (eβ(|k|2 −µ) + 1)2 R2d 2

2

2

−µ)

. (5.7)

We refer to Subsection 5.2 for a qualitative discussion of the resistance, in three dimensions, d = 3. Onsager reciprocity relations. Let us study the interdependence of the ﬂows near equilibrium. The relevant parameters are the diﬀerence of the inverse temperatures, and the diﬀerence of the chemical potentials divided by the temperature. Precisely, we set β I = β; β II = β − ∆β; ν = β I µI ; ∆ν = β I µI − β II µII . We consider the ﬂows to depend on β, ν, ∆β, and ∆ν. One easily checks that ∂ eβ(|k| −µ) ρII (k) − ρI (k) = β(|k|2 −µ) |k|2 , ∂∆β ∆β=∆ν=0 (e + 1)2 2 ∂ eβ(|k| −µ) ρII (k) − ρI (k) = − β(|k|2 −µ) . ∂∆ν ∆β=∆ν=0 (e + 1)2 2

(5.8)

The ﬁrst partial derivative is taken at constant β, ν, and ∆ν; the second partial derivative is at constant β, ν, and ∆β. Then from (5.3) and (5.5) we observe that ∂P2,2 ∂J2,2 =− . ∂∆ν ∆β=∆ν=0 ∂∆β ∆β=∆ν=0

(5.9)

This is an Onsager reciprocity relation and we see that it holds at lowest order.

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Entropy production rate. Recall that P II = −P I (equation (2.77)) and = −J I (equation (2.78)), hence E = (β I − β II )P − (β I µI − β II µII )J .

Using the above expressions for Jk,l and Pk,l , we obtain E1,1

= E1,2 = 0,

E2,2

= (β I − β II )P2,2 − (β I µI − β II µII )J2,2 = 2π dk dl δ(|k|2 − |l|2 )|w 1 ((−k, II), (l, I))|2 ×

R2d I

2

{(β − β II )|k|2 − (β I µI − β II µII )}{eβ (|k| −µ ) − eβ (eβ I (|k|2 −µI ) + 1)(eβ II (|k|2 −µII ) + 1) I

I

I

II

(|k|2 −µII )

}

.

II

The numerator of the fraction is of the form (xI − xII )(ex − ex ), with xr = β r (|k|2 − µr ), hence it is strictly positive unless xI = xII . We assume that dl δ(|k|2 − |l|2 )|w 1 ((−k, II), (l, I))|2 (5.10) Rd

does not vanish for all k ∈ Rd . Then E2,2 is strictly positive unless β I (|k|2 − µI ) = β II (|k|2 − µII ) for all k in the support of (5.10). This shows that E2,2 is strictly positive unless (β I , µI ) = (β II , µII ), in which case E2,2 vanishes. Thermal junctions. The particle current is zero, a thermal junction allows only for an exchange of heat between the two reservoirs. Since W1 = 0, the lowest order term which is nonvanishing is P2,4 . Without loss of generality, we take the coupling function w 2 to be of the form 2 (k1 , k2 , l1 , l2 ). w 2 ((k1 , r1 ), (k2 , r2 ); (l1 , s1 ), (l2 , s2 )) = δr1 ,I δr2 ,II δs1 ,I δs2 ,II w A somewhat lengthy but straightforward calculation yields P2,4 = 2π dk1 dk2 dl1 dl2 δ(|k1 |2 + |k2 |2 − |l1 |2 − |l2 |2 ) ×|w 2 (−k1 , −k2 , l1 , l2 )|2 (|k1 |2 − |l1 |2 )ρI (l1 )ρII (l2 )(1 − ρI (k1 ) − ρII (k2 )), (5.11) from which we obtain the following qualitative discussion. – If (β I , µI ) = (β II , µII ), then ρI = ρII , and by switching l1 ↔ l2 , k1 ↔ k2 in the integral, we see that P2,4 = 0. – By splitting the integral in (5.11) into a sum of two integrals over the regions χ(|k1 |2 > |l1 |2 ) and χ(|k1 |2 < |l1 |2 ), and switching k1 ↔ l1 , k2 ↔ l2 , we

938

J. Fr¨ ohlich, M. Merkli and D. Ueltschi

can rewrite

P2,4 = 2π

Ann. Henri Poincar´e

dk1 dk2 dl1 dl2 δ(|k1 |2 + |k2 |2 − |l1 |2 − |l2 |2 )

×|w 2 (−k1 , −k2 , l1 , l2 )|2 (|k1 |2 − |l1 |2 ) χ(|k1 |2 > |l1 |2 ) × ρI (l1 )ρII (l2 )[1 − ρI (k1 ) − ρII (k2 )] −ρI (l1 )ρII (l2 )[1 − ρI (k1 ) − ρII (k2 )] = 2π dk1 dk2 dl1 dl2 δ(|k1 |2 + |k2 |2 − |l1 |2 − |l2 |2 ) ×|w 2 (−k1 , −k2 , l1 , l2 )|2 (|k1 |2 − |l1 |2 ) χ(|k1 |2 > |l1 |2 ) × ρII (l2 )[1 − ρII (k2 )][ρI (l1 ) − ρI (k1 )] −ρI (k1 )[1 − ρI (l1 )][ρII (k2 ) − ρII (l2 )] . The ﬁrst product in the round brackets { } is strictly positive and tends to zero, as β II → ∞ (because in the limit β r → ∞, ρr (k) tends to the characteristic function χ(|k|2 < µr )). The second term in the round brackets is strictly negative and tends to zero, as β I → ∞. We conclude that P2,4 < 0 if β I < ∞, and β II is large enough; as expected!

5.1

Fourier representation

The creation and annihilation operators in the Fourier representation are deﬁned by dx eikx a∗ (x, r), (5.12) a∗ (k, r) = (2π)−d/2 Rd a(k, r) = (2π)−d/2 dx e−ikx a(x, r), (5.13) Rd

where k ∈ R , r = I, II; compare with (3.16) and (3.22). The dynamics of a∗ (k, r) and a(k, r) is given by d

αrt (a∗ (k, r)) = eiωt a∗ (k, r), αrt (a(k, r)) = e−iωt a(k, r), where

(5.14)

ω = ω(k) = |k|2 .

The operators H r , N r , WN deﬁned in (3.19), (3.20), (3.34) are represented in Fourier space (and in the thermodynamic limit) by Hr = dk ω(k)a∗ (k, r)a(k, r), (5.15) d R Nr = dk a∗ (k, r)a(k, r), Rd WN = dK (N ) dL(N ) a∗ (K (N ) )w N (−K (N ) , L(N ) )a(L(N ) ),

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where we introduce notation analogous to (3.24)–(3.31). For K (N ) = (K1 ,...,KN ), Kj = (kj , rj ) ∈ Rd × {I, II}, we set −K (N ) := (−K1 , . . . , −KN ), where −Kj = (−kj , rj ). The symbol denotes the Fourier transform, i.e. w N (K (N ) , L(N ) ) −dN = (2π) R2dN

dx1 · · · dy N e−i(k1 x1 +···+lN yN ) wN (X (N ) , Y (N ) ).

We recall some properties of the state ω 0 deﬁned in (2.60), which is given by ω 0 = ωβ I ,µI ⊗ ωβ II ,µII ,

(5.16)

where ωβ r ,µr is the equilibrium state of reservoir r = I, II in the thermodynamic limit; see also Theorem 3.1. The two-point function of ωβ r ,µr is ωβ r ,µr (a∗ (k, r)a(l, r)) = δ(k − l)ρr (k), where ρr (k) =

1 eβ r (|k|2 −µr )

+1

.

The average of a monomial in n creation and m annihilation operators is zero unless n = m, in which case it can be calculated recursively from the formula   n n ωβ r ,µr  a∗ (ki , r) a(lj , r) i=1

=

n

j=1



(−1)n−p ωβ r ,µr (a∗ (k1 , r)a(lp , r)) ωβ r ,µr 

p=1

n

a∗ (ki , r)

i=2

n

 a(lj , r) .

j=1,j =p

For details, we refer to [BR]. We are now ready for explicit calculations of the currents.

5.2

Calculations for tunnelling junctions

Particle current and resistance. The particle current J = ωstat (−i[N ⊗ 1l, W ]) = ω 0 (σ+ (−i[N ⊗ 1l, W ]))

(5.17)

has been introduced in (2.86), (2.78), see also (2.82). We set = 1. We see from the Dyson series expansion of σ+ , see (4.1), and the deﬁnition of Jk,l (see after (5.1)), that J1,1

= −iω 0 ([N ⊗ 1l, W1 ]),

J1,2

= −iω 0 ([N ⊗ 1l, W2 ]), ∞ = dt ω 0 ([W1 (t), [N ⊗ 1l, W1 ]]).

J2,2

0

(5.18)

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Ann. Henri Poincar´e

It is not diﬃcult to verify that [N ⊗ 1l, WN ] =

dK (N ) dL(N )

N (δrj ,I − δrj ,I ) j=1

∗

×a (K

(N )

)w N (−K (N ) , L(N ) )a(L(N ) ),

(5.19)

where δ is the Kronecker symbol and L(N ) = (L1 , . . . , LN ), Lj = (lj , rj ). Using that ω 0 is invariant under A → eisN Ae−isN , we ﬁnd that I

I

ω 0 ([N ⊗ 1l, Wl ]) = 0,

for all l.

(5.20)

Thus J1,1 = J1,2 = 0. Next, we calculate J2,2 . Recalling that W1 (t) = α0−t (W ) and equation (5.14), we write 2 2 [W1 (t), [N ⊗ 1l, W1 ]] = (δs,I − δs ,I ) dk dl dk dl e−i(|k| −|l| )t ×w 1 ((−k, r), (l, r

r,r ,s,s =I,II )) w 1 ((−k , s), (l , s ))

R4d

[a∗ (k, r)a(l, r ), a∗ (k , s)a(l , s )].

We expand the commutator on the right side and apply the state ω 0 to obtain ∞ J2,2 = dt dk dl w 1 ((−k, II), (l, I)) w 1 ((−l, I), (k, II)) 0 R2d 2 2 2 2 × e−i(|k| −|l| )t + ei(|k| −|l| )t (ρII (k) − ρI (l)) . Because W1 is selfadjoint, we have the relation w 1 ((−k, II), (l, I)) = w 1 ((−l, I), (k, II)); using this and the formula useful to keep in mind that

∞ −∞

dt eiτ t = 2πδ(τ ), one sees that (5.3) holds. It is 2

2

eβ (|k| −µ ) − eβ (|k| −µ )

. ρII (k) − ρI (k) = β I (|k|2 −µI ) + 1 eβ II (|k|2 −µII ) + 1 e I

I

II

II

The resistance. Let β I = β II = β and µI = µ, µII = µ + ∆µ, with ∆µ small. We expand ρII (k) − ρI (k) = β ∆µ eβ(|k|

2

−µ)

(eβ(|k|

2

−µ)

+ 1)−2 + O((∆µ)2 ).

Retaining only the ﬁrst order in ∆µ in equation (5.3) gives (5.6) and (5.7). Let T = 1/β denote the temperature and assume that µ > 0. We see that R(µ, β) ∼ T , as T → ∞. Next, we examine the dependence of the resistance on T , for small T ,

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in three dimensions, and where w 1 is a radial function in both variables (i.e., w 1 depends only on |k| and |l|). We then have R(µ, β)−1 = 8π 3 β

∞ 0

√ √ eβ(r−µ) dr r|w 1 (( r, II), ( r, I))|2

2 . eβ(r−µ) + 1

The fraction in the integral equals −β −1 ∂r (eβ(r−µ) + 1)−1 and it follows that ∞ 1 −1 dr f (r) β(r−µ) , (5.21) R(µ, β) = e +1 0 where f denotes the derivative of the function √ √ 1 (( r, II), ( r, I))|2 . f (r) := 8π 3 r|w Let us split the integral in (5.21) as µ ∞ 1 1 f (r) β(r−µ) f (r) β(r−µ) + R(µ, β)−1 = e +1 e +1 0 µ µ ∞ 1 1 = f (µ) − f (r) + f (r) β(r−µ) . −β(r−µ) 1 + e e +1 0 µ (5.22) Apply the change of variables t = −β(r − µ) and t = β(r − µ) in the ﬁrst and second integral on the right side of (5.22), respectively. Then one has R(µ, β)−1 = f (µ) 1 ∞ + dt {f (t/β + µ) − f (−t/β + µ)χ(t ≤ βµ)} (et + 1)−1 β 0 and using the mean value theorem, # " ∞ 2 t + O(e−βµ ) , dt f (ξt ) t R(µ, β)−1 = f (µ) + 2 β e +1 0 for some ξt ∈ [−t/β + µ, t/β + µ] and where the exponentially small remainder term comes from removing the cutoﬀ function χ(t ≤ βµ). Retaining the main term (β → ∞) yields π2 R(µ, β)−1 ≈ f (µ) + 2 f (µ) 6β and consequently, R(µ, β) ≈

1 , f (µ) + π 2 T 2 f (µ)/6

T → 0.

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Ann. Henri Poincar´e

At zero temperature, the resistance has the value R(µ, ∞) = f (µ)−1 and it increases or decreases with increasing T according to whether f (µ) < 0 or f (µ) > 0. Energy current. The energy current P = ωstat (−i[H ⊗ 1l, W ]) = ω 0 (σ+ (−i[H ⊗ 1l, W ])) has been introduced in equations (2.85), (2.77), see also (2.82). We set = 1. Using the CAR and expression (5.15) for H, one obtains [H ⊗ 1l, WN ]

=

dK (N ) dL(N )

N

(|kj |2 δrj ,I − |lj |2 δrj ,I )

j=1 ∗

×a (K

(N )

)w N (−K (N ) , L(N ) )a(L(N ) ),

and it is readily veriﬁed that P1,1 = P1,2 = 0, and a similar calculation as for the particle current shows that ∞ P2,2 = dt ω 0 ([W1 (t), [H ⊗ 1l, W1 ]]) (5.23) 0

is given by (5.5).

Acknowledgments J.F. thanks S. Dirren and G.M. Graf for useful discussions during an early stage of his eﬀorts. We are grateful to R. Fern´ andez, B. Nachtergaele and I.M. Sigal for numerous helpful discussions. We have greatly beneﬁtted from studying the works of V. Jakˇsi´c and C.-A. Pillet and of D. Ruelle on related problems. D.U. acknowledges the hospitality of ETH Z¨ urich, where most of this work has been carried out.

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J. Aﬄeck, Fields, strings and critical phenomena, (Les Houches 1988), Amsterdam North Holland, 1990.

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A. Yu. Alekseev, V.V. Cheianov, J. Fr¨ ohlich, Universality of Transport Properties in Equilibrium, the Goldstone Theorem, and Chiral Anomaly, Phys. Rev. Letters 81, 3503–3506 (1998).

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W. Aschbacher, C.-A. Pillet Non-Equilibrium Steady States of the XY Chain, preprint, mp-arc, 2002.

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[Bee]

C.W.J. Beenakker, Random-matrix theory of quantum transport, arXiv:cond-mat/9612179v1, 19 Dec. 1996.

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J. Bellissard, A. van Elst, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall eﬀect. Topology and physics, J. Math. Phys. 35 no. 10, 5373–5451 (1994).

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F. Bonetto, J.L. Lebowitz, L. Rey-Bellet, Fourier’s Law: A Challenge to Theorists, Mathematical physics 2000, 128–150, Imp. Coll. Press, London, 2000.

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D.D. Botvich, V.A. Malishev, Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas, Comm. Math. Phys. 91 no. 3, 301–312 (1983).

[BR]

O. Bratteli, D. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Texts and Monographs in Physics, Springer Verlag, 1987, 1997.

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H.B. Callen, Thermodynamics and introduction to thermostatistics, John Wiley & Sons, second edition, 1985.

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H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger Operators, Texts and Monographs in Physics, Springer Verlag, 1987.

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C. Connaughton, A.C. Newell, Y. Pomeau, Non-stationary spectra of local wave turbulence, preprint (2003).

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A.M. Chang, L.N. Pfeiﬀer, K.W. West, Observation of Chiral Luttinger Behaviour in Electron Tunnelling into Fractional Quantum Hall Edges, Phys. Rev. Letters 77 no. 12, 2538–2541 (1996).

[CWCPW] A.M. Chang, M.K. Wu, C.C. Chi, L.N. Pfeiﬀer, K.W. West, Plateau Behaviour in the Chiral Luttinger Liquid Exponent, Phys. Rev. Letters 86 no. 1, 143–146 (2001). [DFG]

S. Dirren, ETH diploma thesis winter 1998/99, chapter 5; (written under the supervision of J. Fr¨ ohlich and G.M. Graf).

[EPR]

J.-P. Eckmann, C.-A. Pillet, L. Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Stat. Phys. 95 no. 1-2, 305–331(1999).

[FMSU]

J. Fr¨ ohlich, M. Merkli, S. Schwarz, D. Ueltschi, Statistical Mechanics of Thermodynamic Processes, submitted to World Scientiﬁc Press.

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Ann. Henri Poincar´e

[FP1]

J. Fr¨ ohlich, B. Pedrini, New applications of the chiral anomaly, Mathematical physics 2000, 9–47, Imp. Coll. Press, London, 2000.

[FP2]

J. Fr¨ ohlich, B. Pedrini, Axions, quantum mechanical pumping, and primeval magnetic ﬁelds, in A. Cappelli and G. Mussardo (eds.), Statistical Field Theories, 289–303, Kluwer Academic Publishers, 2002.

[FST]

J. Fr¨ ohlich, U.M. Studer, E. Thiran, Quantum theory of large systems of non-relativistic matter, G´eom´etries ﬂuctuantes en m´ecanique statistique et en th´eorie des champs (Les Houches, 1994), 771–912, North-Holland, Amsterdam, 1996.

[Ha]

R. Haag, Local Quantum Physics, Texts and Monographs in Physics, Springer Verlag, 1992.

[He]

K. Hepp, Rigorous results on the s − d model of the Kondo eﬀect, Solid State Communications, 8, 2087–2090 (1970).

[JP1]

V. Jak˘si´c, C.-A. Pillet, On entropy production in quantum statistical mechanics, Comm. Math. Phys. 217 no. 2, 285–293 (2001).

[JP2]

V. Jak˘si´c, C.-A. Pillet, Non-equilibrium steady states of ﬁnite quantum stystems coupled to thermal reservoirs, Comm. Math. Phys., 226 No. 1, 131–162 (2002).

[La]

R.B. Laughlin, Quantized Hall Conductivity in Two Dimensions, Phys. Rev. B23, 5632–5633 (1981).

[LS]

L.S. Levitov, A.V. Shytov, Tunnelling in a bosonized Fermi liquid, arXiv:cond-mat/9510006, 1995, release of September 2002.

[LSH]

L.S. Levitov, A.V. Shytov, B.I. Halperin, Eﬀective action of a compressible QH state edge: application to tunnelling, arXiv:cond-mat/0005016v2, July 2001.

[Ma]

G.D. Mahan, Many Particle Physics, Plenum Press, New York, 1981.

[MO]

T. Matsui, Y. Ogata, Variational principle for non-equilibrium steady states of the XX model, preprint (2003).

[Rob]

D. Robinson, Return to Equilibrium, Comm. Math. Phys. 31, 171–189 (1973).

[RSII, III] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II (Fourier Analysis, Self-Adjointness), III (Scattering Theory), Academic Press, New York 1975, 1979. [Ru1]

D. Ruelle, Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98 no. 1-2, 57–75 (2000).

[Ru2]

D. Ruelle, Entropy production in quantum spin systems, Comm. Math. Phys. 224 no. 1, 3–16 (2001).

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[Ru3]

D. Ruelle, Statistical Mechanics. Rigorous results, Reprint of the 1989 edition. World Scientiﬁc Publishing Co., Inc., River Edge, NJ; Imperial College Press, London, 1999.

[SGJE]

L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Observation of the e/3 Fractionally Charged Laughlin Quasiparticle, Phys. Rev. Lett. 79, 2526–2529 (1997).

[SR1]

M. Sigrist, T.M. Rice, Paramagnetic Eﬀect in High Tc Superconductors – A Hint for d-Wave Superconductivity, Journal of the Physical Society of Japan 61 no. 12, December, 4283–4286 (1992).

[SR2]

M. Sigrist, T.M. Rice, Unusual paramagnetic phenomena in granular high-temperature superconductors – A consequence of d-wave pairing?, Rev. Mod. Phys. 67, no.2, April, 503–513 (1995).

[TKNN] D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405–408 (1982). [vW]

B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, Quantized Conductance of Point Contacts in a two-dimensional Electron Gas, Phys. Rev. Lett. 60, 848– 850 (1988).

J¨ urg Fr¨ ohlich, Marco Merkli† Theoretische Physik ETH-H¨onggerberg CH-8093 Z¨ urich Switzerland email: [email protected] email: [email protected] Daniel Ueltschi Department of Mathematics University of California Davis, CA 95616 USA email: [email protected] Communicated by Gian Michele Graf submitted 15/01/03, accepted 25/02/03

† supported

by an NSERC Postoctoral Fellowship and by ETH-Zurich

Ann. Henri Poincar´e 4 (2003) 947 – 972 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/050947-26 DOI 10.1007/s00023-003-0151-7

Annales Henri Poincar´ e

The AC Stark Eﬀect and the Time-Dependent Born-Oppenheimer Approximation∗ Steven W. Jilcott, Jr.

Abstract. We discuss the physical problem of a molecule interacting with an electromagnetic ﬁeld pulse and model the problem using a time-dependent perturbation of the Born-Oppenheimer approximation to the Schr¨ odinger equation. Using previous results that develop asymptotic series solutions in the Born-Oppenheimer parameter , we derive a formal Dyson series expansion in the perturbation parameter µ, which is proportional to the electromagnetic ﬁeld strength. We then prove that this series is asymptotically accurate in both parameters, provided that the Hamiltonian for the electrons has purely discrete spectrum. Under more general hypotheses, we show that the series is accurate to ﬁrst order in µ.

1 Introduction Physicists have long been familiar with the process by which an atom, when illuminated by electromagnetic radiation, can undergo an electron energy-level transition. When the frequency of the incoming light is such that the energy of a photon is exactly equal to the energy gap between electron energy levels, the electron can be excited from one level to another. At typical light intensities, impinging photons of an energy that does not match an atomic energy level gap have only a small chance of being absorbed and causing an electron transition. In this article we study the quantum physics of a molecule subjected to a very short laser pulse of the type described above, but we restrict attention to a pulse of weak intensity. This particular physical system has been utilized in a very high proﬁle experimental applications. Femtochemists use laser pulses on the order of 10−15 seconds in length to take “snapshots” of the dynamics of molecular formation [14]. A powerful “pump pulse” is used to initiate a molecular reaction, then, a series of “probe pulses” actually allow the observer to study the changing conﬁguration of the molecule as the reaction takes place. The emission spectrum caused by the excitation of a particular molecular conﬁguration by a laser pulse can be calculated using theoretical techniques in quantum mechanics such as those described in this article so that the experimental data can be interpreted. We provide rigorous mathematical justiﬁcation for some common techniques used in studying the physical model used in femtochemistry. ∗ This work was supported by the Cunningham Research Fellowship provided by Virginia Tech and by National Science Foundation Grant DMS-9703751.

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Since the actual equations cannot be solved in closed form, we choose to model the evolution of the molecular motion using the Born-Oppenheimer approximation, which will allow us to ﬁnd an asymptotic series solution. In the Born-Oppenheimer approximation, we treat the nuclei as extremely massive in comparison with the electrons. The electrons “orbit” the nuclei very rapidly, and hence changes in the nuclear conﬁguration are on a long time scale compared to the electronic motion. This approximation allows us to develop the electronic motion adiabatically – as if the electrons adjust their dynamical state immediately to the relatively slow motion of the nuclei. Thus, if the electrons start in a bound state conﬁguration, they will always occupy that bound state conﬁguration as if the nuclei were ﬁxed. The electron conﬁguration determines an eﬀective potential in which the nuclei move. Because of the relatively large nuclear mass, we may describe the nuclear motion semiclassically in this eﬀective potential. To illustrate, when we study the Born-Oppenheimer approximation, the Schr¨ odinger equation looks like i2

∂Ψ 4 ∂ 2 Ψ =− + h(X)Ψ ∂t 2 ∂X 2

(1.1)

where 4 is a small parameter equal to the ratio of the electron and nuclear mass, Ψ(X, x, t) is the wave-function depending on nuclear conﬁguration X, electronic conﬁguration x, and time t. By h(X), we denote a family of electron Hamiltonians parameterized by the nuclear conﬁguration X. If we have a solution Φ(x) to the time-independent Schr¨ odinger equation h(X)Φ(X, x) = E(X)Φ(X, x)

(1.2)

that holds for any X, then the approximation says that to leading order Ψ(X, x, t) = φ(X, t)Φ(X, x)

(1.3)

and that φ(X, t) describes the semiclassical motion of the nuclei as the solution of i2

4 ∂φ = − ∆X + E(X)φ . ∂t 2

(1.4)

The Born-Oppenheimer approximation has been well studied by Hagedorn [4] and also by Combes, et al. [2]. Several results have been rigorously demonstrated. Hagedorn has derived rigorous asymptotic estimates in for low-lying eigenvalues and bound states in the time-independent case [4, see also [2]]. In the time-dependent Born-Oppenheimer approximation (1.1), Hagedorn has also derived rigorous asymptotics for the propagation of wave packets in situations with and without electron energy level crossings [5]. The results we build on are discussed in detail in Section 3. We choose to treat the addition of an electromagnetic interaction by the addition of a perturbation consisting of an AC electric ﬁeld µx cos(ωt/2 ) multiplied by a function f (t/2 ) of compact support to produce a pulse. The factor of −2

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scales the electric ﬁeld oscillation so that this perturbation aﬀects mostly the electrons. In other words, the photon energy is approximately the same size as the energy gap between the diﬀerent electron levels. Since the energy gaps between nuclear conﬁgurations within each electron level are of order 2 , the perturbation itself should have only a minor eﬀect on the nuclear dynamics. Physically, one expects the most signiﬁcant change in the nuclear dynamics to occur via the electron energy level transition, which alters the eﬀective potential felt by the nuclei. In particular, this scaling separates the eﬀect of the electromagnetic pulse on the electrons from its eﬀect on nuclear vibrational or rotational modes. The principal theorems of this paper are an extension of the asymptotics of Hagedorn to equations similar in form to i2

4 ∂ 2 Ψ 1 ∂2Ψ ∂Ψ =− − + V (X, x)Ψ + µxf (t/2 ) cos(ωt/2 )Ψ. ∂t 2 ∂X 2 2 ∂x2

(1.5)

Working from the asymptotic expansion for the solution of the unperturbed time-dependent Born-Oppenheimer problem, we develop another asymptotic expansion for the perturbed problem using the Dyson series technique. Under very general conditions, we show that the technique allows us to calculate the wavefunction up to ﬁrst order in the perturbation parameter µ, but that the possibility of electrons escaping to the continuum prevented a further extension of the series. By placing restrictions on the abruptness of the laser pulse, we extend the series to second order in µ. Under the supposition that our electron Hamiltonian had no continuous spectrum, we develop the Dyson series fully to all orders in µ and the BornOppenheimer parameter . Each term of the Dyson series can be interpreted physically as representing all the possibilities of n separate electron transitions. The nuclei then propagate semiclassically in the eﬀective potential induced by the current electron state. We also summarize the intuitive meaning of each term of the Dyson series. The leading order term is propagation with no electron transitions. The O(µ) term covers all the possible situations where the molecule begins with initial electron energy state E(i) and then at some time s1 makes an electron transition to some other state E(k1 ) . If we sum over all the excited states k1 and integrate over all the possible times s1 when this transition can occur, we get the ﬁrst order term of the Dyson expansion (3.81). In general, the O(µm ) term covers all the possibilities for m electron transitions.

2 Mathematical preliminaries Suppose the molecule under consideration has n1 nuclei of masses M1 −4 , . . . , Mn1 −4 and charges q1 , . . . , qn1 and n2 electrons of mass me and charge qe , making a total of n = n1 + n2 particles. Let X = {ξ1 , . . . , ξn1 , η1 , . . . , ηn2 } ∈ R3n denote the conﬁguration vector of the molecule, where ξi denotes a nuclear coordinate

950

Steven W. Jilcott, Jr.

Ann. Henri Poincar´e

and ηi an electron coordinate. Assume that the particles interact pairwise under a smooth potential V (Xi − Xj , qi , qj ) where i, j = 1, . . . , n. and Xi ∈ R3 is the coordinate vector of the ith particle. Then the Schr¨odinger equation of interest is n1 n2 ∆ηj 4 ∆ξi 1 2 ∂Ψ = − − i ∂t 2 i Mi 2 j me n 2 + V (Xi − Xj , qi , qj ) + µ ck Xk F (t/ ) Ψ . (2.6) i,j

k

The perturbation parameter is µ. The function F is required to be bounded and of compact support, which rules out the uniform AC Stark perturbation. The motivating example, the electromagnetic pulse, is the speciﬁc case n qk Xk f (t/2 ) cos(ωt/2 ) (2.7) µ E· k 2

where f (t/ ) is bounded and of compact support. For a physically interesting application, one typically wants f to be supported over several periods of the cosine function, since f is intended to be the “shape” of a laser pulse. The integrability of f models the condition that our pulse turns “on” at some time in the past, and then turns “oﬀ” at some time in the future. If the masses of the nuclei are all equal, and the charges of the particles all have the same magnitude, one may easily extend our results to include F (t/2 ) ∈ L1 (−∞, ∞).

2.1

Clustered Jacobi coordinates

It is convenient to transform (2.6) into clustered Jacobi coordinates as described in [15, vol. III]. We deﬁne cluster C1 to be the n1 nuclear coordinates and cluster C2 to be the n2 electron coordinates. Then we compute the respective centers of mass of C1 and C2 −1 n1 n1 R1 = Mi Mi ξi , (2.8) i

n2 1 R2 = ηj . n2 j

i

(2.9)

We then let ζ1 = R2 − R1 , ( Mi −4 )R1 + n2 me R2 ζ2 = . Mi −4 + n2 me

(2.10) (2.11)

Note that ζ1 is intuitively the intercluster distance, and ζ2 is the total center of mass of the molecule.

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Deﬁne new nuclear and electron coordinates by using the Jacobi coordinates inside each cluster −1 Xi = ξi+1 − Mj Mk ξk (2.12) i = 1, . . . , n1 − 1 , k≤i

1 ηk xj = ηj+1 − j

k≤i

j = 1, . . . , n2 − 1 .

(2.13)

k≤j

The coordinates {X1 , . . . , Xn1 −1 , ζ1 , ζ2 , x1 , . . . , xn2 −1 } form clustered Jacobi coordinates for the problem in question. After transformation into these coordinates, the total center of mass coordinate ζ2 does not appear in the potential V . Since we are not interested in the center of mass motion of the molecule under the perturbation (the solution to that problem is known), but rather in the energy level transitions of molecule, we drop coordinate ζ2 from the equation entirely and deﬁne ζ1 ≡ ζ. We also use the shorthand X ≡ (X1 , . . . , Xn1 −1 ), and x ≡ (x1 , . . . , xn2 −1 ). Let us abuse notation slightly and call the new vector of coordinates (X, ζ, x) ≡ X also. After this transformation, we now have

i

2 ∂Ψ

∂t

=−

n 1 −1 i

n 2 −1 ∆xj 4 4 ∆Xi + c( )∆ζ + Ψ + V (X, q)Ψ+ 2 2 j 4 4 4 Ci ( )Xi + Cζ ( )ζ + Cj ( )xj F (t/2 ) . (2.14) µ i

j

The Ci and c are easily veriﬁed to be smooth functions of 4 at = 0. They are merely coeﬃcients resulting from the coordinate transformation, hence they depend on the masses and the charges of the particles involved. We will change coordinates once more to put (2.14) in a standard form. The ﬁnal equation 1 =− i ∂t 2 2 ∂Ψ

n 1 −1

4

∆Xi + ∆ζ +

n 2 −1

i

∆xj Ψ + V (X, 4 , q)Ψ + µX · C()F (t/2 )

j

(2.15) is obtained by merely multiplying various coordinates by scale factors depending on 4 , the masses, and the charges. The various Ci have been condensed into a vector C(). Remark. The potential V (X, 4 , q) and the Ci can be expanded in a Taylor series in 4 about 0, valid for > 0. That is, we may substitute V =

4p

V (p) p!

Ci =

4p

(p)

Ci p!

952

Steven W. Jilcott, Jr.

Ann. Henri Poincar´e

and calculate additional corrections to the asymptotic series we will derive in Section 4 at orders of divisible by 4. In the sequel, then, it will be assumed that the V and the Ci are not dependent on for ease in exposition. The reader will observe that the proof given is applicable even if such corrections are added, and the calculational modiﬁcations follow from regular perturbation theory. In (2.15), we identify n 2 −1 1 ∆xj + V (X, q) h(X) = − ∆ζ + 2 j as the electron Hamiltonian depending parametrically on the nuclear variables. Then, we further abbreviate H() ≡ −

n1 −1 4 ∆Xi + h(X), 2 i

(2.16)

so that the unperturbed equation to be solved will be i2

∂ Ψ = H()Ψ . ∂t

(2.17)

Furthermore, H1 (t/ ) ≡ Cζ ζ + Cj xj F (t/2 ) 2

(2.18)

j

will abbreviate the expression for the perturbation. To simplify the exposition, we have left out the Xj terms in the time-dependent perturbation. As will be explained in Section 3, those terms will have little eﬀect. Since the nuclear masses are large, the resonance frequency required for the electric ﬁeld pulse to excite transitions between nuclear bound states within a particular electron energy level is much smaller than any of the signiﬁcant frequency components of F (t/2 ).

2.2

Existence of unitary propagator

Any time-dependent perturbation modiﬁes the propagation operator that develops the solution from its initial conditions. Thus, a common technique in timedependent perturbation theory is to develop an asymptotic series for the new propagator, thereby allowing us to approximate the perturbed solution at any future time. Following Theorem X.69 from [15, vol.II] we construct a Dyson expansion to approximate the perturbed propagator. The Born-Oppenheimer Hamiltonian we are interested in is not bounded, hence, the Dyson series does not converge uniformly to the correct propagator. Instead, we will now show that it is asymptotic to the correct propagator.

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Stark Eﬀect and Time-Dependent Born-Oppenheimer Approximation

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We will denote the propagator for the unperturbed Hamiltonian H() by 2 e−i(t−s)H()/ ≡ U (t, s), and that of the perturbed Hamiltonian H() + µH1 by U,µ (t, s). Because of the singular nature of the time-dependent perturbation H1 , it is not immediately apparent that the propagator U,µ (t, s) exists. We demonstrate its existence by construction using the method of Yajima [16] by introducing a transformation of (2.15) (2.19) Xi → Xi − µCi dt dt F (t/2 ). By dt, we mean integration from some Tl , a value below the compact support of F , up to t. Under this transformation, if Ψ solves (2.15), then ΨD ≡ T (t)−1 Ψ solves i2

1 ∂ΨD =− ∂t 2

n 1 −1

4 ∆Xi + ∆ζ +

i

n 2 −1

∆xj Ψ + V (X + µC dt dt F (t/2 ))Ψ

j

(2.20) where the operator T (t) is unitary and is expressed by 2 T (t)f = exp iµC · X F (t/ )dt + i(µ|C|)2

2 2 2 2 2 F (t/ )dt − i(µ|C|) /2 dt dt dt F (t/ ) F (t/ )dt

2 2 2 dt F (t/ ) dt dt F (t/ ) − i(µ|C|) f (X − µC dt dt F (t/2 )). (2.21) Thus, by a unitary transformation, we can bring the time-dependency inside the argument of the potential V , and we notice that a unitary propagator exists for this new Hamiltonian. Remark. We will always develop the solution from t = Tl , that is, from some ﬁnite time before the pulse turns “on”. Thus, we assume that we know, as an initial condition, the wavepacket solution of the time-dependent molecular Schr¨ odinger equation at that time. odinger equation i ∂Ψ Deﬁnition 2.1 Suppose Ψ0 (t) is a solution of the Schr¨ ∂t = H0 Ψ, where H0 is some unperturbed Hamiltonian, and U (·, ·) is the unitary propagator for H0 . Then a formal Dyson series solution for the perturbed Schr¨ odinger equation i

∂Ψ = (H0 + µH1 )Ψ ∂t

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is given by Ψ(t) = Ψ0 (t) +

N

(−i µ)n

n=1

× H2 (s2 )

t Tl

U (t, sn )H1 (sn )

sn

Tl

U (sn , sn−1 )H1 (sn−1 )

s2

Tl

U (s2 , s1 )H1 (s1 )U (s1 , Tl )Ψ0 (Tl )dsn . . . ds1 . (2.22)

In the next section, we examine the known asymptotic series solution for the unperturbed time-dependent Born-Oppenheimer approximation. We will then use the solution described there as a base from which we will develop a formal Dyson series solution to the perturbed equation.

2.3

Semiclassical wave packets

In order to study the propagation of a wave-packet under the electromagnetic ﬁeld perturbation, we shall utilize a special basis for the nuclear wave-function space. This basis is constructed from the solutions of the generalized n-dimensional quantum harmonic oscillator, giving the basis useful properties for obtaining estimates of the errors made in semiclassical propagation. The theorems quoted here, as well as other properties, are given a complete discussion in [6]. We refer the reader to that source for their proofs. We will assume a,η ∈ Rn1 , and matrices A,B ∈ Cn1 ×n1 satisfying the normalization conditions At B − B t A = 0 , A∗ B + B ∗ A = 2I .

(2.23) (2.24)

These conditions ensure most importantly that 1) A and B are invertible matrices. 2) BA−1 has the form [real symmetric + i real symmetric]. 3) (Re BA−1 )−1 = AA∗ , which is strictly positive deﬁnite. We next let l = (l1 , . . . , ln1 ) be a multi-index, i.e., any ordered n-tuple of nonnegative integers. We interpret |l| =

n1

lj ,

Dl =

l! = l1 !l2 ! . . . ln1 !,

j=1

(∂X1

)l1 (∂X

∂ |l| , ln l 2 ) 2 . . . (∂Xn1 ) 1

l

and ﬁnally, the monomial xl = xl11 . . . xnn11 . For any multi-index l, then ϕl (A, B, , a, η, X) = 2−|l|/2 (l!)−1/2 π −n1 /4 −n1 /4 (det A)−1/2 × Hl (A; −1/2 |A|−1 (X − a)) exp{−(X − a), BA−1 (X − a)/2+ iη, (X − a)/} (2.25) where Hl is the generalized Hermite polynomial deﬁned in [6].

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These functions form a complete orthonormal basis for L2 (Rn1 ). Suppose α(t), β(t), γ(t) are continuous real n1 × n1 matrix-valued functions, δ(t), (t) are continuous Rn1 vector-valued functions, and ζ(t) is continuous and real-valued. Identifying p ≡ −i∇X , we then focus our consideration on the Hamiltonian 1 p p α(t) β(t) + δ(t), p + (t), x + ζ(t) (2.26) H(x, p, t) = · t (t) γ(t) x x β 2 which describes a generalized n1 -dimensional quantum harmonic oscillator system. Now given initial conditions (A(Tl ), B(Tl ), a(Tl ), η(Tl ), S(Tl )), the classical equations of motion a(t) ˙ =

β(t)a(t) + α(t)η(t) + δ(t)

η(t) ˙ = ˙ A(t) = ˙ B(t) =

−γ(t)a(t) − β t (t)η(t) − (t) β(t)A(t) + iα(t)B(t)

˙ S(t) =

(2.27)

t

iγ(t)A(t) − β (t)B(t) α(t)2 η(t)2 − γ(t) − (t)a(t) − ζ(t) α(t) 2 2

have a unique solution (A(t), B(t), a(t), η(t), S(t)). In the classical motion, a(t) represents the position vector, η(t) the momentum vector, and S(t) the classical action. It is helpful to identify these same quantities with the wave packets we will propagate semiclassically. The following theorem is veriﬁed by an elementary induction in [6] using the raising and lowering operators for solutions of the form (2.25). Theorem 2.1 Let α(t), β(t), γ(t), δ(t), (t), ζ(t) be as above and let (A(t), B(t), a(t), η(t), S(t)) be any solution to the system (2.27). Then, for any multi-index k, Ψ(, t) = eiS(t)/ ϕk (A(k), B(t), , a(t), η(t), x) exactly solves the Schr¨ odinger equation i

∂Ψ = H(x, p, t)Ψ ∂t

where H(x, p, t) is the Hamiltonian in (2.26). Of central importance in the estimation of semiclassical propagation errors is the asymptotic semiclassical expansion theorem below.

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Theorem 2.2 Suppose N ≥ 1, and V ∈ C N +2 (Rn1 ) satisﬁes −C1 ≤ V (X) ≤ 2 C2 eMx for some C1 , C2 , and M . Let (A(t), B(t), a(t), η(t), S(t)) be a solution to the following system of equations a(t) ˙ =

η(t)

η(t) ˙ = ˙ A(t) = ˙ B(t) =

−V (1) (a(t)) iB(t)

˙ S(t) =

(2.28)

(2)

iV (a(t))A(t) η(t)2 − V (a(t)) 2

4

and let H() = − 2 ∆X + V (X). Then given coeﬃcients cj (Tl ), there exists timedependent coeﬃcients cj (t, ) such that

−itH()/2

e cj (Tl )ϕj (A(Tl ), B(Tl ), 2 , a(Tl ), η(Tl ), X)

|j|≤J

−e

iS(t)/

2

N cj (t, )ϕj (A(t), B(t), , a(t), η(t), X)

≤ C3 . (2.29) 2

|j|≤J+3N −3

We require the following elementary but crucial lemma (which is also used in the proof of 2.2) elsewhere in the sequel. Lemma 2.1 Suppose H(, t) is a family of possibly time-dependent self-adjoint operators for > 0, and U (·, ·) is the unitary propagator for H(, t). Suppose ψ(t, ) belongs to the domain of H(, t), is continuously diﬀerentiable in t, and approximately solves the Schr¨ odinger equation in the sense that i2

∂ψ (t, ) = H(, t)ψ(t, ) + ξ(t, ) ∂t

(2.30)

where ξ(t, ) satisﬁes ξ(t, ) ≤ µ(t, ). Then, U (t, Tl )ψ(Tl , ) − ψ(t, ) ≤

−2

t

µ(s, ) ds.

(2.31)

Tl

The following lemma will be used later to justify small corrections to the semiclassical propagation resulting from the addition of a time-dependent perturbation in the nuclear variables. Lemma 2.2 Let F (t/2 ) be a bounded function of compact support, with V meeting the conditions of Theorem 2.2. Suppose that we have a solution to the system (2.28).

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If (A(t), B(t), a(t), η(t), S(t)) is a solution to the modiﬁed system of equations a(t) ˙ η(t) ˙ ˙ A(t) ˙ B(t)

= η(t) = −V (1) (a(t)) − µF (t/2 ) = iB(t) = iV (2) (a(t))A(t) η(t)2 − V (a(t)) − µ = Ci (a(t))i F (t/2 ) 2 i

˙ S(t)

(2.32)

with the same initial conditions as the solution to (2.28), then the limit as → 0 of each of the quantities A(t), B(t), a(t), η(t), S(t) exists. Proof. We need only consider the solutions of the coupled equations a(t) ˙ = η(t) ˙ =

η(t) −V (1) (a(t)) − µF (t/2 ).

(2.33)

If a(t) and η(t) have deﬁned limits, then clearly the other quantities also have deﬁned limits from the form of their diﬀerential equations. We assume a solution to the two analogous equations from (2.28) and proceed to develop a formal approximate solution to our revised equations. We then rigorously verify that these approximate solutions give us the limits we desire. Let us set s = t/2 and assume series solutions of the form a(s, t) = a0 (s, t) + 2 a2 (s, t) + 4 a4 (s, t) + · · · 2

4

η(s, t) = η0 (s, t) + η2 (s, t) + η4 (s, t) + · · ·

(2.34) (2.35)

We proceed to ﬁnd a solution now to our equations by the method of multiple scales. Rewriting (2.33) to reﬂect our two independent time scales s and t, a˙ + −2 a η˙ + −2 η

= =

η −V (1) (a) − µF (s).

(2.36)

where ˙ denotes diﬀerentiation with respect to t, and denotes diﬀerentiation with respect to s. By substituting our series (2.34) into (2.36) and expanding V (1) (a) in powers of 2 , we may begin solving for the terms in the series by matching up powers of . Order −2 . We have a0 = 0 , η0 = 0 , so that a0 and η0 depend only on t.

(2.37) (2.38)

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Order 1. The next set of equations is a˙ 0 + a2 = η0 η˙ 0 +

η2

(2.39)

= −V

(1)

(a0 ) − F (s) .

(2.40)

Because a2 = η0 − a˙ 0 , and we wish to eliminate secular behavior in the s variable, we require a2 = 0 and a˙ 0 = η0 . Note that this latter equation is satisﬁed by the a(t) solution to (2.28). Further, we have η2 = −V (1) (a0 )− η˙ 0 −F (s). Because F is of compact support, we avoid secular behavior by requiring that s F (s ) ds + f2 (t) (2.41) η2 (s, t) = − Tl /2

η˙ 0 = −V (1) (a0 ).

(2.42)

The latter equation once again is satisﬁed by the η(t) solution to (2.28). We now have found (a0 , η0 ), which are completely determined by the initial conditions to (2.28). We have also completely determined the s-dependence of (a2 , η2 ). At this point let us reiterate that the initial conditions to our problem are independent of , so that aj (Tl /2 , Tl ) = 0 and ηj (Tl /2 , Tl ) = 0 for j ≥ 2. Order 2 . The last set of equations we will need to solve will be a˙ 2 + a4 = η2 , η˙2 +

η4

= −V

(2.43) (2)

(a0 )a2 .

(2.44)

Using what we have learned before s a4 = − F (s ) ds + f2 (t) − a˙ 2

(2.45)

in which we can avoid secular behavior by choosing ∞ F (s ) ds , a˙ 2 = f2 (t) −

(2.46)

Tl /2

Tl /2

a4 (s, t) = −

s

Tl

/2

Further, we have

ds

s

Tl

/2

F (s ) ds − s

∞

Tl

/2

F (s ) ds + g4 (t) .

η4 = −f˙2 − V (2) (a0 )a2

and so we must choose

f2 (t) = −

0

t

V (2) (a0 )a2 dt .

(2.47)

(2.48)

(2.49)

But putting (2.46) and (2.49) together a ¨2 = −V (2) (a0 )a2

(2.50)

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which can only have the trivial solution because of our initial conditions. Thus we have formally determined an approximate solution a(s, t) = a0 (t) + O(4 ) s η(s, t) = η0 (t) − 2

Tl /2

(2.51) F (s ) ds + O(4 ) + · · ·

For convenience, let us write the system (2.33) as η 0 a˙ − . = µF (t/2 ) η˙ −V (1) (a)

(2.52)

(2.53)

The right-hand side of this equation is a function of a and η whose derivative with respect to (a, η) is 0 1 . (2.54) −V (2) (a) 0 We claim that this derivative is bounded, and therefore that the right-hand side of (2.53) is Lipschitz. To see this, we note that (2.51) is an approximate solution to the ﬁrst two equations of (2.28) and that when we substitute the former into the latter we obtain a truncation error that is O(µ). We appeal to Gronwall’s Lemma in [12, p. 380], and note that the right-hand side of (2.28) is already known to be Lipschitz (because the exact solution (a, η) of (2.28) is contained within a compact set by conservation of energy). We learn from Gronwall’s Lemma that the solutions (2.51) are also contained within a compact set, hence the derivative (2.54) is bounded. We now note that the truncation error resulting from substituting (2.51) into (2.53) is 2 t/ 2 Tl /2 F (s ) ds (2.55) 0 which is O(2 ). By the preceding argument we appeal to Gronwall’s Lemma again to show that the diﬀerence between the exact solutions to (2.33) and (2.51) is in fact of O(2 ). This is enough to show that the limits as → 0 of the exact solutions to (2.33) are in fact the solutions to (2.28), which is enough to prove the lemma.

3 The time-dependent perturbation In this section, we seek to develop a formal Dyson series expansion for the eﬀect of the time-dependent perturbation on the asymptotic series developed in Section 3.1. We then prove that this series is accurate to the required orders in µ and in . The ﬁrst result develops the Dyson series for any smooth potential V , but only up to an accuracy of O(µ). The series will be accurate to any order in .

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We consider the case where there are no electron energy level crossings. In principle, this technique could be used in conjunction with solution to the unperturbed case with crossings developed in [5], but the calculations involved would be daunting. After discussing the problem in that generality, we then restrict attention to smooth potentials such that our unperturbed electron Hamiltonian has purely discrete spectrum. Under this restriction, we can develop the series to any order in µ. The failure of this technique to provide any greater accuracy in the general case will be discussed in detail in section 5.

3.1

Known unperturbed asymptotics

The asymptotics for the unperturbed time-dependent Born-Oppenheimer equation were ﬁrst developed by Hagedorn in 1986 [5]. Because our perturbation in (2.15) can be rendered into a part depending only on nuclear variables X and a part depending only on electronic variables ζ, x, we will choose to treat these parts diﬀerently. The part containing the nuclear variables will be subsumed by the semiclassical mechanics of the nuclear variables, since these equations of motion (2.28) can still be solved in terms of the special basis of semiclassical wave packets described in Section 2. Only the perturbation terms involving the electronic variables will ﬁgure in the Dyson series solution. We summarize here portions of the theorem of Hagedorn [7] respecting the unperturbed asymptotics. The proof in [7] can be modiﬁed to reﬂect the additional time-dependent term µ i Ci Xi F (t/2 ). It can be shown that this part of the timedependent perturbation can safely be treated by means of small corrections to the equations (2.28), but we will drop this term from the exposition entirely in the interest of clarity, and assume that our perturbation includes only the electron variables, as in the deﬁnition of H1 . This simpliﬁcation by no means reduces the mathematical strength of the results – in fact, in the physical problem motivating this study, a molecule containing nuclei of equal masses is perturbed by a term containing only the electron variables. Theorem 3.1 Suppose h(·) is a smooth time-independent electron Hamiltonian and suppose that there is an open set U ∈ Rn1 such that h(X) has discrete, multiplicity 1 eigenvalues E(i) (X) for X ∈ U that depend smoothly on X, and furthermore, do not cross, i.e., E(i) (X) = E(j) (X) for any X ∈ U . Suppose that a(t) and η(t) satisfy (2.28), and assume that a(t) ∈ U for all t. Then we can choose δ > 0, such that {X : |X − a(t)| ≤ 2δ} ⊂ U for all t, and deﬁne U1 (t) = {|X : X − a(t)| ≤ δ} and U2 (t) = {|X : X − a(t)| ≤ 2δ}. We choose a C ∞ function G : Rn1 → [0, 1], such that G(X) = 0 for |X| ≥ 2δ and G(X) = 1 for |X| ≤ δ. Then for small > 0 there exist functions Ψj (, X, ζ, x, t) with the properties (1) For any smooth choice of initial normalized electron eigenfunction Φ(i) (X, Tl ) corresponding to the eigenvalue E(i) (X) for X ∈ U2 (Tl ), there exists a smooth function Φ(i) (X, t) on the set {(X, t) : X ∈ U2 (t), t ∈ [Tl , T ]} that is a normalized eigenfunction corresponding to E(i) (X). Furthermore, we may choose

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the phase such that

∂ Φ(i) (X, t), i + iη(t) · ∇X Φ(i) (X, t) = 0 ∂t for all X ∈ U2 (t) and t ∈ [Tl , T ]. J (2) The function j=0 Ψj (, X, t)j is a Jth order asymptotic approximation to a solution of the Schr¨ odinger equation in the sense that

J J

−itH()/2

j j

e Ψj (, X, ζ, x, Tl ) − Ψj (, X, ζ, x, t) ≤ CJ J+1

j=0

j=0

(3) For each j, Ψj (, X, ζ, x, t) is bounded as 0. (4) The Ψj ’s have support only on the support of G. Remarks 1. The function G merely serves to focus our attention on the propagation of the portion of the wavepacket with nuclear coordinates contained in the set U where the electron eigenvalues E(i) satisfy the hypotheses of the theorem. 2. The proof of the theorem requires three steps. The technique of multiple scales transforms the problem into a higher-dimensional one, which is then formally solved. We then apply Lemma 2.1 to the results of the calculations to obtain the estimates in the ﬁrst conclusion of the theorem. 3. The multiple scales procedure followed in [7] generates an asymptotic series. The partial sum Ψ(i),n (, t) denotes that the series, in which the nuclei propagate semiclassically under the inﬂuence of the eﬀective potential generated by electrons in the eigenstate Φ(i) (X), has been truncated at the n term, plus “perpendicular” terms for the next two orders. That is, Ψ(i),n (, t) ≡

n

n+1 n+2 Ψk (X, t)j + Ψ⊥ + Ψ⊥ . n+1 (X, t) n+2 (X, t)

(3.56)

k=0

The perpendicular terms Ψ⊥ k (X, t) arise in the multiple scales procedure as projections of Ψk (, X, ζ, x, t) orthogonal to the electron eigenfunction Φ(i) (X). 4. The functions Ψj (, X, ζ, x, t) can be calculated explicitly by extending the formal process of the multiple scales expansion.

3.2

Notation

ˆ µ, t) to denote the formal Dyson series expansion, and Ψ(, t) to We will use Ψ(, denote the solution to the unperturbed case discussed in Section 3. As before, the

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ith electron eigenfunction of n 2 −1 1 h(X) = ∆xj + V (X, ζ, x, q) ∆ζ + 2 j is represented by Φ(i) (X), with corresponding eigenvalue E(i) (X). In the development of the solution to the perturbed case, we will discover that the event of an electron transition at time s1 will change the dynamics of the nuclei at that time. Before s1 , the nuclei will propagate under the inﬂuence of the eﬀective potential created by the initial electron energy level, say E(i) (X), and then afterward will propagate under the inﬂuence of the excited electron energy level E(j) . To denote the eﬀect of multiple electron transitions, we will use a convenient shorthand notation. Whenever we wish to designate the semiclassical state ϕj (A(i) (t), B (i) (t), 2 , a(i) (t), η (i) (t), X) propagating using the eﬀective potential from the ith electron energy level, we will (i) (i,k ,...,kn ) (s1 , . . . , sn , t) use the shorthand ϕj (t). Furthermore, by the expression ϕj 1 we will mean the semiclassical state that propagates using the eﬀective potential from the ith electron energy level in the time interval (Tl , s1 ], using the eﬀective potential from the k1 th electron energy level in the time interval (s1 , s2 ], and so on, using the dynamics in the kn th level during the interval (sn , t]. Thus,  (i) ϕj (τ ) −∞ < τ < s1 ,     ϕ(k1 ) (τ ) s1 ≤ τ < s2 , j (i,k ,...,kn ) (3.57) ϕj 1 (s1 , . . . , sn , t) = ..   .    (kn ) ϕj (τ ) sn ≤ τ < t. It is understood that the solutions to (2.28) in each interval will be chosen to make the function continuous on the interval boundaries. We deﬁned Ψ(i),n (, t) in the third remark of Theorem 3.1. When we talk about Ψ(i,k1 ,...,kn ),n (, s1 , . . . , sn , t) we will mean that all the ϕj ’s in the formula (i,k ,...,kn ) (s1 , . . . , sn , t). Finally, note that by our deﬁnition, are in fact of the form ϕj 1 an expression like Ψ(i,k1 ),n (, s1 , s1 ) would mean that the ϕj ’s in the formula are (i) (k ) ϕj ’s up to but not including time s1 , and that at the time s1 , they are ϕj 1 ’s. We will use similar notation with the analogue of the classical action S (i) (t).

3.3

Orthogonality lemma

In our calculation of the formal Dyson expansions, we will require the fact that series determined in Section 3 from diﬀerent electron eigenfunctions Φ(i) (X) are approximately orthogonal in the space of the ζ and x variables.

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Lemma 3.1 For any time t, Ψ(i),n (, t), Ψ(j),n (, t)ζ,x = O(n+1 ) .

(3.58)

Proof. Consider a revised Born-Oppenheimer equation i2

∂Ψ 4 = − f (t)∆X Ψ + h(X)Ψ + µ Ci Xi F (t/2 ) ∂t 2 i

(3.59)

or

∂Ψ = H (, µ, t)Ψ . (3.60) ∂t Here, f (t) ∈ C ∞ is chosen to be identically one on the interval (Tl , 0), and identically zero on the interval (1, ∞). Thus, the asymptotic series solutions to (3.59) are exactly those calculated in Section 3 when t < 0. We will show that the approximate solutions that result from an application of Theorem 3.1 are exactly orthogonal when t > 1, and then apply the propagator for this new equation to show that the approximate solutions were almost orthogonal even when t < 0. Since the propagator is unitary, this procedure will prove the lemma. To ﬁnd asymptotic series solutions to (3.59), one may carry out computations in exactly the same manner as in Section 3. The presence of f (t) requires only minor modiﬁcations in certain formulas there. We will show that all ψj⊥ terms will i2

vanish for t > 1. Since all the ψj terms are multiples of the electron eigenfunctions, the truncated approximations Ψ(i),n (, t) will all be exactly orthogonal for diﬀerent i when t > 1. Now, ψ0 and ψ1 are both completely parallel to the electron eigenfunctions, so we begin our consideration with the equation determining ψ2⊥ . ψ2⊥ (z, y, t) = ig0 (z, y, t)[h(z) − E(i) (z)]−1 (f (t)η(t) · ∇z Φ(z)) .

(3.61)

It is clear that when t > 1, then f (t) = 0, and hence ψ2⊥ vanishes. For ψ3⊥ , we have the equation ψ3⊥ = [h(z) − E(i) (z)]−1 ig1 f (t)η(t) · ∇z Φ(i) (z) + P⊥ f (t)(∇y g0 ) · ∇z Φ(i) (z) . (3.62) Each term contains an f (t), and hence vanishes when t > 1. For the induction step (j ≥ 4), we have that ψj⊥

j k 1 (k) y ⊥ ⊥ f (t)∆y ψj−2 − ψ = [h(z) − E(i) (z)] E(i) 2 k! j−k k=2 1 + i P⊥ f (t)η(t) · ∇z ψj−2 + P⊥ f (t)∇z · ∇y ψj−3 + P⊥ f (t)∆z ψj−4 . (3.63) 2 −1

iP⊥ ψ˙ j−2 −

⊥ for k ≥ 2 are already zero Most terms contain f (t). Terms containing ψj−k ⊥ by the previous parts of the induction. One term contains a derivative of ψj−2 ,

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hence will contain f (t) and its derivatives, which are also identically zero for t > 1. Thus we see that all ψj⊥ terms vanish for t > 1, hence we have exact orthogonality on that interval. Now consider ∂ Ψ(i),n (, t), Ψ(j),n (, t)ζ,x ∂t ∂ ∂ = Ψ(i),n (, t), Ψ(j),n (, t) + Ψ(i),n (, t), Ψ(j),n (, t) ∂t ∂t H () n+1 Ψ(i),n (, t) + O( ), Ψ(j),n (, t) = i2 H () n+1 + Ψ(i),n (, t), Ψ(j),n (, t) + O( ) i2 H () H () n+1 = O( )+ Ψ(i),n (, t), Ψ(j),n (, t) + Ψ(i),n (, t), Ψ(j),n (, t) i2 i2 (3.64) by equation (3.60) and Theorem 3.1. Because the operator H () is self-adjoint, the last two terms in the last step above cancel. The conclusion must be that the lemma holds for the revised equation (3.59). Since for t < 0 the states of the revised equation are identical to those of our original equation, the above argument repeated with H() in place of H () shows that the lemma holds for (2.17).

3.4

General case

Theorem 3.2 Let h(X) satisfy the conditions of Theorem 3.1, and suppose that for any particular X, P˜ (X) is the projection, deﬁned on the electron Hilbert space, determined by the set of eigenvalues E(i) (X) of h(X). Further deﬁne ˜ P= (P˜ (X)) dX . (3.65) R3n1

Then an approximate formal Dyson series solution to O(µ) for wave-functions ˆ µ, t) where Ψ ˆ satisﬁes P˜ Ψ(, i2

∂ ˆ ˆ µ, t) Ψ(, µ, t) = (H() + µH1 )Ψ(, ∂t

(3.66)

is given by iµ t iS (i) (t)/2 ˜ ˆ P Ψ(, µ, t) = Ψ(i),n (, t)e − 2 ds1 × Tl (i,k1 ) (s1 ,t)/2 Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x Ψ(i,k1 ),n (, s1 , t)eiS . k1

(3.67)

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Stark Eﬀect and Time-Dependent Born-Oppenheimer Approximation

This approximation is asymptotic in the sense that it satisﬁes

U,µ (t, Tl )Ψ(, ˆ µ, Tl ) − Ψ(, ˆ µ, t) ≤ Cn n+1 + Kµ2 .

965

(3.68)

ˆ µ, t). Remark. We will compute this approximation only for wave-functions P˜ Ψ(, Thus, we are ignoring the piece of the wave-function where any electrons are removed to the continuum, or ionized. One can physically measure the result of this projection, and hence, in a practical sense, one can aﬀord to ignore the continuum in this expansion. For all s ≤ t, the electronic conﬁguration state in what we are computing is given by some linear combination of the electronic eigenfunctions Φ(i) . Proof. We will ﬁrst verify formula (3.67) from the deﬁnition of the formal Dyson series expansion. The ﬁrst term is given by U (t, Tl )Ψ(i),n (, Tl ) = eiS

(i)

(t)/2

Ψ(i),n (, t)

and introduces an error of only O(n+1 ) by Theorem 3.1. The second term is given by −iµ t ds1 U (t, s1 )H1 (s1 /2 )U (s1 , Tl )Ψ(i),n (, Tl ) . 2 Tl

(3.69)

(3.70)

Of course, in (3.70) U (s1 , Tl )Ψ(i),n (, Tl ) = eiS

(i)

(s1 )/2

Ψ(i),n (, s1 ),

once again making an acceptable error of O(n+1 ). Lemma 3.1 allows us to easily calculate components in the Ψ(k1 ),n direction. So we now express H1 (s1 /2 ) applied to the right-hand side of (3.71) as (i) 2 Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x Ψ(i,k1 ),n (, s1 , s1 )eiS (s1 )/ k1

(3.71) with an error of O(n+1 ). Note that the inner product here is taken in the space of the electron variables so that the expansion is in terms of a complete basis for ˜ the range of P. Now we apply U (t, s1 ) to (3.71), and the propagator acts diﬀerently on the individual terms of the sum, moving the nuclei semiclassically inside the eﬀective potential of the k1 th level. So, in fact, the ﬁrst order term of the Dyson series can be written iµ − 2

t

Tl

ds1

Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x

k1

Ψ(i,k1 ),n (, s1 , t)eiS

(i,k1 ) (s1 ,t)/2

. (3.72)

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At no stage in this calculation have we introduced an error of more than O(n+1 ) by propagating or taking inner products with the truncated approximations Ψ(i),n (, t). It remains to be shown, however, that this calculation has produced an accurate approximation to the solution of the problem perturbed in µ. The appropriate quantity to investigate comes from Lemma 2.1. (i) 2 iµ t ∂ ds1 i2 − (H() + µH1 ) Ψ(i),n (, t)eiS (t)/ − 2 ∂t Tl (i,k1 ) (s1 ,t)/2 Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x Ψ(i,k1 ),n (, s1 , t)eiS . k1

(3.73) We will break this computation into several steps. Step A. Because of Theorem 3.1, we already know that (i) 2 ∂ i2 − (H() + µH1 ) Ψ(i),n (, t)eiS (t)/ ∂t = O(

n+3

2

) − µH1 (t/ )Ψ(i),n (, t)e

(3.74) iS (i) (t)/2

.

Step B. Take the derivative 2 ∂ i Ψ(i,k1 ),n (, t, t)||H1 (t/2 )Ψ(i),n (, t)ζ,x Ψ(i,k1 ),n (, t, t) (3.72) = µ ∂t k1 t (i) 2 iµ eiS (t)/ − 2 ds1 Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x Tl k1 (i,k1 ) ∂ (s1 ,t)/2 × i2 ]. (3.75) [Ψ(i,k1 ),n (, s1 , t)eiS ∂t Step C. We apply −(H() + H1 (t/2 )) to (3.72). If we apply Lemma 2.1 to the sum of the results of the above three steps, we will obtain an estimate on the size of the error in our approximation (3.67). The ﬁrst term of (3.74) gives us an O(n+1 ) error to the wave-function. Now consider the sum of the ﬁrst term on the RHS of (3.75) with the µ term of (3.74). Because of Lemma 3.1, we must conclude that the sum has the form µn+1 ξ(t/2 ) . This term contributes an error bounded by t ds µn+1 ξ(s/2 ) −2

(3.76)

(3.77)

Tl

and once we carry out the integral via the substitution u = s/2 , we see that it is bounded by O(µn+1 ).

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Next, the second term of (3.75) and the ﬁrst term from Step C combine by virtue of Theorem 3.1 to give −

iµ 2

t Tl

ds1

Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 )ζ,x × O(n+1 )

(3.78)

k1

and again, using the substitution u = s1 /2 to do the integral, this term is O(n+1 ). Our calculation leaves us with only the second term from Step C and we choose to use substitution in the integral again to get a term which contributes an error bounded by

−2

t

Tl

2

2

ds iµ H1 (s/ )

s/2

du Tl /2

Ψ(i,k1 ),n (, 2 u, 2 u)||H1 (u)Ψ(i),n (, 2 u)Ψ(k1 ),n (, 2 u, 2 s)eiS

(i,k1 ) (u,s)/2

.

k1

(3.79) We use the substitution v = s/2 to see that this term is O(µ2 ). Hence the conclusion of Theorem 3.2.

3.5

Molecule without continuum

Theorem 3.3 Let h(X) satisfy the conditions of Theorem 3.1, with the additional condition that the spectrum of h(X) contains only the set of eigenvalues E(i) (X). That is, h(X) has purely discrete spectrum. Then an approximate formal Dyson series solution to O(µm ) for the equation i2

∂ ˆ ˆ µ, t) Ψ(, µ, t) = (H() + µH1 )Ψ(, ∂t

(3.80)

is given by iµ t iS (i) (t)/2 ˆ Ψ(, µ, t) = Ψ(i),n (, t)e − 2 ds1 Tl (i,k1 ) (s1 ,t)/2 Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 )Ψ(i,k1 ),n (, s1 , t)eiS k1

m t s2 sm iµ + − 2 dsm dsm−1 . . . ds1 Ψ(i,k1 ,...,km ),n (, s1 , . . . , sm , t) Tl Tl Tl eiS

(i,k1 ,...km ) (s1 ,...,sm ,t)/2

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× Ψ(i,k1 ,...,km ),n (, s1 , s2 , . . . , sm , sm )H1 (sm /2 ) km

×

Ψ(i,k1 ,...,km−1 ),n (, s1 , . . . , sm ) Ψ(i,k1 ,...,km−1 ),n (, s1 , s2 , . . . , sm−1 , sm−1 )H1 (sm−1 /2 )

km−1

Ψ(i,k1 ,...,km−2 ),n (, s1 , . . . , sm−1 ) 2 Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 / )Ψ(i),n (, s1 ) . . . × ··· . (3.81) k1

This approximation is asymptotic in the sense that it satisﬁes

U,µ (t, Tl )Ψ(, ˆ µ, Tl ) − Ψ(, ˆ µ, t) ≤ Cn n+1 + Km µm+1 .

(3.82)

Remarks 1. The zeroth and ﬁrst order terms (in µ) are identical to those found in (3.67). The additional assumption that the electrons have no continuum to which to escape allows us to extend the series. 2. We may summarize the intuitive meaning of each term of the Dyson series. The leading order term is propagation with no electron transitions. The O(µ) term covers all the possible situations where the molecule begins with initial electron energy state E(i) and then at some time s1 makes an electron transition to some other state E(k1 ) . If we sum over all the excited states k1 and integrate over all the possible times s1 when this transition can occur, we get the ﬁrst order term of the Dyson expansion (3.81). In general, the O(µm ) term covers all the possibilities for m electron transitions. As one can see from looking at Deﬁnition 2.22, the process of calculating additional terms is recursive. We show the calculation of the second order term (in µ). The other terms are computed in a completely analogous manner. Let us ˆ 1 (, µ, Tl ). denote the second term of (3.81) evaluated at t = Tl by Ψ The third term of the Dyson expansion would be −iµ t ˆ 1 (, µ, Tl ). ds1 U (t, s2 )H1 (s2 /2 )U (s2 , Tl )Ψ (3.83) 2 Tl Applying U (s2 , Tl ), we have −

iµ 2

s2

Tl

ds1

Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 )

k1

Ψ(i,k1 ),n (, s1 , s2 )eiS while making an error of O(n+1 ).

(i,k1 ) (s1 ,s2 )/2

(3.84)

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We also expand the operator H1 (s2 ) in the approximate basis of Lemma 3.1 to obtain iµ − 2 Ψ(i,k1 ,k2 ),n (, s1 , s2 , s2 )H1 (s2 /2 ) k2 s2 × ds1 Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 ) Tl

k1

× Ψ(i,k1 ),n (, s1 , s2 )eiS

(i,k1 ) (s1 ,s2 )

/2 Ψ(i,k1 ,k2 ),n (, s1 , s2 , s2 ).

(3.85)

After interchanging the order of integration over s1 with the inner product integration and the sum over the electron states (k2 ), we then apply U (t, s2 ) inside the integral sign and, per the Dyson expansion procedure, integrate over s2 to ﬁnd the ﬁnal form of the term t s2 iµ Ψ(i,k1 ,k2 ),n (, s1 , s2 , s2 )H1 (s2 /2 ) − ( 2 )2 ds2 ds1 Tl Tl k2 × Ψ(i,k1 ),n (, s1 , s1 )||H1 (s1 /2 )Ψ(i),n (, s1 )Ψ(i,k1 ),n (, s1 , s2 ) k1

× Ψ(i,k1 ,k2 ),n (, s1 , s2 , t)eiS

(i,k1 ,k2 ) (s1 ,s2 ,t)/2

. (3.86)

At no stage in this calculation have we introduced an error of more than O(n+1 ) by propagating or taking inner products with the truncated approximations Ψ(i,k1 ),n (, s1 , t). It remains to be shown, once again, that this calculation has produced an accurate approximation to the solution of the problem perturbed in µ. The proof of Theorem 3.2 provides us with the anchor to an induction proof in m. We will assume that we have carried out the calculation of the truncation error from Lemma 2.1 on the series up to the (m−1)st term and were left with terms that will produce a solution error of O(n+1 ) and a term of the form t sm −1 (−iµ)m 2 dsm−1 dsm−2 . . . H1 (t/ ) 2(m−1) Tl Tl s2 (i,k1 ,...km −1 ) (s1 ,...,sm −1 ,t)/2 ds1 Ψ(i,k1 ,...,km −1 ),n (, s1 , . . . , sm−1 , t)eiS Tl × Ψ(i,k1 ,...,km−1 ),n (, s1 , s2 , . . . , sm−1 , sm−1 ) km−1

×

km−2

H1 (sm−1 /2 )Ψ(i,k1 ,...,km−2 ),n (, s1 , . . . , sm−1 ) Ψ(i,k1 ,...,km−2 ),n (, s1 , s2 , . . . , sm−2 , sm−2 )

H1 (sm−2 /2 )Ψ(i,k1 ,...,km−3 ),n (, s1 , . . . , sm−2 ) × · · · Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 ) . . . × . (3.87) k1

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We need to calculate the truncation error from just the O(µm ) term of (3.81). If we apply Lemma 2.1 to the sum of the results of the above three steps, we will obtain an estimate on the size of the error in our approximation (3.81). ∂ i2 − (H() + µH1 ) [O(µm ) term of (3.81)] . ∂t

(3.88)

The computation involves two steps. ∂ Step A. Apply the derivative operator i2 ∂t . We notice that the ﬁrst term of the result is just the expansion of the previously left over term (3.87), in the approximate basis of Lemma 3.1, with the opposite sign. The sum of those two terms gives us an O(n+1 ) error to the wave-function by applying the reasoning of (3.77). The second term resulting from the derivative will combine with the ﬁrst term resulting from Step B.

Step B. We apply −(H() + H1 (t/2 )) to the quantity of which we took the derivative in Step A. The ﬁrst term of Step B combines with the second term of Step A to give −

−iµ 2

m

s2 sm dsm dsm−1 . . . ds1 Tl Tl Tl × Ψ(i,k1 ,...,km ),n (, s1 , s2 , . . . , sm , sm ) H1 (sm /2 ) t

km

×

Ψ(i,k1 ,...,km−1 ),n (, s1 , s2 , . . . , sm−1 , sm−1 )H1 (sm−1 /2 ) . . .

km−1

×

Ψ(i,k1 ),n (, s1 , s1 )|| H1 (s1 /2 )Ψ(i),n (, s1 )Ψ(i,k1 ),n (, s1 , s2 ) . . .

k1

× Ψ(i,k1 ,...,km−1 ),n (, s1 , . . . , sm ) × O(n+1 ) . (3.89) By virtue of Theorem 3.1, and again, using the substitution u = s1 /2 to do the integral, this term is O(n+1 ). Our calculation leaves us with only a second term from Step B. By using analogous reasoning to that in the proof of Theorem 3.2 on this term, we obtain the result that this term is O(µm+1 ). Hence the conclusion of Theorem 3.3.

3.6

Conclusion

We began by investigating the perturbation problem resulting from a semiclassical model of a molecule excited by an electromagnetic ﬁeld pulse. Working from the

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asymptotic expansion for the solution of the unperturbed time-dependent BornOppenheimer problem, we developed another asymptotic expansion for the perturbed problem using the Dyson series technique. The results are valid for a perturbation linearly dependent on the molecular variables, with a time-dependency that is a function of compact support. We establish results for smooth potentials V (X, x) and compute asymptotics to any order in , and provided that the electron Hamiltonian has purely discrete spectrum, to any order in µ.

References [1] Peter Atkins, Physical Chemistry, New York: W.H. Freeman 1986. [2] J.-M. Combes, P. Duclos and R. Seiler, The Born-Oppenheimer Approximation, in Rigorous Atomic and Molecular Physics (eds. G. Velo, A. Wightman), pp. 139–159. Wien, New York: Springer 1977. [3] S. Guerin, F. Monti, J.-M. Dupont and H.R. Jauslin, On the relation between cavity-dressed states, Floquet states, RWA, and semiclassical models, J. Phys. A. Math. Gen. 30, 7193–7215 (1997). [4] George A. Hagedorn, High order corrections to the time-dependent BornOppenheimer Approximation I: Smooth Potentials, Ann. of Math. (2) 124, no. 3, 571–590 (1986). [5] George A. Hagedorn, Molecular propagation through electron energy level crossings (Memoirs of the American Mathematical Society 111, No. 536), Providence, RI: American Mathematical Society 1994. [6] George A. Hagedorn, Raising and lowering operators for semiclassical wave packets, Ann. Physics 269, no. 1, 77–104 (1998). [7] Time-reversal invariance and the time-dependent Born-Oppenheimer approximation, in Forty more years of ramiﬁcations: spectral asymptotics and its applications, 161–197, Discourses Math. Appl., 1, Texas A&M Univ., College Station, TX 1991. [8] George A. Hagedorn, Alain Joye, Semiclassical dynamics with exponentially small error estimates, Comm. Math. Phys. 207, no. 2, 439–465 (1999). [9] George A. Hagedorn, Semiclassical quantum mechanics IV. Large order asymptotics and more general states in more than one dimension, Ann. Inst. H. Poincar´e Phys. Th´eor. 42, no. 4, 363–374 (1985). [10] George A. Hagedorn, Sam L. Robinson, Bohr-Sommerfeld quantization rules in the semiclassical limit, J. Phys. A. Math. Gen. 31, 10113–10129 (1998).

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[11] Lee W. Johnson, R. Dean Riess, Numerical Analysis, Reading, Massachusetts: Addison-Wesley Publishing Company, 1982. [12] Serge Lang, Analysis I, London: Addison-Wesley Publishing Company 1969. [13] Harm Geert Muller, Photoionization of Atoms in Strong Radiation Fields, Amsterdam: Vrije Universiteit te Amsterdam 1985. [14] Nord´en, Bengt, 1999 Nobel Prize in Chemistry Press Release, http://www.nobel.se/announcement-99/chemback99.pdf, 1999. [15] M. Reed and B. Simon, Methods of Modern Mathematical Physics, New York, London: Academic Press 1978. [16] K. Yajima, Resonances for the AC-Stark Eﬀect. Comm. Math. Phys. 87, 331– 352 (1982).

Steven W. Jilcott, Jr. ALPHATECH, Inc., 50 Mall Rd. Burlington, MA 01803 USA email: [email protected] Communicated by Vincent Rivasseau submitted 28/10/02, accepted 26/05/03

Ann. Henri Poincar´e 4 (2003) 973 – 999 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/050973-27 DOI 10.1007/s00023-003-0152-6

Annales Henri Poincar´ e

Long Range Scattering and Modified Wave Operators for the Wave-Schr¨ odinger System II∗ J. Ginibre and G. Velo

Abstract. We continue the study of scattering theory for the system consisting of a Schr¨ odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper, we proved the existence of modiﬁed wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the diﬀerent propagation properties of the wave and Schr¨ odinger equations. Here we eliminate that condition by using an improved asymptotic form for the solutions.

1 Introduction This paper is a sequel to a previous paper with the same title ([1], hereafter referred to as I) where we studied the theory of scattering and proved the existence of modiﬁed wave operators for the Wave-Schr¨odinger (WS) system (1.1) i∂t u = −(1/2)∆u − Au A = |u|2

(1.2)

where u and A are respectively a complex-valued and a real-valued function deﬁned in space time R3+1 . We refer to the introduction of I for general background and references and we give here only a general overview of the problem. The main result of I was the construction of modiﬁed wave operators for the WS system, with no size restriction on the solutions. That construction basically consists in solving the Cauchy problem for the WS system with inﬁnite initial time, namely in constructing solutions with prescribed asymptotic behaviour at inﬁnity in time. That asymptotic behaviour is imposed in the form of suitable approximate solutions of the WS system. One then looks for exact solutions, the diﬀerence of which with the given approximate ones tends to zero at inﬁnity in time in a suitable sense, more precisely in suitable norms. The approximate solutions are obtained as low order iterates in an iterative resolution scheme of the WS system. In I we used second order iterates. They are parametrized by data (u+ , A+ , A˙ + ) which play the role of (actually are in simpler cases) initial data at time zero. Those data constitute the asymptotic state for the actual solution. ∗ Work

supported in part by NATO Collaborative Linkage Grant 979341

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An inherent diﬃculty of the WS system is the diﬀerence of propagation properties of the wave equation and of the Schr¨ odinger equation. Because of that diﬃculty, we had to impose in I a support condition on the Fourier transform F u+ of the Schr¨ odinger asymptotic state u+ , saying in eﬀect that F u+ vanishes in a neighborhood of the unit sphere, so that u+ generates a solution of the free Schr¨ odinger equation which is asymptotically small in a neighborhood of the light cone. Such a support condition is unpleasant because it cannot be satisﬁed on a dense subspace of any reasonable space where one hopes to solve the problem, typically with u in F H k for k > 1/2 (H k is the standard L2 based Sobolev space). The theory of scattering and the existence of modiﬁed wave operators can also be studied for various equations and systems including the WS system by a method simpler than that of I, proposed earlier by Ozawa [5]. Contrary to that of I, that method is restricted to the case of small data and small solutions. It has been applied to various systems, in particular to the Klein-Gordon-Schr¨ odinger (KGS) system in dimension 2, which is fairly similar to the WS system in dimension 3 from the point of view of scattering [6]. Similar propagation diﬃculties also appear for that system, thereby again requiring a support condition on F u+ in the treatment given in [6]. A progress on that problem was made recently by Shimomura [7], [8] who was able to remove the previous support condition in the construction of the modiﬁed wave operators by the Ozawa method in the case of the KGS system in dimension 2 [7] and of the WS system in dimension 3 [8]. The key of that progress consists in using an improved asymptotic form for the Schr¨ odinger function, obtained by adding a term depending on (A+ , A˙ + ) which partly cancels the contribution of the asymptotic ﬁeld for A in the Schr¨ odinger equation. Although the method used in I is more complicated than the Ozawa method (so as to accommodate arbitrarily large data and solutions), it turns out that the improved asymptotic form of u used in [8] can be transposed into the framework of the method of I, thereby allowing to remove the support condition on F u+ assumed in I, at least in a restricted interval of values of the parameters deﬁning the regularity of the solutions. The purpose of the present paper is to implement that improvement, namely to rederive the main results of I without assuming the support condition used in I, by using the improved asymptotic form of the solution inspired by that of [8]. In the remaining part of this introduction, we shall brieﬂy review the method used in I in the modiﬁed form used in the present paper. We refer to Section 2 of I for a more detailed exposition. The main result of this paper will be stated in semi-heuristic terms at the end of this introduction. The ﬁrst step in that method consists in eliminating the wave equation (1.2) by solving it for A and substituting the result into the Schr¨ odinger equation, which then becomes both non-linear and non-local in time. One then parametrizes the Schr¨ odinger function u in terms of an amplitude w and a phase ϕ and one replaces the Schr¨ odinger equation by an auxiliary system consisting of a transport equation for the amplitude and a Hamilton-Jacobi equation for the phase. One solves the Cauchy problem at inﬁn-

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ity, namely with prescribed asymptotic behaviour, for the auxiliary system, and one ﬁnally reconstructs the solution of the original WS system from that of the auxiliary system. We now proceed to the technical details. We restrict our attention to positive time, actually to t ≥ 1. We ﬁrst eliminate the wave equation. We deﬁne ω = (−∆)1/2 and we replace (1.2) by

,

K(t) = ω −1 sin ωt

,

˙ K(t) = cos ωt

A = A0 + A1 (|u|2 )

(1.3)

˙ ˙ A0 = K(t)A + + K(t)A+ , ∞ dt K(t − t )|u(t )|2 . A1 (|u|2 ) = −

(1.4)

where

(1.5)

t

Here A0 is a solution of the free wave equation with initial data (A+ , A˙ + ) at time t = 0. The pair (A+ , A˙ + ) is the asymptotic state for A. We next perform the change of variables mentioned above on u. The unitary group U (t) = exp(i(t/2)∆) (1.6) which solves the free Schr¨odinger equation can be written as U (t) = M (t) D(t) F M (t) where M (t) is the operator of multiplication by the function M (t) = exp ix2 /2t ,

(1.7)

(1.8)

F is the Fourier transform and D(t) is the dilation operator D(t) = (it)−n/2 D0 (t)

(1.9)

(D0 (t)f ) (x) = f (x/t) .

(1.10)

where We parametrize u in terms of an amplitude w and of a real phase ϕ as u(t) = M (t) D(t) exp[−iϕ(t)]w(t) .

(1.11)

Substituting (1.11) into (1.1) yields an evolution equation for (w, ϕ), namely i∂t + (2t2 )−1 ∆ − i(2t2 )−1 (2∇ϕ · ∇ + ∆ϕ) + t−1 B + ∂t ϕ − (2t2 )−1 |∇ϕ|2 w = 0 (1.12) where we have expressed A in terms of a new function B by A = t−1 D0 B .

(1.13)

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Corresponding to the decomposition (1.3) of A, we decompose B = B0 + B1 (w, w) where A0 = t−1 D0 B0 and A1 = t−1 D0 B1 . One computes easily ∞ B1 (w1 , w2 ) = dν ν −3 ω −1 sin((ν − 1)ω)D0 (ν)(Re w ¯1 w2 )(νt) .

(1.14)

(1.15)

1

At this point, we have only one evolution equation (1.12) for two functions (w, ϕ). We arbitrarily impose a second equation, namely a Hamilton-Jacobi (or eikonal) equation for the phase ϕ, thereby splitting the equation (1.12) into a system of two equations, the other one of which being a transport type equation for the amplitude w. For that purpose, we split B0 and B1 into long range and short range parts as follows. Let χ ∈ C ∞ (R3 , R), 0 ≤ χ ≤ 1, χ(ξ) = 1 for |ξ| ≤ 1, χ(ξ) = 0 for |ξ| ≥ 2 and let 0 < β0 , β < 1. We deﬁne B0 = B0L + B0S where

,

B1 = BL + BS

F B0L (t, ξ) = χ(ξt−β0 )F B0 (t, ξ) , F BL (t, ξ) = χ(ξt−β )F B1 (t, ξ) .

(1.16)

(1.17)

The splitting (1.16), (1.17) diﬀers from that made in I in two respects. First and more important is the fact that we perform that splitting both on B0 and on B1 , whereas in I it was done only on B1 . Second, we use here a smooth cut-oﬀ χ instead of a sharp one. The smooth cut-oﬀ is actually needed only for B0 . For β = β0 , the splitting is the same for B0 and B1 and can therefore be performed on B without any reference to the asymptotic state (A+ , A˙ + ). The parameters β0 and β will have to satisfy various conditions which will appear later, all of them compatible with β = β0 = 1/3. We split the equation (1.12) into the following system of two equations. ∂t w = i(2t2 )−1 ∆w + t−2 Q(∇ϕ, w) + it−1 (B0S + BS (w, w))w (1.18) ∂t ϕ = (2t2 )−1 |∇ϕ|2 − t−1 B0L − t−1 BL (w, w) where we have deﬁned Q(s, w) = s · ∇w + (1/2)(∇ · s)w

(1.19)

for any vector ﬁeld s. The ﬁrst equation of (1.18) is the transport type equation for the amplitude w, while the second one is the Hamilton-Jacobi type equation for the phase ϕ. Since the right-hand sides of (1.18) contain ϕ only through its gradient, we can obtain from (1.18) a closed system for w and s = ∇ϕ by taking the gradient of the second equation, namely ∂t w = i(2t2 )−1 ∆w + t−2 Q(s, w) + it−1 (B0S + BS (w, w))w (1.20) ∂t s = t−2 s · ∇s − t−1 ∇B0L − t−1 ∇BL (w, w) .

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Once the system (1.20) is solved for (w, s), one recovers ϕ easily by integrating the second equation of (1.18) over time. The system (1.20) will be referred to as the auxiliary system. The construction of the modiﬁed wave operators follows the same pattern as in I. The ﬁrst task is to construct solutions of the auxiliary system (1.20) with suitably prescribed asymptotic behaviour at inﬁnity, and in particular with w(t) tending to a limit w+ = F u+ as t → ∞. That asymptotic behaviour is imposed in the form of a suitably chosen pair (W, φ) and therefore (W, S) with S = ∇φ with W (t) tending to w+ as t → ∞. For ﬁxed (W, S), we make a change of variables in the system (1.18) from (w, ϕ) to (q, ψ) deﬁned by (q, ψ) = (w, ϕ) − (W, φ)

(1.21)

or equivalently a change of variables in the system (1.20) from (w, s) to (q, σ) deﬁned by (q, σ) = (w, s) − (W, S) , (1.22) and instead of looking for a solution (w, s) of the system (1.20) with (w, s) behaving asymptotically as (W, S), we look for a solution (q, σ) of the transformed system with (q, σ) (and also ψ) tending to zero as t → ∞. Performing the change of variables (1.22) in the auxiliary system (1.20) yields the following modiﬁed auxiliary system for the new variables (q, σ)  2 −1 −2 −1   ∂t q = i(2t ) ∆q + t (Q(s, q) + Q(σ, W )) + it B0S q +it−1 BS (w, w)q + it−1 (2BS (W, q) + BS (q, q)) W − R1 (W, S)   ∂t σ = t−2 (s · ∇σ + σ · ∇S) − t−1 ∇ (2BL (W, q) + BL (q, q)) − R2 (W, S) , (1.23) where the remainders R1 (W, S) and R2 (W, S) are deﬁned by R1 (W, S) = ∂t W − i(2t2 )−1 ∆W − t−2 Q(S, W ) − it−1 (B0S + BS (W, W ))W (1.24) R2 (W, S) = ∂t S − t−2 S · ∇S + t−1 ∇B0L + t−1 ∇BL (W, W )

(1.25)

and the dependence of the remainders on B0 has been omitted in the notation. For technical reasons, it is useful to consider also a partly linearized version of the system (1.23), namely  2 −1 −2 −1   ∂t q = i(2t ) ∆q + t (Q(s, q ) + Q(σ, W )) + it B0S q −1 −1 +it BS (w, w)q + it (2BS (W, q) + BS (q, q)) W − R1 (W, S)   ∂t σ = t−2 (s · ∇σ + σ · ∇S) − t−1 ∇ (2BL (W, q) + BL (q, q)) − R2 (W, S) . (1.26) The construction of solutions (w, s) of the auxiliary system (1.20) deﬁned for large time and with prescribed asymptotic behaviour (W, S) proceeds in two steps. The ﬁrst step consists in solving the system (1.23) for (q, σ) tending to zero at inﬁnity under suitable boundedness properties of B0 and (W, S) and suitable time

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decay properties of the remainders R1 (W, S) and R2 (W, S), by a minor variation of the method used in I. That method consists in ﬁrst solving the linearized system (1.26) for (q , σ ) with given (q, σ), and then showing that the map (q, σ) → (q , σ ) thereby deﬁned has a ﬁxed point, by the use of a contraction method. The second step consists in constructing (W, S) with W (t) tending to w+ as t → ∞ and satisfying the required boundedness and decay properties. This is done by solving the auxiliary system (1.20) by iteration to second order as in I and then adding to W an additional term of the same form as that used in [8]. The detailed form of (W, S) thereby obtained is too complicated to be given here and will be given in Section 3 below (see (3.25)–(3.29) and (3.31)). Once the system (1.20) is solved for (w, s), one can proceed therefrom to the construction of a solution (u, A) of the original WS system. One ﬁrst deﬁnes the phases ϕ and φ such that s = ∇ϕ and S = ∇φ and one reconstructs (u, A) from (w, ϕ) by (1.11), (1.3), (1.5), thereby obtaining a solution of the WS system deﬁned for large time and with prescribed asymptotic behaviour. The modiﬁed wave operator for the WS system is then deﬁned as the map Ω : (u+ , A+ , A˙ + ) → (u, A). The main result of this paper is the construction of (u, A) from (u+ , A+ , A˙ + ) as described above, together with the asymptotic properties of (u, A) that follow from that construction. It will be stated below in full mathematical detail in Proposition 4.1. We give here a heuristic preview of that result, stripped from most technicalities. We set β = β0 = 1/3 for deﬁniteness. Proposition 1.1. Let β0 = β = 1/3. Let (u+ , A+ , A˙ + ) be such that w+ = F u+ ∈ H k+ for suﬃciently large k+ , that (A+ , A˙ + ) be suﬃciently regular, and that (F A+ , F A˙ + ) be suﬃciently small near ξ = 0. Let (W, S) be the approximate solution of the system (1.20) deﬁned by (3.25)–(3.29), (3.31). Then (1) There exists T = T (u+ , A+ , A˙ + ), 1 ≤ T < ∞, such that the auxiliary system (1.20) has a unique solution (w, s) in a suitable space, deﬁned for t ≥ T and such that (w − W, s − S) tends to zero in suitable norms when t → ∞. (2) There exists ϕ and φ such that s = ∇ϕ, S = ∇φ, φ(1) = 0 and such that ϕ−φ tends to zero in suitable norms when t → ∞. Deﬁne (u, A) by (1.11), (1.3), (1.5). Then (u, A) solves the system (1.1), (1.2) for t ≥ T and (u, A) behaves asymptotically as (M D exp(−iφ)W , A0 + A1 (|DW |2 )) in the sense that the diﬀerence tends to zero in suitable norms (for which each term separately is O(1)) when t → ∞. The unspeciﬁed condition that (F A+ , F A˙ + ) be suﬃciently small near ξ = 0 can be shown to follow from more intuitive conditions in x-space, consisting of decay conditions at inﬁnity in space, and, depending on the values of the parameters deﬁning the relevant function spaces, of some moment conditions on (A+ , A˙ + ). This paper relies on a large amount of material from I. In order to bring out the structure while keeping duplication to a minimum, we give without proof a shortened logically self-suﬃcient sequence of those intermediate results from I that

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are needed, and we provide a full exposition only for the parts that are new as compared with I. When quoting I, we shall use the notation (I.p.q) for equation (p.q) of I and Item I.p.q for Item p.q of I, such as Lemma, Proposition, etc. The remaining part of this paper is organized as follows. In Section 2 we collect notation and some estimates of a general nature. In Section 3, we study the Cauchy problem at inﬁnity for the auxiliary system (1.20). We recall from I the existence results of solutions under suitable boundedness properties of (W, S) and suitable decay properties of the remainders, with the appropriate modiﬁcations (Proposition 3.1). We then deﬁne (W, S) and prove that they satisfy the previous properties, concentrating on the terms in the remainders that are new as compared with I (Proposition 3.2). We then discuss the assumptions on (F A+ , F A˙ + ) at ξ = 0 mentioned above. Finally in Section 4, we construct the wave operators for the WS system (1.1) (1.2) and we derive the asymptotic properties of the solution (u, A) in their range that follow from the previous estimates (Proposition 4.1).

2 Notation and preliminary estimates In this section we introduce some notation and we collect a number of estimates which will be used throughout this paper. We denote by · r the norm in Lr ≡ Lr (R3 ) and we deﬁne δ(r) = 3/2 − 3/r. For any interval I and any Banach space X we denote by C(I, X) the space of strongly continuous functions from I to X and by L∞ (I, X) the space of measurable essentially bounded functions from I to X. For real numbers a and b we use the notation a ∨ b = Max(a, b) and a ∧ b = Min(a, b). In the estimates of solutions of the relevant equations we shall use the letter C to denote constants, possibly diﬀerent from an estimate to the next, depending on various parameters but not on the solutions themselves or on their initial data. We shall use the notation C(a1 , a2 , · · · ) for estimating functions, also possibly diﬀerent from an estimate to the next, depending on suitable norms a1 , a2 , · · · of the solutions or of their initial data. We shall use the Sobolev spaces H˙ rk and Hrk deﬁned for −∞ < k < +∞, 1 ≤ r ≤ ∞ by H˙ rk = u : u; H˙ rk ≡ ω k u r < ∞ and

Hrk = u : u; Hrk ≡ < ω >k u r < ∞

where ω = (−∆)1/2 and · = (1 + | · |2 )1/2 . The subscript r will be omitted if r = 2. We shall look for solutions of the auxiliary system (1.20) in spaces of the type C(I, X k, ) where I is an interval and X k, = H k ⊕ ω −1 H namely

X k, = (w, s) : w ∈ H k , ∇s ∈ H

(2.1)

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where it is understood that ∇s ∈ L2 includes the fact that s ∈ L6 , and we shall use the notation w; H k = |w|k . (2.2) We shall use extensively the following Sobolev inequalities, stated here in Rn , but to be used only for n = 3. Lemma 2.1 Let 1 < q, r < ∞, 1 n/r. Let σ satisfy j/k ≤ σ ≤ 1 and n/p − j = (1 − σ)n/q + σ(n/r − k) . Then the following inequality holds ω j u p ≤ C u 1−σ ω k u σr . q

(2.3)

The proof follows from the Hardy-Littlewood-Sobolev (HLS) inequality ([2], p. 117) (from the Young inequality if p = ∞), from Paley-Littlewood theory and interpolation. We shall also use extensively the following Leibnitz and commutator estimates. Lemma 2.2 Let 1 < r, r1 , r3 < ∞ and 1/r = 1/r1 + 1/r2 = 1/r3 + 1/r4 . Then the following estimates hold ω m (uv) r ≤ C ( ω m u r1 v r2 + ω m v r3 u r4 )

(2.4)

for m ≥ 0, and [ω m , u]v r ≤ C ω m u r1 v r2 + ω m−1 v r3 ∇u r4

(2.5)

for m ≥ 1, where [ , ] denotes the commutator. The proof of those estimates is given in [3], [4] with ω replaced by ω and follows therefrom by a scaling argument. We next give some estimates of B0L , B0S , BL and BS deﬁned by (1.16) (1.17). It follows immediately from (1.16) (1.17) that m−p m−p ω m B0L 2 ≤ 2tβ0 ω p B0L 2 ≤ 2tβ0 ω p B0 2

(2.6)

for m ≥ p and ω m B0S 2 ≤ tβ0 (m−p) ω p B0S 2 ≤ tβ0 (m−p) ω p B0 2

(2.7)

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for m ≤ p. Similar estimates hold for BL , BS with β0 replaced by β. On the other hand it follows from (1.15) that ω m+1 B1 (w1 , w2 ) 2 ≤ Im ( ω m (w ¯1 w2 ) 2 ) where Im is deﬁned by

(Im (f )) (t) =

∞

1

dν ν −m−3/2 f (νt) .

(2.8)

(2.9)

We ﬁnally collect some estimates of the solutions of the free wave equation A0 = 0 with initial data (A+ , A˙ + ) at time zero, given by (1.4). Lemma 2.3 Let k ≥ 0. Let A+ and A˙ + satisfy the conditions A+ , ω −1 A˙ + ∈ H k

,

∇2 A+ , ∇A˙ + ∈ H1k .

(2.10)

Then the following estimate holds: ω m A0 r ≤ b0 t−1+2/r

for 2 ≤ r ≤ ∞ ,

(2.11)

for 0 ≤ m ≤ k and for all t > 0, where b0 depends on (A+ , A˙ + ) through the norms associated with (2.10). The estimate (2.11) can be expressed in an equivalent form in terms of B0 deﬁned by (1.13), namely ω m B0 r ≤ b0 tm−1/r

for 2 ≤ r ≤ ∞ .

(2.12)

Furthermore, it follows from (1.17) and (2.12) that ω m B0L r ≤ F −1 χ 1 ω m B0 r ≤ Cb0 tm−1/r

(2.13)

where we have used the Young inequality and the fact that the L1 -norm of F −1 χ is invariant under the rescaling of ξ by tβ0 which occurs in (1.17). From (2.12) (2.13) and (1.16) it follows that also ω m B0S r ≤ Cb0 tm−1/r .

(2.14)

In the applications, the estimate (2.12) will be used mostly through its consequence (2.14).

3 Cauchy problem at infinity for the auxiliary system In this section, we solve the Cauchy problem at inﬁnity for the auxiliary system (1.20) in the diﬀerence form (1.23). We ﬁrst solve the system (1.23) for (q, σ) tending to zero at inﬁnity under suitable boundedness properties of (B0 , W, S) and suitable time decay properties of the remainders R1 (W, S) and R2 (W, S). We

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then construct (W, S) with W (t) tending to w+ = F u+ as t → ∞ and satisfying the required boundedness and decay properties. The method closely follows that of Sections 6 and 7 of I. We ﬁrst estimate a single solution of the linearized auxiliary system (1.26) at the level of regularity where we shall eventually solve the auxiliary system (1.20). The following lemma is basically Lemma I.6.1, restricted to the case where 1 < k < 3/2, and sharpened in order to take into account the fact that the W used in this paper is less regular than that used in I (compare (3.1) below with (I.6.1)). Lemma 3.1 Let 1 < k < 3/2 < and β > 0. Let T ≥ 1 and I = [T, ∞). Let B0 satisfy the estimate (2.12) for 0 ≤ m ≤ k. Let (U (1/t))W, S) ∈ C(I, X k+1,+1 ) ∩ C 1 (I, X k, ) and let W satisfy Sup W ∞ ∨ W ; H 3/2 ∨ t1/2−k W ; H˙ k+1 ≤ a < ∞ . (3.1) t∈I

Let (q, σ), (q , σ ) ∈ C(I, X k, ) with q ∈ L∞ (I, H k ) ∩ L2 (I, L2 ) and let (q , σ ) be a solution of the system (1.26) in I. Then the following estimates hold for all t ∈ I : |∂t q 2 | ≤ C t−2 a ∇σ 2 +t−1−β a2 I0 ( q 2 ) +t−1 a I−1 ( q 2 q 3 ) + R1 (W, S) 2 , (3.2)

∂t ω k q 2 ≤ C b0 ω k−1 q 2 +tk−1−δ/3 q r +t−2 a ω k ∇σ 2 +tk−1/2 σ ∞ +t−2 ∇s ∞ + ω 3/2 ∇s 2 ω k q 2 +t−1 a2 Ik−1 ω k−1 q 2 + ω 1/2 q 2 +t−1 a Ik−1 ( ω k q 2 q 3 ) + I1/2 ( ω 1/2 q 2 ) ω k q 2 +t−1 I1/2 ∇q 22 ω k q 2 + ω k R1 (W, S) 2

(3.3)

where s = S + σ and 0 < δ = δ(r) ≤ k. |∂t ω m ∇σ 2 | ≤ C t−2 ∇s ∞ ω m ∇σ 2 + ω m ∇s 2 ∇σ ∞ + ω m ∇σ 2 ∇S ∞ + σ ∞ ω m ∇2 S 2 +C t−1+β(m+1) a I0 ( q 2 ) + t−1+β(m+5/2) I−3/2 q 22 + ω m ∇R2 (W, S) 2

(3.4)

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for 0 ≤ m ≤ , |∂t ∇σ 2 | ≤ C t−2 ∇s ∞ ∇σ 2 + ∇σ 2 ∇S ∞ + ω 3/2 ∇S 2 +C t−1+β aI0 ( q 2 ) + t−1+5β/2 I−3/2 q 22 + ∇R2 (W, S) 2 .

(3.4)0

Proof. (3.2) is identical with (I.6.2) and is proved in the same way. In order to prove (3.3), we start from (cf. I.6.9))

∂t ω k q 2 ≤ t−1 [ω k , B0S ]q 2 + t−2 [ω k , s] · ∇q 2 + (∇ · s)ω k q 2 + ω k ((∇ · s)q ) 2 + ω k Q(σ, W ) 2 t−1 [ω k , BS (w, w)]q 2 + ω k (2BS (W, q) + BS (q, q)) W 2 + ω k R1 (W, S) 2

(3.5)

and we estimate the various terms in the RHS successively. The contribution of B0 is estimated exactly as in I and yields [ω k , B0S ]q 2 ≤ C b0 t ω k−1 q 2 +tk−δ/3 q r .

(3.6)

The contribution of Q(s, q ) is estimated by Lemmas 2.1 and 2.2 as [ω k , s] · ∇q 2 + (∇ · s)ω k q 2 + ω k ((∇ · s)q ) 2 ≤ C ∇s ∞ + ω 3/2 ∇s 2 ω k q 2

(3.7)

in the same way as in I, in the case k < 3/2. The contribution of Q(σ, W ) is estimated by Lemmas 2.1 and 2.2 as ω k Q(σ, W ) 2 ≤ C σ ∞ ω k ∇W 2 + ω k σ 6 ∇W 3 + ω k ∇σ 2 W ∞ + ∇σ r ω k W 3/k

with δ(r) = k, · · · ≤ C σ ∞ ω k ∇W 2 + ω k ∇σ 2 W ∞ + ω k W 3/k (3.8) ≤ C a tk−1/2 σ ∞ + ω k ∇σ 2 by (3.1). The contribution of BS with w = W + q yields a number of terms which we order by increasing powers of q, q . We ﬁrst expand BS (w, w) = BS (W, W ) + 2BS (W, q) + BS (q, q) .

(3.9)

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By Lemmas 2.1 and 2.2, we estimate [ω k , BS (W, W )]q 2 ≤ C ∇BS (W, W ) 3/ε + ω k BS (W, W ) 3/(k−1+ε) × ω k−1+ε q 2 for ε > 0. Taking ε = 3/2 − k yields · · · ≤ C ω k+1 B1 (W, W ) 2 ω 1/2 q 2 ≤ C Ik ω k W 2 W ∞ ω 1/2 q 2 ≤ C a2 ω 1/2 q 2

(3.10)

by Lemma 2.2 again and by (2.8) (3.1). In a similar way, we estimate by Lemmas 2.1, 2.2 and by (2.8) [ω k , BS (W, q)]q 2 ≤ C ∇BS (W, q) 3 ω k−1 q 6 + ω k BS (W, q) 3/k q r with δ(r) = k, · · · ≤ C ω 3/2 B1 (W, q) 2 ω k q 2 ≤ C I1/2 W ∞ ω 1/2 q 2 + ω 1/2 W 6 q 3 ω k q 2 ≤ C a I1/2 ω 1/2 q 2 ω k q 2 . (3.11) In a similar way, we estimate

[ω k , BS (q, q)]q 2 ≤ C ∇BS (q, q) 3 + ω k BS (q, q) 3/k ω k q 2

≤ C ω 3/2 B1 (q, q) 2 ω k q 2 ≤ C I1/2 ω 1/2 q 3 q 6 ω k q 2 ≤ C I1/2 ∇q 22 ω k q 2 .

(3.12)

We next estimate in a similar way

ω k (BS (W, q)W ) 2 ≤ C ω k B1 (W, q) 2 W ∞ + ω k W 3/k ≤ C a ω k B1 (W, q) 2 ≤ C a Ik−1 W ∞ + ω k−1 W 3/(k−1) ω k−1 q 2 ≤ C a2 Ik−1 ω k−1 q 2 .

Finally, we estimate in a similar way

(3.13)

ω k (BS (q, q)W ) 2 ≤ C ω k B1 (q, q) 2 W ∞ + ω k W 3/k ≤ C a Ik−1 ω k−1 |q|2 2 (3.14) ≤ C a Ik−1 ω k q 2 q 3 .

Substituting (3.6)–(3.8) and (3.10)–(3.14) into (3.5) yields (3.3).

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The estimates (3.4) and (3.4)0 of σ are identical with (I.6.4) and (I.6.4)0 and have exactly the same proof. The additional term with B0L in the equation for σ is included in R2 (W, S) and does not appear explicitly at this stage. From there on, the treatment of the Cauchy problem at inﬁnity for the auxiliary system follows that given in I verbatim. We need to estimate the diﬀerence of two solutions of the linearized auxiliary system (1.26), and that estimate, given by Lemma I.6.2, requires no modiﬁcation because it uses regularity properties of W which are weaker than (3.1). We then solve the Cauchy problem ﬁrst for the linearized auxiliary system (1.26) with ﬁnite initial time by Proposition I.6.1, then at inﬁnity by Proposition I.6.2, and then for the auxiliary system (1.20) or (1.23) by a contraction method, by Proposition I.6.3, part (2). The only diﬀerence in the proof of Propositions I.6.2 and I.6.3 is due to the term t−2 a tk−1/2 σ ∞ in (3.3), which did not appear in Lemma I.6.1, and which is due to the fact that the assumption (3.1) is weaker than (I.6.1). That term generates an additional term a Z t−1−λ−3(1−β)/2 in the RHS of (I.6.59), with time decay strictly better than t−1−λ and therefore harmless. We now state the ﬁrst main result of this section, which corresponds to Proposition I.6.3, part (2). Proposition 3.1 Let 1 < k < 3/2 < . Let λ0 , λ and β satisfy the conditions λ>0

(1 <)λ + k < λ0 ,

,

(3.15)

(3.16) 0 < β < 2/3 , β( + 1) < λ0 . k+1,+1 ˙ ) Let (A+ , A+ ) satisfy the conditions (2.10). Let (U (1/t)W, S) ∈ C([1, ∞), X ∩ C 1 ([1, ∞), X k, ) and let (W, S) satisfy the estimates Sup W ∞ ∨ W ; H 3/2 ∨ t1/2−k W ; H˙ k+1 ≤ a < ∞ , (3.17) t≥1

ω m ∇S 2 ≤ b t1−η+β(m−3/2)

(3.18)

for some η > 0 and for 0 ≤ m ≤ + 1,

m

R1 (W, S) 2 ≤ c0 t−1−λ0 ,

(3.19)

ω k R1 (W, S) 2 ≤ c1 t−1−λ ,

(3.20)

ω ∇R2 (W, S) 2 ≤ c2 t

−1−λ0 +β(m+1)

for 0 ≤ m ≤ .

(3.21)

Then there exists T , 1 ≤ T < ∞ and positive constants Y0 , Y and Z, depending on (A+ , A˙ + ) through the norms in (2.10) and depending on k, , β, λ0 , λ, a, b,

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c0 , c1 and c2 such that the auxiliary system (1.20) has a unique solution (w, s) ∈ C(I, X k, ), where I = [T, ∞), satisfying the estimates w − W 2 ≤ Y0 t−λ0 ,

(3.22)

ω k (w − W ) 2 ≤ Y t−λ ,

(3.23)

ω m ∇(s − S) 2 ≤ Z t−λ0 +β(m+1) for 0 ≤ m ≤ ,

(3.24)

for all t ∈ I. We now turn to the construction of approximate solutions (W, S) of the system (1.20) satisfying the assumptions of Proposition 3.1 and in particular the estimates (3.17), (3.18) of (W, S) and the estimates (3.19)–(3.21) of the remainders. In I we took for (W, S) the second order approximate solution of the system (1.20) in an iterative scheme not taking into account the terms containing B0 , thereby ending with an explicit B0 W term in the remainder R1 (W, S). Here, following [8], we improve that asymptotic form by adding one more term in W , so as to partly cancel the B0S W term in R1 (W, S). Thus we deﬁne W = w0 + w1 + w2 ≡ W1 + w2

,

S = s0 + s1

(3.25)

where w0 , s0 , w1 , s1 are the same as in I, namely w0 = U ∗ (1/t)w+ , s0 (t) = −

1

w1 (t) = −U ∗ (1/t) s1 (t) = −

t

∞

t

(3.26)

dt t−1 ∇BL (w0 (t ), w0 (t )) ,

t

∞

dt t−2 U (1/t )Q(s0 (t ), w0 (t )) ,

dt t−2 s0 (t ) · ∇s0 (t ) + 2

t

∞

(3.27) (3.28)

dt t−1 ∇BL (w0 (t ), w1 (t )) . (3.29)

In order to partly cancel B0S W in R1 (W, S), we take w2 = hw0 , thereby obtaining a linear contribution of w2 to R1 (W, S) ∂t − i(2t2 )−1 ∆ w2 = h ∂t − i(2t2 )−1 ∆ w0 − it−2 ∇h · ∇w0 + (∂t h)w0 −i(2t2 )−1 (∆h)w0 .

(3.30)

The ﬁrst term in the RHS is small, actually zero, by the choice of w0 . We use the last term in the RHS to cancel the main contribution B0S w0 of B0S W by making the choice , h = −2t∆−1 B0S . (3.31) w2 = hw0 Note that because of the short range cut-oﬀ in B0S , h is well deﬁned, actually h ∈ C([1, ∞), H k+2 ). With that choice, the remainders become Ri (W, S) = Ri0 (W, S) + Riν (W, S)

i = 1, 2 ,

(3.32)

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where Ri0 (W, s) are the parts not containing w2 or B0L , namely R10 (W, S) =

−t−2 (Q(S, w1 ) + Q(s1 , w0 )) −it−1 (B0S w1 + BS (W1 , W1 )W1 )

R20 (W, S) = −t−2 (s0 · ∇s1 + s1 · ∇s0 + s1 · ∇s1 ) + t−1 ∇BL (w1 , w1 ) ,

(3.33) (3.34)

while Riν (W, S) are the parts containing w2 or B0L , namely R1ν (W, S) =

−t−2 Q(S, w2 ) − it−1 B0S w2 + (∂t h)w0 − it−2 ∇h · ∇w0 −it−1 (BS (W, W )w2 + BS (W + W1 , w2 )W1 ) ,

R2ν (W, S) = t−1 ∇B0L + t−1 ∇BL (W + W1 , w2 ) .

(3.35) (3.36)

The parts Ri0 of the remainders are the remainders occurring in I, up to the replacement of B0 by B0S and the disappearance of the term B0S w0 , precisely the term which was responsible for the support condition in I. Up to a minor point (see below), (W1 , S) and Ri0 (W, S) have been estimated in I as follows (see Lemma I.7.1). Lemma 3.2 Let 0 < β < 1, k+ ≥ 3, w+ ∈ H k+ and a+ = |w+ |k+ . Then the following estimates hold for all t ≥ 1: |w0 |k+ ≤ a+ , ω

m

s0 2 ≤

(3.37)

C a2+ n t

for 0 ≤ m ≤ k+

C a2+ tβ(m−k+ )

for m > k+ ,

|w1 |k+ −1 ≤ C a3+ t−1 (1 + n t) ,

ωm

(3.38) (3.39)

 4 −1 2 for 0 ≤ m ≤ k+ − 1   C a+ t (1 + n t) 4 −1+β(m+1−k+ ) (3.40) s1 2 ≤ (1 + n t) C a+ t   −1 for k+ − 1 < m < k+ − 1 + β ,  −3 3 for 0 ≤ m ≤ k+ − 2   C(a+ ) t (1 + n t)

ω m R20 (W, S) 2 ≤

C(a+ ) t−3+β(m+2−k+ ) (1 + n t)2   for k+ − 2 < m < k+ − 2 + β −1 ,

(3.41) Let in addition 0 ≤ k ≤ k+ −1 and let B0 satisfy the estimate (2.12) for 0 ≤ m ≤ k. Then ω m R10 (W, S) 2 ≤ C(a+ ) t−3 (1 + n t)2 + t−1−β(k+ −m+1) + C b0 a3+ tm−5/2 (1 + n t)

for 0 ≤ m ≤ k .

(3.42)

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Proof. The estimates (3.37)–(3.42) are those of Lemma I.7.1 except for the estimate of the term t−1 B0S w1 in R10 (W, S) which is responsible for the last term in (3.42). We estimate that term by Lemmas 2.1 and 2.2 and by (2.12) (3.39) as ω m B0S w1 2 ≤

for m ≤ 1/2 C ω m B0 2 w1 ∞ + B0S r ω m w1 1/m C ( ω m B0 2 w1 ∞ + B0S ∞ ω m w1 2 )

for m > 1/2

≤ C b0 a3+ tm−3/2 (1 + n t)

(3.43)

for 0 ≤ m ≤ k, and 1/r = 1/2 − m for m ≤ 1/2, which completes the proof of (3.42). We now turn to estimating Riν (W, S), i = 1, 2. We ﬁrst reduce that question to that of estimating h and B0L , assuming for the moment a boundedness property of w2 which is part of (3.17) and which we shall prove later. We deﬁne the auxiliary space (3.44) Y = L∞ [1, ∞), L∞ ∩ H 3/2 and we remark that for k+ > 5/2, it follows from (3.37), (3.39) that W1 = w0 + w1 ∈ Y . We can now state the estimates of Riν (W, S). Lemma 3.3 Let 1 < k < 3/2 and k+ ≥ 3, let w+ ∈ H k+ and let a+ = |w+ |k+ . Assume that w2 ∈ Y and let W1 Y ∨ W Y ≤ a < ∞ .

(3.45)

Let B0 satisfy the estimate (2.12) for 0 ≤ m ≤ k. Then the following estimates hold for all t ≥ 1: R1ν (W, S) 2 ≤ C(a+ )t−2 n t ∇h 2 +C a+ b0 t−1 h 2 + ∂t h 2 +a2 t−1 ( h 2 +I0 ( h 2 )) , (3.46) ω k R1ν (W, S) 2 ≤ C(a+ )t−2 n t ω k ∇h 2 + ω δ ∇h 2 +C a+ b0 tk−1−δ/3 ω δ h 2 +t−1 ω k h 2 + ω k ∂t h 2 + ω δ ∂t h 2 +t−2 ω k+1 h 2 + ω 2 h 2 + t−1 a2 ω k h 2 +Ik−1 ω k−1 h 2 (3.47) for 0 < δ < 1/2, ω m R2ν (W, S) 2 ≤ t−1 ω m+1 B0L 2 +C a a+ t−1+βm I0 ( h 2 ) for all m ≥ 0.

(3.48)

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Proof. We ﬁrst consider R1ν (W, S) 2 . We estimate successively Q(S, w2 ) 2

≤

S ∞ ( w0 ∞ ∇h 2 + ∇w0 3 h 6 ) + ∇S 3 w0 ∞ h 6 ≤ C(a+ ) n t ∇h 2 (3.49)

by (3.37), (3.38), (3.40) and Lemma 2.1, B0S w2 2 ≤ B0S ∞ w0 ∞ h 2 ≤ C b0 a+ h 2 ,

(3.50)

by (2.14) and (3.37), (∂t h)w0 2 ≤ w0 ∞ ∂t h 2 ≤ C a+ ∂t h 2 ,

(3.51)

∇h · ∇w0 2 ≤ ∇w0 ∞ ∇h 2 ≤ C a+ ∇h 2 ,

(3.52)

BS (W, W )w2 2 ≤ BS (W, W ) ∞ w0 ∞ h 2 ≤ C a+ a2 h 2 , (3.53) by estimating BS (W, W ) in a way similar to that in Lemma 3.1, BS (W + W1 , w2 )W1 2 ≤ C W1 3 ωB1 (W + W1 , w2 ) 2 ≤ C W1 3 W + W1 ∞ w0 ∞ I0 ( h 2 ) ≤ Ca+ a2 I0 ( h 2 )

(3.54)

by Lemma 2.1 and by (2.8). Collecting (3.49)–(3.54) yields (3.46). We next consider ω k R1ν (W, S). We estimate successively ω k Q(S, w2 ) 2 ≤ C w0 ∗ S ∗ ω k+1 h 2 + w0 ∗ ω k+1 S 3/(1+δ) + ω k+1 w0 3/(1+δ) S ∗ ω δ ∇h 2 ≤ C(a+ ) nt ω k ∇h 2 + ω δ ∇h 2 (3.55) by (3.37), (3.38), (3.40) and Lemmas 2.1 and 2.2, with f ∗ = f ∞ + ∇f 3 , B0 ∞ w0 ∞ ω k h 2 + ω k B0 3/δ w0 ∞ + B0 ∞ ω k w0 3/δ ω δ h 2 (3.56) ≤ C b0 a+ ω k h 2 +tk−δ/3 ω δ h 2 ω k (B0S w2 ) 2 ≤ C

by Lemmas 2.1 and 2.2 and by (2.14), (3.37), ω k ((∂t h)w0 ) 2 ≤ C w0 ∞ ω k ∂t h 2 + ω k w0 3/δ ω δ ∂t h 2 ≤ C a+ ω k ∂t h 2 + ω δ ∂t h 2 (3.57)

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Lemmas 2.1 and 2.2 and by (3.37), ω k (∇h · ∇w0 ) 2 ≤ C ∇w0 ∞ ω k+1 h 2 + ω k+1 w0 3 ∇h 6 ≤ C a+ ω k+1 h 2 + ω 2 h 2 (3.58) by Lemmas 2.1 and 2.2 and by (3.37). We next estimate ω k (BS (W, W )w2 ) 2 ≤ C BS w0 ∞ + ω k (BS w0 ) 3/k ω k h 2 ≤ C BS ∞ + ω k BS 3/k w0 ∞ + ω k w0 3/k ω k h 2 ≤ C a+ a 2 ω k h 2

(3.59)

where we have omitted the arguments in BS , by Lemmas 2.1 and 2.2 and by (3.37), (3.45), and after estimating BS in a way similar to that in Lemma 3.1. In the same way ω k (BS (W + W1 , w2 )W1 ) 2 ≤ C W1 ∞ + ω k W1 3/k ω k B1 (W + W1 , w2 ) 2 ≤ CaIk−1 ω k−1 h 2 w0 (W + W1 ) ∞ + ω k−1 (W + W1 )w0 3/(k−1) (3.60) ≤ Ca+ a2 Ik−1 ω k−1 h 2 . Collecting (3.55)–(3.60) yields (3.47). We ﬁnally estimate R2ν . From (2.6), (2.8) we obtain ω m R2ν (W, S) 2 ≤ t−1 ω m+1 B0L 2 +C t−1+βm I0 ( W + W1 ∞ w0 ∞ h 2 )

which yields (3.48) by using (3.37), (3.45).

In order to complete the estimate of the parts Riν (W, S), i = 1, 2, of the remainders, we now estimate h and B0L . Those estimates require some restrictions on the behaviour of (F A+ , F A˙ + ) at ξ = 0. Those restrictions are imposed in a dilation homogeneous way through the use of a parameter µ ∈ (−1, 1) in terms of quantities which have the same scaling properties as A+ ; H˙ −3/2−µ and A˙ + ; H˙ −5/2−µ . They will be further discussed at the end of this section. Lemma 3.4 Let 1 < k < 3/2 and −1 < µ < 1. Let (A+ , A˙ + ) satisfy the conditions A+ ∈ H k−1 xA+ ∈ H k−1 xA+ ∈ H˙ −1/2−µ ,

, ,

A˙ + ∈ L2 ,

(3.61)

xA˙ + ∈ L3/2 ,

(3.62)

A+ , xA˙ + ∈ H˙ −3/2−µ ,

Let B0L and h be deﬁned by (1.17) and (3.31).

A˙ + ∈ H˙ −5/2−µ .

(3.63)µ

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Then the following estimates hold : ω m B0L 2 ≤ Ctm−1/2+(β0 −1)(m+3/2+µ) A+ ; H˙ −3/2−µ + A˙ + ; H˙ −5/2−µ (3.64) for all m ≥ 0,

ω m h 2 ≤ 2tm−3/2 1 ∨ t(β0 −1)(m−1/2+µ) A+ ; H˙ ρ + A˙ + ; H˙ ρ−1 (3.65) for all m ≤ k + 1, where ρ = (m − 2) ∨ (−3/2 − µ) = −3/2 − µ + (m − 1/2 + µ) ∨ 0 , ω m ∂t h 2 ≤ C tm−5/2 1 ∨ t(β0 −1)(m−1/2+µ) xA+ ; H˙ ρ+1 + xA˙ + ; H˙ ρ + A+ ; H˙ ρ + A˙ + ; H˙ ρ−1

(3.66)

(3.67)

for all m ≤ k and ρ given by (3.66), h ∞ ≤ C(A+ , A˙ + )

(3.68)

where the constant depends on (A+ , A˙ + ) through the norms in (3.61), (3.63)µ . Proof. (3.64) follows immediately from the deﬁnitions (1.4), (1.13) and (1.17) of A0 and B0L , from (2.6) and from (3.63)µ . In order to derive the estimates of h, it is convenient to come back to the variable A0 . The deﬁnition (3.31) of h can be rewritten as h = 2t2 ω −2 D0−1 A0S = D0−1 f

(3.69)

f = 2ω −2 A0S ,

(3.70)

A0S = t−1 D0−1 B0S = χS A0 ≡ F −1 1 − χ(ξt1−β0 ) F A0

(3.71)

where A0S is deﬁned by

and χ is deﬁned before (1.16). (3.65). We estimate ω m h 2 = tm−3/2 ω m f 2 = 2tm−3/2 ω m−2 A0S 2 ≤ 2tm−3/2 1 ∨ t(β0 −1)(m−1/2+µ) ω ρ A0 2

(3.72)

and the result follows from the assumptions (3.61) (3.63)µ . (3.67). We use in addition the commutation relations t∂t = D0−1 P D0

,

P ω −j = ω −j (P + j) ,

[P, eiωt ] = 0

(3.73)

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where P is the dilation generator P = t∂t + x · ∇ . In particular

∂t h = t−1 t ∂t D0−1 f = t−1 D0−1 P f .

(3.74)

Using the commutation relations (3.73), we compute (1/2)P f

= ω −2 (P + 2)A0S = ω −2 cos ωt (P + 2)χS A+ + ω −3 sin ωt (P + 3)χS A˙ + .

Using the fact that P + 3 = t∂t + ∇ · x and the commutation relation [P, χS ] = −βF −1 ξt1−β0 · ∇χ(ξt1−β0 )F ≡ χ we obtain (1/2)P f

= ω −2 cos ωt ∇ · χS xA+ + ω −3 sin ωt ∇ · χS xA˙ + +ω −2 cos ωt ( χ − χS )A+ + ω −3 sin ωt χ A˙ + .

We then estimate

ω m ∂t h 2 = tm−5/2 ω m P f 2

(3.75)

(3.76)

and we estimate the contribution of the various terms of (3.75) exactly as in the proof of (3.65), with (m, A+ , A˙ + ) replaced by (m − 1, xA+ , xA˙ + ) in the ﬁrst two − χS or by χ in the last two terms. This yields terms, and with χS replaced by χ (3.67). (3.68). By Lemma 2.1, 1/2

1/2

h ∞ ≤ C ω 3/2−ε h 2

ω 3/2+ε h 2

and (3.68) follows from (3.65) with 0 < ε ≤ (k − 1/2) ∧ (µ + 1).

(3.77)

We now collect the results of Lemmas 3.2, 3.3 and 3.4 in order to exhibit a set of assumptions which imply those of Proposition 3.1 Proposition 3.2. Let 1 < k < 3/2 < . Let µ, λ0 , λ, β0 , β and k+ satisfy the conditions −1/4 < µ ≤ 1/2 (3.78) λ>0

(1 <)λ + k < λ0 < 7/6 + 2µ/3(≤ 3/2)

,

0 < β0 ≤ β < 2/3

k+ ≥ k + 2

,

,

β( + 1) < λ0

(3.79) (3.80)

β0 (1/2 − µ) > λ0 − 1 − µ

(3.81)

β0 (µ + 5/2) < 2 + µ − λ0

(3.82)

β(k+ + 1) ≥ λ0

,

β( + 3 − k+ ) < 1 .

(3.83)

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Let w+ ∈ H k+ and let (A+ , A˙ + ) satisfy (2.10), (3.62), (3.63)µ . Let (W, S) be deﬁned by (3.25)–(3.29), (3.31). Then (W, S) satisfy the estimates (3.17), (3.18), (3.19), (3.20), (3.21) (with 0 < η < 1−3β/2, in (3.18)) and all the assumptions of Proposition 3.1 are satisﬁed. Proof. The contribution of the terms not containing w2 or B0L in (W, S) and in the remainders are estimated by Lemma 3.2 in the same way as in I. We concentrate on the remaining terms. The terms containing w2 are estimated by Lemma 3.3 in terms of h, and h and B0L are estimated by Lemma 3.4. The condition (3.17) restricted to w2 = hw0 follows from the fact that it holds for h by (3.65), (3.68) and trivially for w0 , and that it is multiplicative. Together with (3.45) for W1 , it implies (3.45) for W . We next consider R1ν (W, S). Its L2 norm is estimated by (3.46). By Lemma 3.4, it satisﬁes the estimate (3.19) provided (1 − β0 ) ((1/2 − µ) ∨ 0) < 3/2 − λ0

(3.84)

which reduces to (3.81) for µ ≤ 1/2. Similarly, R1ν (W, S) is estimated in H˙ k norm by (3.47) and satisﬁes the estimate (3.20) for δ suﬃciently small under the condition (3.84) because the time decay of (3.47) is worse than that of (3.46) at worst by a factor tk+2δ/3 which is better than the allowed tλ0 −λ for 0 < 2δ/3 ≤ λ0 − λ − k. We now turn to R2ν (W, s). The contribution of B0L is estimated by (3.64) and satisﬁes the estimate (3.21) provided m + 1/2 + (β0 − 1)(m + 7/2 + µ) ≤ −1 − λ0 + β(m + 1)

(3.85)

which is implied by (3.82) for β0 ≤ β. The term containing w2 is estimated by (3.48) and satisﬁes the estimate (3.21) by (3.65) under the condition (3.84). We remark here that the upper bound on λ0 in (3.79) is the compatibility condition of (3.81), (3.82). The remaining conditions in (3.78)–(3.83) come from I. We now comment brieﬂy on the various parameters that occur in Proposition 3.2 and on the conditions (3.78)–(3.83) that they have to satisfy. The parameters k and characterize the regularity of the spaces of resolution for (w, s). As a consequence, k also characterizes the regularity of (A+ , A˙ + ) as given by (2.10). ˙ ) = (F A , F A˙ ) at ξ = + , A The parameter µ characterizes the behaviour of (A + + + 0 through the condition (3.63)µ . The parameters λ0 and λ are the time decay exponents of the norms of q in L2 and in H˙ k . The µ dependent upper bound on λ0 in (3.79) ranges over (1, 3/2] when µ ranges over (−1/4, 1/2]. Since the condition (3.84) saturates at λ0 < 3/2 for µ ≥ 1/2, there is no point in considering values of µ > 1/2. The parameters β0 and β characterize the splitting of B0 and B1 respectively into short range and long range parts, and therefore the splitting

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of the Schr¨ odinger equation into transport and Hamilton-Jacobi equations. The parameter β should not be too large and can be taken equal to β0 . The parameter β0 satisﬁes two inequalities (3.81) and (3.82) in opposite directions, depending on λ0 and µ, and expressing the fact that B0S and B0L are not too large. The upper bound on λ0 in (3.79) is the compatibility condition of those inequalities. Whenever it is satisﬁed, the value β0 = 1/3 is allowed. Actually both (3.81) and (3.82) reduce to that upper bound for β0 = 1/3. Finally k+ characterizes the regularity of w+ and should be suﬃciently large, depending on k, , λ0 and β. Remark 3.1. For µ = 1/2 the short range restriction is no longer needed in the estimates of h and ∂t h in Lemma 3.4, and therefore the splitting of B0 into short range and long range parts is no longer needed, namely B0 can be kept entirely in the equation for w. We ﬁnally discuss the condition (3.63)µ of Lemma 3.4. That condition restricts the behaviour of the relevant functions for small |ξ| in Fourier transformed variables. Let A be any of the functions A+ , A˙ + , xA+ , xA˙ + and deﬁne A< by < (ξ) = χ(ξ)A(ξ). A Then the conditions on A< contained in (3.63)µ all take the form (3.86) A< ∈ H˙ −3/2−ν for ν = µ, µ±1. We ﬁrst remark that in the proof of Lemma 3.4, all such conditions can be replaced by < ∈ L∞ |ξ|−ν A (3.87) at the expense of inserting an additional factor ( n t)1/2 in (3.65) in the case of equality, namely for m = 1/2 − µ. This follows from the fact that m−3/2

|ξ|

S< 2 ≤ A

< ∞ 1 ∨ t(β0 −1)(m+ν) for m = −ν C |ξ|−ν A < ∞ ( n t)1/2 C |ξ|−ν A

for m = −ν . (3.88) The occurrence of the factor ( n t)1/2 is harmless for the applications. The condi< away from zero, tion (3.86) is weaker than (3.87) as regards the behaviour of A 2 ∞ since it requires only A< ∈ Lloc instead of A< ∈ Lloc . Furthermore (3.86) almost follows from (3.87), up to a change of ν into ν + ε. In fact < 2 ≤ C ε−1/2 |ξ|−ν−ε A < ∞ |ξ|−3/2−ν A

(3.89)

for ε > 0. In addition, under the short range condition S< 2 ≤ C( n t)1/2 |ξ|−ν A < ∞ |ξ|−3/2−ν A

(3.90)

which is the special case m = −ν of (3.88). ˙ ) at ξ = 0 expressed by (3.86) have the unpleasant + , A The restrictions on (A + feature that for ν ≥ 0 they cannot be ensured by imposing decay of (A+ , A˙ + ) at

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inﬁnity in space and that they require in addition some moment conditions. For instance even for A ∈ S one has ω −3/2−ν A< = C|x|−3/2+ν ∗ A< for |ν| < 3/2 [9], which behaves as −3/2+ν

|x|

A dx

as |x| → ∞ and therefore cannot be in L2 for ν ≥ 0 unless A dx = 0. More generally when ν increases, vanishing of the n-th moment of A is necessary as soon as ν ≥ n. Actually the parameter µ in (3.63)µ has been introduced in order to minimize the number of such conditions by taking µ small. We now give suﬃcient conditions on (A+ , A˙ + ) in terms of space decay and vanishing of suitable moments so as to ensure the low frequency part of (3.63)µ . Lemma 3.5 Let −1 < µ < 1. Let (A+ , A˙ + ) satisfy (3.61), (3.62) and in addition 1+µ+ε ˙ A˙ + dx = 0 , x A+ ∈ L1 , (3.91) x A+ ∈ L3/(2+µ)∨2 ,

A+ dx =

A+ , xA˙ + ∈ L3/(3+µ) x A˙ + dx = 0

,

x

µ+ε

for µ < 0 , A+ ∈ L1

(3.92) for µ ≥ 0 .

(3.93)

Then (3.63)µ holds. Proof. The high frequency part of (A+ , A˙ + ) is controlled by (3.61), (3.62) and it is suﬃcient to consider (A+< , A˙ +< ), although in some cases the high frequency parts are also controlled by (3.91), (3.92). We ﬁrst consider xA+ . For −1/2 ≤ µ < 1, we estimate ω −1/2−µ x A+ 2 ≤ C x A+ 3/(2+µ)

(3.94)

by Lemma 2.1. For µ ≤ −1/2, we estimate simply ω −1/2−µ x A+< 2 ≤ C x A+ 2 .

(3.95)

We next consider A+ and xA˙ + together and we use A to denote either of them. For −1 < µ < 0, we estimate ω −3/2−µ A 2 ≤ C A 3/(3+µ)

(3.96)

by Lemma 2.1. For µ ≥ 0, we estimate

|ξ|−µ−ε |A(ξ)| = (2π)−3/2 |ξ|−µ−ε

dx (exp(−ixξ) − 1) A(x)

≤ C |x|µ+ε A 1

(3.97)

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for 0 ≤ µ + ε ≤ 1. The required estimate then follows from (3.61), (3.62), (3.91) and (3.93). We ﬁnally consider A˙ + . For µ < 0, we apply the previous result with A replaced by A˙ + and µ replaced by µ − 1. For µ ≥ 0, we estimate −1−µ−ε ˙ −3/2 −1−µ−ε |ξ| |A(ξ)| = (2π) |ξ| dx (exp(−ixξ) − 1 − ixξ) A˙ + (x) ≤ C |x|1+µ+ε A˙ + 1

(3.98)

for 0 ≤ µ + ε ≤ 1. The required estimate then follows from (3.61), (3.91), (3.93).

4 Wave operators and asymptotics for (u, A) In this section we complete the construction of the wave operators for the system (1.1), (1.2) and we derive asymptotic properties of solutions in their range. The construction relies in an essential way on Propositions 3.1 and 3.2. So far we have worked with the system (1.20) for (w, s) and the ﬁrst task is to reconstruct the phase ϕ. Corresponding to S = s0 + s1 , we deﬁne φ = ϕ0 + ϕ1 where t dt t−1 BL (w0 (t ), w0 (t )) , (4.1) ϕ0 = − ϕ1 = −

t

1

∞

2 −1

dt (2t )

2

|s0 (t )| + 2

t

∞

dt t−1 BL (w0 (t ), w1 (t )) ,

(4.2)

so that s0 = ∇ϕ0 and s1 = ∇ϕ1 . Let now (w, s) be the solution of the system (1.20) constructed in Proposition 3.1 and let (q, σ) = (w, s) − (W, S). We deﬁne ∞ dt (2t2 )−1 (σ · (σ + 2S) + s1 · (s1 + 2s0 )) (t ) ψ=− t ∞ dt t−1 (BL (q, q) + 2BL (W, q) + BL (w1 , w1 )) (t ) (4.3) + t

which is taylored to ensure that ∇ψ = σ, given the fact that s0 , s1 and σ are gradients. The integral is easily seen to converge in H˙ 1 (see I.8.4), and to satisfy ∇ψ 2 = σ 2 ≤ C t−λ0 .

(4.4)

Finally we deﬁne ϕ = φ + ψ so that ∇ϕ = s, and (w, ϕ) solves the system (1.18). For more details on the reconstruction of ϕ from s, we refer to Section 8 of I. We can now deﬁne the wave operators for the system (1.1), (1.2) as follows. We start from the asymptotic state (u+ , A+ , A˙ + ) for (u, A). We deﬁne w+ = F u+ , we deﬁne B0 by (1.4) (1.13), namely ˙ A0 = K(t) A+ + K(t) A˙ + = t−1 D0 B0 , and we deﬁne (W, S) by (3.25)–(3.29), (3.31).

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We next solve the system (1.20) with inﬁnite initial time by Propositions 3.1 and 3.2 and we reconstruct ϕ from s as explained above, namely ϕ = ϕ0 + ϕ1 + ψ with ϕ0 , ϕ1 and ψ deﬁned by (4.1), (4.2), (4.3) with (q, σ) = (w, s) − (W, S). We ﬁnally substitute (w, ϕ) thereby obtained into (1.11), (1.3) thereby obtaining a solution (u, A) of the system (1.1), (1.2). The wave operator is deﬁned as the map Ω : (u+ , A+ , A˙ + ) → (u, A). In order to state the regularity properties of u that follow in a natural way from the previous construction, we introduce appropriate function spaces. In addition to the operators M = M (t) and D = D(t) deﬁned by (1.8), (1.9), we introduce the operator J = J(t) = x + it ∇ , (4.5) the generator of Galilei transformations. The operators M , D, J satisfy the commutation relation iM D ∇=J M D . (4.6) For any interval I ⊂ [1, ∞) and any k ≥ 0, we deﬁne the space u : D∗ M ∗ u ∈ C(I, H k ) X k (I) = = u :< J(t) >k u ∈ C(I, L2 )

(4.7)

where λ = (1 + λ2 )1/2 for any real number or self-adjoint operator λ and where the second equality follows from (4.6). We now collect the information obtained for the solutions of the system (1.1), (1.2) and state the main result of this paper as follows. Proposition 4.1. Let 1 < k < 3/2 < . Let µ, λ0 , λ, β0 , β and k+ satisfy the conditions (3.78)–(3.83). Let u+ ∈ F H k+ , let w+ = F u+ and a+ = |w+ |k+ . Let (A+ , A˙ + ) satisfy (2.10), (3.62), (3.63)µ . Let (W, S) be deﬁned by (3.25)–(3.29), (3.31). Then (1) There exists T , 1 ≤ T < ∞ such that the auxiliary system (1.20) has a unique solution (w, s) ∈ C([T, ∞), X k, ) satisfying w(t) − W (t) 2 ≤ C a+ , A+ , A˙ + t−λ0 , (4.8) (4.9) ω k (w(t) − W (t)) 2 ≤ C a+ , A+ , A˙ + t−λ , ω m (s(t) − S(t)) 2 ≤ C a+ , A+ , A˙ + t−λ0 +βm for 0 ≤ m ≤ + 1 , (4.10) for all t ≥ T , where the constants C(a+ , A+ , A˙ + ) depend on (A+ , A˙ + ) through the norms associated with (2.10), (3.62), (3.63)µ . (2) Let φ = ϕ0 + ϕ1 be deﬁned by (4.1), (4.2), let ϕ = φ + ψ with ψ deﬁned by (4.3) and (q, σ) = (w, s) − (W, S). Let u = M D exp(−iϕ)w

(1.11) ≡ (4.11)

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and deﬁne A by (1.3), (1.4), (1.5). Then u ∈ X k ([T,∞)), (A,∂t A) ∈ C([T,∞), H k ⊕ H k−1 ), (u, A) solves the system (1.1), (1.2) and u behaves asymptotically in time as M D exp(−iφ)W in the sense that it satisﬁes the following estimates: u(t) − M (t) D(t) exp(−iφ(t))W (t) 2 ≤ C(a+ , A+ , A˙ + )t−λ0 ,

(4.12)

|J(t)|k (exp(iφ(t, x/t))u(t) − M (t) D(t) W (t)) 2 ≤ C(a+ , A+ , A˙ + )t−λ , (4.13) u(t) − M (t) D(t) exp(−iφ(t)) W (t) r ≤ C(a+ , A+ , A˙ + )t−λ0 +(λ0 −λ)δ(r)/k (4.14) for 0 ≤ δ(r) = (3/2 − 3/r) ≤ k, for all t ≥ T . Deﬁne in addition (4.15) A2 = A − A0 − A1 (|DW |2 ) . Then A behaves asymptotically in time as A0 + A1 (|DW |2 ) in the sense that A2 satisﬁes the following estimates: A2 (t) 2 ≤ C(a+ , A+ , A˙ + ) t−λ0 +1/2

(4.16)

∇A2 (t) 2 ≤ C(a+ , A+ , A˙ + ) t−2λ0 −1/2+(λ0 −λ)3/2k

(4.17)

ω

2k−1/2

A2 (t) 2 ≤ C(a+ , A+ , A˙ + ) t−2λ−2k+1

(4.18)

for all t ≥ T . (3) The solution (u, A) also behaves asymptotically as (M D exp(−iφ)W1 , A0 + A1 (|DW1 |)2 in the sense that the estimates (4.12)–(4.14) and (4.16)–(4.18) also hold with W replaced by W1 (see (3.25)). Sketch of proof. Part (1) is a restatement of the conclusions of Proposition 3.1 supplemented by (4.4) and follows from Propositions 3.1 and 3.2. Part (2) follows from Part (1) and is proved in exactly the same way as Part (2) of Proposition I.8.1. Part (3) is proved in the same way as Part (2). It follows from the fact that the only estimates of W and q = w − W that are used in the proof of Part (2) are (3.45) which also holds for W1 and (4.8) (4.9) which also hold for w2 . In fact, the latter estimates hold for h by Lemma 3.4, especially (3.65), under the assumptions of Proposition 3.2 and follow therefrom for w2 in a trivial way. Remark 4.1. It may seem surprising that the improved asymptotic form W for w does not give rise to better asymptotic estimates than the simpler form W1 in the norms (4.12)–(4.14) and (4.16)–(4.18). The reason is that the additional term w2 is small and gives rise to small contributions in terms of those norms. This does not prevent that term to give a large contribution to the time derivative ∂t w in (1.20) through the derivative term t−2 ∆w2 . That contribution is essential to allow for the solution of the system (1.20) without assuming the support condition. The same phenomenon appears in [8].

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Acknowledgments We are grateful to Dr. A. Shimomura for enlightening discussions.

References [1] J. Ginibre, G. Velo, Long range scattering and modiﬁed wave operators for the Wave-Schr¨ odinger system, Ann. H.P. 3, 537–612 (2002). [2] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators, Vol I, Springer, Berlin, 1983. [3] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 891–907 (1988). [4] C. Kenig, G. Ponce, L. Vega, The initial value problem for a class of nonlinear dispersive equations, in Functional-Analytic Methods for Partial Diﬀerential Equations, Lect. Notes Math., 1450, 141–156 (1990). [5] T. Ozawa, Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Commun. Math. Phys., 139, 479–493 (1991). [6] T. Ozawa, Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schr¨ odinger equations, in Spectral and Scattering Theory and Applications, Adv. Stud. in Pure Math., Jap. Math. Soc., 23, 295–305 (1994). [7] A. Shimomura, Wave operators for the coupled Klein-Gordon-Schr¨ odinger equations in two space dimensions, Funkcial. Ekvac., in press. [8] A. Shimomura, Modiﬁed wave operators for the coupled Wave-Schr¨ odinger equations in three space dimensions, Disc. Cont. Dyn. Syst., in press. [9] E. Stein, Singular integrals and diﬀerentiability properties of functions, Princeton Univ. Press, Princeton 1970. J. Ginibre Laboratoire de Physique Th´eorique UMR 8627 du CNRS Universit´e de Paris XI Bˆatiment 210 F-91405 ORSAY Cedex, France email: [email protected] Communicated by Bernard Helﬀer submitted 20/02/03, accepted 24/06/03

G. Velo Dipartimento di Fisica Universit` a di Bologna and INFN Sezione di Bologna, Italy email: [email protected]

Ann. Henri Poincar´e 4 (2003) 1001 – 1013 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/051001-13 DOI 10.1007/s00023-003-0153-5

Annales Henri Poincar´ e

The Heat Kernel Expansion for the Electromagnetic Field in a Cavity F. Bernasconi, G.M. Graf and D. Hasler Abstract. We derive the ﬁrst six coeﬃcients of the heat kernel expansion for the electromagnetic ﬁeld in a cavity by relating it to the expansion for the Laplace operator acting on forms. As an application we verify that the electromagnetic Casimir energy is ﬁnite.

1 Introduction The modes of an electromagnetic ﬁeld in a cavity, taken together with their unphysical, longitudinal counterparts, can be mapped onto the eigenstates of the Laplacian acting on the de Rham complex of a 3-manifold with boundary. The electric and magnetic ﬁelds are thereby associated to forms of degree p = 1 and p = 2 respectively. In this correspondence transverse modes are associated with coexact, resp. exact forms, which permits to further map longitudinal modes to forms of degree p = 0 and p = 3. We will use this observation, which is explained in detail in Section 2 below, to compute the ﬁrst six coeﬃcients of the heat kernel expansion for the electromagnetic ﬁeld in a cavity. The result is used to show in a simple way that the Casimir energy in an arbitrary cavity with smooth boundaries is ﬁnite, a conclusion which has been reached previously [3]. In an appendix the derivation of the numerical coeﬃcients of the expansion is presented. We shall present a Hilbert space formulation of the classical Maxwell equations in a cavity Ω ⊂ R3 . In a preliminary Hilbert space L2 (Ω, R3 ) we deﬁne the dense subspaces R = V ∈ L2 (Ω, R3 ) | rot V ∈ L2 (Ω, R3 ) , R0 = {V ∈ R | U, rot V = rot U, V, ∀U ∈ R} and the (closed) operator R = rot

with domain D(R) = R0 .

Its adjoint is then given as R∗ = rot with D(R∗ ) = R. We remark that R, resp. R∗ , is also the closure of rot deﬁned on smooth vector ﬁelds V with boundary condition V = 0 on the smooth boundary ∂Ω, resp. without boundary conditions. This is what is meant when we later simply say that a diﬀerential operator is deﬁned with (or without) a certain boundary condition.

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The subspace

Ann. Henri Poincar´e

H = V ∈ L2 (Ω, R3 ) | div V = 0

(1)

and its orthogonal complement in L2 (Ω, R3 ) are preserved by R and, therefore, by R∗ . We will thus view them as operators on the physical Hilbert space H. The Maxwell equations with boundary condition E = 0 on the ideally conducting shell ∂Ω can now be written as ∂ E E i =M (2) B B ∂t with

M=

iR∗ 0

0 −iR

= M∗

on H ⊕ H ,

cf. [12]. Since no boundary condition has been imposed on B, we have M (0, B) = 0 for all B = ∇ψ with ψ harmonic, and hence dim Ker M = ∞ .

(3)

We shall compute the heat kernel trace 2

TrH⊕H (e−tM ) =

2

e−tωk ,

k

where means that the contributions of zero-modes, i.e., of eigenvalues ωk = 0 of M , have been omitted. This is necessary in view of (3), but a more physical justiﬁcation, tied to the application to the Casimir eﬀect to be discussed later, is that zero-modes are not subject to quantization. The square of M is M2 =

R∗ R 0

0 RR∗

=

−∆E 0

0 −∆B

,

(4)

where ∆E , resp. ∆B , is the Laplacian on H with boundary conditions E = 0 ,

resp. (rot B) = 0 .

(5)

The operators RR∗ and R∗ R have the same spectrum, including multiplicity, except for zero-modes. Incidentally, we note that eigenfunctions (E, B) corresponding to ωk = 0 satisfy B = −iωk−1 rot E and hence, by Stokes’ theorem, the boundary condition B⊥ = 0, which we did not impose, but which is usually also associated with ideally conducting shells. Since ∂t2 + M 2 = (i∂t − M )(−i∂t − M ), each pair of non-zero eigenvalues of R∗ R and RR∗ corresponds to a single oscillator mode

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The Heat Kernel Expansion for the Electromagnetic Field in a Cavity

for (2). We will thus discuss the heat kernel asymptotics for TrH et∆E 1 −tM 2 TrH⊕H (e )= 2 TrH et∆B ∼ =

∞

an t

n−3 2

,

1003

(6) (7) (t ↓ 0) .

(8)

n=0

The coeﬃcients an are known, see, e.g., [5], for general operators of Laplace type. The direct application of such results is prevented by the divergence constraint in H, see (1). In the next section we indicate how to remove it. First however we present the main result. Let Lab = (∇ea eb , n) ,

(a, b = 1, 2) ,

be the second fundamental form on the boundary ∂Ω with inward normal n and local orthonormal frame {e1 , e2 , n}. We denote by |Ω| the volume of Ω and set f (y)dy , f [∂Ω] = ∂Ω

where dy is the (induced) Euclidean surface element on ∂Ω. The corresponding Laplacian on ∂Ω is denoted by ∇2 . Theorem 1 Let Ω ⊂ R3 an open, connected domain with compact closure and smooth boundary ∂Ω consisting of n components of genera g1 , g2 , . . . , gn . Then 3

a0

= 2(4π)− 2 |Ω| ,

a1

= 0, 3 4 = − (4π)− 2 (tr L)[∂Ω] , 3 n 1 1 −1 2 (4π) 3(tr L) − 4 det L [∂Ω] − = (1 + gi ) + 1 , 64 2 i=1

a2 a3 a4

=

a5

=

(9)

3 16 (4π)− 2 2(tr L)3 − 9 tr L · det L [∂Ω] , 315 1 (4π)−1 2295(tr L)4 − 12440(tr L)2 det L + 122880 +13424(det L)2 + 1200 tr L · ∇2 tr L [∂Ω] .

We will give two partially independent proofs, based on (6), resp. (7). Their agreement is related to the index theorem, as it may be seen from (4). A further, partial check of these coeﬃcients has been made on the basis of general cylindrical domains and of the sphere, where a separation into TE and TM modes is possible. The coeﬃcient a0 was computed in [13] (except for the factor 2 replaced by 3, as the divergence condition (1) was ignored), a1 , a2 in [1]. The coeﬃcient a3 is closely related to a result of [3], as discussed in Section 3.

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2 Proofs

n We consider the space of (square integrable) forms, Λ(Ω) = p=0 Λp (Ω), on the manifold Ω with boundary, together with the exterior derivative dp+1 : Λp (Ω) → Λp+1 (Ω) deﬁned with relative boundary condition ([11], Section 2.7.1) ω ∂Ω = 0 , as a form ω|∂Ω ∈ Λp (∂Ω). For later use we recall that by the de Rahm theorem for manifolds with boundary ([9] or [11], Thm. 2.7.3) we have Hrp (Ω) ∼ = Hn−p (Ω) ∼ = Hp (Ω, ∂Ω) ,

(10)

where Hrp (Ω) = Ker dp+1 /Im dp is the p-th relative cohomology group, Hp (Ω) is the p-th homology group, and Hp (Ω, ∂Ω) is the p-th relative homology group, i.e., the homology based on chains mod ∂Ω. We shall henceforth restrict to Ω ⊂ R3 as in Theorem 1. Using either homology (10), the dimension of Hrp (Ω) is seen to be 0 n−1 n gi

(p = 0) , (p = 1) , (11)

(p = 2) ,

i=1

1

(p = 3) .

These are also the dimensions of the spaces of harmonic p-forms.

3 The space Λ(Ω) = p=0 Λp (Ω) may be identiﬁed as Λ(Ω) = L2 (Ω) ⊕ L2 (Ω, R3 ) ⊕ L2 (Ω, R3 ) ⊕ L2 (Ω) (φ, E, B, ψ) , where d : Λ(Ω) → Λ(Ω) acts as d : L2 (Ω) −→ L2 (Ω, R3 ) −→ L2 (Ω, R3 ) −→ L2 (Ω)−→0 rot

grad

div

with boundary conditions φ = 0, E = 0, B⊥ = 0 on ∂Ω. Then d∗ : 0 ←− L2 (Ω) ←− L2 (Ω, R3 ) ←− L2 (Ω, R3 ) ←− L2 (Ω) −div

rot

−grad

without any boundary conditions. The Laplace-Beltrami operator on forms, −∆ =

3 (−∆p ) = dd∗ + d∗ d , p=0

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is seen to correspond to the Euclidean Laplacian with boundary conditions φ=0

(p = 0) ,

E = 0 , div E = 0 B⊥ = 0 , (rot B) = 0

(p = 1) , (p = 2) ,

(grad ψ)⊥ = 0

(p = 3) .

(12)

Each of the four problems admits a heat kernel expansion, TrΛp (Ω) e∆p t ∼ =

∞

a(p) n t

n−3 2

,

(13)

n=0

whose coeﬃcients have been computed (n = 0, . . . , 3) [4] or can be computed using existing results (n = 4, 5) [5]. To this end we note that the boundary conditions for p = 1, 2 can be formulated equivalently as E = 0 , B⊥ = 0 ,

∂E⊥ − (tr L)E⊥ = 0 ∂n ∂B − LB = 0 ∂n

(p = 1) , (14) (p = 2) .

First approach. We will compute (6). We observe that −∆E is just the restriction of −∆1 to its invariant subspace H = E ∈ L2 (Ω, R3 ) | div E = 0 = Ker d∗1 . Hence

TrH et∆E = TrL2 (Ω,R3 ) et∆1 − TrH⊥ et∆1 ,

where the orthogonal complement of H in L2 (Ω, R3 ) is H⊥ = Ran d1 = Ran d1 = ∇φ ∈ L2 (Ω, R3 ) | φ = 0 on ∂Ω , (Ran d is closed by the Hodge decomposition, see, e.g., [8, 11]). By d∆ = ∆d, the operators (−∆1 ) H⊥ and −∆0 have the same spectrum (in fact ∇φ = 0 implies φ = 0 by the boundary condition). Thus, using also (11), we ﬁnd TrH et∆E = TrL2 (Ω,R3 ) et∆1 − TrL2 (Ω) et∆0 = TrL2 (Ω,R3 ) et∆1 − TrL2 (Ω) et∆0 − (n − 1) , i.e., (1)

(0)

(1) a3

(0) a3

ak = ak − ak , a3 =

−

(k = 3) ,

−n+1.

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Ann. Henri Poincar´e

(p)

These relations, together with the values of ak computed in the appendix, yield the values of the coeﬃcients stated in the Theorem 1. In particular, we will obtain 1 (4π)−1 3(tr L)2 + 28 det L [∂Ω] . 64 This matches the stated value of a3 because of (1)

(0)

a3 − a3 =

1 1 (1 + gi ) + (1 − gi ) 2 i=1 2 i=1 n

n=

n

and of the Gauss-Bonnet theorem, 1 1 (1 − gi ) = (4π)−1 (det L)[∂Ω] . 2 i=1 2 n

(15)

Second approach. We now compute (7). As it has been noted in the introduction, eigenmodes of −∆B , except for zero-modes, satisfy the boundary condition B⊥ = 0, and are thus eigenmodes of −∆2 belonging to its invariant subspace H, cf. (5, 12). The converse is obvious. We conclude that TrH et∆B = TrL2 (Ω,R3 ) et∆2 − TrH⊥ et∆2 . Since H = {B ∈ L2 (Ω, R3 ) | div B = 0} = Ker d3 , we have

H⊥ = Ran d∗3 = Ran d∗3 = −∇ψ ∈ L2 (Ω, R3 ) | ψ ∈ L2 (Ω) .

Using d∗ ∆ = ∆d∗ , we see that (−∆2 ) H⊥ and −∆3 have the same spectrum, except for a single zero-mode (in fact, −∇ψ = 0 implies ψ = const ). We thus ﬁnd, using (11), TrH et∆B = TrL2 (Ω,R3 ) et∆2 − TrL2 (Ω) et∆3 = TrL2 (Ω,R3 ) et∆2 − TrL2 (Ω) et∆3 −

n

gi − 1 ,

i=1

i.e., (2)

(3)

ak = ak − ak , (2)

(3)

a3 = a3 − a3 −

(k = 3) , n

gi + 1 .

i=1

From these relations and from the results of the appendix we again recover Theorem 1. In particular, (2)

(3)

a3 − a3 =

1 (4π)−1 3(tr L)2 − 36 det L [∂Ω] 64

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1007

leads to the claim for a3 , because of n

1 1 gi = (1 + gi ) − (1 − gi ) 2 i=1 2 i=1 i=1 n

n

and of (15).

3 Application to the Casimir eﬀect For the purpose of this discussion we simply deﬁne the Casimir energy by the mode summation method, see, e.g., [3]. In particular, we do not address the issue [6] of whether it is the most appropriate physically. We shall however observe that the Casimir energy is ﬁnite – a conclusion obtained in [3], but questioned in [10]. Consider the cavity Ω ⊂ R3 enclosed in a large ball Ω0 . As usual we compare the vacuum energy of the electromagnetic ﬁeld in the domains Ω∪(Ω0 \Ω) with that of the reference domain Ω0 . Each eigenmode of either domain contributes a zero1/2 point energy ωk /2, resp. ωk0 /2. As a regulator for the eigenfrequencies ωk = λk , −γλk , (γ > 0). The corresponding deﬁnition of the Casimir energy is we choose e

1 1 0 12 −γλ0k 2 −γλk lim lim λk e − (λk ) e . EC = 2 Ω0 →∞ γ↓0 k

k

We shall prove that the limit γ ↓ 0 is ﬁnite. It will also be clear that the subsequent limit Ω0 → ∞ exists, though we shall not make the eﬀort to prove that (see however, e.g., [8], Section 12.7 for the necessary tools). Using 1 1 λk2 = − √ π

0

∞

1

dt t− 2

d −tλk e dt

and (8) we ﬁnd for the regularized sum of the eigenfrequencies k

1 2

λk e

−γλk

δ 4 n−5 n−3 1 √ an ≈− dt t− 2 (t + γ) 2 2 π 0 n=0

as γ ↓ 0. Here δ > 0 is arbitrary, but ﬁxed, and “≈” means up to terms O(1). Using  4 −2  (n = 0) ,  3γ   3 π −   (n = 1) , δ 2γ 2 n−5 − 12 −1 2 dt t (t + γ) ≈ 2γ (n = 2) ,  0  πγ − 12 (n = 3) ,    − log γ (n = 4) ,

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Ann. Henri Poincar´e

we ﬁnd k

√ 1 π 3 1 2 1 1 a1 γ − 2 + √ a2 γ −1 + 0 · a3 γ − 2 + √ a4 log γ . λk2 e−γλk ≈ √ a0 γ −2 + 2 π π 2 π

Hence a ﬁnite Casimir energy requires (cf. [7]) that a0 , a1 , a2 , a4 (but not necessarily a3 !) agree for Ω ∪ (Ω0 \ Ω) and for the reference domain Ω0 . This is indeed 3 so for a0 = 2(4π)− 2 |Ω0 | and for a1 = 0, but also for a2 , a4 as the contributions from the two sides of ∂Ω cancel. The same conclusion is obtained if the regulator 1/2 e−γλk is replaced by e−(γλk ) (see [7], Eq. (27)): k

1

λk2 e−(γλk )

1/2

3 1 24 2 1 ≈ √ a0 γ −2 + 4a1 γ − 2 + √ a2 γ −1 + 0 · a3 γ − 2 + √ a4 log γ . π π π

Since no renormalization is necessary, the value of EC agrees with that obtained by means of the zeta function. In the rest of this section we compare our results with those of [2, 3]. To the extent the comparison is done we will ﬁnd agreement. An important tool there is the mode generating function, Eq. (4.5) in [2], −∆E −∆B . 1 + Φ(k) = Tr 2 −∆E − k 2 −∆B − k 2 (16) 2 . k 2 −1 2 −1 , (k ∈ C \ R) , Tr (−∆E − k ) + (−∆B − k ) = 2 . where “=” means equality “within addition of some polynomial in k 2 ”. Since the resolvents in (16) are not trace class, ∞ but their squares are, we ﬁrst consider that replacement. Using (A + µ)−2 = 0 dt t e−t(A+µ) we obtain, as µ → ∞, ∞ ∞ n−3 1 −2 −2 ∼ Tr (−∆E + µ) + (−∆B + µ) an dt · t 2 e−tµ = 2 0 n=0 =

∞

− Γ( n+1 2 )an µ

n+1 2

n=0

with coeﬃcients an given in Theorem 1. Integrating w.r.t. µ we ﬁnd ∞ n−1 1 . Γ( n−1 )an µ− 2 − a1 log µ Tr (−∆E + µ)−1 + (−∆B + µ)−1 = 2 2 n=0 n =1

and hence, with µ1/2 = −ik, √ √ . √ Φ(k) = 2 πa0 ik 3 − πa1 k 2 ln(−k 2 ) + i πa2 k − a3 + O(k −1 ) .

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Upon insertion of the mentioned values for a0 , . . . , a3 this agrees with Eq. (4.40) in [2], except for a3 which is there replaced by its local part, see (9), 1 (4π)−1 3(tr L)2 − 4 det L [∂Ω] 64 3 1 = dσ (κ21 + κ22 ) − κ1 κ2 , 64 ∂Ω 4

a ˜3 =

where κ1 , κ2 are the principal curvatures. Note however that this discrepancy is . implicit in the deﬁnition of “=”. It is resolved in [3] by ﬁrst considering δΦ(k), i.e., the diﬀerence of the mode generating functions corresponding to the conﬁgurations Ω ∪ (Ω0 \ Ω) and Ω0 . Thus δΦ(k) = −2˜ a3 + O(k −1 ) , ˜3 double the value. Not since the contributions to a0 , a2 cancel, and those to a ambiguous then is “the number of additional modes of ﬁnite frequency created by introducing the conducting surface ∂Ω”: C = ψ(0+) − ψ(∞) , where ψ(y) = δΦ(iy). For a connected boundary ∂Ω of genus g the value of ψ(0+) has been established as ψ(0+) = −g (see [3], Eq. (5.8)), resulting in C = 2˜ a3 − g .

(17)

This result agrees with Theorem 1: the non-local terms in (9) take the values − 12 (g − 1), − 12 g, 12 for Ω, Ω0 \ Ω and Ω0 respectively. Thus, a3 − g , δa3 = 2˜ in agreement with (17).

A

Appendix

In this appendix we compute the heat kernel coeﬃcients in (13) for p = 0, . . . , 3 and n = 0, . . . , 5 on the basis of Theorems 1 and 4 in [5]. We use the same notation, together with P = n ⊗ n denoting the normal projection at the boundary. The vector bundle is V = Ω × R for p = 0, 3, resp. V = T Ω for p = 1, 2, equipped with the Euclidean connection. The decompositions of V |∂Ω = VN ⊕ VD (φN , φD ) N (with projections Π+ , resp. Π− ) and boundary conditions φN = 0, resp. ;n + Sφ

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φD = 0, are speciﬁed as follows, cf. (14) and [5]: Π+ = 0 , p=0: Π− = 1 , Π+ = P , S = −Laa P , p=1: Π− = 1 − P , Π+ = 1 − P , S = −L , p=2: Π− = P , Π+ = 1 , S=0, p=3: Π− = 0 .

(18)

The result is 3

a0 = (4π)− 2 c0 |Ω| , 1 (p) (p) a1 = (4π)−1 c1 |∂Ω| , 4 3 (p) 1 (p) a2 = (4π)− 2 c2 (tr L)[∂Ω] , 3 (p) 1 (p) (p) (4π)−1 c31 (tr L)2 + c32 (det L) [∂Ω] , a3 = 384 3 (p) 1 (p) (p) (4π)− 2 c41 (tr L)3 + c42 tr L · det L [∂Ω] , a4 = 315 (p) 1 (p) (p) (p) a5 = (4π)−1 c51 (tr L)4 + c52 (tr L)2 det L + c53 (det L)2 245760 (p) + c54 tr L · ∇2 tr L [∂Ω] (p)

(p)

with coeﬃcients given in Table 1. The computation of the table is based on the general result of [5], which has been applied to (18) using the following identities: Tr(P:a P:b ) = 2(L2 )ab , Tr(P:a P:a P:b P:b ) = (L4 )aa + (L2 )aa (L2 )bb , Tr(P:a P:b P:a P:b ) = 2(L4 )aa , Tr(P:aa P:bb ) = 2Lac:a Lbc:b + 4(L4 )aa + 4(L2 )aa (L2 )bb , Tr(P:ab P:ab ) = 2Lab:c Lab:c + 6(L4 )aa + 2(L2 )aa (L2 )bb . They can be derived by using ∇ea n = −Lab eb , so that P:a = −Lac (ec ⊗ n + n ⊗ ec ) ,

Vol. 4, 2003

The Heat Kernel Expansion for the Electromagnetic Field in a Cavity

(p)

c0

(p)

c1

(p)

c2

(p)

c31

(p)

c32

(p)

c41

(p)

c42

(p)

c51

(p)

c52

(p)

c53

(p)

c54

p=0 1 −1 1 3 −20 4 −18 555 −2840 2224 120

p=1

p=2

p=3

3

3

1

−1

1

1

−3

−3

1

21

33

15

148

−220

−4

36

60

28

−162

−186

−42

5145

8625

4035

−27720

−35720

−10840

29072

29712

2864

2520

4680

2280

1011

Table 1: These values imply Theorem 1, as explained in its proof. and by assuming without loss that ∇ea eb has no component parallel to Tp ∂Ω at the point p of evaluation, i.e., ∇ea eb = Lab n. Then P:ab = −Lac:b (ec ⊗ n + n ⊗ ec ) − 2(L2 )ab P + (Lac Lbd + Lad Lbc )ec ⊗ ed , from which the above traces follow. In turn they allow the computation of similar traces with P replaced by χ = Π+ − Π− , i.e., by χ = ±(2P − 1) in the cases p = 1, 2. In these two cases we also have Tr S:a = −Lbb:a , Tr S:ab = −Lcc:ab , and, moreover, for p = 1, Tr(S:a S:a ) = Lbb:a Lcc:a + 2Lbb Lcc (L2 )aa , Tr(P:a S:b ) = −2(L2 )ab Lcc , Tr(P S:a S:a ) = Lbb:a Lcc:a + Lbb Lcc (L2 )aa ,

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resp. for p = 2, Tr(S:a S:a ) = Lab:c Lab:c + 2(L4 )aa , Tr(P:a S:a ) = 2(L3 )aa , Tr(P S:a S:a ) = (L4 )aa . Furthermore, traces of Lk , (k ≥ 2), were reduced to tr L, det L by means of L2 − (tr L)L + det L = 0. Finally, we used the Codazzi equation, Lab:c = Lac:b , as well as Lab:ca − Lab:ac = Laa (L2 )bc − (L2 )aa Lbc , which follows from the Gauss equation.

Acknowledgment We thank M. Levitin and G. Scharf for discussions. The research of D. Hasler was supported in part under the EU-network contract HPRN-CT-2002-00277.

References [1] R. Balian, C. Bloch, Distribution of eigenfrequencies for the wave equation in a ﬁnite domain. II. Electromagnetic ﬁeld. Riemannian spaces, Ann. Phys. (N.Y.) 64, 271 (1971); Errata, ibid. 84, 559 (1974). [2] R. Balian, B. Duplantier, Electromagnetic waves near perfect conductors, I. Multiple scattering expansions. Distribution of modes, Ann. Phys. (N.Y.) 104, 300 (1977). [3] R. Balian, B. Duplantier, Electromagnetic waves near perfect conductors, II. Casimir eﬀect, Ann. Phys. (N.Y.) 112, 165 (1978). [4] N. Blaˇzi´c, N. Bokan, P.B. Gilkey, Spectral geometry of the form valued Laplacian for manifolds with boundary, Indian J. Pure Appl. Math. 23, 103 (1992). [5] T.P. Branson, P.B. Gilkey, K. Kirsten, D.V. Vassilevich, Heat kernel asymptotics with mixed boundary conditions, Nucl. Phys. B 563, 603 (1999). [6] P. Candelas, Vacuum energy in the presence of dielectric and conducting surfaces, Ann. Phys. (N.Y.) 143, 241 (1982). [7] G. Cognola, L. Vanzo, S. Zerbini, Regularization dependence of vacuum energy in arbitrarily shaped cavities, J. Math. Phys. 33, 222 (1992). [8] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger operators, Springer (1987).

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[9] G.F.G. Duﬀ, Diﬀerential forms in manifolds with boundary, Ann. Math. 56, 115 (1952). [10] D. Deutsch, P. Candelas, Boundary eﬀects in quantum ﬁeld theory, Phys. Rev. D 20, 895 (1978). [11] P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC (1995). [12] R. Leis, Initial boundary value problems in mathematical physics, Teubner/Wiley (1986). ¨ [13] H. Weyl, Uber die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetzte, J. f. reine u. angew. Math. 143, 177 (1913). F. Bernasconi Department of Mathematics ETH-Zentrum CH-8092 Z¨ urich Switzerland email: [email protected] G.M. Graf Theoretische Physik ETH-H¨onggerberg CH-8093 Z¨ urich Switzerland email: [email protected] D. Hasler Department of Mathematics University of Copenhagen DK-2100 Copenhagen Denmark email: [email protected] Communicated by Vincent Rivasseau submitted 17/02/03, accepted 04/07/03

Ann. Henri Poincar´e 4 (2003) 1015 – 1050 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061015-36 DOI 10.1007/s00023-003-0154-4

Annales Henri Poincar´ e

Plancherel Inversion as Uniﬁed Approach to Wavelet Transforms and Wigner Functions S. Twareque Ali, Hartmut F¨ uhr and Anna E. Krasowska

Abstract. We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, uniﬁed perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We ﬁrst prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L2 -function on an appropriately chosen group while the Wigner function is deﬁned on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L2 -functions on a group unitarily to ﬁelds of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modiﬁed Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very diﬀerent perspectives, are naturally uniﬁed within our approach. Explicit computations on a number of groups illustrate the theory.

1 Introduction The continuous wavelet transform is used extensively in image processing and signal analysis and its group theoretical origin is well known [1, 12]. The Wigner function has also been employed in the analysis of signals as well as in numerous quantum optical and quantum statistical computations [8, 9, 20, 33, 34]. It has been argued before that both the wavelet transform and the Wigner function owe their origin to the square-integrability of certain group representations. This point was discussed extensively in [4], where the square-integrability of a single unitary irreducible representation of a group was exploited to build both a generalized wavelet transform and a class of Wigner functions. It is the purpose of this paper to show, quite generally, how both these concepts can be uniﬁed using the Plancherel transform for Type-I groups. The Plancherel transform sets up a unitary isomorphism between the Hilbert space of square-integrable (with respect to the Haar measure) functions on the group and the direct integral Hilbert space (with respect to the Plancherel measure) built out of the spaces of Hilbert-Schmidt operators on the Hilbert spaces of unitary irreducible representations of the group. It is the inverse of this unitary map which, when appropriately restricted, leads to

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a generalized wavelet transform. On the other hand, taking the inverse Plancherel transform and following it up with a Fourier type of transform leads to functions on the dual of the Lie algebra of the group. The ﬁnal function, when restricted to appropriate coadjoint orbits, then yields a wide class of generalized Wigner functions, which share many of the interesting properties of the original Wigner function [33], but now is deﬁnable for a vast array of groups and representations. Generalized wavelet transforms can also be seen as coherent state transforms of vectors in the Hilbert spaces of group representations [1]. However, in most cases one deﬁnes the coherent state transform on Hilbert spaces carrying a single unitary irreducible representation of the group. This requires that the representation in question be square-integrable, or in other words, that it belong to the discrete series of representations of the group. If, on the other hand, the group in question does not admit square-integrable representations, the above construction of coherent states and related transforms clearly fails. In such cases, in speciﬁc examples, it has been demonstrated [23] how the use of direct integral representations, over some convenient subset of the unitary dual of the group, leads once more to the existence of a coherent state transform. We show here that this situation is generic and is again a simple consequence of the Plancherel transform. The rest of this paper is organized as follows: In Section 2 we brieﬂy recall the Plancherel transform and its inverse for Type-I groups. In Section 3 we derive explicit expressions for the inverse Plancherel transform and demonstrate how it can be used to deﬁne coherent states and a generalized wavelet transform. We carry out the construction explicitly in Section 4 for the case of the Poincar´e group in a two-dimensional space-time. Section 5 is devoted to a deﬁnition and construction of the generalized Wigner function. This function is deﬁned on the coadjoint orbits, foliating the dual of the Lie algebra of the group, and we introduce a modiﬁed Fourier transform on the range of the Plancherel transform to arrive at it. We also discuss general properties of the Wigner function, which follow immediately from the deﬁnition. As examples, we compute in Section 6 Wigner functions for the cases of three commonly used groups: the two-dimensional Poincar´e group, the aﬃne Poincar´e group (the Poincar´e group including dilations) and the WeylHeisenberg group, which leads us back to the original quasi-probability distribution function introduced by Wigner. Finally, in the appendix we collect together a few results, of a computational nature, used in working out the examples.

2 Plancherel measure Let us ﬁrst ﬁx some notation: G denotes a second countable, locally compact group. All representations will be understood to be unitary and strongly contin we denote the set of equivalence classes of irreducible representations uous. By G of G, equipped with the Mackey Borel structure(see, e.g., [16]). It will often be of representations and necessary to distinguish between an equivalence class σ ∈ G a speciﬁc realization of a representation Uσ , in this equivalence class and acting

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Plancherel Inversion, Wavelet Transforms and Wigner Functions

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on a particular Hilbert space Hσ . In the direct integrals below, a measurable realization of the representations Uσ used is provided by the theory [31, Theorem 10.2]. µG denotes the left Haar measure, and Lp (G) is the corresponding Lp -space. Cc (G) denotes the space of compactly supported continuous functions on G. The modular function of G is denoted by ∆G , the convention being, dµG (x) = ∆G (x) dµr (x) ,

(1)

where µr is the right invariant Haar measure. For a function f on G, we write f(x) := f (x−1 ). For a given Hilbert space H, B2 (H) denotes the space of Hilbert-Schmidt operators. It is a Hilbert space, endowed with the scalar product A|B2 = tr(A∗ B); the corresponding norm shall be denoted by · 2 . Furthermore, B1 (H) denotes the subspace of trace class operators, endowed with the norm A1 := tr(|A|), where |A| := (AA∗ )1/2 . Elements of special interest in both spaces are the rank-one operators, denoted by |ηφ|, (for η, φ ∈ H), which are deﬁned by |ηφ|(ψ) = φ|ψη, for any ψ ∈ H. We have |ηφ|1 = |ηφ|2 = ηφ. The usual operator norm is denoted by · ∞ . If a densely deﬁned operator A has a bounded extension, we denote the extension by [A]. A simple and often used fact is that for linear operators A, B, T with A, B bounded, such that [AT ] and [T B] exist, [AT B] = [AT ]B = A[T B]. The central object of interest in this paper is the left regular representation λG of G, acting on L2 (G) via (λG (x)f )(y) := f (x−1 y). Another representation acting on L2 (G) is the right regular representation ρG , deﬁned by (ρG (x)f )(y) := ∆G (x)1/2 f (yx). The left and the right regular representations commute and are unitarily equivalent. Finally, the two-sided representation is denoted by λG × ρG . This is a representation of the product group G × G, deﬁned by (λG × ρG )(x, y) := λG (x)ρG (y). The Plancherel theory can be seen as the theory of a direct integral decomposition of the two-sided representation into irreducibles, where the intertwining operator is given by the operator-valued Fourier transform. In this paper, we shall only be concerned with groups G such that λG is Type-I, i.e., the von Neumann algebra generated by the left (right) regular representation is Type-I (see [13, 14] for more details). This includes in particular the Type-I groups. Recall that the operator-valued Fourier transform on G maps each f ∈ L1 (G) to the family {Uσ (f )}σ∈G of operators, where each Uσ (f ) is deﬁned by the weak operator integral f (x)Uσ (x)dµG (x) . (2) Uσ (f ) := G

This deﬁnes a ﬁeld of bounded operators, in fact, we have Uσ (f )∞ ≤ f 1

(3)

Another feature of the operator-valued Fourier transform, reminiscent of the wellknown Fourier transform over the reals, is that convolution becomes operator

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Ann. Henri Poincar´e

multiplication on the Fourier side, more precisely, Uσ (f ∗ g) = Uσ (f ) ◦ Uσ (g). In order to invert this transform, we have to ﬁnd a Hilbert space H, such that f → {Uσ (f )}σ∈G extends from L1 (G)∩L2 (G) to a unitary equivalence L2 (G) → H. To see the relationship of this deﬁnition to the usual Fourier transform (over the reals, say), let us suppose for a moment that G is Abelian. Then each Uσ (f ) is a scalar, since each Uσ is a character, and the above mapping yields the usual Fourier transform f (except that in the generally used deﬁnition one integrates is a locally compact Abelian group, over Uσ∗ rather than Uσ ). Also, in this case, G and the Abelian Plancherel theorem states that we may take the Haar measure on in the stated unitary equivalence. as the Plancherel measure, i.e., H = L2 (G), G Returning to the general case, let us try to motivate the construction of the Hilbert space H. The Fourier transform {Uσ (f )}σ∈G forms a ﬁeld of bounded Furthermore, this ﬁeld is measurable, as follows from the deﬁnition operators on G. It is thus reasonable to expect H to be the of the Mackey Borel structure on G. νG ), where each ﬁber is some Hilbert space direct integral over a measure space (G, of operators, and the measure νG is to be determined. The natural choice for the ﬁbres is given by the Hilbert-Schmidt operators on the representation spaces Hσ . At this point, the Plancherel theory splits into the unimodular and the nonunimodular cases: In the unimodular case, Uσ (f ) is automatically Hilbert-Schmidt, for every In the nonunimodular case we f ∈ L1 (G) ∩ L2 (G) and almost every σ ∈ G. have to employ a family (Cσ )σ∈G of densely deﬁned unbounded operators Cσ on Hσ , with densely deﬁned inverses, such that Uσ (f )Cσ−1 is Hilbert-Schmidt (more precisely: for almost all σ (with respect to the measure νG ), the closure [Uσ (f )Cσ−1 ] is Hilbert-Schmidt). These operators can indeed be constructed in such way that the operator Fourier transform extends to a unitary map. Let us now give the exact statement of the Plancherel theorem in the form we are going to use [15]. Theorem 2.1 Let G be a second countable locally compact group having a Type-I called the Plancherel regular representation. Then there exists a measure νG on G, measure, and a measurable ﬁeld (Cσ )σ∈G of self-adjoint positive operators with densely deﬁned inverses, such that the following hold: the closure of the operator (i) For f ∈ L1 (G) ∩ L2 (G) and νG -almost all σ ∈ G, Uσ (f )Cσ−1 is a Hilbert-Schmidt operator on Hσ . (ii) The map L1 (G) ∩ L2 (G) f → {[Uσ (f )Cσ−1 ]}σ∈G extends to a unitary equivalence ⊕ P : L2 (G) → B2 (Hσ )dνG (σ) . (4) G

This unitary operator is called the Plancherel transform of G. It has the intertwining property P(λG (x)ρG (y)f )(σ) = Uσ (x) (P(f )(σ)) Uσ (y)∗ .

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(iii) There exists a subspace D(G) ⊂ L1 (G) ∩ L2 (G), dense in L2 (G), such that the operator for all f ∈ D(G) and νG -almost all σ ∈ G, [Uσ (f )Cσ−2 ] = [[Uσ (f )Cσ−1 ]Cσ−1 ] is densely deﬁned and has a trace class extension, and we have the Fourier inversion formula, f (x) = tr [Uσ (x)∗ Uσ (f )Cσ−2 ] dνG (σ) . (5) G

(iv) The Plancherel measure is essentially unique: The covariance relation Uσ (x)Cσ Uσ (x)∗ = ∆G (x)1/2 Cσ

(6)

ﬁxes each Cσ uniquely up to multiplication by a scalar, and once these are ﬁxed, so is νG . Conversely, one can ﬁx νG (which is a priori only unique up to equivalence) and thereby determine the Cσ uniquely. (v) G is unimodular if and only if for νG -almost all σ, Cσ is a multiple of the identity Iσ on Hσ . In this case we require that Cσ = Iσ , which then determines νG completely. If G is nonunimodular, Cσ is an unbounded operator for (νG -almost all) σ ∈ G. Remark 2.2 The inversion formula (5) was shown in [15] to hold for the space of Bruhat functions introduced in [11]. It can be written as D(G) = {Cc∞ (G/K) : K ⊂ G compact such that G/K is a Lie group } , where Cc∞ (G/K) is the space of arbitrarily smooth functions on G/K with compact support, canonically embedded into Cc (G). 2 In the following, we suppose that G is a second countable group with TypeI regular representation. We use to denote the Plancherel transform. So, for f ∈ L1 (G) ∩ L2 (G) we have (Pf )(σ) = f(σ) := [Uσ (f )Cσ−1 ]. The direct integral space of Hilbert-Schmidt spaces in (4) will be denoted by B2⊕ . The scalar product of two elements, Ai ∈ B2⊕ , i = 1, 2, consisting of the measurable ﬁelds {Ai (σ) ∈ B2 (Hσ )}σ∈G , is given by tr [A1 (σ)∗ A2 (σ)] dνG (σ). A1 |A2 B⊕ = 2

G

3 The wavelet transform as inverse Plancherel transform Let us quickly recall the group-theoretical formalism for the construction of wavelet transforms: Suppose we are given a unitary (not necessarily irreducible) representation U of G, on a Hilbert space H and a vector η ∈ H. We can then deﬁne the

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(generalized) wavelet transform of φ ∈ H as the function Vη φ on G, deﬁned by (Vη φ)(x) := U (x)η|φ .

(7)

Generally, this construction gives an injective operator Vη : φ → Vη φ, whenever η is a cyclic vector (which means that the orbit U (G)η is total in H). However, in order to have an eﬃcient way of inverting Vη , we require more, viz , that Vη : H → L2 (G) be an isometry (possibly up to a scalar factor cη ). Note that generally, even the well-deﬁnedness, that is Vη (H) ⊂ L2 (G), is not guaranteed. However, if Vη is an isometry (in which case we say that η is an admissible vector), we can rewrite the isometry property in the form of an inversion formula, 1 φ= (Vη φ)(x) U (x)η dµG (x) , (8) cη G where the integral is understood in the weak operator sense or, equivalently, as a resolution of the identity: 1 IdH = U (x)|ηη|U (x)∗ dµG (x) . (9) cη G The relationship to the left regular representation is quite obvious: Besides being an isometry, the wavelet transform Vη is easily seen to intertwine the representation U of G with its left regular representation λG . Hence wavelet transforms fall quite naturally in the domain of the Plancherel theory. In fact, as will become clear below, if U is given as a direct integral of irreducible representations, a wavelet transform is just the inverse Plancherel transform, applied to certain operator ﬁelds. In order to motivate the last statement, let us take a closer look at the case where U = Uσ is an irreducible, square-integrable representation of G. This means that, there exist explicit admissibility conditions involving Cσ , which are usually cited in the following form [15, Theorem 3]: Theorem 3.1 Let Uσ be an irreducible subrepresentation of λG . (i) For η, φ ∈ Hσ , the wavelet transform Vη φ is square-integrable (i.e., is an element of L2 (G)) iﬀ η ∈ dom(Cσ ). (ii) For η1 , η2 ∈ dom(Cσ ) and φ1 , φ2 ∈ Hσ , we have the orthogonality relation, Vη1 φ1 |Vη2 φ2 L2 (G) = Cσ η2 |Cσ η1 H φ1 |φ2 H .

(10)

To see the relationship with the Plancherel transform, let us consider the rank-one operators Ai = |φi Cσ ηi | (i = 1, 2). Then Vηi φi (x) = Uσ (x)ηi |φi = tr(|φi ηi |Uσ (x)∗ ) = tr(|φi Cσ ηi |Cσ−1 Uσ (x)∗ ) = tr [Ai Cσ−1 Uσ (x)∗ ], which is essentially the Plancherel inversion formula (5) (up to an ordering of operators), with the operator ﬁelds supported only at the point σ. Here we have taken account of

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Plancherel Inversion, Wavelet Transforms and Wigner Functions

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the fact that Uσ is a subrepresentation of λG iﬀ νG ({σ}) > 0, and hence, without loss of generality, we may take νG ({σ}) = 1. Assuming that the inversion formula holds for both Vηi φi , i = 1, 2, we obtain (P(Vηi φi ))(π) = (V ηi φi )(π) = Ai , for π = σ and 0 elsewhere. Thus, the orthogonality relations, and in particular the isometry property of the generalized wavelet transform Vη , are immediate consequences of the unitarity of the Plancherel transform. While this way of showing the isometry property of Vη , using the Plancherel transform, is much too complicated in the irreducible case (which is easily dealt with using Schur’s lemma), it has the advantage of being readily generalizable to direct integral representations, once we have extended the inversion formula (5) to a wider class of functions. Let us ﬁrst establish a few preliminary facts. The ﬁrst lemma deals with the operators Cσ and their relation to convolution. Lemma 3.2 Let f ∈ Cc (G). (i) For νG -almost every σ, we have f(σ)∗ = Cσ−1 Uσ (∆−1 G f ). In particular the right-hand side is everywhere deﬁned and bounded. (ii) For νG -almost every σ, we have −1/2

[Uσ (f )Cσ−1 ] = Cσ−1 Uσ (∆G

f) ,

in particular the right-hand side is everywhere deﬁned and bounded. (iii) For all g ∈ L2 (G), we have −1/2

(g ∗ f )(σ)

= g(σ)Uσ (∆G

(f ∗ g)(σ)

g (σ) . = Uσ (f )

f) ,

Proof. For part (i) we invoke [32, Theorem 13.2], to ﬁnd that, since Cσ−1 is selfadjoint and Uσ (f ) is bounded, (Uσ (f )Cσ−1 )∗ = Cσ−1 Uσ (f )∗ . Moreover, since f(σ) is bounded, the right-hand side of the last equation is everywhere deﬁned. Calculating Uσ (f )∗ is routine. −1/2 For (ii) we ﬁrst note that by (i), applied to ∆G f ∈ Cc (G), the right-hand side is bounded and everywhere deﬁned. Moreover, the left-hand side is bounded since f ∈ L1 (G) ∩ L2 (G). It thus remains to show that the equality holds on the dense subspace dom(Cσ−1 ): for φ, η ∈ dom(Cσ−1 ) the deﬁnition of the weak operator integral yields φ|Uσ (f )Cσ−1 η = φ|Uσ (x)Cσ−1 ηf (x)dµG (x) G = φ|∆G (x)−1/2 Cσ−1 Uσ (x)ηf (x)dµG (x) G

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Ann. Henri Poincar´e

= = =

Cσ−1 φ|Uσ (x)η∆G (x)−1/2 f (x)dµG (x) G −1/2 Cσ−1 φ|Uσ (∆G f )η −1/2 φ|Cσ−1 Uσ (∆G f )η ,

where the second equality uses the covariance relation (6), and the self-adjointness of Cσ−1 (at several points). This proves (ii). Part (iii) is then immediate from (i) and (ii), at least for g ∈ L1 (G) ∩ L2 (G). It extends by continuity to all of L2 (G): the left-hand sides are continuous operators, being convolution operators with f ∈ Cc (G), and the right-hand sides are continuous because of inequality (3). The next lemma deﬁnes the space B1⊕ , which arises very naturally when dealing with inversion formulae. In fact, there is a natural representation-theoretic interpretation of B1⊕ as the space of Fourier transforms of the Fourier algebra A(G). This was noted for the unimodular case by Lipsman [27], but the arguments go through for the non-unimodular case as well. Lemma 3.3 Let B1⊕ be the space of measurable ﬁelds {B(σ)}σ∈G of trace class operators, for which the norm B(σ)1 dνG (σ) BB⊕ := 1

G

is ﬁnite. Here we identify operator ﬁelds which agree νG -almost everywhere. Then (B1⊕ , ·B⊕ ) is a Banach space and the set of measurable ﬁelds of rank-one operators 1 in B1⊕ spans a dense subspace. The proof consists of standard arguments and is omitted here. Now we can show that the inversion formula holds almost everywhere, whenever it makes sense (i.e., whenever all quantities involved can be expected to converge). This is the nonabelian analogue of the well-known Fourier inversion formula for an L2 -function whose Plancherel transform is in L1 . The statement for the unimodular case was in fact given in [27]. A(σ)C −1 extends to Theorem 3.4 Let A ∈ B2⊕ be such that for almost all σ ∈ G, σ a trace class operator and such that {[A(σ)Cσ−1 ]}σ∈G ∈ B1⊕ . Let a ∈ L2 (G) be the inverse Plancherel transform of A. Then we have (for almost every x ∈ G) a(x) = tr(Uσ (x)∗ [A(σ)Cσ−1 ]) dνG (σ) . (11) G

If we assume that Cσ−1 A(σ) is trace-class, and that {[Cσ−1 A(σ)]}σ∈G ∈ B1⊕ , then σ → tr(|[Cσ−1 A(σ)]|) is integrable and we obtain (almost everywhere) a(x) = tr([Cσ−1 Uσ (x)∗ A(σ)])dνG (σ) . (12) G

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Proof. Let (fn )n∈N ⊂ Cc (G) be a sequence with decreasing supports, satisfying the following requirements: fn ≥ 0, f n L1 = 1, and supp(fn ) runs through a neighborhood base at unity. Then (fn )n∈N is a bounded approximate identity with respect to right convolution, i.e., for all g ∈ L2 (G) we have g ∗ fn → g in L2 (G), and the operator norms of g → g ∗ fn are bounded by a constant. In −1/2 1/2 addition, (∆G fn )n∈N has the same properties, since ∆G f n L1 → 1. By passing to a subsequence, if necessary, we may assume that a ∗ fn → a pointwise almost everywhere. We ﬁrst evaluate the convolution using the unitarity of the Plancherel transform, obtaining for almost every x ∈ G: (a ∗ fn )(x)

= = = = =

λ(x)f n |a

∗ −1 dνG (σ) tr A(σ) Uσ (x)Uσ (f n )Cσ G ∗ ∗ tr(A(σ)[Cσ−1 Uσ (f n ) Uσ (x) ]dνG (σ) G ∗ tr([A(σ)Cσ−1 Uσ (∆−1 G fn )Uσ (x) ])dνG (σ) , G ∗ tr([A(σ)Cσ−1 ]Uσ (∆−1 G fn )Uσ (x) )dνG (σ) . G

(13)

(14)

∗ Here we have used the fact that Cσ−1 Uσ (∆−1 G fn )Uσ (x) is bounded, by Lemma 3.2 −1 (i), as well as the existence of [A(σ)Cσ ], as assumed. From the deﬁnition of B1⊕ , it is clear that (B(σ))σ∈G → tr(B(σ))dνG (σ) G

deﬁnes a bounded linear functional; this was our motivation for introducing the space. Comparing the right-hand side of (11) with (14), we ﬁnd that it suﬃces to show that the sequence of operators (n)

T1

: B1⊕

(B(σ))σ∈G

→ B1⊕ → B(σ)Uσ (∆−1 G fn ) σ∈G (n)

converges strongly to the identity operator. For this purpose, let us write T2 : B2⊕ → B2⊕ for deﬁned operators on B2⊕ . Let us ﬁrst note that the identically −1 −1 Uσ (∆ fn ) ≤ ∆ fn 1 ≤ K, with K independent of n, thus both sequences G G ∞ L of operators are norm-bounded. Applying Lemma 3.2 (ii), we ﬁnd that the Plancherel transform conjugates (n) −1/2 (T2 )n∈N with the family of convolution operators S (n) : g → g ∗ (∆G fn ), which strongly converges to the identity operator. Moreover, it strongly converges with respect to the B1⊕ -norm on the subspace generated by the ﬁelds of rank-one operators: let B = {|φ(σ)η(σ)|}σ∈G be such a ﬁeld. We may assume φ(σ) =

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∗

(n) (n) with Tσ(n) = Uσ (∆−1 fn ). η(σ). Then (T1 B)(σ) = |φ(σ)Tσ η(σ)|, σ ∈ G, G Hence ∗ (n) φ(σ) Tσ(n) η(σ) − η(σ) dνG (σ) T1 B − B ⊕ = B1

G

≤

1/2 φ(σ) dνG (σ) 2

G

1/2 2 (n)∗ × η(σ) − η(σ) dν (σ) , Tσ G G

by the Cauchy-Schwarz inequality. Here we have used that |φ(σ)η(σ)|1 = φ(σ)η(σ) = φ(σ)2 = η(σ)2 , hence all integrals converge. Picking any measurable family {ξ(σ)}σ∈G of unit vectors, we can deﬁne the operator ﬁeld B = {|ξ(σ)η(σ)|}σ∈G ∈ B2⊕ , and ﬁnd that 2 2 (n) (n)∗ Tσ η(σ) − η(σ) dνG (σ) = T2 B − B ⊕ B2

G

converges to zero. (n) Thus (T1 )n∈N is a bounded sequence of operators converging strongly on a dense subspace, which entails strong convergence on B1⊕ . Hence, ∗ limn→∞ tr([A(σ)Cσ−1 ]Uσ (∆−1 G fn )Uσ (x) )dνG (σ) G = tr([A(σ)Cσ−1 ]Uσ (x)∗ )dνG (σ) , G

and the ﬁrst equation is proved. The second formula is proved by modifying the argument for the ﬁrst: we employ

∗

∗ −1 −1 = tr Uσ (x)Uσ (f tr A(σ) Uσ (x)Uσ (f A(σ) n )Cσ n )Cσ in equation (13). After using −1/2

∗ [Cσ−1 Uσ (∆−1 G fn )Uσ (x) A(σ)] = Uσ (∆G

fn )[Cσ−1 Uσ (x)∗ A(σ)] ,

−1/2

the fact that (∆G fn )n∈N is a bounded approximate identity with respect to left convolution now gives the desired convergence of the traces, and we are done. Remark 3.5 Before we apply the theorem to general direct integral representations, let us ﬁrst consider the relevance of the two inversion formulae, (11) and Uπ a representation in this class (12), for the irreducible case. So let π ∈ G, and assume it to be an irreducible subrepresentation of λG . Let φ, η ∈ Hπ with

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η ∈ dom(Cπ ). The rank-one operator |φCπ η| fulﬁlls the requirement for the ﬁrst inversion formula, hence Vη φ is the inverse Plancherel transform of this operator. But we can also consider the operator |Cπ ηφ|, suitable for the second inversion formula, which gives tr(Cπ−1 |Uπ (x)∗ Cπ ηφ|)

=

φ|Cπ−1 Uπ (x)∗ Cπ η

=

φ|∆G

−1/2

−1/2 Vη φ)(x)

(x)Uπ (x)∗ η = (∆G

.

This reveals the general relationship between the two inversion formulae: the op−1/2 erators f → ∆G f and {A(σ)}σ∈G → {A(σ)∗ }σ∈G are conjugate under the Plancherel transform, hence an inversion formula for f gives rise to an inversion −1/2 2 formula for ∆G f, and vice versa. Now let Uπ be a multiplicity-free subrepresentation of λG . Since G has a Type-I regular representation, we may assume ⊕ Uπ = Uσ dνG (σ) , Σ

A simple method for the construction of for some measurable subset Σ ⊂ G. admissible vectors is then given in the following corollary: Corollary 3.6 Let φ = {φ(σ)}σ∈Σ , {η(σ)}σ∈Σ ∈ Hπ be given. Assume, moreover, that η(σ) ∈ dom(Cσ ), and that the ﬁeld A(σ) := |φ(σ)Cσ η(σ)|, extended trivially outside Σ, is in B2⊕ . Then Vη φ ∈ L2 (G), with (V η φ) = A, and hence 2 2 2 φ(σ) Cσ η(σ) dνG (σ) . Vη φ = Σ

Thus, η is admissible iﬀ {η(σ)}σ∈Σ can be chosen such that Cσ η(σ) = 1, for νG -almost every σ ∈ Σ. Proof. Let a ∈ L2 (G) be the inverse Plancherel transform of A. Then, observing that A(σ)Cσ−1 = |φ(σ)η(σ)|, we see that as a function of σ, tr(|A(σ)Cσ−1 |) = φ(σ)η(σ) is integrable, since φ and η are square-integrable vector ﬁelds. Hence all requirements of Theorem 3.4 are met, and we obtain almost everywhere a(x) = tr(A(σ)Cσ−1 Uσ (x)∗ )dνG (σ) G = Uσ (x)η(σ)|φ(σ)dνG (σ) G

=

(Vη φ)(x) .

The equality of norms is then immediate, since the right-hand side is the norm squared of A in B2⊕ ; and the admissibility condition is an immediate corollary.

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The construction of admissible vectors for representations with multiplicities can be a subtle task. For instance, it is known that λG has admissible vectors, for G non-unimodular with Type-I regular representation [18], but a direct construction of such vectors, without the use of Plancherel transform, could not be given. By contrast, the admissibility condition of the corollary is fairly easy to handle, once the direct integral decomposition of the representation is obtained. One important class of representations which fall under this category are the quasi-regular representations of certain semidirect product groups, see [17]. Remark 3.7 At the moment, we do not know whether the requirement ησ ∈ dom(Cσ ) is necessary for admissibility, i.e., for the ﬁniteness of Vη φ2 , though we expect it to be true. 2 A criterion for the existence of admissible vectors is given in the following theorem. The proof for the unimodular part is a straightforward consequence of Corollary 3.6, for the non-unimodular case see [18], where in fact all representations having admissible vectors are classiﬁed. Theorem 3.8 The unitary representation Uπ has admissible vectors iﬀ G is nonunimodular or G is unimodular and 0 < νG (Σ) < ∞.

4 Example of the (1+1)-Poincar´e group In this section we want to calculate the Plancherel measure of the Poincar´e group ↑ (1, 1) = R2 SO0 (1, 1) in (1 + 1)-dimensional space-time. This is the group P+ (connected part of SO(1, 1)) and we shall explicitly construct admissible vectors for some of its representations. Note that SO0 (1, 1) is the proper Lorentz group in a space-time of (1 + 1)-dimensions. In computing the Plancherel measure, we follow the procedure given by Kleppner and Lipsman [24], which employs the Mackey machinery for this purpose. Recall that it follows from Mackey’s theory of induced representations [29, 30], that (almost all of) the unitary irreducible representations of R2 SO0 (1, 1) are in one-to-one correspondence with the orbits of SO0 (1, 1) in 2 (which we identify here with R2 itself). Here we have used the the dual space R 2 \{0}, such that each dual orbit contributes fact that SO0 (1, 1) operates freely on R precisely one irreducible representation, and we have dropped the one-dimensional representation arising from the dual orbit {0}. We parametrize the Lorentz group by

cosh θ sinh θ . (15) R θ → Λθ = sinh θ cosh θ In this parametrization, dθ is invariant, under both left and right actions, and we choose this for the Haar measure on SO0 (1, 1). We write a generic element of G x0 as (x, h), with x = ∈ R2 and h a matrix of the form (15). As Haar measure x

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↑ on P+ (1, 1) we may take dµ(x, h) = dxdθ, dx being the Lebesgue measure on R2 , and note that this group is unimodular. The ﬁrst step for the calculation of the Plancherel measure is the computation of the dual orbits. They are conveniently represented by the set

1 0 ±1 0 yv | v ∈ { , }, y ∈ R∗ , 0 1 ±1 0

(16)

where R∗ = R \ {0}. The last ﬁve points represent Lebesgue null sets; and hence the set of representations arising from these orbits will have Plancherel measure zero (see below). We therefore drop them from further discussion. (It ought to be pointed out, however, that the ﬁrst four of these orbits correspond, physically, to zero-mass systems. Thus, while they do not play any role in the Plancherel theory, they are by no means physically negligible.) On any of the remaining orbits, Ov,y = ySO0 (1, 1)v, the Lorentz group operates freely. Hence we obtain the parametrization

k0 (17) R θ → k = := y Λθ v ∈ Ov,y , k of Ov,y , and the measure dθ is the image of the Haar measure of SO0 (1, 1), under 2 is parametrized by this parametrization. Hence, up to a null set, R R×{

1 0 cosh θ , } × R∗ (θ, v, y) → y 0 1 sinh θ

sinh θ cosh θ

v ,

(18)

where θ parametrizes Ov,y , and (v, y) parametrizes the orbit space. By Mackey’s theory of induced representations [29, 30], each Ov,y contributes ↑ exactly one representation class σv,y ∈ P+ (1, 1). Denoting the corresponding induced representation in this class by Uv,y , its action on L2 (Ov,y , dθ) is given by (Uv,y (x, h)f )(k) = eik , x f (h−1 k) , 2 , which we take (following the , denoting the dual pairing between R2 and R physicists’ convention) as k , x = k0 x0 − kx. (Note that this choice of dual pairing, as opposed to the more conventional mathematician’s choice, k , x = k0 x0 + kx, does not change the dual action of SO0 (1, 1).) ↑ Hence the Plancherel measure νP , of the semidirect product group P+ (1, 1) 2 2 canbe viewed as a measure on the orbit space R /SO0 (1, 1), or, = R SO0 (1, 1), 1 0 equivalently, on { , } × R∗ . It is obtained by decomposing the Lebesgue 0 1 2 measure

of R along the orbits; in other words, we are looking for a measure λ 1 0 2 , the Lebesgue on { , } × R∗ such that in the parametrization (18) of R 0 1

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measure is given by dθdλ(v, y). By computing the Jacobian of (18), we obtain dνP (σv,y ) = dλ(v, y) = dv y dy ,

(19) 1 0 where dv is just the counting measure on the two-element set , . 0 1 That we have indeed computed the Plancherel measure is due to [17, Theorem 3.3]. (An alternative argument could be derived from [24, II, Theorem 2.3], or rather, the proof of that result.) Generally, the procedure for the computation of the Plancherel measure of semidirect products Rk H following Kleppner and Lipsman [24], involves three steps, which in this (unimodular) setting may be roughly sketched as follows: First compute invariant measures on the orbits (in our case, this was the measure dθ). Then compute a unique measure on the orbit space (our νP ) such that ﬁrst integrating along the orbits and then integrating over the orbit space gives the Lebesgue measure on the dual. Finally the Plancherel measure of the little ﬁxed group and the measure on the orbit space can be combined to give the Plancherel measure of the semidirect product. In our case, the little ﬁxed groups are trivial, and in this case the last step reduces to a – still somewhat subtle – normalization issue. (This is discussed at length in [17].) The role played by the decomposition of Lebesgue measure for the construction of the Plancherel measure also justiﬁes dropping the ﬁve orbits from our discussion: They constitute a Lebesgue null set, hence the representations arising from the orbits are a null set with respect to the Plancherel measure. The Poincar´e group above also provides us with an easy example of the use of ↑ (1, 1) have Plancherel measure Theorem 3.8. Since the individual points σv,y ∈ P+ zero, none of the (irreducible) representations Uv,y is by itself square-integrable and hence does not have admissible vectors. However, it is known [1, 2] that if one ↑ (1, 1)/T , where T = {(x0 , 0) | x0 ∈ R} is the works on the homogeneous space P+ subgroup of time translations, it is possible to again obtain admissible vectors for these irreducible representations. On the other hand, it should also be possible, according to Theorem 3.8, to take sets of these representations, of ﬁnite Plancherel measure such that the corresponding (reducible) direct integral representations possess admissible vectors. A similar construction was done previously in [23]. However, here we work it out again, as an example of an application of Theorem 3.8. Further applications, e.g., to the case of the SU(1,1) group will be presented elsewhere. Let v = (1, 0) and Σ be any Borel subset of R∗ for which y dy < ∞. 0 < νP (Σ) = Σ

Consider the direct integral Hilbert space and the direct integral representation on it, ⊕ ⊕ 2 L (Ov,y , dθ) y dy, UΣ (x, h) = Uv,y (x, h) y dy. HΣ = Σ

Σ

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Elements φ ∈ HΣ are ﬁelds of vectors φv,y ∈ L2 (Ov,y , dθ), y ∈ Σ, representable by functions on R2 of the type, φv,y (k) = φv,y (k0 , k) = φv,y (y cosh θ, y sinh θ),

y=

k0 |k0 |

k02 − k2 ∈ Σ.

Explicitly, the representations Uv,y (x, h) act on the Hilbert spaces L2 (Ov,y , dθ) in the manner, (Uv,y (x, h)φv,y )(y cosh θ, y sinh θ)

= exp [iy(x0 cosh θ − x sinh θ)] × φv,y (y cosh(θ − ξ), y sinh(θ − ξ)), (20)

where we have written h = Λξ

cosh ξ sinh ξ x0 = , x= , sinh ξ cosh ξ x = k0 x0 − kx = x0 cosh θ − x sinh θ.

If we use the variables (k0 , k) rather than (y, θ) to designate points in the orbits, then y dy dθ = dk0 dk, (21) and (Uv,y (x, h)φv,y )(k) = exp[i(k0 x0 − kx)] φv,y (h−1 k),

y=

k0 |k0 |

k02 − k2 . (22)

1 Let η = {ηv,y | y ∈ Σ} ∈ HΣ be a vector such that ηv,y = 2π , for almost all y. Then UΣ (x, h)η|φ = e−i(k0 x0 −kx) ηv,y (h−1 k) φv,y (k) dk0 dk, (23) Σ×R

and a straightforward computation shows that I := |UΣ (x, h)η|φ|2 dµ(x, h) = φ2 ,

φ ∈ HΣ .

(24)

G

To indicate how (24) is obtained, note that using (20) and (21) we may write, UΣ (x, h)η | φ = e−iy(x0 cosh θ−x sinh θ) Σ×R

×

ηv,y (y cosh(θ − ξ), y sinh(θ − ξ))

×

φv,y (y cosh θ, y sinh θ) y dy dξ .

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Thus, I

ei(y cosh θ−y

= Σ×R

Σ×R

cosh θ )x0

ei(y sinh θ−y

sinh θ )x

R3

×

ηv,y (y cosh(θ − ξ), y sinh(θ − ξ)) ηv,y (y cosh(θ − ξ), y sinh(θ − ξ))

×

φv,y (y cosh θ, y sinh θ) φv,y (y cosh θ , y sinh θ )

×

y dy dθ y dy dθ dx0 dx dξ .

Starting out with suﬃciently smooth vectors η, φ, so that the distributional identity ei(y cosh θ−y cosh θ )x0 ei(y sinh θ−y sinh θ )x dx0 dx R2

= (2π)2 δ(y cosh θ − y cosh θ ) δ(y sinh θ − y sinh θ )

can be used (and later using standard arguments to extend to other vectors), we may perform the integrations in y and θ to obtain, |ηv,y (y cosh(θ − ξ), y sinh(θ − ξ))|2 I = (2π)2 Σ×R2

× |φv,y (y cosh θ, y sinh θ)|2 y 2 dy dθ dξ . Using Fubini’s theorem to change the order of integration, integrating over ξ and using the fact that ηv,y = 1/2π, this reduces to 1 2 I = (2π) ηv,y 2 |φv,y (y cosh θ, y sinh θ)|2 y 2 dy dθ Σ×R y = |φv,y (y cosh θ, y sinh θ)|2 y dy dθ Σ×R

= φ2 . This proves (24) and hence the vector η is admissible for the representation UΣ (x, h). Deﬁning coherent states, η(x,h) = UΣ (x, h)η, we get the resolution of the identity on HΣ , G

|η(x,h) η(x,h) | dµ(x, h) = IdΣ .

(25)

Before leaving this section, it is worthwhile looking also at the aﬃne Poincar´e ↑ (1, 1) just considered, together with dilagroup, which is the Poincar´e group P+ tions. Writing this group as PAﬀ (1, 1) = R2 H, where H now consists of matrices of the type

a cosh θ a sinh θ , a > 0, a Λθ = a sinh θ a cosh θ

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2 consist of the four open cones, we see that the orbits of H in R

±1 0 ↑, ↓ ↑, ↓ ↑ ↓ C± = Hv± , v± = , v± = , 0 ±1

1031

(26)

the four semi-inﬁnite lines, ± ±

=

± Hv± ,

± v±

=

±1 , ±1

(27)

and the singleton consisting of the origin. The ﬁrst four are open free orbits, which ↑ are unions of orbits of the Poincar´e group P+ (1, 1) (see (16)–(17)). For example,

1 0 ↑ ↓ C+ = Ov,y , v = , C− = Ov,y , v = , (28) 0 1 y>0

y<0

↑ etc. The remaining ﬁve orbits of PAﬀ (1, 1) coincide with the ﬁve orbits of P+ (1, 1) which have Plancherel measure zero. The Plancherel measure of PAﬀ (1, 1) is just the counting measure on the ﬁrst four orbits, the last ﬁve orbits again having Plancherel measure zero. The unitary irreducible representations corresponding to ↑, ↓ are again induced representations (from the subgroup of H consistthe orbits C± ing of the identity element alone) and are square-integrable. However, the group PAﬀ (1, 1) is nonunimodular and hence not every vector in these representations is admissible (see, e.g., [1, 10]).

5 Wigner functions Wigner functions are a class of transforms associated to elements of the direct integral Hilbert space appearing in (4). We denoted this space by B2⊕ in Section 2. The Wigner map associates its elements isometrically to square-integrable functions on the dual of the Lie algebra of G. This dual space foliates into orbits under the coadjoint action of the group, the invariant components being often identiﬁable with phase spaces of physical systems. Motivated by the properties of such a function, originally introduced in the context of quantum statistical mechanics by Wigner [33], a general procedure for constructing analogous maps (applicable to a class of groups admitting square-integrable representations) was introduced in [4] and further discussed in a speciﬁc context in [3]. Here we extend the deﬁnition of a Wigner function given in [4] to representations which are not necessarily square-integrable, using the Plancherel transform. This will also bring into focus the fact that the Wigner function, like the wavelet transform, owes its existence to the Plancherel transform. It will ﬁrst be necessary to set out a few details about Lie groups and their duals. Again, let G be a Lie group with a Type-I regular representation, g its Lie algebra and g∗ the dual space of g. We make the assumption that the range of the exponential map, g X → eX ∈ G, is a dense set in G, and such that its complement

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has Haar measure zero. By an exponential group we mean a simply connected, connected solvable Lie group for which the exponential map is a homeomorphism. A nilpotent group is understood to be a simply connected, connected nilpotent Lie group. In particular, nilpotent groups are exponential. A Lie group has a natural action on its Lie algebra, the adjoint action, X → Adx0 X, x0 ∈ G, deﬁned by, X [Adx0 X] x−1 . The dual of this map, acting on g∗ , deﬁnes the coadjoint 0 e x0 = e ∗ ∗ action, g X → Adx0 X ∗ , x0 ∈ G, via Adx0 X ∗ ; X = X ∗ ; Adx−1 X, 0 where ; denotes the dual pairing between g and g∗ . Orbits of vectors in g∗ under the coadjoint action are the coadjoint orbits of the group G. The corresponding orbit space, denoted O(G), has a natural quotient topology, and according to the Kirillov theory [21, 22] for nilpotent groups, later extended to exponential of the group, via groups [26], this space is homeomorphic to the unitary dual, G, ↑ (1, 1). For this the so-called Kirillov map. One example is the Poincar´e group P+ group, each coadjoint orbit can be naturally identiﬁed with the cotangent bundle of a corresponding dual orbit. More generally speaking, it is known that coadjoint orbits have the structure of symplectic manifolds and carry natural invariant measures under the coadjoint action, making them resemble physical phase spaces. The collection of coadjoint orbits exhausts g∗ , allowing for a foliation of the type Oλ , g∗ = λ∈J

where Oλ ∈ O(G) denotes an orbit, parametrized by an index (or collection of indices) λ, and J is the corresponding index set. We make the assumption that the orbit space is a countably separated Borel space, in which case the Lebesgue measure on g∗ can be decomposed along these orbits, i.e., if dX ∗ denotes this Lebesgue measure, then it is possible to write, dX ∗ = σλ (Xλ∗ ) dκ(λ) dΩλ (Xλ∗ ),

Xλ∗ ∈ Oλ ,

(29)

where σλ is a positive density deﬁned on the orbit Oλ and dΩλ the (coad)-invariant measure on Oλ . Note that the assumption on the coadjoint orbits entails that also is a countably separated Borel space, which is equivalent to the the dual space G Type-I property of G (and thus of λG ). The measure κ on the orbit space could be continuous or discrete; whenever it has an atom, it is in fact supported on ﬁnitely many of them. It is only necessary to assume that the above disintegration holds on an open dense set of g∗ , such that its complement has Lebesgue measure zero. (Such a decomposition, which is sort of a regularity condition, certainly holds for nilpotent groups [5] and semi-direct product groups admitting open free orbits [25], and in these cases, the measure κ is essentially the Plancherel measure.) For each orbit Oλ , consider the Hilbert space L2 (Oλ , dΩλ ) and denote by H the direct integral Hilbert space, ⊕ H = L2 (Oλ , dΩλ ) dκ(λ) L2 (g∗ ), J

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where we used the measure disintegration (29) to canonically identify L2 (g∗ ) with the direct integral. Elements in H are ﬁelds of vectors, Φ = {Φλ ∈ L2 (Oλ , dΩλ )}λ∈J with the norm, Φ2 = Φλ 2 dκ(λ), J

the norm inside the integral being taken in L2 (Oλ , dΩλ ). If the measure κ is discrete, then clearly the integral would just be a sum. The Wigner map will be deﬁned as a linear isometry, W : B2⊕ −→ H . Let N0 ∈ g be the maximal symmetric set (i.e., N0 includes the origin and X ∈ g ⇒ −X ∈ g) such that its image under the exponential map is dense in G and such that the complement of this image set has Haar measure zero. For any f ∈ L2 (G), f (eX ) deﬁnes a function on N0 . We transfer the left Haar measure µG to N0 , using the exponential map and write, X ∈ N0 , (30) dµG (g) = dµG (eX ) = m(X) dX, where dX is the Lebesgue measure of g and m an appropriate, positive density function. It is not hard to see that m(X) = |det [−F (adX)]|,

(31)

where F is the function (59) deﬁned in the appendix and adX the linear transformation on g deﬁned by adX(Y ) = [X, Y ], Y ∈ g. Let us next deﬁne a modiﬁed Fourier transform, F : L2 (G) −→ H as 1

(F f )λ (Xλ∗ ) =

[σλ (Xλ∗ )] 2 n (2π) 2

∗

1

e−iXλ ;X f (eX )[m(X)] 2 dX,

(32)

N0

at least on L1 (G) ∩ L2 (G), and extend by continuity. (Note, we are assuming the dimension of the group G, and hence of its Lie algebra, to be n.) Since the complement of the set N0 is of (Haar) measure zero, this map is easily seen to be an isometry. For nilpotent groups (32) simpliﬁes considerably since all the density functions involved (i.e., σλ , m) are identically equal to one. Deﬁnition 5.1 The composite transformation W := F ◦ P −1 : B2⊕ −→ H ,

(33)

where P is the Plancherel transform in (4), is called the Wigner map and for any A ∈ B2⊕ , the function W (A|Xλ∗ ) := (WA)λ (Xλ∗ ),

Xλ∗ ∈ Oλ

is called the Wigner function of A, restricted to the orbit Oλ .

(34)

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For any A ∈ H which satisﬁes the conditions of Theorem 3.4, using (11) we obtain the following explicit expression for its Wigner function: W (A|Xλ∗ )

= ×

1 ∗ [σλ (Xλ∗ )] 2 e−iXλ ;X n (2π) 2 N0 1 −X −1 2 tr (Uσ (e )[A(σ)Cσ ]) [m(X)] dνG (σ) dX,

G

(35)

provided the inverse Plancherel transform of A is integrable. The inverse of this transform can be computed using (11) and standard Fourier transform methods. We get, ∗ 1 A(σ) = eiXλ ; X W (A|Xλ∗ ) Uσ (eX ) n (2π) 2 N0 J Oλ

1 ∗ −1 × [σλ (Xλ ) m(X)] 2 dΩλ dκ(λ) dX Cσ , (36) the extreme pair of square brackets implies taking the closure of the operator involved. A few properties of the Wigner map can easily be established from its deﬁnition. We collect these into the theorem below. The proof involves straightforward computations, similar to those done to obtain analogous results in [4], and we omit it. On B2⊕ and H we deﬁne the two unitary representations, U ⊕ and U , of the group G: (U ⊕ (x)A)(σ) = Uσ (x)A(σ)Uσ∗ (x), x ∈ G, (37) (w.r.t. νG ), and the above relations holding for almost all σ ∈ G (U (x)Φ)λ (Xλ∗ ) = Φλ (Adx−1 Xλ∗ ),

x ∈ G,

(38)

holding for almost all λ ∈ J (w.r.t. κ). Theorem 5.2 The Wigner map is a linear isometry, which intertwines the representation U ⊕ of G with the representation U . The corresponding Wigner function satisﬁes the overlap condition, W (A1 |Xλ∗ ) W (A2 |Xλ∗ ) dΩλ dκ(λ) = A1 |A2 B⊕ , A1 , A2 ∈ B2⊕ , J

2

Oλ

(39)

and the covariance condition, W (U ⊕ (x)A|Xλ∗ ) = W (A| Adx−1 Xλ∗ ),

(40)

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for all x ∈ G, and almost all Xλ∗ ∈ Oλ (w.r.t. Ωλ ). If A = A∗ is self-adjoint, its Wigner function is real, i.e., W (A|Xλ∗ ) = W (A|Xλ∗ ),

(41)

almost everywhere. Now suppose that A1 , A2 ∈ B2⊕ satisfy the conditions of the validity of the inversion formula (11) in Theorem 3.4. Then the unitarity of the Plancherel transform implies the orthogonality relation tr([Uσ (x)∗ A1 (σ)Cσ−1 ])dνG (σ) G G −1 ∗ 2 × tr([Uσ (x) A (σ )Cσ ])dνG (σ ) dµ(x) = A1 | A2 B⊕ , (42) 2

G

which is equivalent to the overlap condition (39). Here the absolute convergence of the integral above follows from the assumptions guaranteeing absolute convergence has positive Plancherel measure, then the above relation in (11). If now σ ∈ G implies the restricted orthogonality relation, tr([Uσ (x)∗ A1 (σ)Cσ−1 ]) tr([Uσ (x)∗ A2 (σ)Cσ−1 ]) dµ(x) G

= tr [A1 (σ)∗ A2 (σ)],

familiar from the theory of square-integrable group representations [1, 19]. This equation was the basis for the construction of Wigner functions, for square-integrable group representations, in [4]. Remarks 5.3 A few comments are in order here: (a) Generally, the range of the Wigner map W is a closed, proper subspace of H , which we denote by HW . If we restrict the Wigner map to a subspace of B2⊕ of the type B2Σ =

⊕

B2 (Hσ ) dνG (σ),

Σ

such that νG (Σ) = 0, then clearly its range W(B Σ ) is a where Σ is a subset of G, 2 in (35) has to be replaced closed subspace of HW . In this case, the integral over G by an integral over Σ, however the expressions (36) and (39) remain unchanged. In particular, if Σ is a discrete subset, the representations σ ∈ Σ are square-integrable and we recover the results of [4]. (b) Suppose that the group G is exponential and assume, moreover, that it is a Type-I group. Note that the homeomorphism property of the Kirillov map entails that the coadjoint orbit space is a countably separated Borel space; in particular the measure disintegration (29) exists.

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Then we have on the one hand a mapping ⊕ ⊕ B2 (Hσ )dνG (σ) → L2 (Oλ , dΩλ ) dκ(λ) L2 (g∗ ), W: G

J

between the direct integral spaces, and on the other hand the inverse of the Kirillov → J, σ → λσ . It is thus a natural question map, which gives rise to a bijection G to ask whether the Wigner map is decomposable, i.e., if there exists a ﬁeld of operators Wσ : B2 (Hσ ) → L2 (Oλσ , dΩλσ ) and almost all X ∗ ∈ Oλσ , such that for almost all σ ∈ G λσ W (A|Xλ∗σ ) = [Wσ A(σ)] (Xλ∗σ ) . The existence of such a ﬁeld of operators is not just of mathematical interest, but also desirable from a physical point of view: The coadjoint orbits have a natural interpretation as phase spaces of physical systems, but the dual g∗ , as a disjoint union of such phase spaces, does not usually have a natural interpretation, except in some cases, where one might look upon a set of orbits as constituting the phase space of a composite physical system. Correspondingly, the space L2 (Oλσ , dΩλσ ) has a simpler interpretation than L2 (g∗ ). A similar reasoning applies to the representations. A related question concerns the supports of the Wigner functions. Even when the Wigner map is restricted to a subspace such as B2Σ as in part (a), the corresponding Wigner functions could in general have supports on orbits which are not associated to the representations in Σ (see example of the Poincar´e group below). This is possible even for representations which arise from semidirect product groups admitting open free orbits [25]. It is obvious that whenever the Wigner map is decomposable, the supports of the Wigner functions of elements in B2Σ are contained (up to a null set) in the coadjoint orbits corresponding to Σ; we expect the converse of this statement to hold as well. It turns out that these questions have been addressed, and to a large extent solved, in the context of star products: ﬁrst of all, the nilpotent Lie groups for which the Wigner transform is decomposable are precisely those for which almost all coadjoint orbits are aﬃne subspaces [28]. If a nilpotent Lie group does not fulﬁll this condition, the modiﬁed Fourier transform (32) can be replaced by an adapted Fourier transform. Following [5, 7], the adaptation consists in constructing a suitable mapping α : g × V → R, polynomial in the elements of g and rational in the elements of a suitably chosen open conull subset V ⊂ g∗ . The speciﬁc construction of α ﬁrst ensures that deﬁning ∗ 1 e−iα(X,X ) f (eX ) dX , (43) Fad (f )(X ∗ ) = n 2 (2π) g for f in the Schwartz space of the group, induces a unitary map L2 (G) → L2 (g∗ ). Secondly, the adapted Wigner map Wad = Fad ◦ P −1 has all the properties of the

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Wigner map collected in Theorem 5.2, and is in addition decomposable. Extensions of this construction to certain solvable groups exist [6]. It seems worthwhile to explicitly work out adapted Wigner transforms for concrete examples. This might also provide additional criteria for the choice of α, which is apparently not unique.

6 Some examples ↑ Let us go back to the Poincar´e groups P+ (1, 1) and PAﬀ (1, 1), studied in Section 4, and explicitly compute the Wigner functions for them.

6.1

↑ The Poincar´e group P+ (1, 1)

↑ We start by writing a general element of P+ (1, 1) as a 3 × 3 matrix,

h x (x, h) = , 0T = (0, 0), 0T 1

where x and h are as deﬁned earlier (in Section 4). The Lie algebra p is generated by the three elements,       0 1 0 0 0 1 0 0 0 (44) Y 1 = 1 0 0 , Y 2 = 0 0 0 , Y 3 = 0 0 1 , 0 0 0 0 0 0 0 0 0 which satisfy the commutation relations, [Y 1 , Y 2 ] = Y 3 ,

[Y 1 , Y 3 ] = Y 2 ,

[Y 2 , Y 3 ] = 0.

A general element X ∈ p can be written as (see (61) in the appendix),

0 θ ξ1 Xq xp := X (0, θ) = , θ ∈ R, x = X= , X ∈ R2 , q q p θ 0 0T 0 ξ2 so that eX =

X e q 0T

F (Xq )xp , 1

F (Xq ) being the matrix function deﬁned in (63) in the appendix. Following (30), the Haar measure dµ(x, h) = dx dθ, expressed in terms of the Lie algebra variables xp , θ, becomes dµ(x, h) = m(xp , θ) dxp dθ,

dxp = dξ1 dξ2 ,

and the density m(xp , θ) is easily calculated to be (see (63) in the appendix), θ m(xp , θ) = det [F (Xq )] = sinch 2 ( ). 2

(45)

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↑ The adjoint action of P+ (1, 1) on p, given by X −→ X = (x, h)X(x, h)−1 , leads to the transformation

xp h σ1 x xp , M (x, h) = = M (x, h) , θ 0T θ 1

of the variables xp , θ, where σ1 is the 2 × 2 matrix deﬁned in the appendix. Let p∗ denote the dual space of p. We write elements X ∗ ∈ p∗ in terms of the dual basis {Y1∗ , Y2∗ , Y3∗ } as X ∗ = γY1∗ + k0 Y2∗ + kY3∗ and compute the coadjoint action, in terms of a matrix M (x, h) acting on the variables k, γ,

k k k k0 −1 −1 T = M (−h x, h ) ∈ R2 , = M (x, h) , k= γ γ k γ to obtain, k γ

= h−1 k = γ − xT σ1 k ,

xT = (x0 , x).

(46)

↑ Using these relations, all the coadjoint orbits of P+ (1, 1) in R3 p∗ can now be calculated. Indeed, introducing the vectors yv, deﬁned in (16), we get the coadjoint orbits

v ↑ ∗ (47) Ov,y = yM (x, h) (x, h) ∈ P+ (1, 1) , 0

which, taken together for all y, v, exhaust R3 . It is also clear from (46) that these ∗ = T ∗ Ov,y , of the orbits Ov,y = orbits are precisely the cotangent bundles, Ov,y ySO0 (1, 1)v computed in Section (4). Explicitly, let us take the set

1 ↑ Σ = {yv | y > 0} ⊂ P+ (1, 1), v= . (48) 0 Points in the corresponding coadjoint orbits can then be parametrized as ∗ Ov,y = {(k, γ) ∈ R3 | k = (k0 , k), k0 > 0, y = k02 − k2 }, and the invariant measure under the coadjoint action calculated to be dΩv,y (k, γ) =

dk dγ. k0

∗ Of course, we could also use the alternative coordinates (θ, −γ) for Ov,y , where −1 θ = − tanh (k/k0 ), which are actually the Darboux coordinates for this orbit, and then the invariant measure would simply be dθ dγ. However it will be more useful, for the purposes of computing the Wigner function, to use the (k, γ) coordinates. The Lebesgue measure on p∗ in the (k, γ) coordinates is dk0 dk dγ, and making

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the change of variables, (k0 , k, γ) −→ (y, k, γ) we get the measure disintegration along the coadjoint orbits in Σ (see (29)), dk0 dk dγ

=

with

dk y dy dγ = σv,y (k, γ) dκ(v, y) dΩv,y (k, γ), k2 + y 2 σv,y (k, γ) ≡ 1, dκ(v, y) = y dy.

(49)

Thus, in this case, the measure κ is precisely equal to the Plancherel measure νP , restricted to Σ (see (19)). We are now in a position to explicitly compute the Wigner map W : B2Σ −→ H , ↑ ↑ for P+ (1, 1), restricted to the subset Σ deﬁned in (48). Since P+ (1, 1) is unimodular, the Duﬂo-Moore operators Cσ are trivial and in fact, using relations such as (23), it can be seen that for almost all yv ∈ Σ, the corresponding Duﬂo-Moore operator is Cv,y = 2π IdH on the representation space H = L2 (Ov,y , dθ). Let us consider elements in B2Σ which are of the type

A = {A(v, y)}yv∈Σ , A(v, y) = |φv,y ψv,y |,

φv,y , ψv,y ∈ L2 (Ov,y , dθ).

A tedious but straightforward manipulation, after inserting the various quantities into the expression (35) for the Wigner function and using relations such as (63), yields the ﬁnal expression, W (A|kv,y , γ) = ×

e− θσ21 σ k 1 3 v,y 1 θ sinch ( 2 ) sinch ( θ2 ) (2π) 2 R e θσ21 σ k y 3 v,y dθ, y(θ) = . φv,y(θ) sinch ( θ2 ) sinch ( θ2 ) 1

eiθγ ψv,y(θ)

(50)

Here, kv,y ∈ Ov,y , and the point (kv,y , γ) ∈ T ∗ Ov,y and hence the above expression is for the Wigner function restricted to the orbit T ∗ Ov,y . However, it ought to be noted that its value on any orbit receives contributions from vectors φv,y(θ) , ψv,y(θ) coming from representations associated to all the orbits in Σ. Hence we see that the Wigner map is not decomposable. (Completely the opposite situation is true for the aﬃne Poincar´e group, as will be shown in Theorem 6.1 below.) Using (46) and (64) we directly verify the covariance condition (40),

kv,y ⊕ −1 kv,y , = M (x, h) W (UΣ (x, h)A|kv,y , γ) = W (A|kv,y , γ ), γ γ ↑ (x, h) ∈ P+ (1, 1), with

(UΣ⊕ (x, h)A)(v, y) = Uv,y (x, h)|φv,y ψv,y |Uv,y (x, h)∗ ,

yv ∈ Σ,

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and the overlap condition, Ov,y ×R+

W (A1 |kv,y , γ)W (A2 |kv,y , γ) dΩv,y (kv,y , γ) y dy = R+

2 1 ψv,y |ψv,y φ1v,y |φ2v,y y dy,

i where, Ai = {|φiv,y ψv,y |}yv∈Σ , i = 1, 2.

↑ (see (28), Consider now the open forward light cone C+ ↑ C+

k0 2 2 2 = ∈ R | k0 > 0, k0 > k . k

Using the coordinates (y, θ) = ( k02 − k2 , − tanh−1 (k/k0 )), the invariant measure under the action of SO0 (1, 1) is clearly y dy dθ and the Hilbert space ↑ L2 (C+ , y dy dθ) is naturally isomorphic to the direct integral Hilbert space HΣ =

⊕

L2 (Ov,y , dθ) y dy.

Σ

The corresponding direct integral representation UΣ can thus be expressed by its ↑ action on L2 (C+ , y dy dθ) in the manner (UΣ (x, h)φ)(y, θ) = exp[iy(x0 cosh ξ − x sinh ξ)] φ(y, θ − ξ), ↑ φ ∈ L2 (C+ , y dy dθ) and ξ being the hyperbolic angle of the transformation h. Thus, the representation UΣ is precisely the Fourier transform of the quasi-regular ↑ ↑ (1, 1), restricted to the Hilbert space L2 (C+ , y dy dθ), of representation of P+ functions with support in the forward light cone. The Wigner function (50) can ↑ now be thought of as a function on C+ × R y>0 T ∗ Ov,y and, written in terms of the variables (y, θ, −γ) (the invariant measure under the coadjoint action being y dy dθ dγ), it becomes

ξ 1 y , θ − 1 ξ 2 sinch ( ξ2 ) sinch ( 2 ) (2π) 2 R ξ y , θ+ dξ, θ, γ ∈ R, y > 0, (51) × φ ξ 2 sinch ( 2 )

W (ψ, φ | θ, γ; y) =

1

e−iξγ ψ

↑ φ, ψ ∈ L2 (C+ , y dy dθ). It ought to be noted, however, that although in this way of writing the Wigner function, W (ψ, φ | k, γ) is sesquilinear in ψ, φ ∈ ↑ L2 (C+ , y dy dθ), and can be extended to the linear span of rank-one operators ↑ , y dy dθ)) |φψ|, it cannot be used to deﬁne an isometric map between B2 (L2 (C+

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↑ and L2 (C+ × R, y dy dθ dγ), since |W (ψ, φ | θ, γ; y)|2 y dy dθdγ

1041

=

↑ C+ ×R

R+ ×R2

×

|ψ(y, θ)|2 |φ(y, ξ)|2

y dy dθ dξ = φ2 ψ2 .

Thus W (φ, ψ | θ, γ; y) is not a Wigner function for the operator |φψ| ∈ ↑ , y dy dθ)), in the sense of our deﬁnition (hence the use of the altered B2 (L2 (C+ notation W ). On the other hand, physically the representation UΣ refers to systems of relativistic particles of all possible (positive) masses and W (ψ, ψ | y, θ, γ) can serve as the Wigner function for the state of a system consisting of a cluster of masses. Furthermore, this form of the Wigner function is particularly simple looking and bears a striking resemblance to the original Wigner function [33] (see (56) below).

6.2

The aﬃne Poincar´e group PAﬀ (1, 1)

Elements, (x, ah) ∈ PAﬀ (1, 1), matrices of the form

a > 0, h ∈ SO0 (1, 1) can be represented by

ah x (x, ah) = . 0 1

The group is nonunimodular, with left and right Haar measures, dµ (a, h, x) =

1 dx0 dx da dθ, a3

dµr (a, h, x) =

1 dx0 dx da dθ. a

As discussed at the end of Section 4, there are four irreducible representations of ↑, ↓ (see (26)) which are PAﬀ (1, 1), corresponding to the four open free orbits C± square-integrable, and these are the only ones which contribute to the Plancherel measure. It will be enough to work out the Wigner function for the one orbit ↑ , for the other three are entirely similar. The Hilbert space of the irreducible C+ ↑ ↑ representation U+ , associated to this orbit, is L2 (C+ , dk0 dk) and ↑ (U+ (x, ah)φ)(k) = aeik ,x φ(ah−1 k).

The Duﬂo-Moore operator C for this representation is unbounded, acting on ↑ L2 (C+ , dk0 dk) in the manner (see [1, 10]), (Cφ)(k) =

2π 1

|k02 − k2 | 2

φ(k).

↑ Recall that the orbit C+ is characterized by k0 > 0, k02 > k2 and the invariant 0 dk . measure on it under the action of the group elements ah is dk k02 −k2 The Lie algebra pAﬀ is four-dimensional, being generated by the three el↑ ements (44) of the Lie algebra of P+ (1, 1) together with I3 , the 3 × 3 identity

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matrix. Computation of the coadjoint orbits is routine. The one which concerns ↑ ↑ us here is the cotangent bundle, T ∗ C+ , of the orbit C+ . Denoting its elements by γ1 ↑ (γ, k), γ = ∈ R2 , k ∈ C+ , using relations such as (64)–(65) the coadjoint γ2 action is computed to be, 1 −1 h k, a

k −→ k

=

γ −→ γ

1 = γ + Xq (x0 , x) h−1 k. a

(52)

↑ under this action is The invariant measure on T ∗ C+

dΩ↑+ (k, γ) =

dk0 dk dγ1 dγ2 , k02 − k2

and thus the densities σλ and m appearing in the Wigner function (see (29), (30) and (35) become in this case, ↑ σ+ (k, γ) = k02 − k2 ,

m(λ, θ) =

2(cosh λ − cosh θ) . eλ (λ2 − θ2 )

The ﬁnal expression for the Wigner function is obtained after a routine computation, starting with (35) and using expressions such as (63)–(65) in the appendix. We get, W (ψ, φ | k, γ) = ×

1 4π

∞

dλ −∞

!

∞

dθ e −∞

−i(γ1 λ+γ2 θ)

ψ

σ3 eXq (λ,θ)/2 k sinch (Xq (λ, θ)/2) "

! (k02 − k2 )(λ2 − θ2 ) σ3 e−Xq (λ,θ)/2 φ k , cosh λ − cosh θ sinch (Xq (λ, θ)/2)

"

(53)

1 (where we have written for [ sinch A]−1 ). The above expression should sinch A be compared with (50) and (51). Also, by virtue of Lemma 7.1 and (62) in the ↑ appendix, if k ∈ C+ then so also are the arguments of the functions ψ and φ in the above expression for the Wigner function. Thus we have the important result:

Theorem 6.1 The Wigner function W (ψ, φ | k, γ) in (53) has support inside the ↑ ↑ coadjoint orbit, T ∗ C+ , associated to the representation U+ .

6.3

The Weyl-Heisenberg group GWH

The Wigner function arising from the Weyl-Heisenberg group is the original phase space distribution introduced by Wigner [33] in 1932 (see also [16]). Although this function is well known, to our knowledge, it has not been obtained by the methods

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introduced in this paper, linking it to the Plancherel transform. We, therefore, give a somewhat detailed derivation of it. Since this function was the original motivation for developing our general analysis, it is also worthwhile to put it in this context. (A somewhat diﬀerent derivation, based on the theory of square-integrability of group representations, modulo subgroups, was given in [4].) The group GWH consists of 4 × 4 matrices  

1 12 ζ T ω θ 0 1   ω= , g(θ, ξ, η) = 0 I2 ζ , −1 0 1 0 0T with θ ∈ R,

η ζ= ∈ R2 , ξ

0 0= . 0

This group is unimodular and nilpotent. The Lie algebra gWH is generated by the three elements,  

T

1 T

1 T 0 0 − 2 e2 0 1 e 0 X0 = , X1 = 2 3 , X2 = , 0 = 0 , O 0 O e1 O e2 0 O being the 3 × 3 zero matrix and e1 , e2 , e3 the canonical basis vectors in R3 . They satisfy the commutation relations, [X 0 , X 1 ] = [X 0 , X 2 ] = 0,

[X 1 , X 2 ] = X 0 .

Writing a general element of gWH as Y = −θX 0 − ηX 1 + ξX 2 , and noting that (Y )2 is the null matrix, we see that eY = g(−θ, ξ, −η) = I4 + Y. Thus, the group and the Lie algebra have essentially the same parametrization; the Haar measure of GWH is dθ dζ = dθ dξ dη and the density m(X) = 1 (see (30)), almost everywhere. Let {X0∗ , X1∗ , X2∗ } be the dual basis in g∗WH and denote a general element in it by X ∗ = γ 0 X0∗ + γ 1 X1∗ + γ 2 X2∗ . The computation of the coadjoint action of g(θ, ξ, η) ∈ GWH on g∗WH is now a routine matter. The coordinates γ i , i = 1, 2, 3, transform under this action in the manner γ 0 −→ γ 0 γ 1 −→ γ 1

γ 2 −→ γ 2

= =

γ0, γ1 − ξγ 0 ,

=

γ2 − ηγ 0 .

The (physically) non-trivial coadjoint orbits are the planes,

γ1 γ0 2 γ = , γ = 0 , ∈ R Oγ0 = 0 γ γ2

(54)

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which carry the (coad-)invariant measures dΩγ0 (γ) = dγ = dγ1 dγ2 , and comparing with (29) we get, dκ(γ0 ) = dγ0 and σγ0 (γ) = 1, for all γ0 = 0 and almost all γ = R2 . The (physically) non-trivial unitary irreducible representations of GWH are in one-to-one correspondence with the orbits Oγ0 and thus the unitary dual GWH is identiﬁable with R\{0}. We choose the realization, Uλ , of the UIR (corresponding to λ ∈ GWH ), which is carried by the Hilbert space Hλ = L2 (R, dx) and acts in the manner, ξ

(Uλ (θ, ξ, η)φλ )(x) = eiλθ eiλη(x− 2 ) φλ (x − ξ). Since GWH is unimodular, the Duﬂo-Moore operator Cλ is a multiple of the identity, which we denote by Nλ (> 0). Writing the Plancherel measure as dνGW (λ) = ρ(λ) dλ, where ρ is some density function, we may compute both Nλ and ρ by noting that the orthogonality condition (42) leads in this case to the explicit relation, 1 Nλ2

R3

R\{0}

ψλ |Uλ (θ, ξ, η)φλ ρ(λ) dλ

× ρ(λ ) dλ

R\{0}

dθ dξ dη = (2π)2 R\{0}

ψλ |Uλ (θ, ξ, η)φλ

φλ 2 ψλ 2

(ρ(λ))2 λ

dλ,

1 and ρ(λ) = |λ|, almost everywhere, for all φλ , ψλ ∈ Hλ . We easily obtain, Nλ = 2π so that the Plancherel measure of GWH is |λ| dλ. The Wigner function is now obtained after a routine computation, using (35):

W (A | γ1 , γ2 ; λ) =

1

γ γ x x 1 1 φλ − + dx, eixγ2 ψλ − − λ 2 λ 2 R

1

(2π) 2

(55)

for A ∈ B2⊕ such that A = {|φλ ψλ | ∈ B2 (Hλ )}λ∈R\{0} and γ1 , γ2 ∈ Oλ . Note that again, in this case, the support of the Wigner function is concentrated on the orbit Oλ which corresponds to the UIR Uλ . The above formula for the Wigner function is particularly interesting, since for ﬁxed λ, |W (A | γ1 , γ2 ; λ)|2 dγ1 dγ2 = |λ| φλ 2 ψλ 2 . R2

This means that, for ﬁxed λ, the expression (55) can be used to deﬁne a function W (Aλ | γ1 , γ2 ; λ) on Oλ R2 for any Aλ ∈ B2 (Hλ ), such that the map Aλ −→

1 1

|λ| 2

W (Aλ | · ; λ) ∈ L2 (Oλ , dγ),

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is an isometry. Indeed, writing WQM (ψλ , φλ | q, p; ) = =

q p 1 W Aλ − , − ; 1 (2π) 2 xp x x 1 φ q+ dx, e−i ψλ q − 2π R 2 2 1

(56)

we recover the well-known function originally introduced by Wigner [33]. Thus, eﬀectively, in this case Wigner functions can be deﬁned for each UIR, Uλ , that appears in the direct integral decomposition of the regular representation and the support of the Wigner function is concentrated on the corresponding coadjoint orbit.

7 Conclusion The procedure outlined and illustrated in this paper is general enough to cover most groups of practical importance, for constructing wavelet transforms and Wigner functions. There are still, however, group representations which are used in practical applications, but which are not amenable to the present technique. Representations which are not in the support of the Plancherel measure fall into this category. For example, in the case of the two Poincar´e groups discussed here, the representations corresponding to the boundaries of the cones (the “mass zero representations”) fall outside of our scheme. One interesting direction for further research concerns the use of adapted Fourier transforms for the construction of decomposable Wigner maps. Also, the precise relationship between Wigner maps and deformation quantization (which is where the adapted Fourier transforms originate) should be worked out explicitly.

Acknowledgments We would like to thank M. Cahen and S. Gutt for stimulating discussions and pointing out various references to us. The authors would like to acknowledge ﬁnancial support from the Natural Sciences and Engineering Research Council (NSERC), Canada and the Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche (FCAR), Qu´ebec. HF would like to thank the Department of Mathematics and Statistics of Concordia University, Montr´eal, for their hospitality. STA would also like to thank G. Schlichting for hospitality at the Zentrum Mathematik der Technischen Universit¨ at M¨ unchen, where part of this work was completed.

Appendix We collect in this appendix a few formulae and results for the for the various special matrix functions which appear in this paper. We begin by deﬁning three

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real-valued functions on R: sinch x

:= =

cosinch x

:= =

F (x)

:= =

x4 x6 x2 + + + ... 3! 5! 7! sinh x , if x = 0, x x x3 x5 x7 + + + + ... 2! 4! 6! 8! cosh x − 1 , if x = 0, x

1+

(57)

(58)

x x2 x3 x x sinch x + cosinch x = e 2 sinch ( ) = 1 + + + + ... 2 2! 3! 4! x e −1 , if x = 0. (59) x

Note that sinch x is an even function of x, while cosinch x is an odd function. The inverse of F (x) has an interesting series expansion: 2k x # e− 2 k−1 Bk x = 1 − + (−1) sinch ( x2 ) 2 (2k)! x

F (x)−1

= e−x F (−x)−1 =

k≥1

=

x , ex − 1

if x = 0

(60)

where the Bk are the Bernoulli numbers, B1 = 16 , B2 = etc., and generally, ∞ (2k)! # 1 . Bk = 2k 2k−1 π 2 n2k n=1

1 30 ,

B3 =

1 42 ,

B4 =

1 30 ,

If A is an n×n matrix, then using the series expansions, we can deﬁne the matrix versions sinch A, cosinch A, F (A) of these functions. In addition, if det A = 0, then we can also write, sinch A = A−1 sinh A, cosinch A = A−1 (cosh A − 1), etc. In particular, for the matrix

1 0 0 1 Xq (λ, θ) = λI2 + θσ1 , λ, θ ∈ R, I2 = , σ1 = , (61) 0 1 1 0 we easily compute, eXq (λ,θ) F (Xq (λ, θ)) det [F (Xq (λ, θ))]

= eλ cosh θ I2 + eλ sinh θ σ1 ,

Xq (λ,θ) Xq (λ, θ) = e 2 sinch , 2 =

sinch Xq (0, θ) = F (Xq (0, θ))

=

det [eXq (λ,θ) ] = e2λ , (62)

2eλ (cosh λ − sinh θ) θ , det [F (Xq (0, θ))] = sinch 2 ( ), λ2 − θ2 2 sinch θ I2 , cosinch Xq (0, θ) = cosinch θ σ1 , θ Xq (0,θ) sinch θ I2 + cosinch θ σ1 = sinch ( ) e 2 , (63) 2

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To these we add the useful relationships, σ3 eXq (λ,θ) σ3

=

eXq (λ,−θ) ,

σ1 eXq (λ,θ) σ1

=

eXq (λ,θ) ,

σ3 F (Xq (λ, θ)) σ3 = F (Xq (λ, −θ)),

1 0 σ3 = , σ12 = σ32 = I2 . (64) 0 −1

Note that the matrices Xq (λ, θ) span the Lie algebra of the SOAﬀ (1, 1), which is the group SO0 (1, 1) together with dilations. Let us denote this Lie algebra by soAﬀ (1, 1) and note that the group SOAﬀ (1, 1) itself consists of all invertible elements of this algebra. The set soAﬀ (1, 1) is closed under ordinary matrix multiplication and under this multiplication it is a commutative set. Furthermore, are elements of soAﬀ (1, 1), for all λ, θ ∈ R. For any two the matrices F (X q (λ, θ)) u0 k0 , u= ∈ R2 , and any 2 × 2 matrix A, vectors, k = k u Xq (u0 , u)k

= Xq (k0 , k)u,

Ak , σ3 Au = det A k , u = det A (k0 u0 − ku). Diagonalizing the matrices Xq (λ, θ),

λ+θ 0 V Xq (λ, θ)V T = , 0 λ−θ

1 V = √ 2

(65)

1 1 , −1 1

we get, V F (Xq (λ, θ))V

T

det [F (Xq (λ, θ))] Writing ζ=

F (λ + θ) = 0

0 , F (λ − θ)

= F (λ + θ) F (λ − θ) > 0.

(66)

ζ1 k0 = V k, =V ζ2 k

↑ we see that the condition that k ∈ C+ (i.e., k02 > k2 , k0 > 0), is equivalent to having ζ1 , ζ2 > 0. Thus we have the result, ↑ ↑ Lemma 7.1 If k ∈ C+ , then F (Xq (λ, θ))k ∈ C+ , for all λ, θ ∈ R.

Proof. By (66), V F (Xq (λ, θ))V T ζ =

F (λ + θ)ζ1 , F (λ − θ)ζ2

and since both F (λ + θ), F (λ − θ) > 0, the condition ζ1 , ζ2 > 0 is preserved under ↑ the action of V F (Xq (λ, θ))V T . Hence the condition k ∈ C+ is preserved under the action of F (Xq (λ, θ)).

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References [1] S.T. Ali, J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations, Springer, New York, 2000. [2] S.T. Ali, J.-P. Antoine and J.-P. Gazeau, Square integrability of group representations on homogeneous spaces. II. Coherent and quasi-coherent states. The case of the Poincar´e group, Ann. Inst. Henri Poincar´e 55, 857–890 (1991). [3] S.T. Ali, A.E. Krasowska and R. Murenzi, Wigner functions from the twodimensional wavelets group, J. Opt. Soc. Am. A17, 1–11 (2000). [4] S.T. Ali, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, The Wigner function for general Lie groups and the wavelet transform, Ann. H. Poincar´e 1, 685–714 (2000). [5] D. Arnal and J.C. Cortet, Nilpotent Fourier transform and applications, Letters in Math. Phys. 9, 25–34 (1985). [6] D. Arnal, J.C. Cortet and J. Ludwig, Moyal product and representations of solvable Lie groups, J. Funct. Anal. 133, 402–424 (1995). [7] D. Arnal and S. Gutt, D´ecomposition de L2 (G) et transformation de Fourier adapt´ee pour un groupe G nilpotent, C.R. Acad. Sci. Paris, t. 306, S´erie I, 25–28 (1988). [8] M.J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Opt. Comm. 25, 26–30 (1978), and Wigner distribution functions and its application to ﬁrst-order optics, J. Opt. Soc. Am. 69, 1710–1716 (1979). [9] J. Bertrand and P. Bertrand, Repr´esentations temps-fr´equence des signaux, C.R. Acad. Sc. Paris 299, S´erie I, 635–638 (1984), and A class of Wigner functions with extended covariance properties, J. Math. Phys. 33, 2515–2527 (1992). [10] D. Bernier and K. Taylor, Wavelets from square-integrable representations, SIAM J. Math. Anal. 27, 594–608 (1996). [11] F. Bruhat, Distributions sur un groupe localement compact et applications a l’´etude des repr´esentations des groupes p-adiques, Bull. Soc. math. France ` 89, 43–75 (89). [12] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61, 1992. [13] J. Dixmier, Von Neumann-Algebras, North Holland, Amsterdam, 1981. [14] J. Dixmier, C ∗ -Algebras, North Holland, Amsterdam, 1977.

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[15] M. Duﬂo and C.C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal. 21, 209–243 (1976). [16] G.B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995. [17] H. F¨ uhr and M. Mayer, Continuous wavelet transforms from semidirect products: Cyclic representations and Plancherel measure, J. Fourier Anal. Appl. 8, 375–398 (2002). [18] H. F¨ uhr, Admissible vectors for the regular representation, Proc Am. Math. Soc. 130, 2959–2970 (2002). [19] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations I: General Results, J. Math. Phys. 26, 2473–2479 (1985). [20] M. Hillery, R.F. O’Connel, M.O. Scully, and E.P. Wigner, Distribution functions in physics: fundamentals, Phys. Rep. 106, 121–167 (1984). [21] A.A. Kirillov, in Representation Theory and Noncommutative Harmonic Analysis I, Springer Verlag, New York, 1994. [22] A.A. Kirillov, Elements of the Theory of Representations, Springer Verlag, Berlin, 1976. [23] J.R. Klauder and R.F. Streater, A wavelet transform for the Poincar´e group, J. Math. Phys. 32 , 1609–1611 (1991). [24] A. Kleppner and R.L. Lipsman, The Plancherel formula for group extensions, I and II, Ann. Sci. Ecole Norm. Sup. 5, 459–516 (1972); ibid. 6, 103–132 (1973). [25] A.E. Krasowska and S.T. Ali, Wigner functions for a class of semi-direct product groups, J. Phys A36, 2801–2820 (2003). [26] H. Leptin and J. Ludwig, Unitary representation theory of exponential Lie groups., de Gruyter, Berlin, 1994. [27] R.L. Lipsman, Non-abelian Fourier analysis, Bull. Sci. Math. 98, 209–233 (1974). [28] J. Ludwig, On the Hilbert-Schmidt semi-norms of L1 of a nilpotent Lie group, Math. Ann. 273, 383–395 (1986). [29] G.W. Mackey, Induced representations of groups and quantum mechanics. W.A. Benjamin Inc., New York, 1968.

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[30] G.W. Mackey, Induced representations of locally compact groups, I., Ann. of Math. 55, 101–139 (1952). [31] G.W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85, 134–165 (1957). [32] W. Rudin, Functional Analysis. McGraw-Hill, New York, 1973. [33] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932). [34] K.B. Wolf, Wigner distribution function for paraxial polychromatic optics, Opt. Comm. 132, 343–352 (1996). S. Twareque Ali, Anna E. Krasowska Department of Mathematics and Statistics Concordia University, 7141 Sherbrooke Street West Montr´eal, Quebec, Canada H4B 1R6 email: [email protected] email: [email protected] Hartmut F¨ uhr Institut f¨ ur Biomathematik und Biometrie GSF D-85764 Neuherberg, Deutschland email: [email protected] Communicated by Gian Michele Graf submitted 05/06/01, accepted 19/09/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 1051 – 1082 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061051-32 DOI 10.1007/s00023-003-0155-3

Annales Henri Poincar´ e

Algebraic Quantum Theory of the Josephson Microwave Radiator Thomas Gerisch, Reinhard Honegger and Alfred Rieckers Abstract. A closed, entirely quantum mechanical Josephson oscillator model is considered, consisting of two superconductors in tunneling contact, which weakly interact with the photon ﬁeld. For each superconductor we use, for notational simplicity, the BCS model in the strong coupling approximation and restrict ourselves to Anderson’s quasi-spin formulation. The thermodynamic limit of the global (nonequilibrium) dynamics is formulated for a large variety of states. It arises a generalization of previously developed cocycle equations, connecting the collective behaviour of the two superconductors with the photon ﬁeld dynamics. In the physically most interesting situations, where the two superconductors are in thermal equilibrium (below the critical temperature) at voltage diﬀerence V , we show, that for arbitrary initial states the outgoing multi-photon states are quantum optically all-order coherent and constitute an almost monochromatic radiation of frequency 2eV /. Furthermore, we deduce in detail, how the collective behaviour of the superconductors inﬂuences the collective (that are the optical) features of the emitted microwave photons.

1 Introduction Starting from a microscopic quantum description and performing the thermodynamic limit, one has obtained in previous works many aspects of the usual phenomenological descriptions of the Josephson eﬀect and the Josephson oscillator [1], [2], [3], [4]. The thermodynamic limit, by letting the electron number N tend to inﬁnity with ﬁxed particle density, is performed at various stages of mathematical rigor and generality. In a series of theoretical treatments of the BCS model and the Josephson eﬀect, the thermodynamic limit has been perfected within the formalism of inﬁnite mean-ﬁeld quantum lattice systems, starting from usual physical models in the ﬁnite lattice regions [5], [6], [7], [1], [2], [8], [9], [10], [11], [12], [13]. The Josephson oscillator model in terms of a closed system in [4], which is the paradigm for our present investigation, had not yet been formulated by means of a general, algebraic mean-ﬁeld setup. The state-dependent collective phenomena of the superconducting material are there approached by a special (resolvent) limit for selected observables, which is to picture the ground state situation only. This technique is to be considered as an improvement of Davies’ treatment of Dicke models [14], to which it refers, and in which the collective dynamics had not at all been microscopically founded. In our present investigation we aim to extend the theoretical insights into the model for a Josephson oscillator, where the (condensed) Cooper pair tunneling of the two superconductors is coupled to

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the quantized electromagnetic ﬁeld, by applying global algebraic techniques, which cover in principle all kinds of states, the participating systems may assume. The general non-equilibrium dynamics for an extended class of weakly interacting mean-ﬁeld lattice systems, coupled to the boson resp. photon ﬁeld, has been constructed in [15]. The treated class covers arbitrary powers of the collective mean-ﬁeld operators but only linear expressions of the boson annihilation and creation operators. The interacting dynamics provides in the thermodynamic limit cocycle equations, which connect the classical dynamical behaviour of the mean-ﬁeld system with the quantum mechanical (quasi-free) boson resp. photon ﬁeld dynamics. For this intertwined dynamics the outgoing states, restricted to the boson ﬁeld, are evaluated in [16] for the inﬁnite time limit (t → ∞). Due to the inﬂuence of the interaction one partially rediscovers within the radiation states the collective ordering of the inﬁnite quantum lattice system in terms of a coherent classical part (cf. also [17], [18]). So far, the results of [15] and [16] have only been applied to the weakly coupled inﬁnite Dicke model in [19], resp. anticipated in [20], [21], [22]. The present investigation of the Josephson oscillator, with its physically most interesting collective structure, constitutes a new application of the general formalism, in which the treatment of the cocycle dynamics can be carried through as far to get reﬁned formulas for experimentally conﬁrmed laws. We use for each superconductor the homogeneous BCS model [23], [24], [25] in the strong coupling approximation [26]. As in Anderson’s quasi-spin formulation we restrict the observables to the algebra A, generated by the Cooper pairs [26], [27], [28], [5], [29]. The BCS model is introduced in Section 2, where we employ techniques of the algebraic mean-ﬁeld theory (cf. [30], [31], [32], [33], [34], [35], [36], and especially [37] for details of the homogeneous BCS model). As observable algebra for the quantized electromagnetic ﬁeld we take in Section 3 the C*-Weyl algebra W(E) over an appropriate one-photon testfunction space E, which includes all modes of the ﬁeld and ﬁts to the quantization in the total position space R3 . The one-photon Hamiltonian is just the square root of the negative Laplacian (setting = 1), which gives rise to the photon dynamics in terms of the associated group of Bogoliubov transformations on W(E). Thus, for the combined system we arrive at the C*-algebra C := Al ⊗ Ar ⊗ W(E) as observable algebra for the Josephson oscillator, where Al and Ar are copies of the inﬁnite product algebra A for the left and right superconductor. In Subsection 4.2 we formulate the cocycle methods for the weak coupling between the two superconductors and the photon ﬁeld of the Josephson radiator, which leads to a non-equilibrium dynamical description on a very large folium of states F of the oscillator C*-algebra C. The photon states of the Josephson oscillator, which are obtained after a suﬃciently long time, are investigated in Subsection 4.3. These time-asymptotic photon states represent the emitted radiation in dependence of the initial states of the whole system. From the shape of the time-asymptotic photon states one recognizes how the classical, macroscopic ordering of the two superconductors inﬂuences the photon ﬁeld.

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In Section 5 we restrict ourselves to the physically most interesting situations: The left and the right superconductors are prepared into thermal equilibrium below the critical temperature. The superconductors are brought then into weak tunneling contact and the photon ﬁeld starts with few excitations. The superconductors are submitted to a voltage diﬀerence V = u/e, which is expressed by the diﬀerence of their chemical potentials u = µr − µl in the corresponding grand canonical ensembles. Mathematically, this is expressed by specifying the initial states of the system from the subfolium Feq,F of F on C. This folium is physically characterized by the equilibrium properties of the superconductors and by very low photon intensities, i.e., it consists of the grand canonical sectors of the BCS systems and of the vacuum (Fock) sector of the photon ﬁeld. We obtain the result, that for an arbitrary initial state in the equilibriumplus-Fock sector Feq,F , the emitted multi-photon states are to a good approximation all-order coherent and correspond to a nearly monochromatic radiation of frequency 2|u|, the Josephson frequency. The optical coherence properties of the outgoing photon states are discussed in terms of their operator algebraic version [38], [39], [40], [41], [42], [43], which is a mathematical reﬁnement of Glauber’s original deﬁnition [44], [45], [46] and an extension to the frequent non-Fock representations in quantum optical models. Moreover, the relationship between the collective phase diﬀerence between the two superconductors with the collective, macroscopic phase of the outgoing multi-photon states, that is what is called ‘the optical phase of the microwave radiation’, is explicitly evaluated. In the conclusions of Section 6 we discuss again the physical and fundamental aspects of the fully quantum mechanical radiation model, which could be realized by the use of operator algebraic notions. A possible generalization of the present Josephson oscillator model could have been achieved, without principal diﬃculties, in terms of inhomogeneous BCS or bipolaronic Hubbard models coupled to the photon ﬁeld. For this, the techniques from [10], [47], [48], [49], [50], [51], where the homogeneous mean-ﬁeld theory is perturbed with certain classes of momentum dependent inhomogeneities, would apply. The global operator algebraic formalism We start our exposition with a brief introduction to the operator algebraic description of a many-particle quantum system in terms of folia of states and representations. Let A be an arbitrary C*-algebra with state space S(A) describing a physical system. Denote by Mu ≡ Mu (A) the universal enveloping W*-algebra of A associated with the universal representation Πu , and its center by Z u [52], [53], [54]. We do not distinguish between a state on A and its unique normal extension to Mu . The notion of a folium was introduced by Haag et al. [55] (cf. also [56]). A folium F of A is a norm-closed, convex subset of S(A), which is invariant under

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; B . B “perturbations” in the sense that ω ∈ F implies ωB ∈ F, with ωB ; . = ω ω ; B ∗ B , for all B ∈ A with ω ; B ∗ B = 0. Folia arise naturally as the set of the Π-normal states of a representation Π of A. More exactly, we have the following well-known results (see, e.g., [54], [55]).

Proposition 1.1 There is a one-to-one correspondence between the folia F of A, the quasiequivalence classes of representations Π, and the orthogonal projections c in the center Z u , which preserves the partial orderings, respectively. That is, if for j ∈ {1, 2} Fj , Πj , and cj ∈ Z u are in correspondence, then F1 ⊆ F2 ⊆ S(A)

⇐⇒

Π1 ≤ Π2 ≤ Πu

⇐⇒

c1 ≤ c2 ≤ 1 u .

Moreover, if F , ΠF , and cF are in correspondence one has MF := ΠF (A) = cF Mu , ZF := MF ∩ MF = cF Z u (i.e., cF agrees with the unit 1F in MF ), and ω ; cF = 1 for all ω ∈ F. F consists just of the ΠF -normal states, that is, MF = F of Mu with Π F (Mu ) = MF , LH(F )∗ . ΠF extends to a normal representation Π u ΠF (Z ) = ZF . A folium F expresses classical, macroscopic aspects of the physical system, which one also reﬁnds in the center ZF of the associated W*-algebra MF . Thus, dealing with a subfolium of F instead of F itself, means a restriction of the considered collective, macroscopic aspects of the physical system. Disjoint folia give macroscopically totally diﬀerent aspects and deﬁne diﬀerent superselection sectors, which also is demonstrated by the orthogonality of the associated central projectors. A dynamical description in the Schr¨ odinger picture of our physical system is given as the tripel (A, F , ν), where F is a folium of the C*-algebra A, and ν ≡ {νt | t ∈ R} is a one-parameter group of aﬃne bijections νt on F deﬁning the time evolutions νt (ω) of each state ω ∈ F, and such that we have the timecontinuity of the expectation mappings R t → νt (ω) ; A for every observable A ∈ A and each state ω ∈ F [57], [15]. This dynamical concept generalizes the one of Kadison [58] and the concepts of C*- resp. W*-dynamical systems (e.g., [59, Deﬁnition 2.7.1]). The dynamics in the Heisenberg picture is obtained by the duality on MF , νt (ω) ; A = ω ; τt (A), for all ω ∈ F and all A ∈ MF , where each τt is assumed to be an automorphism on MF . If F ⊆ F is a ν-invariant subfolium, that is νt (F ) ⊆ F for all t ∈ R, then we arrive at the dynamical subdescription (A, F , ν) of (A, F , ν). For more details we refer to [15, Section 1].

2 The BCS superconductor model In this section we introduce the BCS model and sketch its dynamics as given above in the introduction. Finally, we summarize the limiting Gibbs states.

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The local BCS Hamiltonians

The local Hamiltonian of the BCS model is given for a ﬁnite number of momenta in a region Λ with energy near the Fermi energy by [26], [27], [28] 1 εk (c∗k↑ ck↑ + c∗−k↓ c−k↓ ) − gk1k2 c∗k ↑ c∗−k ↓ c−k2 ↓ ck2 ↑ . 1 1 |Λ| k∈Λ

k1 , k2 ∈Λ

The operators ckσ and c∗kσ are annihilation resp. creation operators of Bloch states with momentum k and spin σ. Due to the strong coupling approximation we replace εk and gk1k2 by their averages ε and g > 0. In the quasi-spin approximation the annihilation and creation operators of a Cooper pair, c−k1 ↓ ck1 ↑ resp. c∗k ↑ c∗−k ↓ , 1 1 are replaced by the pair annihilation and creation operators, bk resp. b∗k , which are characterized by the following relations [6]: b∗2 = b2k = 0 , k

{b∗k ; bk } = 1 ,

(∗) (∗) [b ; b ] = 0, for k = k , k

k

(2.1)

[b∗k ; bk ] = 2 nk − 1 ,

where nk := b∗k bk is the number operator of the Cooper pair with momentum k. Let us introduce a numbering of all admissible momenta and denote the vector k by the corresponding natural number k ∈ N. The set of all local regions then is deﬁned by L := {Λ ⊂ N | |Λ| < ∞} with the number |Λ| of elements in Λ. (∗) The local C*-algebras AΛ are generated by the pair operators bk , k ∈ Λ. With the usual set inclusion we construct the C*-inductive limit algebra A of the lattice system [60]. It is well known that A is realized as the inﬁnite tensor product M2 (2.2) A = k∈N

of the complex 2 × 2-matrices M2 . Let be k−1 ∞ ak := 12 ⊗ a ⊗ 12 ∈ A j=1

j=k+1

the embedding of a ∈ M2 at the kth lattice point. Then the Cooper pair annihilation operator bk for the lattice point k ∈ N is the embedding of b := 12 (σ1 − iσ2 ), and the Cooper pair number operator nk = b∗k bk is the embedding of n := b∗ b = 1 1, 2, 3, denote the usual Pauli spin 2 (σ3 + 12 ) at the kth lattice site (the σj , j = M2 is embedded into A by admatrices). For Λ ∈ L the local algebra AΛ = k∈Λ AΛ joining the unit 12 of M2 at each lattice point k ∈ Λ. In this sense A0 := Λ∈L

forms a norm dense subset of A. In the quasi-spin approximation, the local Hamiltonian HΛ , Λ ∈ L, writes up to an additive constant as g ∗ HΛ = 2 ε nk − bk bk . (2.3) |Λ| k∈Λ

k,k ∈Λ

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The HΛ , Λ ∈ L, form a net of local Hamiltonians deﬁning the physical dynamics. The reduced Hamiltonians HΛµ , Λ ∈ L, which are relevant for thermodynamic calculations in the grand canonical ensemble are obtained by replacing ε in equation (2.3) by εµ := ε − µ. The physical and the reduced local Hamiltonians, HΛ resp. HΛµ , are connected by the local number operator nk , (2.4) NΛ = 2 k∈Λ

which counts the electrons in the local lattice region Λ ∈ L, times the chemical potential µ ∈ R HΛµ

2.2

=

HΛ − µ N Λ ,

Λ ∈ L.

(2.5)

The limiting dynamics

2.2.1 The kinematical structure For the physical states we use the folium F p of A generated by the homogeneous states S p (A) of A (these states are invariant under permutations of lattice sites). F p is the smallest folium containing S p (A). Moreover, let Πp be the representation of A uniquely associated with the folium F p and the corresponding von Neumann algebra Mp := Πp (A) (see Proposition 1.1). Observe, that the algebra M2 of a single Cooper pair is generated by the unit 12 and the Lie algebra G = {a ∈ M2 | a = a∗ , tr(a) = 0} corresponding to the Lie group SU(2). The limit 1 ∈ Zp m(a) := s-lim Πp |Λ| k∈Λ ak Λ∈L

of the average of a ∈ M2 over the lattice exists in the strong operator topology and is an element of the center Z p of Mp . Let us denote by Ncl the sub-C*-algebra of the center Z p , which is generated by the set {m(a) | a ∈ M2 }, respectively by {m(a) | a ∈ G}. Ncl is a commutative C*-algebra, which we identify with the C*algebra C(P ) of continuous functions on the state space P ≡ S(M2 ) of the matrix algebra M2 (the density operators on C2 ). We also regard P as a compact, convex subset of the vector space dual G ∗ of the Lie algebra G, which is a real vector space of dimension 3. The duality relation x ; a for x ∈ G ∗ and a ∈ G ⊂ M2 is a restriction of the duality relation on M2 .

G a → exp i m(a) ∈ Z p is a unitary representation of the additive group G. Thus by the SNAG theorem there exists a unique projection-valued measure P ∗ from the Borel subsets of G into the set of orthogonal projections of the center p Z such that m(a) = G ∗ x ; adP(x) for each a ∈ G, respectively for each a ∈ M2 . It follows that P is the support of P. Now the spectral calculus deﬁnes the desired isomorphism

P : C(P ) −→ Ncl , η −→ η(x) dP(x) . P

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Ncl ∼ = C(P ) is interpreted as the C*-algebra of classical observables for the superconductor in the thermodynamic limit. 2.2.2 The limiting dynamics The (physical) local time evolutions are deﬁned as the group of automorphisms on A, given by

(2.6) αΛ t (A) := exp itHΛ A exp −itHΛ , for A ∈ A, t ∈ R, and Λ ∈ L. The Schr¨ odinger picture For each t ∈ R there exists an aﬃne bijection νt on the folium F p such that νt (ω) ; A = lim ω ; Πp (αΛ t (A)) , Λ∈L

∀A ∈ A ,

∀ω ∈ F p .

The group ν ≡ {νt | t ∈ R} induces the dynamical descriptions (A, F p , ν) in the sense of the introduction [30]. The Heisenberg picture By duality we get the dynamics (Mp , α) in the Heisenberg picture, which is not a W*-dynamical system in the sense of [59, Deﬁnition 2.7.1] (cf., e.g., [15]). The group α ≡ {αt | t ∈ R} of automorphisms on Mp does not leave the represented algebra Πp (A) invariant. Nevertheless, α leaves invariant the sub-C*-algebra of Mp , which is generated by Πp (A) and Ncl and which is isomorphic to the tensor product A ⊗ C(P ). Moreover, α lifts to a C*-dynamical system (A ⊗ C(P ), α) [30], [31]. 2.2.3 The classical (sub-)dynamics The classical phase space manifold The Pauli matrices {σ1 , σ2 , σ2 } deﬁne an orthogonal base of the Lie algebra G (with respect to the Hilbert-Schmidt scalar product), which allows to identify P ≡ S(M2 ) with the ball {x = (x1 , x2 , x3 ) ∈ R3 | x ≤ 1/2} by the well-known aﬃne homeomorphism x ∈ P −→ x ; 12 σ1 , x ; 12 σ2 , x ; 12 σ3 =: (x1 , x2 , x3 ) . (2.7) 1 The state space of the commutative C*-algebra C(P ) is the convex set M+ (P ) p of probability measures on P . Thus the restriction ω|Ncl of ω ∈ S (A) from Mp to Ncl gives the probability measure

−1 ω|Ncl ◦ P

1 =: ρω ∈ M+ (P ) ,

(2.8)

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1 which implies the aﬃne bijection ω → ρω from S p (A) onto M+ (P ). It holds p ρω (.) = ω ; P(.) for all ω ∈ S (A) with the above SNAG measure P on P . The point measure at x ∈ P agrees with the product state ωx := x, x∈P, (2.9) k∈N

on A. Consequently, the permutation invariant states S p (A) of A form a Bauer simplex with the extreme boundary ∂e S p (A) = {ωx | x ∈ P }. The probability mea 1 (P ) gives the central decomposition of ω ∈ S p (A), ω = P ωx dρω (x), sure ρω ∈ M+ [61], [33]. P is the classical phase space manifold of the BCS superconductor. The associated Poisson bracket is constructed in the usual manner by means of the Lie product on G (i times the commutator [., .]). However, since the underlying pre-symplectic geometry is degenerate, the Poisson bracket here is not quite of the type commonly dealed with in classical Hamiltonian mechanics [62] (for more details see [31], [63], [64], [15]). The classical Hamiltonian phase space flow Using equation (2.6), one ﬁnds that the strong limit of the local Hamiltonian densities exists in the representation Πp and is given by 1

P(q) := s-lim Πp |Λ| HΛ = 2 ε m(n) − g m(b∗ ) m(b) ∈ Ncl . Λ∈L

Here q is the polynomial in C(P ), which is given by q(x1 , x2 , x3 ) =

2 ε x3 − g (x21 + x22 ) + ε ,

if one uses the parameterization from equation (2.7). With the Poisson bracket and the polynomial q, considered as Hamiltonian function on C(P ), the Hamiltonian vector ﬁeld λq : P → G ∗ is introduced the standard way. Then the associated Hamiltonian phase space ﬂow ϕ ≡ {ϕt | t ∈ R} on the phase space manifold P is d ϕt (x) = λq (ϕt (x)) for all x ∈ P . With obtained from the diﬀerential equation dt the parameterization (2.7) of the classical phase space P the ﬂow ϕ expresses as      x1 x1 cos(2(ε + gx3 )t) − sin(2(ε + gx3 )t) 0 ϕt x2  =  sin(2(ε + gx3 )t) cos(2(ε + gx3 )t) 0 x2  (2.10) 0 0 1 x3 x3 for all t ∈ R and all x = (x1 , x2 , x3 ) ∈ P , which is just the rotation around the x3 -axis with x3 -dependent angle velocity 2(ε + gx3 ). The Schr¨ odinger picture The dynamical group ν leaves ∂e S p (A) and S p (A) invariant. More exactly, for the product states (2.9) we ﬁnd νt (ωx ) =

ωϕt (x) ,

∀x ∈ P ,

∀t ∈ R ,

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with the above classical phase space ﬂow ϕ. Then the time evolution νt (ω) for ω ∈ S p (A) is given in terms of the associated probability measures, equation (2.8), ρνt (ω)

=

ρω ◦ ϕ−t =: ϕ∗∗ t (ρω ) ,

ω ∈ S p (A) .

(2.11)

1 (P ), ϕ∗∗ ) to be the dynamical descripThis implies (Ncl , S p (A), ν) ∼ = (C(P ), M+ tion of the classical part of the BCS superconductor system in the Schr¨odinger picture.

The Heisenberg picture We have αt (Ncl ) = Ncl for all t ∈ R, and via the isomorphism P the restriction

◦ ϕ∗t ◦ P

−1 , where ϕ∗t (η) := η ◦ ϕt for η ∈ C(P ). Thus the is given by αt |Ncl = P C*-dynamical system (Ncl , α) is isomorphic to (C(P ), ϕ∗ ), the latter is just the familiar form of the algebraic dynamics in classical mechanics.

2.3

The limiting Gibbs states in the grand canonical ensemble

The equilibrium state on A for the grand canonical ensemble of the ﬁnite system with local Hamiltonian HΛ is given uniquely by

µ tr ; exp −βHΛµ A βHΛ

, ω ; A = ∀A ∈ A . tr ; exp −βHΛµ Here, tr is the trace state on A and β > 0 is the inverse temperature. The chemical potential µ has to be chosen such that (in the thermodynamic limit) a certain particle density is ﬁxed. In the thermodynamic limit Λ → N for ﬁxed µ and β these local states converge in the weak*-topology to the so-called limiting Gibbs state ω β,µ , which is an element of S p (A) [65], [35], [36], [5], [10], [11]. More correctly, a chemical potential µΛ has to be chosen for each local region Λ ∈ L, depending on the given particle density. In [10], [11] it is shown that the µΛ converge towards µΛ µ µ and that weak*-limΛ→N ω βHΛ = weak*-limΛ→N ω βHΛ . −1 Below the critical temperature βc the gauge symmetry of the limiting Gibbs state is broken and the system is in the superconducting state, that is for the 1 artanh( 2(ε−µ) ). The central decomposition of inverse temperature β > βc = ε−µ g the gauge invariant limiting Gibbs state ω β,µ is given with the product states from (2.9) by [9] ω β,µ = ωx dhβ,µ (x) , P

where hβ,µ is the Haar measure on the circle line P β,µ ⊂ P , where the latter is deﬁned with (2.7) by

2 2 2 x = (x1 , x2 , x3 ) ∈ P | x3 = µ−ε (2.12) P β,µ = g , x1 + x2 = ∆β,µ .

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Here ∆β,µ > 0 is the gap parameter of the superconductor obtained from the solution of 2E = tanh(βE) , with E = (ε − µ)2 + g 2 ∆2β,µ . (2.13) g For β ≤ βc (normal phase), the limiting Gibbs state is a factor state and the only solution of the gap equation (2.13) is ∆β,µ = 0. In the sequel we are interested in the superconducting phase, β > βc , only. Let us denote by F β,µ the smallest folium of A containing the limiting Gibbs state ω β,µ . It holds F β,µ ⊂ F p and νt (F β,µ ) = F β,µ for all t ∈ R, and thus (A, F β,µ , ν) is a dynamical subdescription of (A, F p , ν) in the sense of the introduction. Let be cβ,µ ∈ Z p the central projection, Πβ,µ ≤ Πp the representation, and Mβ,µ = Πβ,µ (A) = cβ,µ Mp the W*-algebra uniquely associated with the subfolium F β,µ according to Proposition 1.1. The folium F β,µ corresponds to the macroscopic parameter of the ﬁxed inverse temperature β and the ﬁxed chemical potential µ for our BCS superconductor system. Thus, considering F β,µ instead of F p , the relevant classical phase space manifold reduces to the support of the projection-valued measure cβ,µ P in the center Z β,µ = cβ,µ Z p of Mβ,µ , which is just the circle line P β,µ from equation (2.12). Obviously, P β,µ is invariant with respect to the classical ﬂow ϕ on P . The set of all measures ρω (.) := ω ; P(.), ω ∈ F β,µ , agrees with the set of all probability measures on P β,µ absolutely continuous to the Haar measure hβ,µ . For later purposes let us choose the parameterization for P β,µ in terms of the classical equilibrium phase parameter z

:=

x1 + ix2 . ∆β,µ

(2.14)

3 The photon field Originally, the quantization procedure in the Coulomb gauge for the whole position space R3 leads to a one-photon testfunction space, which is dense in the complex Hilbert space of square-integrable functions f : R3 → C3 , satisfying the transversality condition ∇ • f = 0 in the distributional sense (cf., e.g., [45], [66], [17], and [22, Section 5]), and including all directions of polarization. However, for the present investigation it suﬃces to consider only one direction of polarization. As in [22], the results obtained here for one direction extrapolate componentwise to all polarization directions. For the one-photon testfunction space in the position space R3 let us take a (complex) dense subspace E of the complex Hilbert space L2 (R3 ) of squaref : R3 → C with the canonical scalar product f | g = integrable functions 3 R3 f (y)g(y)d y. Then the C*-algebra of the photon system is taken as the Weyl algebra W(E) over E, which is generated by the unitary Weyl operators W (f ),

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f ∈ E, satisfying the Weyl relations (e.g., [59, Theorem 5.2.8])

W (f ) W (g) = exp − 2i Imf | g W (f + g) , W (f )∗ = W (−f ) , ∀f, g ∈ E . √ The one-photon Hamiltonian −∆ ≥ 0 acts selfadjoint on L2 (R3 ), where ∆ is the usual Laplacian on R3 . For introducing the free photon dynamics in the case √ it −∆ where E is not e -invariant, we may follow [67], [15], [16], and [19]. However, here we choose for simplicity a testfunction space E which is invariant with respect √ to the one-photon time evolutions eit −∆ . Let us denote by F the (unitary) Fourier transformation on L2 (R3 ), −3/2

(Ff )(k) = f (k) = (2π) e−ik•y f (y) d3 y , ∀k ∈ R3 . R3

and by C n for n ∈ N ∪ {∞} the n-times continuously diﬀerentiable functions.

√ The Fourier transform Fexp it −∆ F−1 isthe multiplication operator with the continuous function R3 k → exp itk , which is C ∞ except at the origin k = 0. Thus, in momentum space the one-photon time evolutions leave invariant ∞ (R3 ) of all complex-valued C ∞ -functions on R3 with compact support the space Cc,0 and vanishing in a neighborhood of the origin. This leads to our choice of the testfunction space E

:=

∞ F−1 (Cc,0 (R3 )) .

By means of the one-photon evolutions we construct the group β ph ≡ {βtph | t ∈ R} of Bogoliubov automorphisms on the Weyl algebra W(E), satisfying βtph (W (f ))

√

= W (eit

−∆

f) ,

∀f ∈ E ,

∀t ∈ R .

(3.1)

By duality we obtain the group ν ph ≡ {νtph | t ∈ R} of aﬃne bijections νtph on the state space S(W(E)). Since W (f ) − W (g) = 2 for all f = g, the group β ph does not deﬁne a C*-dynamical system on W(E), and thus R t → νtph (ω) ; Y = ω ; βtph (Y ) is non-continuous for some states ω ∈ S(W(E)) and some observables Y ∈ W(E). Because of the lack of the time-continuity, the group ν ph on S(W(E)) is not appropriate for the description of the free photon dynamics according to the introduction. The time-continuity of the expectations t → νtph (ω) ; Y , Y ∈ W(E), is only possible for states ω from suitable folia of W(E). But before we proceed with the formulation of the free photon dynamics, we need an enlargement of the testfunction space E by the following reason. The C 2 -functions φj : R3 → C, j = 1, 2, in the local and the limiting interaction operators of the Josephson oscillator (see the equations (4.3) and (4.5) below) in general are no elements of the relevant testfunction space E. However, the coupling modes φj are contained in an extended testfunction space Ee with E ⊆ Ee ⊆ L2 (R3 ). Following [67], [15], we link E and Ee with a locally convex Hausdorﬀ topology τ on Ee , which is

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stronger than (or equal to) the scalar product topology, so that E is τ -dense in Ee and the one-photon dynamics is τ -continuous. According to [67], [15], the time-continuous free photon dynamics now is τ of the τ -continuous states on W(E), immediately constructed on the folium Fph τ Fph

= {ω ∈ S(W(E)) | Cω is τ -continuous} ,

where Cω : E → C, f → ω ; W (f ) is the characteristic function of the state ω ∈ τ )= S(W(E)). Then the τ -continuity of the one-photon dynamics yields νtph (Fph τ τ . Fph for all t ∈ R, and the time-continuity of the expectations for all states ω ∈ Fph τ ph Consequently, the tripel (W(E), Fph , ν ) constitutes a dynamical description of the photon ﬁeld, the free photon dynamics in the Schr¨ odinger picture. The topology τ is stronger (or equal) than the scalar product topology. Thus the vacuum state ωF , which is given by the characteristic function

∀f ∈ E , CF (f ) = ωF ; W (f ) = exp − 14 f 2 , τ is an element of Fph . The smallest folium containing ωF is denoted by FF . Clearly τ FF ⊂ Fph , and FF consists just of those states on W(E) which are normal to the Fock representation. Thus in the Fock folium FF there are contained those states on W(E), which have a ﬁnite absolute mean number of photons, describing few photon excitations. Furthermore, (W(E), FF , ν ph ) is a dynamical subdescription τ of (W(E), Fph , ν ph ). Since τ is a vector space topology, the representation Πτph , uniquely associated τ with the folium Fph by Proposition 1.1, is regular (that is, for each f ∈ E the map τ R t → Πph (W (tf )) is a strongly continuous unitary one-parameter group). Thus for each testfunction f ∈ E the ﬁeld observable Φτ (f ) arises from Stone’s theorem d Φτ (f ) := −i Πτph (W (tf )) , dt t=0

which is a selfadjoint operator aﬃliated to the W*-algebra Mτph = Πτph (W(E)) corresponding to the representation class Πτph . The creation and annihilation operators, a∗τ (f ) resp. aτ (f ), are constructed as the well-deﬁned closed operators aτ (f ) := √12 Φτ (f ) + iΦτ (if ) , a∗τ (f ) := √12 Φτ (f ) − iΦτ (if ) , (3.2) which for each f ∈ E are adjoint to each other [59]. Note that Φτ (f ), and a∗τ (f ) resp. aτ (f ), as well as the represented Weyl operators Wτ (f ) ≡ Πτph (W (f )) extend as operator-valued distributions to the enlarged testfunction space Ee [67], [15]. τ Let us turn to the ﬁeld expectations. For an (entire) analytic state ω ∈ Fph , e.g., the vacuum ωF , there exist the n-point ﬁeld correlation functions n ∂n n ω ; W ( tk fk ) ω ; Φτ (f1 ) · · · Φτ (fn ) := (−i) ∂t1 · · · ∂tn k=1 t1 =...=tn =0 =

Ωω | Φτ (f1 ) · · · Φτ (fn )Ωω ,

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where n ∈ N resp. the fk ∈ E are arbitrary, and the tk are real diﬀerentiation parameters. The second equality is valid, because the GNS representation Πω is a sub-representation of Πτph , and Ωω is the associated cyclic vector. Forming suitable linear combinations of the ﬁeld n-point functions and taking into account equation (3.2) leads to the normally ordered ﬁeld correlations ω ; a∗τ (f1 ) · · · a∗τ (fm ) aτ (g1 ) · · · aτ (gn ) ,

∀fk , gl ∈ E ,

(3.3)

which we evaluate in the large time limit for certain states in Subsection 5.2.

4 The Josephson oscillator 4.1

The basic observable algebra

The Josephson oscillator consists of two superconductors which are coupled to the quantized electromagnetic ﬁeld. Each of the two superconductors is described by the BCS model of Section 2. The quantities from Section 2 for the superconductor on the left are indexed by l, and for the right by r. We set εl = εr =: ε > 0 and gl = gr =: g > 0 for the averaged Bloch energies and the Cooper pair coupling in the local Hamiltonians (2.3). We form, as the basic observable algebra for the total system, the C*-algebra C

:=

Al ⊗ Ar ⊗ W(E),

(4.1)

where Al and Ar are the left and the right copy of the quasilocal C*-algebra A from equation (2.2). Let us mention, that the subscripts ‘left’ and ‘right’ for the electron observables could have been introduced in terms of a thermodynamic limit for the electron algebra along two inﬁnite, separated space regions. Quite generally, more reﬁned types of a thermodynamic limit are fundamental and commonly accepted in non-equilibrium thermodynamics and hydrodynamics. In the present context, a split thermodynamic limit would elucidate the nature of the Josephson contact as a composite, quantum mechanical macro-object. As will be illustrated by the subsequent discussions, the qualiﬁcations ‘left’ and ‘right’ are connected with central observables, that are classical collective variables, which contradict in no way the concepts of quantum theory. The statement, that an electron creation operator is ‘left’ or ‘right’, is to be compared with the statement, that it belongs to a temperature representation. In both cases the classical parameter belongs to the assemblage of electrons and does not interfer with the indistinguishability of the single microscopic particles (cf. also [68]). The combination of micro- and macro-structures is what makes the following discussion much more involved than a treatment of a homogeneous system of quantum particles on the one side, or a purely macroscopic-classical model on the other side.

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The dynamics of the Josephson oscillator

4.2.1 The local dynamics Using the equations (2.6) and (3.1) we have the local free time evolutions of the combined system in the Heisenberg picture as γtΛ

Λr l αΛ ⊗ βtph t ⊗ αt

=

(4.2)

for the local regions Λ = (Λl , Λr ) ∈ L × L of the left and the right BCS superconductors. Neglecting terms which are quadratic in the photon ﬁeld, we arrive at the following local interaction operators QΛ , Λ = (Λl , Λr ), for our Josephson oscillator model λ1 ∗ bkl ⊗ bkr ⊗ a∗τ (φ1 ) + bkl ⊗ b∗kr ⊗ aτ (φ1 ) QΛ = |Λl ||Λr | kl ∈Λl kr ∈Λr kl ∈Λl kr ∈Λr λ2 ∗ ∗ + bkl ⊗ bkr ⊗ aτ (φ2 ) + b∗kl ⊗ bkr ⊗ aτ (φ2 ) , |Λl ||Λr | kl ∈Λl kr ∈Λr

kl ∈Λl kr ∈Λr

(4.3) (∗)

with the pair operators bk from equation (2.1). As mentioned in Section 3 the functions φj : R3 → C, j = 1, 2, are assumed to be C 2 and to be elements of the extended testfunction space Ee . They have to be calculated by physical arguments for the speciﬁc Josephson contact. For the present purposes let the functions φj be of unspeciﬁed nature. 4.2.2 The limiting dynamics Obviously, the limit of the free local dynamics from equation (4.2) is γt0 := αlt ⊗ αrt ⊗ βtph on the W*-tensor product algebra M = Mpl ⊗ Mpl ⊗ Mτph , i.e., in the representation Π := Πpl ⊗ Πpr ⊗ Πτph of the composite C*-algebra C from equation (4.1). In the Schr¨ odinger picture we obtain νt0 := νtl ⊗ νtr ⊗ νtph on the folium F

τ := Flp ⊗ Frp ⊗ Fph

(4.4)

of C, which leads to the dynamical description (C, F , ν 0 ). The above tensor product of folia is a symbolic notation, it means the normal states on the W*-tensor product M of the W*-algebras corresponding to the folia. Let us turn to the interacting system. The limit Λl,r → N of the local interactions QΛ from (4.3) exists in the representation Π := Πpl ⊗ Πpr ⊗ Πτph and deﬁnes the limiting interaction operator Q

=

λ1 [m(b∗ ) ⊗ m(b) ⊗ a∗τ (φ1 ) + m(b) ⊗ m(b∗ ) ⊗ aτ (φ1 )] ∗

+ λ2 [m(b) ⊗ m(b ) ⊗

a∗τ (φ2 )

∗

+ m(b ) ⊗ m(b) ⊗ aτ (φ2 )] .

(4.5)

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Applying the cocycle techniques from [15] (in direct generalization from a single mean-ﬁeld lattice to the doubled BCS system here; cf. the introduction), we construct explicitly the interacting dynamics of the Josephson oscillator, where the free dynamics consists of the above group γ 0 of automorphisms on the W*-algebra M (Heisenberg picture), respectively of the dynamical description (C, F , ν 0 ) (Schr¨ odinger picture), and the interaction operator Q is taken from equation (4.5). The interacting dynamics γtQ on M is of the form γtQ (Z) = Γt γt0 (Z)Γ∗t ∀Z ∈ M, where the unitaries Γt ∈ M fulﬁll the cocycle relations Γt+s = Γt γt0 (Γs ). This resembles the form of the perturbed dynamics of a C*- resp. W*-dynamical system in the case of a bounded coupling operator (e.g., [59, Proposition 5.4.1]). Because of m(b) ∈ Ncl , the operator Q couples the classical part of the two BCS models to the photon ﬁeld. That is, only the Cooper pair tunneling between the two superconductors interacts with the quantized electromagnetic ﬁeld. Thus we make the ansatz Γt = V (ψt ) with suitable functions ψt : Pl × Pr → Ee . Here V (η) is the spectral integral [15] V (η) := 1pl ⊗1pr ⊗Wτ (η(xl , xr )) d(Pl (xl )⊗Pr (xr )⊗1τ ) ∈ Zlp ⊗Zrp ⊗Mτph , Pl ×Pr

where η : Pl × Pr → Ee is τ -continuous, and Pl resp. Pr are the left and right copy of the SNAG measure P from Subsection 2.2.1. the cocycle equations The cocycle relations V (ψt+s ) = V (ψt )γt0 (V (ψs )) yield √ for the functions ψt , that is ψs+t (xl , xr ) =ψs (xl , xr )+eis −∆ ψt (ϕls xl , ϕrs xr ), which l r ,x ) obey the initial value condition ∂ψt (x = φ(xl , xr ). These equations interre ∂t t=0 late the classical part of the superconductors and the one-photon time evolution. Here ϕl and ϕr are the left and the right copy of the classical phase space ﬂow ϕ from Subsection 2.2.3, and φ : Pl × Pr → Ee is the coupling function uniquely associated with Q, √ φ(xl , xr ) = (4.6) 2 λ1 ξ(xl , xr ) φ1 + λ2 ξ(xl , xr ) φ2 with the function ξ on the classical phase space Pl × Pr (see Subsection 2.2.3 and the parameterization (2.7)), ξ(xl , xr ) :=

(xl1 + ixl2 )(xr1 − ixr2 ) ,

xl ∈ Pl , xr ∈ Pr .

(4.7)

By [15] the searched solution ψt , t ∈ R, uniquely is given by t

√ ψt (xl , xr ) = exp is −∆ φ(ϕls xl , ϕrs xr ) ds , (xl , xr ) ∈ Pl × Pr . (4.8) s=0

Summarizing, the interacting dynamical group γ Q = {γtQ | t ∈ R} of automorphisms on M has the form γtQ (Z) = V (ψt ) γt0 (Z) V (ψt )∗ ,

∀Z ∈ M ,

∀t ∈ R ,

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where ψt is from (4.8). The (pre-) dual group ν Q = {νtQ | t ∈ R} of γ Q determines the dynamical description (C, F , ν Q ) in the Schr¨ odinger picture. It is shown in [17], [15] for suitable subrepresentations of Π, that γ Q is indeed the limiting dynamics of the local interacting dynamics γ Q,Λ from Subsection 4.2.1.

4.3

The time-asymptotic photon states

The restriction ω|ph of a state ω on the C*-algebra C, equation (4.1), to the photon ﬁeld algebra W(E) is given by ω|ph ; Y :=

ω ; 1l ⊗ 1r ⊗ Y ,

∀ Y ∈ W(E) .

We follow the strategy of [16] (generalizing from a single mean-ﬁeld lattice to the two BCS superconductors involved; cf. the introduction) for the investigation of the asymptotic behaviour of the restricted time evolution t ∈ R → νtQ (ω)|ph for t → +∞, using the dynamical description (C, F , ν Q ) from above. Time asymptotic states do not exist for arbitrary ω ∈ F, but for suitable subfolia of F . Let Fϕ be the smallest folium of W(E) containing an arbitrary τ photon state ϕ ∈ Fph , which fulﬁlls the asymptotic product property √

lim Cϕ (eit

−∆

t→∞

f + g) = Cϕ (f ) Cϕ (g) ,

∀ f, g ∈ E .

(4.9)

Especially, the Fock vacuum state ωF fulﬁlls (4.9). We select now the subsequent subfolium of the folium F on C deﬁned in (4.4), Fϕp

:=

Flp ⊗ Frp ⊗ Fϕ ⊂ F .

(4.10)

For ω ∈ F let us deﬁne by ρω (Bl , Br ) := ω ; Pl (Bl ) ⊗ Pr (Br ) ⊗ 1τ , with the Borel subsets Bl ⊆ Pl and Br ⊆ Pr of the left and right classical phase space manifolds, the probability measure ρω on Pl × Pr . Theorem 4.1 For each (xl , xr ) ∈ Pl × Pr let us consider the linear form L(xl ,xr ) : E → C given by 1 (k) f (k) δ † (k − 2g(xr − xl )) d3 k φ L(xl ,xr ) (f ) = λ1 ξ(xl , xr ) 3 3 3 R 2 (k) f (k) δ † (k + 2g(xr − xl )) d3 k , + λ2 ξ(xl , xr ) φ 3 3 R3

where the distribution δ † (k − κ) is deﬁned for arbitrary κ ∈ R by  h(k)   d3 k , if κ > 0 , h(k) dS(k) − pv  iπ 3 k − κ k=κ R h(k) δ † (k − κ) d3 k =  h(k) R3   d3 k , if κ ≤ 0 . − k − κ 3 R

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Here h is a continuously diﬀerentiable and integrable function h : R3 → C vanishing in a neighborhood of the origin, whereas dS(k) means the usual surface measure, and pv denotes the principal value with respect to the radial integral. Then the following assertions are valid: For each ω ∈ Fϕp the function

√ R(ω) ; W (f ) := Cϕ (f ) exp i 2 Re(L(xl ,xr ) (f )) dρω (xl , xr ) , ∀f ∈ E , Pl ×Pr

(4.11) deﬁnes in fact a state R(ω) on W(E). Moreover, with νtSC := νtl ⊗ νtr ⊗ 1ph – the free time evolution on F for the two BCS superconductors alone – we have the time-asymptotic relations in the weak*-topology on S(W(E)), SC weak*-lim νtQ (ν−t (ω))|ph t→∞

= R(ω) ,

∀ ω ∈ Fϕp .

(4.12)

Proof. We give here a sketch of the proof, for details we refer to [16]. With equation (2.10) one immediately checks that dynamics of the function ξ from (4.7) is given by

ξ(ϕlt xl , ϕrt xr ) = exp −it2g(xr3 − xl3 ) ξ(xl , xr ) , ∀(xl , xr ) ∈ Pl × Pr , (4.13) for all times t ∈ R, leading to an explicit expression for ψ−t (xl , xr√) | f , where ψ−t is taken from equation (4.8). Since the one-photon Hamiltonian −∆ has an absolutely continuous spectrum, it is known from the Riemann-Lebesgue lemma that √ ∀f, g ∈ L2 (R3 ) . lim f | eit −∆ g = 0 , t→∞

With a reﬁned treatment of the approximation of the δ † -distribution, δ † (y − κ) = t i lim κ=0 eiτ (y−κ) dκ for y ∈ R, one then may show that the limit L(xl ,xr ) (f ) = t→∞

i lim ψ−t (xl , xr ) | f exists uniformly in (xl , xr ) ∈ Pl × Pr for f ∈ E. t→∞

Now the limit (4.12) is ﬁrst performed for the characteristic functions of the states, involving one Weyl operator only, SC (ω)) ; 1l ⊗ 1r ⊗ W (f ) = R(ω) ; W (f ) , lim νtQ (ν−t

t→∞

∀f ∈ E ,

which then is extended by standard arguments to arbitrary elements of the Weyl algebra. Corollary 4.2 Applying the free dynamics ν SC to the initial state ω, we have instead of (4.12) the relation weak*-lim νtQ (ω)|ph − Rt (ω) = 0, ∀ ω ∈ Fϕp , (4.14) t→∞

where the states Rt (ω) := R(νtSC (ω)) diﬀer from R(ω) merely by the replacement of the measure ρω in (4.11) with its time evolved version ρtω := ρνtSC (ω)

= ρω ◦ (ϕl−t × ϕr−t ) ,

which is the analogue to equation (2.11).

ω∈F,

(4.15)

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Due to (4.14), when starting at time zero with the state ω ∈ Fϕp as initial state, the states Rt (ω) for large times t > 0 are considered as those photon states, which represent the emitted radiation for our Josephson oscillator after suﬃciently time having elapsed. In the sequel the states Rt (ω) are called the time-asymptotic photon states.

5 The Josephson oscillator in thermodynamic equilibrium The present section is devoted to the following situation: The left and the right superconductors are in thermal equilibrium at inverse temperature β > βc , exhibiting their homogeneous superconducting phase structure. They are brought together at t = 0, whereby Cooper pair tunneling starts and the Josephson radiation is forming. The voltage diﬀerence V = u/e is expressed by the diﬀerence of the chemical potentials [24] u

=

µr − µl

(5.1)

of the two superconductors. The initial electromagnetic ﬁeld is assumed to consist of some photons only and is thus an element of the purely quantum mechanical vacuum sector, the Fock folium FF . For the initial state of the superconductors we take weak perturbations of the grand canonical ensembles, the latter being characterized by the macroscopic parameters β, µl , and µr . That means, the initial states ω for the total system are elements of the folium FFβ,µl ,µr

:=

F β,µl ⊗ F β,µr ⊗ FF ⊂ FωpF ⊂ F .

(5.2)

Let us remark that the tripel (C, FFβ,µl ,µr , ν Q ) is a dynamical subdescription of (C, F , ν Q ).

5.1

The time-asymptotic photon states

Obviously, for every ω ∈ FFβ,µl ,µr the probability measures ρtω on Pl × Pr , which are deﬁned before Theorem 4.1 and in equation (4.15), are concentrated on the cartesian product P β,µl × P β,µr ,

(5.3)

where P β,µ is given in (2.12). Let us mention that the set (5.3) is invariant under the classical phase space ﬂow ϕlt × ϕrt . In the following we use equation (2.14) for the parameterization of the set P β,µl × P β,µr in terms of two phases (z l , z r ) ∈ T × T – the classical equilibrium phase parameters of the left and right superconductor –, where the one-dimensional torus group T = {z ∈ C | |z| = 1} represents the phases. It is seen from Theorem 4.1 and Corollary 4.2 that for the time-asymptotic photon states Rt (ω), where

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ω is taken from the (sub-) folium FFβ,µl ,µr of equation (5.2), there appears only the phase diﬀerences v(xl , xr ) := z l z r =

(xl1 + ixl2 )(xr1 − ixr2 ) ξ(xl , xr ) = , (xl , xr ) ∈ P β,µl × P β,µr ∆β,µl ∆β,µr ∆β,µl ∆β,µr

of the left and right BCS superconductor, where ξ is deﬁned in (4.7). Lemma 5.1 For each probability measure ρ on P β,µl × P β,µr there exists a unique probability measure ρ on T such that h(v(xl , xr )) dρ(xl , xr ) = h(v) d ρ(v) P β,µl ×P β,µr

T

for every continuous function h : T → C.

β,µl ,µr t =ρ and all t ∈ R. Moreover, it holds ρ ω ◦ exp i2ut for all ω ∈ FF ω Proof. Let us deﬁne the unitary operator U on the Hilbert space K := L2 (P β,µl × P β,µr , ρ) to be the multiplication operator (U g)(xl , xr ) = v(xl , xr ) g(xl , xr ) for ρ-almost all (xl , xr ) and every g ∈ K. The constant function w(xl , xr ) ≡ 1 is an element of K. With the spectral decomposition for the unitary, U = T v dE(v) with the Kprojection-valued measure E on T (cf., e.g., [69] or [70]), we deﬁne the probability measure ρ on T by setting d ρ(v) := w | dE(v) wK . Then for every continuous function h : T → C the spectral calculus implies h(v(xl , xr )) dρ(xl , xr ) = w | h(U ) wK = h(v) d ρ(v) . P β,µl ×P β,µr

T

Now (2.12) and (4.13) imply v(ϕlt xl , ϕrt xr ) = exp −i2ut v(xl , xr ) for the phase diﬀerences, where (xl , xr ) ∈ P β,µl × P β,µr . Together with (4.15) we ﬁnally arrive at the result. Now, as an immediate consequence of Theorem 4.1 and Corollary 4.2 we obtain Proposition 5.2 For each ω ∈ FFβ,µl ,µr the time-asymptotic photon states Rt (ω) from Subsection 4.3 are given by the characteristic functions Rt (ω) ; W (f ) (5.4)

√ t (v) , = CF (f ) exp i 2 ∆β,µl ∆β,µr (λ1 Re(v G1 (f )) + λ2 Re(v G2 (f ))) dρ ω T

where the linear forms Gj : E → C, j = 1, 2, are given via the (ﬁxed) potential diﬀerence u = µr − µl ∈ R from (5.1) by 1 (k) f (k) δ † (k − 2u) d3 k , G1 (f ) = (5.5) φ R3 2 (k) f (k) δ † (k + 2u) d3 k , G2 (f ) = (5.6) φ R3

where the distribution δ † (k − κ) for κ ∈ R is deﬁned in Theorem 4.1.

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According to Lemma 5.1 the time-dependence R t → Rt (ω) is a rotation of the

t (v) = d phase diﬀerence with velocity 2u, dρ ρω (exp i2ut v), which arises from ω the classical dynamical ﬂow ϕl × ϕr for the two superconductors. Obviously, the singular behaviour of the distributions δ † (k ± 2u) depends on those momenta k ∈ R3 which are in resonance with the doubled potential diﬀerence 2u. In the following let us denote by

k ∈ R3 | k = 2|u| , (5.7) Su := the resonant (momentum) modes of the photon ﬁeld for u ∈ R. Since the initial states ω for the whole system are taken in the present Section from the folium FFβ,µl ,µr on C (cf. equation (5.2)), it follows that for ﬁnite times t ∈ R the accessible photon states νtQ (ω)|ph are elements of the folium FF of W(E). The Fock folium FF , however, is norm-closed (cf. the introduction), which contrasts the approximation (4.14) in terms of the weak*-topology. Thus, the timeasymptotic photon states Rt (ω) may leave the Fock folium FF . Lemma 5.3 Let ω ∈ FFβ,µl ,µr and t ∈ R. Then Rt (ω) ∈ FF , if and only if both linear forms G1 : E → C and G2 : E → C are bounded with respect to the norm topology on E. Moreover, the following assertions are valid: (a) Let u > 0. Then the linear √ form G2 : E → C from equation (5.6) is bounded and given by G2 (f ) = −( −∆ + 2u)−1 φ2 | f for all f ∈ E. Furthermore, the following equivalences hold: √ 1 (k) = 0 ∀k ∈ Su ⇐⇒ φ1 ∈ D(( −∆ − 2u)−1 ) ⇐⇒ G1 is bounded , φ √ in which case G1 (f ) = −( −∆ − 2u)−1 φ1 | f for all f ∈ E. (b) Let u < 0. Then the linear √ form G1 : E → C from equation (5.5) is bounded and given by G1 (f ) = −( −∆ − 2u)−1 φ1 | f for all f ∈ E. Furthermore, the following equivalences hold: √ 2 (k) = 0 ∀k ∈ Su ⇐⇒ φ2 ∈ D(( −∆ + 2u)−1 ) ⇐⇒ G2 is bounded , φ √ in which case G2 (f ) = −( −∆ + 2u)−1 φ2 | f for all f ∈ E. (c) For u = 0 and for j = 1 resp. j = 2 the following equivalences hold: √ −1 φj (0) = 0 ⇐⇒ φj ∈ D( −∆ ) ⇐⇒ Gj is bounded , √ −1 in which case Gj (f ) = − −∆ φj | f for all f ∈ E. Consequently, the only possibility, that both linear forms G1 and G2 are unbounded, is the case u = 0.

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Proof. First assume G1 and G2 to be bounded. Then for j = 1, 2 there exists gj ∈ L2 (R3 ) with Gj (f ) = gj | f for all f ∈ E, and the state Rt (ω) extends to the larger Weyl algebra W(L2 (R3 )). Let K be the linear span of {g1 , g2 } and K ⊥ its orthogonal complement in L2 (R3 ). According to W(L2 (R3 )) ∼ = W(K)⊗W(K ⊥ ) our state Rt (ω) decomposes into the tensor product Rt (ω) = Rt (ω)|W(K) ⊗ ωF |W(K ⊥ ) with the Fock vacuum ωF . Now use the Stone-von Neumann uniqueness result [59, Corollary 5.2.15] to ensure that Rt (ω)|W(K) is Fock normal, which ﬁnally implies the Fock normality of Rt (ω) on W(E). Conversely, if G1 or G2 is unbounded, then the characteristic function of Rt (ω) is not norm-continuous on E, which is immediately checked with [38, Lem. ma 2.4]. Thus Rt (ω) is not an element of the folium Fph of norm-continuous states .

on W(E). But FF ⊂ Fph . The equivalences follow from [22, Proposition 4.2].

Our discussion has demonstrated, that the unboundedness resp. boundedness of j : Gj depends on the behaviour of the Fourier transformed interaction function φ R3 → C at the resonant modes k ∈ Su . By physical arguments one has in general j (k) = 0 for some resonant momenta k ∈ Su and for j = 1, 2, which we will φ always assume in the following.

5.2

Quantum optical coherence of the emitted radiation

The time-asymptotic states Rt (ω) on W(E) from equation (5.4) with the initial states ω from the equilibrium-Fock sector FFβ,µl ,µr are those photon states, which represent the characteristic features of the emitted radiation for large times t > 0 of the Josephson contact in the thermal equilibrium with potential diﬀerence (5.1). In experimental situations the normally ordered expectation values, that is Rt (ω) ; a∗τ (f ) aτ (g) from formula (3.3) are measured with localized testfunctions f, g ∈ E. Lemma 5.4 For each ω ∈ FFβ,µl ,µr and every t ∈ R the time-asymptotic photon state Rt (ω) is an entire-analytic state on W(E) with Rt (ω) ; a∗τ (f ) aτ (g) = ∆β,µl ∆β,µr λ21 G1 (f ) G1 (g) + λ22 G2 (f ) G2 (g) 2 t (v) + G (f ) G (g) t (v) . v d ρ + ∆β,µl ∆β,µr λ1 λ2 G1 (f ) G2 (g) T v 2 dρ 1 2 ω ω T Proof. Let f ∈ E be arbitrary. It holds Rt (ω) ; W (sf ) = CF (sf ) N (s, s; f ) for all s ∈ R, where (w, z) → N (w, z; f ) is the entire-analytic map on C × C deﬁned

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by N (w, z; f ) :=

exp i2−1/2 ∆β,µl ∆β,µr v(wλ1 G1 (f ) + zλ2 G2 (f )) + v(zλ1 G1 (f ) + wλ2 G2 (f )) T

t (v). dρ ω

Thus R s → Rt (ω) ; W (sf ) = CF (sf ) N (s, s; f ) has an entire analytic extension to C, and, by deﬁnition, Rt (ω) is an entire analytic state on W(E). On the other side, a formal application of the Baker-Hausdorﬀ formula, taking into account the CCR [aτ (f ), a∗τ (f )] = f 21τ , gives

Wτ (sf ) = exp isΦτ (f ) = exp is2−1/2 (a∗τ (f ) + aτ (f ))

= CF (sf )exp is2−1/2 a∗τ (f ) exp is2−1/2 aτ (f ) for all s ∈ R. Hence N (s, s; f ) is the so-called normally ordered characteristic function of the state Rt (ω). Indeed, the rigorous treatment in [38] leads to the following decomposition of N (w, z; f ) into normally ordered ﬁeld correlations of the state Rt (ω), "k ! "l ∞ ! iw iz 1 1 √ √ N (w, z; f ) = Rt (ω) ; a∗τ (f )k aτ (f )l , k! l! 2 2 k,l=0

∀w, z ∈ C .

Thus it holds for all m, n ∈ N0 and each f ∈ E that Rt (ω) ;

a∗τ (f )m

n

aτ (f ) =

√ m+n 2 ∂ m+n N (., .; f ) (0, 0) . i ∂wm ∂z n

Now the assertion follows with m = n = 1 and the polarization identity.

Let us ﬁrst assume the potential diﬀerence u from (5.1) to be positive, u > 0. Then by Lemma 5.3 the linear form G1 : E → C is unbounded and G2 : E → C is bounded, and given with the integral kernels in momentum space k ∈ R3 , 1 (k) δ † (k − 2u) , G1 (k) = φ

G2 (k) = −

2 (k) φ , k + 2u

as Gj (f ) = R3 Gj (k)f (k)d3 k, j = 1, 2. The unboundedness of G1 arises from the resonant momentum modes k ∈ Su . From Lemma 5.4 we obtain (formally) with the delta-functions in momentum space as testfunctions the normally ordered expectations of the creation and annihilation operators of the momenta k ∈ R3 , a∗k resp. ak . That is, we arrive at

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the following two-point Wightman distribution for our state Rt (ω): Rt (ω) ; a∗k ak = ∆β,µl ∆β,µr λ21 G1 (k) G1 (k ) + λ22 G2 (k) G2 (k ) (5.8) 2 t (v) + G (k) G (k ) t (v) . + ∆β,µl ∆β,µr λ1 λ2 G1 (k) G2 (k ) T v 2 dρ v d ρ 1 2 ω ω T Obviously, our two-point function R3 × R3 (k, k ) → Rt (ω) ; a∗k ak is singular for the resonant momenta k ∈ Su . The emitted radiation Rt (ω) of the Josephson oscillator thus has the highest intensity at (and in the narrow surroundings of) the resonant momentum modes k ∈ Su . Consequently, the emitted radiation may be approximately regarded as being monochromatic with frequency 2u. This reﬂects the experimental facts. Since the singularity appears only at G1 we may neglect G2 , which is done by setting λ2 = 0. Arguing with testfunctions, we are interested in the testing of the expectations Rt (ω) ; a∗τ (f ) aτ (f ) with those testfunctions f ∈ E whose momentum support supp(f ) has a non-trivial intersection with the sphere Su of resonant modes. Then the term in Lemma 5.4, containing the unbounded linear form G1 quadratically, is much bigger than the other terms, which also shows that G2 may be dropped by setting λ2 = 0. For testing the higher order correlation functionals, Rt (ω) ; a∗τ (f )m aτ (f )m with m ≥ 2, at the resonant modes k ∈ Su the argumentation is analogous. Hence, for a testing of the resonance it suﬃces to consider the approximation λ2 = 0, which is in accordance with the rotating-wave-approximation of the Dicke model in [19]. This contrasts the case of testing the non-resonant modes, that are those modes k ∈ R3 with k = 2u. If the considered testfunctions f ∈ E satisfy # supp(f ) U = ∅ for a ﬁxed open neighborhood U of Su the values G1 (f ) and G2 (f ) (resp. G1 (k) and G2 (k)) are of the same order. Thus in this case the approximation λ2 = 0 is not allowed. However, because of the low intensity of the emitted radiation at the momenta k far from the resonance, there is no interest in considering other modes than the resonant ones. For λ1 = λ2 = 1 Lemma 5.4 and [38], [39], and [40] show that Rt (ω) is not coherent in any order. Because of the above arguments we assume for the remaining part of the present subsection λ2 = 0. Then the time-asymptotic photon states from equation (5.4) reduce to

√ t (v) exp i 2 Re(v L(f )) dρ (5.9) Rt (ω) ; W (f ) = CF (f ) ω T

with the linear form L(f ) := ∆β,µl ∆β,µr λ1 G1 (f ) ,

(5.10)

t (v) = d where G1 (f ) is deﬁned in (5.5), and the phase rotation dρ ρω (exp i2ut v) ω by Lemma 5.1. Let us remark that the linear form L : E → C only depends on

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the inverse temperature β > βc and the chemical potentials µl resp. µr for the left and the right superconductor, i.e., on the selection of the equilibrium-Fock folium FFβ,µl ,µr . L does neither depend on the chosen initial state ω ∈ FFβ,µl ,µr nor on the time t ∈ R. By [38], [39], [40] Rt (ω) is an all-order coherent state on W(E), for which the normally ordered correlation functionals factorize according to Rt (ω) ; a∗τ (f1 ) · · · a∗τ (fm ) aτ (g1 ) · · · aτ (gm ) = L(f1 ) · · · L(fm ) L(g1 ) · · · L(gm ) (5.11) for all m ∈ N0 and all f1 , . . . , fm , g1 , . . . , gm ∈ E. Ranging over all initial states ω ∈ FFβ,µl ,µr , for each time point t ∈ R the t range over those probability measures on the torus group associated measures ρ ω T , which are absolutely continuous to the Haar measure. Consequently the set {Rt (ω) | ω ∈ FFβ,µl ,µr } is weak*-dense in the Bauer simplex SL∞ of the all-order coherent states on W(E) which factorize with respect to our linear form L from equation (5.10) [38]. The integration parameter v ∈ T appearing in (5.9) originally arises from the classical phase diﬀerence of the two BCS superconductors in thermal equilibrium, and now gives the macroscopic phase of the all-order cohert . According to Lemma 5.1 the ent state Rt (ω) ∈ SL∞ , distributed according to ρ ω collective, classical dynamical phase space ﬂow ϕl × ϕr restricted to the thermal classical phase space P β,µl × P β,µr from (5.3) ﬁnally leads to the time evolution t → Rt (ω) of the time-asymptotic states, via the time evolution of the measures,

t = ρ ρ ω ◦ exp i2ut , a rotation of the phase diﬀerences with velocity 2u. Thus ω the time-dependent, ordered, collective structure of the two superconductors is mirrored by the all-order coherence, with time-dependent coherence function, of the emitted radiation. If u < 0, then by Lemma 5.3 the linear form G1 is bounded and G2 is unbounded. The same arguments as above show that now the quantities with the index 2 are the important ones, and those with the index 1 may be neglected, setting λ1 = 0. Here λ1 = 0 leads to the following approximation of the timeasymptotic states

√ t (v) ˜ )) dρ exp i 2 Re(v L(f Rt (ω) ; W (f ) = CF (f ) ω T

with the linear form ˜ ) = L(f

∆β,µl ∆β,µr λ2 G2 (f ) ,

(5.12)

where G2 : E → C is deﬁned in (5.6).

6 Conclusions In order to draw some general conclusions of our model discussion let us recapitulate some basic steps.

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The unperturbed dynamics of the total system is given by the Bloch dynamics plus pairing interactions for the electrons near the Fermi surface in the separated superconductors in Section 2 and by the free photon dynamics in Section 3. The coupling between matter and radiation, set up in Section 4, is the quantized minimal coupling in the linear approximation with respect to the vector potential. The rotating wave approximation, popular in quantum optics, is not applicable in the non-equilibrium regime, since what are non-resonant terms is only deﬁned for a ﬁxed diﬀerence of the chemical potentials. Since the material system is a macroscopic one, a weak coupling is not characterized by a small coupling constant, but by its asymptotic decrease to zero in the thermodynamic limit. This averages out the photon interactions with the individual electronic excitations and it leaves the couplings to the collective modes of the superconductors. The latter are not necessarily in thermodynamic equilibrium and undergo ﬂuctuations. Since the collective variables are operators in the center of the von Neumann algebra, associated with a folium representation of the quasilocal C*-algebra of observables, the previously described cocycle techniques apply and the coupled dynamics becomes almost explicitly solvable. Thus we succeed, without leaving the microscopic quantum mechanical formalism, in evaluating the emitted photon ﬁeld states Rt (ω) of the Josephson radiator model in the long time limit, see especially Theorem 4.1. The initial state ω of the total system ‘junction-plus-photons’ at time t = 0 is arbitrarily distributed with respect to the combined superconductor system, whereas, on the photon side, there have been imposed asymptotic factorization conditions, leading to the folia Fϕ . There is, in the present frame of a corpuscular and statistical photon formalism, a weak resemblance to Sommerfeld’s classical condition on outgoing radiation. It requires some experience in the theory of smeared ﬁelds to analyze the shape of the time-asymptotic photon states Rt (ω) in Equation (4.11) resp. Corollary 4.2. These photon states are given in terms of their expectation values for the Weyl operators W (f ), which are (in regular representations of the photon Weyl algebra) the exponentials of the quantized electromagnetic ﬁelds, smeared with ⊥ ), −0 E the test function f . (In the Coulomb gauge the ﬁeld corresponds to (A, smeared with the real and imaginary part of a 3-component complex testfunction f ∈ E. Such a real-valued smeared ﬁeld is only a real -linear functional of the testfunction.) In this characteristic function Equation (4.11), which determines uniquely the state R(ω), the ﬁrst factor refers to the arbitrary initial state ϕ, which we consider mostly a few-photon state from the Fock folium. It may, however, also be chosen from a folium with macroscopically many photons. The second factor is, so to speak, the ﬁnal result of the interaction. It is purely classical, being a positive-deﬁnite characteristic function over the testfunction space E. It describes functions of the statistical distribution, with measure dρω , over the (characteristic √ the) sharp ﬁeld states with the (smeared) classical ﬁelds 2 Re(L(xl ,xr ) (f )). The latter – the real part from the smeared complex signal function L(xl ,xr ) (f ), which is complex-linear in f – depend parametrically on the values (xl , xr ) of the material

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collective variables. The probability measure dρω gives originally the distribution in the total initial state ω for the (xl , xr ) ∈ Pl × Pr . The coherence properties of the emitted radiation depend essentially on dρω , and only a sharp peak of the (xl , xr ) distribution would lead to a deterministic outgoing classical ﬁeld. In Section 5 we restrict ourselves to the physically most interesting situation of a macroscopic preparation of the Josephson oscillator: The left and the right superconductors are prepared into thermal equilibrium sectors below the critical temperature, with a well-speciﬁed voltage diﬀerence V = u/e = g(xr3 − xl3 ), g the pairing interaction constant, arising from the diﬀerence u of their chemical potentials in Equation (5.1). The initial electromagnetic ﬁeld is assumed to consist of few-photon excitations, only, determining the Fock sector Fϕ ≡ FF . This means physically, that the Josephson system is initially prepared with negligible radiation and the initial state ω is taken from the equilibrium-Fock folium. Only for such kinds of initial states, the familiar result of the Josephson radiator is obtained: The emitted photon ﬁeld corresponds to a nearly monochromatic radiation of frequency 2|u|. Beside this well-known result we show in Subsection 5.2 that for every initial state ω from the equilibrium-Fock sector (5.2) and each t ∈ R the time-asymptotic photon state Rt (ω) approaches an all-order coherent state on the photonic ﬁeld algebra W(E). It factorizes according to (5.11) with the linear ˜ : E → C from form L : E → C from (5.10) for u > 0, resp. with the linear form L (5.12) for u < 0, independently from other features of the initial equilibrium-Fock state ω. That means, that only a ﬂuctuating collective behaviour of the two weakly coupled thermal superconductors, which respects the voltage diﬀerence, may lead to a coherent collective behaviour of the emitted photons, and may be partially reconstructed from the latter: The Josephson frequency is nowadays a high-quality voltage standard. The foregoing discussion refers to the resonant part of the radiation, which is deﬁned in reference to a ﬁxed voltage diﬀerence. The non-resonant photon modes (with frequencies far from 2|u|), are in a regime of very low intensities compared with the resonant intensity. Thus we conﬁrm the applicability of the rotating wave approximation, but only for the classical part of the radiation. For the nonresonant photon modes the above-mentioned all-order coherence does not hold, and the more complex structure of the emitted radiation states Rt (ω), as described in Lemma 5.4 and Equation (5.8), applies. Concerning the relation to a laser resp. maser model, pumping is replaced here by the voltage diﬀerence. The pertinent electrostatic energy drives via the Josephson eﬀect the equilibrium oscillations of the condensate and injects little energy into the radiation. These collective oscillations may be compared with the collective (de-) excitations of two-level atoms during resonant induced emission. Let us emphasize, that the arise of a classical electromagnetic ﬁeld within a fully quantized radiation model (without performing the classical limit → 0) bears some implications of fundamental importance. It means nothing less than that the classical electromagnetic ﬁelds are to be considered collective modes of the photons. Then Maxwell’s theory should be part of QED – in the more comprising

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operator algebraic formulation of the latter, which is not restricted to the Fock representation. We shall demonstrate at another occasion, that the dynamics of the outgoing photonic collective modes is, in fact, Maxwell’s dynamics in a disguised form.

Acknowledgment This work has been supported by the Deutsche Forschungsgemeinschaft.

References [1] A. Rieckers, M. Ullrich, On the Microscopic Derivation of the FiniteTemperature Josephson Relation in Operator Form, J. Math. Phys. 27, 1082– 1092 (1986). [2] T. Unnerstall, Dynamics of the Current-Driven Josephson Junction, J. Stat. Phys. 54, 379–403 (1989). [3] A. Barone, G. Patern` o, Physics and Applications of the Josephson Eﬀect, New York, John Wiley & Sons, 1982. [4] K. Hepp, Two Models for Josephson Oscillators, Annals of Physics 90, 285– 294 (1975). [5] W. Fleig, On the Symmetry Breaking Mechanism of the Strong-Coupling BCS Model, Acta Phys. Austr. 55, 135–153 (1983). [6] A. Rieckers, M. Ullrich, Extended Gauge Transformations and the Physical Dynamics in a Finite Temperature BCS-Model, Acta Phys. Austr. 56, 131–152 (1985). [7] A. Rieckers, M. Ullrich, Condensed Cooper Pairs and Quasi Particles in a Gauge Invariant Finite Temperature BCS-Model, Acta Phys. Austr. 56, 259– 274 (1985). [8] A. Rieckers, Condensed Cooper Pairs and Macroscopic Quantum Phenomena, Proceedings of NATO ASI, Large-Scale Molecular Systems: Quantum and Stochastic Aspects. Beyond the Simple Molecular Picture, Maratea, Italy, 1990. Eds.: W. Gans, A. Blumen and A. Amann. Plenum Press, New York, 1991. [9] T. Gerisch, Internal Symmetries and Limiting Gibbs States in Quantum Lattice Mean Field Theories, Physica A 197, 284–300 (1993). [10] T. Gerisch, A. Rieckers, Limiting Dynamics, KMS-States, and Macroscopic Phase Angle for Inhomogeneous BCS-Models, Helv. Phys. Acta 70, 727–750 (1997).

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[11] T. Gerisch, A. Rieckers, Limiting Gibbs States and Phase Transitions of a Bipartite Mean-ﬁeld Hubbard-Model, J. Stat. Phys. 91, 759–786 (1998). [12] T. Gerisch, A. Rieckers, H.-J. Volkert, Thermodynamic Formalism and Phase Transitions of Generalized Mean-Field Quantum Lattice Models, Z. Naturforsch. 53a, 179–207 (1998). [13] T. Gerisch, A. Rieckers, M.P.H. Wolﬀ, Heisenberg Generators and Arveson Spectra of Perturbed Meanﬁeld Models, Preprint, T¨ ubingen, 2003. [14] E.B. Davies, Exact Dynamics of an Inﬁnite-Atom Dicke Maser Model I, Commun. Math. Phys. 33, 187–205 (1973); The Inﬁnite Atom Dicke Maser Model II, Commun. Math. Phys. 34, 237–249 (1973). [15] R. Honegger, Global Cocycle Dynamics for Inﬁnite Mean Field Quantum Systems Interacting with the Boson Gas, J. Math. Phys. 37, 263–282 (1996). [16] R. Honegger, Time-Asymptotic Boson States from Inﬁnite Mean Field Quantum Systems Coupled to the Boson Gas, Lett. Math. Phys. 27, 191–203 (1993). [17] R. Honegger, Globale Quantentheorie der Strahlung. T¨ ubingen: Thesis, 1991. [18] R. Honegger, Quantized Radiation from Collectively Ordered Atoms, in “Classical and Quantum Systems-Foundation and Symmetries”, Proceedings of the II. International Wigner Symposium, Goslar, Germany, July 1991. World Scientiﬁc, 1992. [19] R. Honegger, The Weakly Coupled Inﬁnite Dicke Model, Physica A 225, 391– 411 (1996). [20] R. Honegger, The Dynamical Generation of Macroscopic Coherent Light, Proceedings of NATO ASI, Large-Scale Molecular Systems: Quantum and Stochastic Aspects. Beyond the Simple Molecular Picture, Maratea, Italy, 1990. Eds.: W. Gans, A. Blumen and A. Amann. Plenum Press, New York, 1991. [21] R. Honegger, A. Rieckers, T. Unnerstall, Perturbational Dynamics of the Inﬁnite Dicke Model, Z. Naturforsch. 48 a, 1043–1053 (1993). [22] R. Honegger, A. Rieckers, Quantized Radiation States from the Inﬁnite Dicke Model, Publ. RIMS Kyoto Univ. 30, 111–138 (1994). [23] J. Bardeen, L.N. Cooper, J.R. Schrieﬀer, Theory of Superconductivity, Phys. Rev. 108, 1175–1204 (1957). [24] M. Tinkham, Introduction to Superconductivity, Tokyo: Mc Graw-Hill, 1975. [25] G. Rickayzen, Theory of Superconductivity, New York: J. Wiley, 1965.

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[26] D.J. Thouless, The Quantum Mechanics of Many Body Systems, New York, London: Academic Press, 1961. [27] P.W. Anderson, Random-Phase Approximation in the Theory of Superconductivity, Phys. Rev. 112, 1900–1916 (1958). [28] Y. Wada, F. Takano, N. Fukuda, Exact Treatment of Bardeen’s Theory of Superconductivity in the Strong Coupling Limit, Progr. Theor. Phys. (Kyoto) 19, 597–598 (1958). [29] W. Thirring, A. Wehrl, On the Mathematical Structure of the BCS Model II, Commun. Math. Phys. 7, 181–189 (1968). [30] E. Duﬀner, A. Rieckers, On the Global Quantum Dynamics of Multi-Lattice Systems with Non-Linear Classical Eﬀects, Z. Naturforsch. 43 a, 521–532 (1988). [31] P. B´ ona, The Dynamics of a Class of Quantum Mean-Field Theories, J. Math. Phys. 29, 2223–2235 (1988). [32] T. Unnerstall, Phase-Spaces and Dynamical Descriptions of Inﬁnite MeanField Quantum Systems, J. Math. Phys. 31, 680–688 (1990). [33] T. Unnerstall, Schr¨ odinger Dynamics and Physical Folia of Inﬁnite MeanField Quantum Systems, Commun. Math. Phys. 130, 237–255 (1990). [34] N.G. Duﬃeld, R.F. Werner, Local Dynamics of Mean-Field Quantum Systems, Helv. Phys. Acta 65, 1016–1054 (1992). [35] G.A. Raggio, A.F. Werner, Quantum Statistical Mechanics of General Mean Field Systems, Helv. Phys. Acta 62, 980–1003 (1989). [36] T. Gerisch, A. Rieckers, The Quantum Statistical Free Energy Minimum Principle for Multi-Lattice Mean Field Theories, Z. Naturforsch. 45a, 931–945 (1990). [37] T. Gerisch, R. Honegger, A. Rieckers, Limiting Dynamics of Generalized Bardeen-Cooper-Schrieﬀer Models Beyond the Pair Algebra, J. Math. Phys. 34, 943–967 (1993). [38] R. Honegger, A. Rieckers, The General Form of Non-Fock Coherent Boson States, Publ. RIMS Kyoto Univ. 26, 397–417 (1990). [39] R. Honegger, A. Rieckers, First Order Coherent Boson States, Helv. Phys. Acta 65, 965–984 (1992). [40] R. Honegger, A. Rieckers, On Higher Order Coherent States on the Weyl Algebra, Lett. Math. Phys. 24, 221–225 (1992).

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[41] R. Honegger, The General Form of the Microscopic Coherent Boson States, Physica A 198, 179–209 (1993). [42] R. Honegger, A. Rieckers, Construction of Classical and Non-Classical Coherent Photon States, Annals of Physics 289, 213–231 (2001). [43] R. Honegger, A. Rapp, General Glauber Coherent States on the Weyl Algebra and their Phase Integrals, Physica A 167, 945–961 (1990). [44] R.J. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130, 2529–2539 (1963); Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131, 2766–2788 (1963); In “Quantum Optics and Electronics”, Les Houches 1964, edited by C. de Witt, A. Blandin and C. Cohen-Tannoudji, New York: Gordon and Breach, 1965. [45] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons & Atoms, Introduction to QED, New York, Toronto, Singapore: Wiley & Sons, 1989. [46] R. Loudon, The Quantum Theory of Light, Oxford: Clarendon Press, 1979. [47] T. Gerisch, A. Rieckers, Comparsion of Weakly Inhomogeneous BCS- and Hubbard-Models, Physica A 242, 439–466 (1997). [48] T. Gerisch, R. M¨ unzner, A. Rieckers, Canonical Versus Grand-Canonical Free Energies and Phase Diagrams of a Bipolaronic Superconductor Model, J. Stat. Phys. 93, 1021–1049 (1998). [49] T. Gerisch, R. M¨ unzner, A. Rieckers, Global C*-Dynamics and Its KMSStates of Weakly Inhomogeneous Bipolaronic Superconductors, J. Stat. Phys. 97, 751–779 (1999). [50] T. Gerisch, R. M¨ unzner, A. Rieckers, Spectral Phase Diagrams in Diﬀerent Ensembles for Bipolaronic Superconductors, Helv. Phys. Acta 72, 419–444 (1999). [51] R. M¨ unzner, Operator Algebraic Formulation and Numerical Evaluation of ubingen, 2000. Inhomogeneous High Tc Superconductor Models, Thesis, T¨ [52] G.K. Pedersen, C*-Algebras and their Automorphism Groups, New York, London: Academic Press, 1979. [53] M. Takesaki, Theory of Operator Algebras I, Berlin, Heidelberg, New York: Springer, 1979. [54] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras I, II, New York, London: Academic Press, 1983, 1986. [55] R. Haag, R.V. Kadison, D. Kastler, Nets of C*-Algebras and Classiﬁcation of States, Comm. Math. Phys. 16, 81–104 (1970).

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[56] G.L. Sewell, States and Dynamics of Inﬁnitely Extended Physical Systems, Commun. Math. Phys. 33, 43–51 (1973). [57] H. Roos, KMS Condition in a Schr¨ odinger Picture of the Dynamics, Physica A 100, 183–195 (1980). [58] R.V. Kadison, Topology 3, Suppl. 2, pp. 177–198, Pergamon Press, 1965. [59] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Berlin, Heidelberg, New York: Springer, 1979, 1981. [60] S. Sakai, C*-Algebras and W*-Algebras, Berlin: Springer-Verlag, 1971. [61] E. Størmer, Symmetric States of Inﬁnite Tensor Products of C*-Algebras, J. Funct. Analysis 3, 48–68 (1969). [62] V.I. Arnold, Mathematical Methods of Classical Mechanics, Berlin, Heidelberg, New York: Springer, 1985. [63] T. Unnerstall, Makroskopische Quantenph¨ anomene in Quantengittermodellen, T¨ ubingen: Thesis, 1990. [64] N.G. Duﬃeld, R.F. Werner, Classical Hamiltonian Dynamics for Quantum Hamiltonian Mean Field Systems, in Proceedings of Swansea Conference on Stochastic and Quantum Mechanics, 1990. [65] M. Fannes, H. Spohn, A. Verbeure, Equilibrium States for Mean Field Models, J. Math. Phys. 21, 355–358 (1980). [66] H.M. Nussenzveig, Introduction to Quantum Optics, London, New York, Paris: Gordon and Breach, 1973. [67] R. Honegger, On the Continuous Extension of States on the CCR-Algebra, Lett. Math. Phys. 42, 11–25 (1997); and, Enlarged Testfunction Spaces for the Global Free Folia Dynamics on the CCR-Algebra, J. Math. Phys. 39, 1153–1169 (1998). [68] A. Rieckers, Macroscopic Quantum Phenomena at the SQUID, in H. At¨ller-Herold [Hrsg.], On Quanta, manspacher, A. Amann, und U. Mu Mind and Matter; Hans Primas in Context, Fundamental Theories of Physics 102, Kluwer Academic Publishers, Dordrecht, 1999. [69] M. Reed, B. Simon, Methods of Modern Mathematical Physics I, New York, London, Toronto, Sydney, San Francisco: Academic Press, 1980. [70] J. Weidmann, Linear Operators in Hilbert Spaces, Berlin, Heidelberg, New York: Springer, 1980.

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Thomas Gerisch, Reinhard Honegger, Alfred Rieckers Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen D-72076 T¨ ubingen, Germany email: [email protected] email: [email protected] email: [email protected] Communicated by Joel Feldman submitted 14/02/03, accepted 04/03/03

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e

Ann. Henri Poincar´e 4 (2003) 1083 – 1099 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061083-17 DOI 10.1007/s00023-003-0156-2

Annales Henri Poincar´ e

On the Positivity of the Jansen-Heß Operator for Arbitrary Mass A. Iantchenko and D.H. Jakubaßa-Amundsen Abstract. The Jansen-Heß operator is an approximate (pseudo-)relativistic no-pair Hamiltonian in the Furry picture which is used in the physics literature to describe heavy atoms. Within the single-particle Coulomb model we prove that their energy, and thus the resulting self-adjoint operator and its spectrum, is positive for Z ≤ 114.

1 Introduction Consider a relativistic electron in the Coulomb ﬁeld V , described by the Dirac operator (in relativistic units, = c = 1) H = D0 + V,

D0 := −iα ∂/∂x + βm,

V (x) := −

γ x

(1)

acting on the Hilbert space L2 (R3 ) ⊗ C4 , where γ := Ze2 , Z the nuclear charge number, e2 = (137.04)−1 the ﬁne structure constant, α and β the Dirac matrices and x := |x|. It is well known that H is not bounded from below. As long as pair creation is neglected, the conventional way to circumvent this deﬁciency is the introduction of the semibounded operator P+ HP+ where P+ projects onto the positive spectral subspace of H (Furry picture, see Sucher [10] and [11] for a review). Jansen and Heß [8], based on work by Douglas and Kroll [3] suggested an approximate operator which is derived from a Foldy-Wouthuysen-type transformation scheme. It is a second-order operator in the potential strength γ. It can be written in the form Λ+ (D0 + V + 2i [W1 , B1 ]) Λ+ on L2 (R3 ) ⊗ C4 where Λ+ projects onto the positive spectral subspace of the free Dirac operator D0 while W1 and B1 are operators linear in γ [7]. Alternatively, as in [4, 2], it can be reduced to an operator acting on two-spinors ϕ ∈ L2 (R3 ) ⊗ C2 ˜ bm := b0m + b1m + b2m := B + γ 2 K

(2)

˜ is the where b0m + b1m := B is the Brown-Ravenhall operator, and b2m := γ 2 K second-order term in γ. For the massless case (m = 0) Brummelhuis, Siedentop and Stockmeyer [2] could prove positivity, i.e., (ϕ, bm ϕ) ≥ 0

for γ ≤ γc

(3)

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with γc = 1.006 (corresponding to nuclear charge numbers Z ≤ 137), where γc 2 was found to be solution of 1 − γ2 ( π2 + π2 ) + γ8 ( π2 − π2 )2 = 0. For m = 0, they could prove boundedness from below for γ ≤ γc which they obtained from the relative boundedness of the massive Jansen-Heß operator with respect to the massless one. From their proof, positivity was found to hold for Z ≤ 25. The aim of the present work is to show positivity of bm for higher coupling constants. We will choose the momentum representation and we set ϕ(p) ˆ := −ip·x e ϕ(x)dx/(2π)3/2 for the Fourier transform of ϕ. Following [4] and [2] we R3

expand ϕ in terms of spherical spinors Ων ϕ(p) ˆ = p−1 fˆν (p)Ων (ˆ p),

ν = (l, M, s)

(4)

ν∈I

such that (ϕ, bm ϕ) =

ν∈I

(fν , blsm fν ) ≥ 0

is equivalent to proving positivity

for each component ν. Here, the index set I := {ν = (l, M, s)| l ∈ N0 , M = ˆ := p/p, p := |p|, and −l − 12 , . . . , l + 12 , s = ± 12 , l + s > 0, Ων = 0}, p

∞ 0

ν∈I

|fˆν (p)|2 dp =

R3

2 |ϕ(p)| ˆ dp.

(5)

In this partial wave decomposition we deﬁne according to (2) (1)

(2)

blsm := b0m + blsm + blsm ,

(6)

where explicitly ([2], [6]) b0m := e(p) := (1)

blsm (p, p ) := − (2) blsm (p, p )

γ2 := 2 2π

0

p 2 + m2

γ p p [ql ( ) + h(p)h(p )ql+2s ( )] A(p)A(p ) π p p

∞

dp N (p, p , p )A(p)A(p ) (F1 + F2 − F3 − F4 )

(7) (8)

1 1 + N (p, p , p ) := A2 (p ) e(p ) + e(p ) e(p) + e(p )

with

h(p) := F1 := ql ( F3 := ql (

p , e(p) + m

p p )ql ( ) h2 (p ), p p

p p )ql+2s ( ) h(p )h(p ), p p

A2 (p) := F2 := ql+2s (

e(p) + m 2e(p)

p p )ql+2s ( ) h(p)h(p ) p p

F4 := ql+2s (

p p )ql ( ) h(p)h(p ). p p

(9)

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Here we have introduced reduced Legendre functions ql (x) := Ql ( 12 (x + x1 )), Ql being the Legendre function of the second kind (see Stegun, pages 331–353, in [1]). From the properties of Ql (see [4]) it follows that Fi ≥ 0, i = 1, . . . , 4. Moreover, (2) we show that F1 +F2 −F3 −F4 ≥ 0, i.e., the kernel blsm (p, p ) is positive (Section 2). Then one can prove the following. Proposition 1 Let γ < γc1 = 0.5929 (Z ≤ 81). Then blsm > 0 for all l ∈ N0 , s = ± 12 and all masses m = 0. (The proof is analytical). For the massless case it was shown [2] that l = 0, s = 12 is the ground-state conﬁguration of the Jansen-Heß operator. When m = 0 and s is ﬁxed, the lowestenergy conﬁgurations are found to be l = 0 (s > 0) and l = 1 (s < 0), respectively (Section 3). This leads to 2

Proposition 2 Let γc2 be the solution of 1 − γ2 ( π2 + π2 ) − γ8 ( π2 − π2 )2 = 0 (γc2 = 0.8368, Z ≤ 114). Then one has blsm > 0 for γ < γc2 and all l ∈ N0 , s = ± 12 and all masses m = 0. (The proof is numerical). The plan of the paper is as follows: we start by considering some properties (2) of the kernel of the operator blsm in (6). In Section 2 we prove that it is positive. In (2) Section 3 we study the monotonicity properties of the kernel of blsm with respect to the orbital quantum number l. These properties are used in Sections 4 and 5, where we prove the positivity of the Jansen-Heß operator (Propositions 1 and 2).

2 Positivity of the kernel of the Jansen-Heß operator (2)

Proposition 3 For all l ∈ N0 , s = ± 21 and p, p > 0 we have blsm (p, p ) > 0, and (2) (2) (2) blsm (0, p ) = blsm (p, 0) = blsm (0, 0) = 0. Proof. We write the sum F1 + F2 − F3 − F4 as a product in the following way: F1 + F2 − F3 − F4 p p p p 2 = ql (p ) + q h h(p)h(p ) ql q l+2s l+2s p p p p p p p p )h(p ) − q − ql h(p h(p)h(p ) ql+2s ql l+2s p p p p p p p p = ql ) − q ) . h(p h(p h(p ) − ql+2s h(p) · ql l+2s p p p p With gl,s (p, p ) := ql

p p

h(p ) − ql+2s

p p

h(p)

(10)

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A. Iantchenko and D.H. Jakubaßa-Amundsen

we have (2) blsm (p, p )

γ2 = 2 2π

∞

0

Ann. Henri Poincar´e

dp N (p, p , p )A(p)A(p ) · gl,s (p, p ) · gl,s (p , p ).

(11)

Proposition 3 follows from the following lemma: Lemma 1 For all p, p > 0 and m ≥ 0, gl,s (p, p ) > 0 for s = 1/2; gl,s (p, p ) < 0 for s = −1/2. For any p, p ≥ 0 and m > 0, we have gl,s (0, p ) = gl,s (p, 0) = gl,s (0, 0) = 0. Proof of Lemma. Using the deﬁnitions (9), we write explicitly p p p p − q , gl,s (p, p ) = ql l+2s p e(p ) + m p e(p) + m

p 1 p 1 1 Pl (s) p ds, t= + = Ql (t) := , (12) p 2 −1 t − s 2 p p where the Pl are Legendre polynomials. We consider ﬁrst the limit case: either p = 0 and p > 0 or p = 0 and p > 0. Taking the limit t → ∞, i.e., either p → 0 and p > 0 or p → 0 and p > 0, we get p ql → 0, and thus gl,s (0, p ) = 0, gl,s (p, 0) = 0, for all p, p > 0. p If p = p > 0, then p (ql (1) − ql+2s (1)) gl,s (p, p) = (13) e(p) + m p = (Ql (1) − Ql+2s (1)) > 0 for s = 1/2 and < 0 for s = −1/2. e(p) + m

where

ql

This follows from the following formulae proven in the appendix, 1 >0 l+1 1 Ql (1) − Ql−1 (1) = − < 0 l Ql (1) − Ql+1 (1) =

∀ l = 0, 1, 2, . . . and ∀ l = 1, 2, . . . ,

(14) (15)

using the representation of Ql in terms of hypergeometric functions [5, p. 999]. In the limit p = p → 0, we get gl,s (p, p ) → 0 if m = 0. Thus the limit case in Lemma 1 is proved. If p = p , p, p > 0, we can use the following representation of the Legendre function of the second kind (as in [4]) : ∞ p z −l−1 1 p p √ dz, t= + ql = Ql (t) = , (16) p 2 p p 1 − 2tz + z 2 t+(t2 −1)1/2 which is valid for t > 1.

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Let x = p/p > 0. Then

2 1

1 1

1 2 2 t −1= , t − 1 = x − , x− 2 x 2 x 

 x,

x > 1 ⇔ 0 1 for all p = p , p, p > 0.

Then,

for s > 0, we have ql+2s

p p

≤ ql

p p

−2s , t + (t2 − 1)1/2

(17)

(18)

(19)

as for s > 0 we have: ∞ −2s p p z −l−1 √ ql+2s t + (t2 − 1)1/2 · z −2s dz ≤ ql ; = p p 1 − 2tz + z 2 t+(t2 −1)1/2 2s p p t + (t2 − 1)1/2 for s < 0, we have ql , (20) ≤ ql+2s p p since for s < 0 we have ∞ 2s p p z −l−1−2s 2s 2 1/2 √ = t + (t ql · z dz ≤ q − 1) . l+2s p p 1 − 2tz + z 2 t+(t2 −1)1/2 For s > 0, using equation (19), we get p p (e(p ) + m) p p (21) ql − ql+2s gl,s (p, p ) = e(p ) + m p p (e(p) + m) p −2s (e(p ) + m) p p p 2 1/2 ≥ q − 1) · · 1 − t + (t l e(p ) + m p (e(p) + m) p p p ql =: θs (p, p ). e(p ) + m p Suppose that p > p > 0. Then according to (17) −2s p f (p) (e(p ) + m)p p2s+1 e(p ) + m θs (p, p ) = 1 − =1− , · = 1 − · p (e(p) + m)p f (p ) p 2s+1 e(p) + m p2s+1 . Since f (0) = 0 and e(p) + m p2s (2s + 1)(e(p) + m)e(p) − p2 f (p) = >0 e(p)(e(p) + m)2

where f (p) :=

for p > 0,

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A. Iantchenko and D.H. Jakubaßa-Amundsen

we have f (p ) > f (p) > 0 for p > p > 0. This implies for s > 0 and 0 0.

We have thus proved the ﬁrst statement in Lemma 1 for p > p > 0 and s > 0. Suppose now that p > p > 0. The ﬁrst statement in Lemma 1 then follows using (17): −2s 2s−1 p p (e(p ) + m)p e(p ) + m > · = 1 − · θs (p, p ) =1 − p (e(p) + m)p p e(p) + m 2s−1 p >1− = 0, for s = 1/2. p Let now s < 0. Then we get, using the bound (20), p p (e(p) + m) p p −gl,s (p, p ) = ql+2s − ql (22) e(p) + m p p e(p ) + m p 2s (e(p) + m) p p p ql+2s ( ) · 1 − t + (t2 − 1)1/2 · ≥ e(p) + m p (e(p ) + m) p p p ql+2s ( )θ−s (p , p) > 0, = e(p) + m p using the bound on θ−s for −s > 0. The proof of Lemma 1 and therefore the proof of Proposition 3 is ﬁnished.

3 The lowest energy configurations In this section we prove a useful pointwise bound on the kernel of the Jansen-Heß part of the operator in (6): Lemma 2 For all p, p > 0, l ∈ N0 and s = ± 12 we have (2) (2) blsm (p, p ) < b0sm (p, p ) for s = 1/2, l > 0; (2) (2) blsm (p, p ) < b1sm (p, p ), for s = −1/2, l > 1. (2)

Note that, if either p or p is zero, then all blsm (p, p ) = 0, by Proposition 3. Proof. Let ﬁrst p = p , p, p > 0 and s = 12 . By Lemma 1 we know that gl,s (p, p ) > 0 and by equation (11) it is enough to prove that gl,s (p, p ) < g0,s (p, p ) for all p = p , p, p > 0, l ∈ N and s = 12 . As gl,s is a C ∞ function of l ≥ 0 we prove that (gl,s (p, p ))l < 0. ql

We use ∞ p p z −l−1 1 p + > 1, (− ln(z)) · √ dz < 0, for t = = p 2 p p 1 − 2tz + z 2 t+(t2 −1)1/2 (23)

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where ql means derivative with respect to l, and we get the bound −2s p p −ql+2s t + (t2 − 1)1/2 , s > 0, t > 1, < −ql p p in the same way as the bound (19). Writing gl,s (p, p ) as in (21) and taking the derivative with respect to l we get as in the previous section p p (e(p ) + m) p p (gl,s (p, p ))l = (24) q − ql+2s e(p ) + m l p p (e(p) + m) p −2s (e(p ) + m) p p p · ql · <− < 0. · −1 + t + (t2 − 1)1/2 e(p ) + m p (e(p) + m) p We have ∞ γ2 (2) blsm (p, p ) = 2 N (p, p , p )A(p)A(p ) (25) 2π 0 l · (gl,s (p, p ))l · gl,s (p , p ) + gl,s (p, p ) · (gl,s (p , p ))l dp < 0 and thus the statement of Lemma 2 for p = p , p, p > 0, s = 1/2 and l > 0. When s = − 12 , we have gl,s (p, p ) < 0 and as in (22) we can prove that −(gl,s (p, p ))l < 0. From this it follows that |gl,s (p, p )| < |g1,s (p, p )| for l > 1. (2) (2) Equation (25) then shows again that blsm (p, p ) < 0. This leads to blsm (p, p ) < l

(2)

b1sm (p, p ) for p = p , p, p > 0. When p = p , we use |gls (p, p )| < |gλs (p, p )| for p = p and p, p > 0 where λ = 0, l > 0 for s = 12 and λ = 1, l > 1 for s = − 21 . Insertion into (11) shows (2) (2) that blsm (p, p) < bλsm (p, p). The proof of Lemma 2 is thus ﬁnished. Lemma 2 provides some information on the lowest energy conﬁguration which we formulate in a proposition below. Note that this proposition will not be used in the proof of our main results Propositions 1 and 2 in the next sections. Proposition 4 We have inf (ϕ, bm ϕ)| (1 + p1/2 )|ϕ| ∈ L2 (R3 ), ϕ = 1 ≥ f ) + (g, b− g)| (1 + p1/2 )|f |, (1 + p1/2 )|g| ∈ L2 (R+ ), inf (f, b− 0, 1 ,m 1,− 1 ,m 2

2

(26)

f 2 + g2 = 1 ,

where the last infimum can in addition be restricted to positive functions f, g, and where (1) (2) b− (27) lsm := b0m + blsm − blsm .

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Proof. For any given f ∈ L2 (R+ ) we have the following bound from below: (1)

(2)

(1)

(2)

(f, blsm f ) = (f, (b0m + blsm + blsm )f ) ≥ (f, b0m f ) − (f, −blsm f ) − |(f, blsm f )| (1)

(2)

≥ (|f |, b0m |f |) − (|f |, −blsm |f |) − (|f |, blsm |f |) = (|f |, b− lsm |f |), (28) (1)

where we have used that the kernel (7) of −blsm is positive, and that according to (2) Proposition 3 the kernel of blsm is positive as well, allowing the bound ∞ ∞

(2)

|(f, blsm f )| ≤

0

0

(2) dp dp |fˆ(p)| blsm (p, p ) |fˆ(p )|.

(29)

Note that the operator b− lsm deﬁned in (27) diﬀers from blsm by a minus sign of the last term. Therefore in contrast to the Brown-Ravenhall case [4], the inequalities (28) do not assure a positive ground-state conﬁguration for the original Hamiltonian bm . However, applying Lemma 2 to the right-hand side of (28) and using [4], we have the bound from below − (|f |, b− lsm |f |) ≥ (|f |, bλsm |f |),

(30)

with λ = 0 for s = 12 and λ = 1 for s = − 12 . Hence we may follow the argumentation of [4] by assuming that the coeﬃcients fν in (4) are zero unless ν = (0, 12 , 12 ) or ν = (1, 12 , − 12 ) when minimizing. According to (28) and (4), (5) we get inf{(ϕ, bm ϕ)} = inf{

ν∈I

(fν , blsm fν )} ≥ inf{|f |, b− |f |) + (|g|, b− |g|)}, 0, 1 ,m 1,− 1 ,m 2

2

(31) where ϕ, f and g obey the restrictions given in (26). Equation (31) shows that as in [4] we may and shall restrict ourselves to positive functions when evaluating the inﬁmum.

4 Proof of Proposition 1 Let us consider the following estimate of the energy in a state characterized by ν. Then from (28) and (29) (1)

(2)

(fν , (b0m + blsm + blsm )fν ) ≥ (fν , b0m fν ) −

(1) (|fν |, −blsm |fν |) (2) blsm (p, p )

−

0

∞ ∞ 0

(32)

(2) dp dp |fˆν (p)| blsm (p, p ) |fˆν (p )|.

Thereby positivity of allows for keeping the four terms Fi from (8) with their respective sign. In the following we will estimate the last term in (32) by

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means of the Lieb and Yau formula [9] for a symmetric and nonnegative kernel k(p, p )

f (p) 2

dp dp |ϕ(p)| ˆ k(p, p ) |ϕ(p ˆ )| ≤ dp |ϕ(p)| ˆ dp k(p, p )

f (p )

0 0 0 0 (33) with a convergence generating function f (p) > 0 for p > 0. Below, we will always 1 use f (p) = p 2 . Factors of the kernel which depend symmetrically on p and p may be absorbed into the functions ϕ(p) ˆ and ϕ(p ˆ ), respectively. For the proof of Proposition 1 we will use strong estimates that allow for an analytical evaluation of the integrals. One has 1 1 e(p ) + m 1 1 N (p, p , p ) ≤ + = ≤ . (34) m + e(p ) m + e(p ) 2e(p ) e(p ) p

∞ ∞

∞

2

∞

Moreover, the negative terms −F3 , −F4 in (8) are estimated by zero. Then (2) applying (33) with ϕˆ := A(p)fˆν (p), the contribution to blsm (p, p ) from F1 is estimated by γ2 2π 2

∞ ∞ 0

γ2 ≤ 2π 2

0

0

∞

dp dp |fˆν (p)| A(p) dp |fˆν (p)|2 A2 (p)

∞

0

0

∞

dp N (p, p , p ) F1 A(p ) |fˆν (p )|

p dp p

∞

0

(35)

dp p p 2 )q q ( ( ) h (p ). l l p p p

us ﬁrst consider states with s = 12 . Then we estimate h2 (p ) ≤ 1 and make of the fact that 0 ≤ ql (x) ≤ · · · ≤ q1 (x) ≤ q0 (x) ∀l ≥ 1, x ∈ R+ (see [4] for

Let use x = 1 and the appendix for x = 1) to get ∞ p ∞ dp p p dp ql ( )ql ( ) h2 (p ) I1 (p) := p p p p 0 0 ≤

∞

0

dp p q0 ( ) p p

∞

0

p p dp q0 ( ). p p

Successively, we substitute z := p /p for p and then ζ := p /p for p and use the formula (noting that ql (z) = ql (1/z) for l ≥ 0) 0

∞

dz q0 (z) = 2 z

0

1

(36)

1 + z

.

with q0 (z) = ln

1 − z

dz π2 q0 (z) = z 2

(37)

Then the two integrals decouple and one obtains I1 (p) ≤ p

π2 2

2 .

(38)

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(2)

According to (35) the second term of blsm (p, p ) resulting from F2 is estimated by

γ2 2π 2

∞

0

dp |fˆν (p)|2 A2 (p)h2 (p)

∞

0

p dp p

0

∞

dp p p ql+1 ( )ql+1 ( ). p p p

(39)

Estimating ql+1 by q1 and using (as in [4])

∞ 0

dx q1 (x) = 2 x

1

0

1 + x

1 1

−1 with q1 (x) = (x + ) ln

2 x 1 − x

(40)

dx q1 (x) = 2 x

one obtains I2 (p) :=

∞ 0

dp p ) q ( l+1 p p

0

∞

p p dp q ( ) ≤ 4p. l+1 p p

(41)

(1)

The ﬁrst-order term blsm is estimated in a similar way, following [4]. According to (1) (32) and (33), since −blsm (p, p ) > 0 for p, p > 0,

∞ 0

≤

γ π

∞ 0

∞

0

(1)

dp dp |fˆν (p)| − blsm (p, p ) |fˆν (p )|

dp |fˆν (p)|2 A2 (p)

∞ 0

p p dp ql ( ) + h2 (p) p p

0

∞

(42)

p p dp q ( ) . l+2s p p

We restrict ourselves to s = 12 and estimate by setting l = 0 as before. Then, making the substitution z := p /p and evaluating the integrals by means of (37) and (40) we ﬁnd ∞ ∞ p p p p 2 dp ql ( ) + h (p) dp ql+1 ( ) (43) I0 := p p p p 0 0 ≤

0

∞

p p dp q0 ( ) + h2 (p) p p

0

∞

p p π2 + h2 (p) · 2) . dp q ( ) = p ( 1 p p 2

Collecting results, the expectation value of the Jansen-Heß operator is estimated by ∞ dp |fˆν (p)|2 e(p) · G0 12 (p), (44) (fν , bl 12 m fν ) ≥ G0 12 (p) := 1 −

γ p A2 (p) π e(p)

0

4 π2 π γ2 p 2 2 + 2h2 (p) − A + 4h (p) (p) . 2 2π 2 e(p) 4

Following the argumentation at the end of Section 3, the minimizing function fν can be chosen with ν = (0, 12 , 12 ). m-invariance of G0 12 (p) is provided by means of introducing p := mx (for m = 0). Then, using the deﬁnition (9) of A(p) and h(p) one obtains with

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√ e(p) = m x2 + 1 γ G0 12 (x) = 1− x π

√

γ x2 1+ + √ . π ( x2 + 1 + 1)(x2 + 1) (45) > 0. One easily derives G0 12 (x) = 1 for x = 0 and

x2 + 1 + 1 x2 + 1

If G0 12 (x) > 0 then bl 12 m 2

π2 γπ 3 + 4 16

3

G0 12 (x) → 1 − πγ (1 + π4 + πγ + γ π16 ) for x → ∞ which is positive for suﬃciently small γ. Our strategy is to look for min G0 12 (x) as a function of γ and subsequently x∈R+

determine γc1 by requiring that this minimum is zero. The requirement G0 1 (x) = 0 gives the following equation for the minimum 2 value x0 3x20 x20 + 1 + x40 + 3x20 2 2 αx0 = α (1 + x0 + 1 ) + β (46) ( x20 + 1 + 1)2 2

3

with α := π4 + γπ and β := 1 + πγ . Deﬁning z0 := 16 quadratic equation for z0 ,

x20 + 1 this results in a

(z0 − 2) (z0 + 1) α = β (z0 − 1) (z0 + 2)

(47)

with the solution (since z0 ≥ 1 and α > β) z0 =

9α2 + 9β 2 − 14αβ . 2(α − β)

(48)

1 γ ! x0 2 [α (z0 + 1) + β (z0 − 1)] = 0 π z0

(49)

α+β+

From this one can calculate G0 12 (x0 ) = 1 −

resulting in γc1 = 0.5929. In the second step of the proof of Proposition 1, we have to investigate the s = − 21 states. For these states, one can again use ql−1 (x) ≤ q0 (x) to estimate (1) (2) the expectation values of blsm and blsm by those for l = 1 and s = − 21 . The subsequent method of calculation is the same as for the states with l = 0, s = 12 , only that q0 (x) and q1 (x) are interchanged. Instead of (44) one now obtains (fν , bl− 12 m fν ) ≥

G1− 12 (p) := 1 −

0

∞

dp |fˆν (p)|2 e(p) · G1− 12 (p),

(50)

γ p π2 2 γ2 p π4 2 A2 (p) 2 + h (p) − 2 A2 (p) 4 + h (p) . π e(p) 2 2π e(p) 4

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We will show that (with p := mx) √ √ x2 + 1 + 1 x2 + 1 − 1 π 2 γ γπ 3 γ (1 + ) + ( + ) G1− 12 (x) = 1 − x π x2 + 1 π x2 + 1 4 16

(51)

is monotonically decreasing, attaining its inﬁmum at x → ∞, namely G1− 12 (x) → 2

2

4

1 − πγ (1 + π4 ) − πγ 2 (1 + π16 ). This limit value is again strictly decreasing with γ, and at γ = γc1 = 0.5929, it equals 0.0932 > 0. This shows that (fν , bl,− 12 ,m fν ) > 0 for γ ≤ γc1 such that we have ﬁnally proved (fν , blsm fν ) > 0 for γ < γc1 . The derivative of G1− 12 (x) can be cast into the form

2 π 1 γ π2 π2 2 2+1 1+ − 1 + (52) x x +1− 2 2 2 π (x + 1) 4 4 4 3 1 π 1 π3 π3 1 2 2 + γ x − + + x +1 + − 16 π π 16 π 16 √ The right-hand side of (52) is positive for all x ∈ R+ since x2 + 1 ≥ 1, showing that G1− 12 (x) is monotonically decreasing. −G1− 1 (x) =

5 Proof of Proposition 2 (2)

In order to improve on γc1 , all contributions to the expectation value of blsm are retained. Also, the estimates introduced after the application of the Lieb and Yau formula are not made. Moreover, for the Brown-Ravenhall operator, an improved estimate for the l = 0, s = 12 states provided by Tix [12] is used (p = mx) ∞ (1) (fν , (b0m + blsm ) fν ) ≥ dp |fˆν (p)|2 e(p) · T0 12 (x), (53) γ T0 12 (x) := 1 − 2

0

√ 2 + 1 − 1)(x − arctan x) ( x arctan x + ( x2 + 1 + 1) . x (x2 + 1) arctan x − x

valid for all l, s according to [4]. Together with Lemma 2 this allows for the following representation of (32) for s = 12 , (1)

(2)

(fν , (b0m + blsm + blsm )fν ) ∞ ∞ (1) (2) ≥ (fν , (b0m + blsm )fν ) − dpdp |fˆν (p)|blsm (p, p )|fˆν (p )| 0 0 ∞ ∞ ∞ (2) 2 ˆ dp |fν (p)| e(p) · T0 12 (x) − dp dp |fˆν (p)| b0 1 m (p, p ) |fˆν (p )| . ≥ 0

0

0

2

(54)

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The second-order term is estimated by means of the Lieb and Yau formula (33) ∞ ∞ ∞ ∞ γ2 p (2) ˆ 2 2 ˆ ˆ dp dp |fν (p)| b0 1 m (p, p ) |fν (p )| ≤ dp |fν (p)| A (p) dp 2 2 2π p 0 0 0 0 ∞ p p p p · dp N (p, p , p ) q0 h2 (p ) + q1 h(p)h(p ) q0 q1 p p p p 0 p p p p − q0 h(p h(p)h(p )h(p ) − q ) . (55) q1 q 1 0 p p p p Again, the two successive substitutions z := p /p for p and ζ := p /p for p are made. Inserting (9) for A2 (p) and h(p) and setting p = mx as before, (54) with (55) is cast into the form ∞ ˜ 1 (x) dp |fˆν (p)|2 e(p) · G (56) (fν , blsm fν ) ≥ 02 0

√ x2 + 1 + 1 ˜ γ 1 4 ˜ G0 12 (x) := T0 12 (x) − I˜2 (x) I1 (x) + √ x 2 2 2 8π x +1 x +1+1 1 I˜4 (x) , − I˜3 (x) − √ x2 + 1 + 1 where we have deﬁned ∞ ∞ dz ˜ ζ2 I˜1 (x) := q0 (ζ) N q0 (z) dζ (57) 2 2 2 2 z x ζ + 1 ( x ζ + 1 + 1) 0 0 ∞ ∞ x2 ζ 2 + 1 + 1 dz ˜ ˜ q1 (z) I2 (x) := q1 (ζ) N ζdζ 2 2 2 2 x ζ +1 x ζ z2 + 1 + 1 0 0 ∞ ∞ dz ζ2 ˜ q1 (z) I˜3 (x) := q0 (ζ) N dζ 2 2 2 2 2+1+1 x ζ + 1 x ζ z 0 0 ∞ ∞ dz ˜ 1 ˜ I4 (x) := N q0 (z) . ζdζ q1 (ζ) 2 2 z x ζ +1 0 0 2

with 1 1 ˜ := N + √ . 2 2 2 2 2 2 x ζ z +1+ x ζ +1 x + 1 + x2 ζ 2 + 1 For the numerical evaluation, the integration interval is reduced to [0, 1] by means of splitting the integrals at 1 and making a variable substitution z → 1/z. It ˜ 0 1 (x) is a monotonically decreasing function of x, is found numerically that G 2 attaining its inﬁmum at x → ∞. From (56) and (57) one derives 2 4 ˜ 0 1 (x) = 1 − γ [ π + 2 ] − γ [ π + 4 − π 2 − π 2 ] . inf G 2 x∈R+ 2 2 π 8π 2 4

(58)

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The limit x → ∞ of Tix’s [12] approximation (53) is the same as for the estimate (45) of the linear term in γ introduced in the previous section. The critical value of γ is obtained from 1 −

γ 2

π 2 + 2 π

−

γ2 8

π 2 − 2 π

2 = 0

(59)

and is given by γc2 = 0.8368. For γ < γc2 , the left-hand side of (59) is positive. From (56) it therefore follows that bl, 12 ,m > 0 for γ < γc2 . For s = − 21 , we have in place of (54) (1)

(2)

(fν , (b0m + blsm + blsm ) fν )

(1)

≥ (fν , (b0m + b1,− 1 ,m ) fν ) − 2

∞ ∞

0

0

(2)

dp dp |fˆν (p)| b1,− 1 ,m (p, p )|fˆν (p )| . (60) 2

For the linear term, again a better estimate than the one given in (50) is needed. In contrast to (42), the factor h(p)h(p ) is kept in the kernel when applying the Lieb and Yau formula (33). Then ∞ ∞ γ ∞ (1) dpdp |fˆν (p)| − b1,− 1 ,m (p, p ) |fˆν (p )| ≤ dp |fˆν (p)|2 A2 (p) (61) 2 π 0 0 0 ∞ ∞ p p p p · dp q1 dp h(p ) q0 + h(p) . p p p p 0 0 The ﬁrst of the integrals over p equals 2p as before, and for the second one the substitution z := p /p and p = mx are used. One ﬁnds ∞ ˜ 1 (x), (fν , blsm fν ) ≥ dp |fˆν (p)|2 e(p) G (62) 1− 2 0

√ √ 2 2 2+1+1 2 x x γ ˜ 1− 1 (x) := 1 − x ˜0 (x) − γ x4 x + 1 + 1 G + J 2 2 2 2 π x +1 2(x + 1) 8π x2 + 1 1 1 J˜2 (x) − J˜3 (x) − √ J˜4 (x) , · J˜1 (x) + √ x2 + 1 + 1 x2 + 1 + 1 where J˜i (x), i = 1, . . . , 4 is obtained from I˜i (x) by interchanging q0 with q1 everywhere, and ∞ 1 ˜ J0 (x) := . dz q0 (z) √ 2 2 x z +1+1 0 ˜ 1− 1 (x) is numerically found to decrease monotonically in x with its inﬁmum G 2 (at ∞) again given by the right-hand side of (58). Moreover, one always has ˜ 1− 1 (x) > G ˜ 0 1 (x). Thus G ˜ 1− 1 (x) > 0 if γ < γc2 where γc2 is determined G 2 2 2

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˜ 1 (x) = lim G ˜ 1 (x) = 0. This shows by means of (62) that from inf G 1− 2 1− 2 x→∞

x∈R+

bl,− 12 ,m > 0 for γ < γc2 . Collecting results, we have blsm > 0 for s = ± 12 and γ < γc2 , which proves Proposition 2. The present proof of positivity by means of the Lieb and Yau formula cannot be extended to provide critical coupling constants beyond γc2 . This is lower than the Brown-Ravenhall critical coupling constant [4] γ˜c = 0.906 (Z ≤ 124), derived from (58) by dropping the quadratic term. A comparison of (59) with the deﬁning equation of the critical coupling constant γc = 1.006 for m = 0 [2] reveals that these equations only diﬀer in the sign of the quadratic term. This sign, however, has been made negative by force in the course of our proof in order to allow for the subsequent estimates. Hence we conjecture that also for m = 0, positivity holds for γ < γc and not just for γ < γc2 .

Appendix We derive an analytical expression for the diﬀerence Ql (x) − Ql+2s (x) of the Legendre functions of the second kind for the limit x → 1 from above. From the representation of Ql (x) in terms of hypergeometric functions 2 F1 one has [5, p. 999] Γ(l + 1)Γ( 12 ) −l−1 3 1 l+2 l+1 , ,l + , 2) x 2 F1 ( 3 l+1 x→1 2 2 2 2 x Γ(l + 2 )

Ql (1) = lim = lim

Γ(l + 1)Γ( 12 )

z→1

[2ψ(1) − ψ(

l+1 2l+1 Γ( l+2 2 )Γ( 2 )

l+1 l+2 ) − ψ( ) − ln(1 − z)] 2 2

with z := 1/x2 where the continuation of the hypergeometric function near z = 1 in terms of Euler’s psi function has been used [1, p. 559]. From this representation, one obtains with the help of the functional equation for the gamma function, Γ(x + 1) = xΓ(x), in the case of s = 12 , Ql (1) − Ql+1 (1) =

Γ(l + 1)Γ( 12 )

l+1 2l+1 Γ( l+2 2 )Γ( 2 )

[−ψ(

l+3 l+1 ) + ψ( )] 2 2

since the logarithmic terms drop out. With the help of the functional equation for the psi function [5, p. 945], ψ(x + 1) − ψ(x) = 1/x, and the product formula for the gamma function [5, p. 938] one ﬁnds Ql (1) − Ql+1 (1) =

Γ(l + 1)Γ( 12 )

l+1 2l Γ( l+2 2 )Γ( 2 )

·

1 1 = . l+1 l+1

Reducing l by 1 one recovers from (63) the result for s = − 21 , Ql (1) − Ql−1 (1) = − which proves the assertions (14) and (15).

1 l

(63)

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Acknowledgment The authors would like to thank Heinz Siedentop and Edgardo Stockmeyer for stimulating discussions.

References [1] Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965. [2] Raymond Brummelhuis, Heinz Siedentop, Edgardo Stockmeyer, The ground state energy of relativistic one-electron atoms according to Jansen and Heß, Doc. Math. 7, 167–182 (2002). [3] Marvin Douglas and Norman M. Kroll, Quantum electrodynamical corrections to the ﬁne structure of helium, Annals of Physics 82, 89–155 (1974). [4] William Desmond Evans, Peter Perry, Heinz Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter, Comm. Math. Phys. 178, 733–746, (1996). [5] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products. Academic Press, New York, 1965. [6] D.H. Jakubaßa-Amundsen, The essential spectrum of relativistic one-electron ions in the Jansen-Heß model, Math. Phys. Electr. Journal 8(3), 1–30 (2002). [7] D.H. Jakubaßa-Amundsen, Analysis of the projected one-electron Dirac operator with the help of pseudodiﬀerential operator techniques, Submitted to Doc. Math., (2003). [8] Georg Jansen and Bernd Heß, Revision of the Douglas-Kroll transformation, Physical Review A 39, 6016–6017 (1989). [9] Elliott H. Lieb and Horng-Tzer Yau, The stability and instability of relativistic matter, Commun. Math. Phys. 118, 177–213, (1988). [10] J. Sucher, Foundations of the relativistic theory of many-electron atoms, Phys. Rev. A 22, 348–362 (1980). [11] J. Sucher, Relativistic many-electron Hamiltonians. Phys. Scripta 36, 271–281 (1987). [12] C. Tix, Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall, Bull. London Math. Soc. 30, 283–290, (1998).

Vol. 4, 2003

Jansen-Heß Operator

A. Iantchenko Malm¨ o University School of Technology and Society S-20506 Malm¨ o, Sweden email: [email protected] D.H. Jakubaßa-Amundsen Mathematics Institute University of Munich Theresienstr. 39 D-80333 Munich, Germany email: [email protected] Communicated by Rafael D. Benguria submitted 14/04/03, accepted 03/06/03

To access this journal online: http://www.birkhauser.ch

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Ann. Henri Poincar´e 4 (2003) 1101 – 1136 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061101-36 DOI 10.1007/s00023-003-0157-1

Annales Henri Poincar´ e

Binding Conditions for Atomic N -electron Systems in Non-Relativistic QED Jean-Marie Barbaroux, Thomas Chen and Semjon Vugalter

Abstract.We examine the binding conditions for atoms in non-relativistic QED, and prove that removing one electron from an atom requires a positive energy. As an application, we establish the existence of a ground state for the Helium atom.

Dedicated to Professor G. Zhislin, on the occasion of his seventieth birthday.

1 Introduction One of the most fundamental results in the spectral theory of multiparticle Schr¨ odinger operators is the proof of the existence of a ground state for atoms and positive ions. It was accomplished for the Helium atom by T. Kato in 1951 [9], and for an arbitrary atom by G. Zhislin in 1960 [12] (cf. the Zhislin theorem in [11]). The standard approach to the proof of these results consists of two main parts. The ﬁrst key ingredient is the HVZ – (Hunziker-van Winter-Zhislin) theorem, which establishes the location of the essential spectrum, and gives a variational criterion for the existence of a bound state. The latter can be referred to as “binding conditions”. The statement is that the bottom of the essential spectrum of the whole system is deﬁned by its decomposition into two clusters. If the inﬁmum of the spectrum of the entire system is, for all non-trivial cluster decompositions, less than the sum of the inﬁma of the spectra of the subsystems, it follows that the whole system possesses a ground state. For an atom with inﬁnite nuclear mass, this condition can be written as E V (N ) < E V (N ) + E 0 (N − N ) for all N < N,

(1)

where E V (N ) is the inﬁmum of the spectrum of the atom, E V (N ) is the inﬁmum of the spectrum of the same atom without (N − N ) electrons, and E 0 (N − N ) is the inﬁmum of the spectrum of the system of (N − N ) electrons, which do not interact with the nucleus. Obviously, in the case of Schr¨ odinger operators (in Quantum mechanics) E 0 (N − N ) = 0, and according to the HVZ theorem, it suﬃces to consider only the decompositions with N = N − 1 in (1). The second key ingredient consists of the construction of a trial state for the Hamiltonian of the whole atom with energy less than E V (N − 1). As noted

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above, this step was accomplished by T. Kato for Helium, and by G. Zhislin for the general case. The problem of the existence of the ground states of atoms has attracted new attention in the context of non-relativistic quantum electrodynamics in the more recent literature. Bach, Fr¨ ohlich and Sigal [2] ﬁrst established the existence of the ground state for the ultraviolet regularized Pauli-Fierz Hamiltonian of an atom, for suﬃciently small values of some constants in the theory. It was subsequently established in [8] that the criterion for the existence of the ground state of multiparticle Schr¨ odinger operators can be extended to hold for Pauli-Fierz Hamiltonians in non-relativistic QED, for arbitrary values of the parameters of the theory. 1 The problem, however, of devising a mathematically rigorous proof of the fact that the binding conditions are fulﬁlled for atoms apart from the one electron case, which was covered by [8], has turned out to be very complicated. To clarify the main obstacles, let us recall the basic idea underlying the proofs of the Kato and Zhislin theorems. If the system is separated into a pair of clusters, one of which contains N − 1 electrons close to the nucleus, and the other comprises a single electron far away, there is an attractive Coulomb potential that acts on the separated particle. If the latter is localized in a ball of radius R centered at some point with distance bR from the origin, and the subsystem with N − 1 electrons is localized in a ball of radius R centered at the origin, the intercluster Coulomb interaction can be estimated as CR−1 with C < 0 for b > N. At the same time, localizing the subsystems in these balls requires an energy CR−2 in the case of Schr¨ odinger operators. For large R, the Coulomb term is obviously dominant, and the binding condition is fulﬁlled. This is contrasted by the situation in non-relativistic QED, where the particles have to be localized together with the quantized radiation ﬁeld. One can expect, on the basis of dimensional analysis [8], that such a localization requires an energy CR−1 , which makes it impossible to establish the dominance of the Coulomb interaction by scaling arguments. In the work at hand, it is demonstrated how this obstacle can be overcome. We prove that if the self-energy operator T0 , restricted to states with total momentum 0, possesses a ground state, it is possible to construct a state consisting of an electron coupled to a photon ﬁeld, localized in a ball of radius R with energy Σ0 + o(R−1 ), where Σ0 is the self-energy of an electron. Hence, similarly as for Schr¨ odinger operators, the localization term o(R−1 ) can again be compensated by the attractive Coulomb potential. This implies that the binding condition is fulﬁlled for decompositions into clusters with N − 1 and 1 particles. Existence of the ground state of T0 has been recently established for suﬃciently small values of the ﬁne structure constant [3]. It was proved earlier in [8] that for the decomposition into clusters with zero electrons and N electrons, the 1 A detailed review of numerous further results connected to the existence of ground states, mostly in Nelson-type models, can be found in [8]. Furthermore, also cf. [7]

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binding condition is also fulﬁlled. Thus, if an atom or a positive ion has only two electrons, the ground state exists. If an atom has more than two electrons, one must also verify the binding conditions for 1 < N − N < N . We note that in contrast to the quantum mechanical case, a system of K electrons coupled to a photon ﬁeld may have an energy smaller than the self-energy of an electron multiplied by a factor K. To control this case, it would be suﬃcient to combine a straightforward modiﬁcation of the method developed in this paper with a generalization of the results of [8], and to apply it to the case of a system without external potential, after separating the center of mass motion. This generalization is, however, beyond the scope of the present work. The ﬁrst proof of the existence of the ground states for all atoms in nonrelativistic QED has, besides numerous other important results, been accomplished by Bach, Fr¨ ohlich and Sigal in [2], by a completely diﬀerent approach. To compare the results in [2] for Helium to the results of the work at hand, we remark that the units used in our paper correspond to those in [8], which diﬀer from the ones in [2]. Furthermore, we emphasize that while the ultraviolet cutoﬀ in the quantized vector potential employed in [2] is, in our units, incorporated at a value Λ ∼ α, where α denotes the ﬁne structure constant, we are studying the corresponding case for an ultraviolet cutoﬀ at Λ ∼ 1. The parameter that accounts for the strength of the perturbation produced by the photon ﬁeld is in [2] assumed to be much smaller than a constant that depends on the ionization energy of the atom, the latter being computed for the Schr¨ odinger operator of the electron subsystem. One of the key issues in the work at hand is to devise a proof that also encompasses the strongly nonperturbative regime, where this parameter is allowed to be much larger than the ionization energy. This is achieved mainly based on the parameter independence of the results of [8], as well as of the methods developed in the present paper, in addition to exploiting the existence of the ground state of T0 for small α.

2 Definitions and main results We consider the Pauli-Fierz Hamiltonian HN for a system of N electrons in an external electrostatic potential, coupled to the quantized electromagnetic radiation ﬁeld, HN =

N 2 √ √ −i∇x ⊗ If + αAf (x ) + ασ · Bf (x ) + V (x ) ⊗ If =1

+

1 2

(2) W (|xk − x |) ⊗ If + Iel ⊗ Hf .

1≤k,≤N

el el The operator HN acts on the Hilbert space H := HN ⊗F, where HN , for N < ∞, is the Hilbert space of N non-relativistic electrons, given by the totally antisymmetric

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wave functions in (L2 (R3 ) ⊗ C2 )N , where R3 is the conﬁguration space of a single electron, and C2 accommodates its spin. We will describe the quantized electromagnetic ﬁeld by use of the Coulomb gauge condition. Accordingly, the one-photon Hilbert space is given by L2 (R3 )⊗C2 , where R3 denotes either the photon momentum or conﬁguration space, and C2 accounts for the two independent transversal polarizations of the photon. The photon Fock space is then deﬁned by Fs(n) , F= n∈N

n 2 3 (n) 2 where the n-photons space Fs = is the symmetric tensor s L (R ) ⊗ C product of n copies of L2 (R3 ) ⊗ C2 . We use units such that = c = 1, and where the √ mass of the electron equals m = 1/2. The electron charge is then given by e = α, with α ≈ 1/137 denoting the ﬁne structure constant. As usual, we will consider α as a parameter. The operator that couples an electron to the quantized vector potential is given by

ζ(|k|) ελ (k) eikx ⊗ aλ (k) + e−ikx ⊗ a∗λ (k) dk =: D(x) + D∗ (x), Af (x) = 1/2 3 2π|k| λ=1,2 R where by the Coulomb gauge condition, divAf = 0. The operators aλ , a∗λ satisfy the usual commutation relations [aν (k), a∗λ (k )] = δ(k − k )δλ,ν ,

[aν (k), aλ (k )] = 0,

and there exists a unique unit ray Ωf ∈ F, the Fock vacuum, which satisﬁes aλ (k)Ωf = 0 for all k ∈ R3 and λ ∈ {1, 2}. The vectors ελ (k) ∈ R3 are the two orthonormal polarization vectors perpendicular to k, (k2 , −k1 , 0) ε1 (k) = 2 k1 + k22

and

ε2 (k) =

k ∧ ε1 (k). |k|

The function ζ(|k|) describes the ultraviolet cutoﬀ on the wavenumbers k. We assume ζ to be of class C 1 , with compact support. The operator that couples an electron to the magnetic ﬁeld Bf = curlAf is given by

ζΛ (|k|) ikx −ikx ∗ k × iε (k) e ⊗ a (k) + e ⊗ a (k) dk Bf (x) = λ λ λ 1/2 R3 2π|k| λ=1,2

=: K(x) + K ∗ (x). In Equation (2), σ = (σ1 , σ2 , σ3 ) is the 3-component vector of Pauli matrices

0 1 0 −i 1 0 , σ2 = , σ3 = . σ1 = 1 0 i 0 0 −1

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The photon ﬁeld energy operator Hf is given by |k|a∗λ (k)aλ (k)dk. Hf = R3

λ=1,2

The potentials V and W are relatively −∆ bounded with relative bound zero and satisfy for positive γ, γ0 and r0 the following conditions: γ0 , |x|

|x| > r0 ,

(3)

γ1 , |x|

|x| > r0 .

(4)

V (x) ≤ − W (x) ≤

One of the main assumptions of the work at hand is the existence of a ground state of the one electron self-energy operator with total momentum P = 0. For its precise formulation, let us consider the case of a free electron coupled to the quantized electromagnetic ﬁeld. The self-energy operator T is given by 2 √ √ T = −i∇x ⊗ If + αAf (x) + ασ · Bf (x) + Iel ⊗ Hf . We note that this system is translationally invariant, that is, T commutes with the operator of total momentum Ptot = pel ⊗ If + Iel ⊗ Pf , ∗ kaλ (k)aλ (k)dk denote the electron and the photon λ=1,2

where pel and Pf = momentum operators. Let HP ∼ = C2 ⊗ F denotes the ﬁbre Hilbert space corresponding to conserved total momentum P . For any ﬁxed value P of the total momentum, the restriction of T to the ﬁbre space HP is given by (see, e.g., [3]) √ √ (5) T (P ) = (P − Pf + αAf (0))2 + ασ · Bf (0) + Hf . We denote Σ = inf σ(T ) and Σ0 = inf σ(T (0)). The following assumptions will be used to formulate the main result Condition C1 . i) Σ = Σ0 ii) Σ0 is an eigenvalue of T (0), with associated eigenspace EΣ0 . iii) There exists Ω0 ∈ EΣ0 with a ﬁnite expectation number of photons, i.e.,

Nf Ω0 , Ω0 < c, where Nf =

λ=1,2

a∗λ (k)aλ (k)dk.

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iv) The above eigenfunction Ω0 fulﬁlls, for λ = 1, 2 and some p0 ∈ (6/5, 2] ∇k aλ (k)Ω0 ∈ Lp0 (R3 ) + L2 (R3 ) . Condition i) was studied by Fr¨ ohlich for a spinless Pauli-Fierz model, [6], who proved that in this case, it is fulﬁlled for all α > 0. For the case including the σ · B term, it was proved in [4] that for small α, the condition is also fulﬁlled. The existence of the eigenspace EΣ0 in ii) was recently proved for suﬃciently small α [3], [4]. Finally, it will be proved in the present paper that for small α, the function Ω0 possesses the properties iii) and iv). Thus, we conclude that there exists a number α0 , such that at least for all α ≤ α0 , condition C1 is fulﬁlled. The second main set of assumptions required for our analysis is given as follows. For M ∈ N, let HM denote the Pauli-Fierz Hamiltonian for M electrons deﬁned in (2). Condition C2 . i) The operator HM has a ground state el ⊗ F, Υ ∈ H = HM

(6)

with a ﬁnite expectation number of photons. ii) For λ = 1, 2 and some p0 ∈ (6/5, 2], (Iel ⊗ ∇k aλ (k))Υ ∈ Lp0 (R3 ) + L2 (R3 ). iii) Let xi i = 1, . . . , M be the position vectors of the electrons. Then, M |xi | ⊗ If Υ ∈ H. i=1

For M ∈ N, let EM = inf σ(HM ). The main result of this article is the following Theorem 2.1 For N ∈ N, let the Conditions C1 and C2 with M = N −1 be fulﬁlled, and assume that the potentials V and W satisfy (3) and (4), with γ0 /γ1 > (N −1). Then, (7) EN < EN −1 + Σ. Remark 2.1 If one assumes that the system with M electrons satisﬁes the binding condition of [8], it was shown in [8] that this system possesses a ground state which satisﬁes all the conditions of C2 . In particular, the ground state of the hydrogen atom fulﬁlls C2 .

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This theorem shows that under the above-stated conditions, removing one electron from the system costs energy. In this sense, the system is stable with respect to the given type of ionization. The conditions on the potential V (x) and W (x) cover a large number of models in atomic and molecular physics. In particular, for V (x) = −βZ/|x| and W = β/|x|, the operator HN describes an atom or ion with N electrons. In the physical case, β is equal to the Sommerfeld ﬁne structure constant α. However, we would like to emphasize that the proof of the Theorem is valid for all values of β > 0, even in the strongly nonperturbative regime 0 < β α. Theorem 2.1 states that as long as the number of electrons N is less than Z + 1 (neutral atoms and positive ions), ionization by separation of one electron is energetically disadvantageous. If was earlier proved in [8] that removal of all electrons from the atom also leads to an increase of the energy. Combining these two results for the case N = 2, and the binding condition in [8, Theorem 3.1], yields Theorem 2.2 The Pauli-Fierz Hamiltonian for Helium H2 =

2 =1

√ √ 2 (−i∇x ⊗ If + αAf (x )) + ασ · Bf (x ) −

2α |x |

⊗ If

α + |x1 −x ⊗ If + Iel ⊗ Hf 2|

has a ground state for all α ≤ α0 . Notice that the conditions on the potential V (x) require only some type of behaviour at inﬁnity. Therefore, instead of one nucleus with Coulomb potential of charge Z, one can consider a system of nuclei V (x) =

k i=1

αZi |x − Ri |

with the same total charge, in the inﬁnite mass approximation. In particular, for Hydrogen molecules as well as for all molecular ions with two electrons, Theorem 2.1 implies the existence of a ground state for all α ≤ α0 .

3 Properties of the ground state of T (0). This section addresses the main properties of the self-energy operator T (0) that are required for the present analysis. In particular, existence of a ground state Ω0 ∈ C2 ⊗ F, ﬁniteness of the expected photon number with respect to Ω0 , and regularity of aλ (k)Ω0 are discussed.

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Existence theorem

In the following theorem, existence of a ground state of T (0), and bounds on the associated expected photon kinetic energy are established. Theorem 3.1 For α suﬃciently small, Σ0 = infσ(T (0)) is a degenerate eigenvalue, bordering to absolutely continuous spectrum, which satisﬁes |Σ0 | ≤ cα . Let EΣ0 = ker(T (0) − Σ0 ) ⊂ C2 ⊗ F denote its eigenspace. Then, dimC EΣ0 = 2, and for any Ω0 ∈ EΣ0 , normalized by Ω0 , Ωf = 1, the estimate √ Ω0 ≤ 1 + c α is satisﬁed. Furthermore, √ 1/2 Af (0)Ω0 , Hf Ω0 ≤ c α

(8)

hold. All constants are uniform in α. For the spinless case, both results are proved in [3] by use of the operatortheoretic renormalization group based on the smooth Feshbach map, cf. [1]. For the case including spin, an outline of the proof is given in the appendix of [4], while a publication containing the detailed proof is in preparation. The bound on Af (0)Ω0 follows straightforwardly from the one on Hf Ω0 . 3.1.1 Expected photon number Using Theorem 3.1, we may next bound the expected photon number with respect to Ω0 . Theorem 3.2 For α suﬃciently small, and Ω0 ∈ EΣ0 deﬁned as in Theorem 3.1, 1/2 Ω0 ∈ D(Nf ), where Nf = λ=1,2 a∗λ (k)aλ (k)dk is the photon number operator, and √ 1/2 Nf Ω0 2 < c α . In particular,

√ χ(|k| < 1)aλ (k)Ω0 ≤ c α|k|−1 .

All constants are uniform in α. Proof. We ﬁrst remark that the integral dk aλ (k)Ω0 2 is ultraviolet ﬁnite, since 2 χ(|k| ≥ 1)|k| aλ (k)Ω0 2 dk χ(|k| ≥ 1) aλ (k)Ω0 dk < ≤

Ω0 , Hf Ω0

≤

cα ,

(9)

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using ( 8). We may thus assume that the domain of the integral is the unit ball B1 (0). For |k| < 1, we employ a similar argument as in [6, 2, 8]. Using : T (0) : −Σ0 aλ (k)Ω0 = : T (0) : , aλ (k) Ω0 , where : ( · ) : denotes Wick ordering, and Σ0 := Σ0 − Af (0)2 Ωf = inf σ(: T (0) :) , we obtain aλ (k)Ω0

where

=

√ ζ(|k|) αR(k) k · Af (0) + 1/2 λ (k) · Pf |k| √ ζ(|k|) ζ(|k|) + 1/2 ik ∧ λ (k) · σ + α 1/2 λ (k) · Af (0) Ω0 , |k| |k|

−1 1 R(k) := Hf + |k| + (Pf + k)2 − Σ0 . 2

(10)

(11)

Clearly, Ωf , : T (0) : Ωf = 0, and a standard variational argument shows that Σ0 < 0 for α > 0. Hence, 0 < R(k) < (Hf + |k|)−1 , and R(k)Pf ≤ R(k)Hf ≤ 1 . Thus, using R(k)|k| ≤ 1 and Theorem 3.1, χ(|k| < 1)aλ (k)Ω0

√ ≤ c αχ(|k| < 1) Af (0)Ω0 + 2|k|−1/2 Ω0 √ + α|k|−1 Af (0)Ω0 √ (12) ≤ c α|k|−1 .

The right-hand side is in L2 (B1 (0)), and the assertion is established.

For the case of a conﬁned electron, it was proved in [8] that the corresponding estimate exhibits a |k|−1/2 singularity instead of |k|−1 as present here, owing to the exponential decay of the particle wave function. Furthermore, if the conserved momentum P is non-zero, there exists a ground state ΩP (κ) for a regularized version of the model, which includes an infrared cutoﬀ below 0 < κ 1 in Af (0) (some requirements on the cutoﬀ function are necessary, cf. [3]). Then, with all modiﬁcations implemented, the additional term √ ζ(|k|) αR(k) 1/2 P · λ (k) ΩP (κ) |k| enters the right-hand side of (10). Therefore, ΩP (κ), Nf ΩP (κ) is logarithmically infrared divergent in the limit κ → 0, for all |P | > 0, and in fact, ΩP (κ) does not converge to an element in Fock space.

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3.1.2 Regularity properties of the ground state Next, we derive a result about the regularity of aλ (k)Ω0 in momentum space, which is, in our further discussion, used for photon localization estimates in position space. Theorem 3.3 For α suﬃciently small, let Ω0 ∈ EΣ0 . Then, ∇k aλ (k)Ω0 ∈ Lp (R3 ) + L2 (R3 ) , for 1 ≤ p < 32 . Proof. We proceed similarly as in [8]. To begin with, we diﬀerentiate the right-hand side of (10) with respect to k, and observe that |∇k R(k)| ≤ (1 + Hf + |k|)R2 (k) ,

(13)

since |Pf | ≤ Hf . Let us ﬁrst bound the ultraviolet part of ∇k aλ (k)Ω0 . For |k| ≥ 1, χ(|k| ≥ 1)∇k aλ (k)Ω0 = ≤ ≤

√ α χ(|k| ≥ 1)∇k R(k)k · Af (0)Ω0 χ(|k| ≥ 1)(1 + Hf + |k|)R(k) + |k|−1 √ αχ(|k| ≥ 1)R(k)k · Af (0)Ω0 √ 2 α χ(|k| ≥ 1)aλ (k)Ω0 , (14)

and consequently, by Theorem 3.2, ∇k aλ (k)Ω0 2 dk ≤ cα .

(15)

|k|≥1

We may thus restrict our discussion to the case |k| < 1. Diﬀerentiating with respect to k, the photon polarization vectors satisfy c |∇k λ (k)| ≤ 2 . k1 + k22

(16)

Recalling that the cutoﬀ function ζ is of class C 1 , and using Theorem 3.2, one straightforwardly deduces that there exists a constant c which is uniform in α, such that √ 1 1 + χ(|k| < 1)∇k aλ (k)Ω0 ≤ c α |k|2 |k| k12 + k22 √ c α ≤ . (17) |k| k12 + k22

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1111

Here, one again uses R(k)Pf ≤ R(k)Hf ≤ 1, and R(k)|k| ≤ 1, in addition to ( 13). Thus, by the H¨ older inequality, 1/p ∇k aλ (k)Ω0 p dk |k|<1

√ ≤ C α

|k|<1

1/r 1 dk |k|r/2 (k12 + k22 )r/2

|k|<1

1 |k|r∗ /2

dk

1/r∗

(18) ,

with p1 = 1r + r1∗ . The integrals on the right-hand side of (18) are bounded for the choices 1 ≤ r∗ < 6, and 1 ≤ r < 2, which implies that 1 ≤ p < 32 , corresponding to the exponent expected from scaling. In the case of a conﬁned electron, [8], the bound analogous to ( 17) is The reason for the fact that it is by a factor |k|1/2 less singular

√ c α √ 2 2. 1/2 |k| k1 +k2

is stated in a previous remark. Consequently, in [8], the inequality corresponding to (18) likewise requires the choice r < 2, but in contrast, r∗ can be chosen arbi1 trarily large. Therefore, the result proved in [8] holds for p1 > 12 + ∞ = 12 , that is, 1 ≤ p < 2.

4 Self-energy of localized states with total momentum P = 0 The goal of this chapter is to arrive at a sharp upper bound on the inﬁmum of the quadratic form of the operator T (0), when restricted to states where all photons are localized in a ball of radius R centered at the origin. To this end, we recall that for the Schr¨ odinger operator −∆ corresponding to a free electron, the inﬁmum of the spectrum on the whole space is zero, whereas the inﬁmum on functions supported in a ball of radius R, with Dirichlet boundary conditions, is C/R2 . The main result of this section is the following. Theorem 4.1 For all R > 0, there exists a function ΦR ∈ D(T (0)), such that i) The n photonic components ΦR n (y1 , . . . , yn ; λ1 , . . . , λn ) fulﬁll suppΦR n ⊂ {(y1 , . . . , yn ; λ1 , . . . , λn ) | sup |yi | < R} i

ii)

T (0)ΦR , ΦR ≤

c(R) Σ0 + ΦR 2 , R

(19)

where c(R) tends to zero as R tends to inﬁnity. iii) The function ΦR has the following additional properties. For all ε > 0 and all |x| > 2R, c(|x|) Φ 2 , | D(x)ΦR , ΦR | ≤ (20) |x|

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| D(x)2 ΦR , ΦR | ≤

c(|x|) Φ 2 , |x|2

| D∗ (x)D(x)ΦR , ΦR | ≤ and | K(x)ΦR , ΦR | ≤

Ann. Henri Poincar´e

c(|x|) Φ 2 , |x|2

c(|x|) Φ 2 |x|

(21) (22)

(23)

where c(|x|) tends to zero, uniformly in R, as |x| tends to inﬁnity. Before addressing the proof of Theorem 4.1, we shall ﬁrst demonstrate how it can be employed to construct a state in H1 ⊗ F that accounts for a system consisting of an electron coupled to a photonic ﬁeld, localized in a ball of radius R centered at a ﬁxed point b, with energy close to the self-energy Σ0 . For that purpose, let us, for given x ∈ R3 , deﬁne the shift operator τx : F → F, which, for φ = (φ0 , φ1 , . . . , φn , . . .) ∈ F, is given by τx φn (y1 , . . . , yn ; λ1 , . . . , λn ) = φn (y1 − x, . . . , yn − x; λ1 , . . . , λn ). Theorem 4.2 Let f be a real-valued function in C02 (R3 ) ⊗ C2 , supported in the unit ball centered at the origin. For R > 0 and b ∈ R3 , we deﬁne ΘR,b ∈ H1 ⊗ F by ΘR,b =

R f ( x−b R ) ⊗ τx Φ . x f ( R ) ⊗ ΦR

(24)

Then, for all ε > 0 and R large enough independent of b, we have √ √ ε (i∇x ⊗ If + αAf (x))2 + ασ.Bf (x) + Iel ⊗ Hf ΘR,b , ΘR,b ≤ Σ0 + . (25) R Proof of Theorem 4.2. For a real-valued function f , let f R,b (x) := f ((x − b)/R). Obviously, √ √ (i∇x ⊗ If + αAf (x))2 + ασ.Bf (x) + Iel ⊗ Hf ΘR,b , ΘR,b (26) 1 R,b R,b R 2 2 R R = f , f Φ + f

T (0)Φ , Φ .

−∆ x x f ( R ) ⊗ ΦR 2 According to Theorem 4.1, the second term on the right-hand side can be estimated by c(R) f 2 T (0)ΦR , ΦR . ≤ Σ0 + R,b R 2 f (x) ⊗ Φ R For the ﬁrst term on the right-hand side of (26), we have

−∆x f R,b , f R,b c ≤ 2 , f R,b (x) 2 R which completes the proof of the Theorem.

Binding Conditions for Atomic N -electron Systems

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4.1

1113

Localization estimates

In order to prove Theorem 4.1, we consider the ground state Ω0 of the self-energy operator T (0) at zero momentum, and act on it with two spatial localization functions U R and V R , which constitute a partition of unity (U R )2 + (V R )2 = 1 on F . This yields a state for which all photons are inside the ball of radius R, and another state for which all photons are outside the ball of radius R/2. Clearly, the expectation of T (0) with respect to Ω0 is not equal to the sum of the expectations with respect to the two localized states. The diﬀerence, which is usually called the localization error, must be estimated to obtain an upper bound on the self-energy of the localized state. In the present subsection, we estimate the localization errors for diﬀerent terms in the operator T (0). Let us begin with deﬁning spatial cutoﬀ functions u and v as follows. We pick u ∈ C0∞ (R+ ) such that u(x) =

1 0

if x ∈ [0, 1/2] if x ≥ 1

,

(27)

√ 0 ≤ u ≤ 1 and v := 1 − u2 ∈ C 2 (R+ ). For Y = (y1 , y2 , . . . , yn ) ∈ Rn , we denote Y ∞ = max1≤i≤n |yi |. For n ∈ N Y ∞ R 2 1 − uR and all Y ∈ Rn , we also deﬁne uR n (Y ) . n (Y ) = u( R ) and vn (Y ) = R R Next, we introduce a pair of operators U and V on F by R U R ψ = ψ0 , uR 1 (y1 )ψ1 (y1 ), . . . , un ((y1 , . . . , yn ))ψn (y1 , . . . , yn ), . . . ,

(28)

V R ψ = ψ0 , v1R (y1 )ψ1 (y1 ), . . . , vnR ((y1 , . . . , yn ))ψn (y1 , . . . , yn ), . . . ,

(29)

and

where we have omitted the polarization indices from the notation. 4.1.1 Localization error for the field energy Hf Lemma 4.1 There exists c < ∞ such that for all ε > 0, and all R large enough, R

R

R

R

Hf U ψ, U ψ + Hf V ψ, V ψ − Hf ψ, ψ ≤ Nf ψ, ψ

ε c V R/2 ψ 2 + R εR ψ 2 (30)

holds for ψ ∈ Q(Hf ) ∩ Q(Nf ). Proof. Since Hf maps each n-photon sector of the Fock space F into itself, it suﬃces to estimate the localization error for the n-photon component of ψ. Fur(n) thermore, since Hf acts on a function in Fs as n|∇y1 |, the statement of the Lemma follows straightforwardly from Lemma 4.2.

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Ann. Henri Poincar´e

Lemma 4.2 There exists c < ∞ such that for all ε > 0, all R large enough,

|∇|u(

|y| |y| |y| |y| )φ, u( )φ + |∇|v( )φ, v( )φ − |∇|φ, φ R R R R

ε c φχ(|y| > R) 2 ≤ + . R εR φ 2

(31)

holds for all φ ∈ C0∞ (R3 ). Proof. By [10, Theorem 9], we have |y| |y| |y| |y|

|∇|φ, φ − |∇|u( )φ, u( )φ − |∇|v( )φ, v( )φ R R R R 2 2 |y| 1 |z| |z| |φ(y)||φ(z)| |y| = u( R ) − u( R ) + v( R ) − v( R ) dydz. 2π 2 |y − z|4 (32) Let us consider

I=

|φ(y)||φ(z)| |y − z|4

2 |y| u( ) − u( |z| ) dydz. R R

(33)

The term with the function v can be estimated similarly. By symmetry, it suﬃces to estimate this integral in the region where |y| ≤ |z|. We split the integral I into three parts I1 , I2 , and I3 , respectively, corresponding to the regions R1 = {|z| < R/2}, R2 = {|z| > R/2, |y − z| > R/4} and R3 = {|z| > R/2, |y − z| < R/4}. Since |y| ≤ |z|, we have, in the region R1 , |y| ≤ |z| < R/2. Thus, in R1 , we |z| have u( |y| R ) − u( R ) = 0. Therefore, I1 = 0 . Now, for all ε > 0

|φ(y)|2 |φ(z)|2 1 dydz + dydz 4 ε R2 |y − z|4 R2 |y − z|

1 1 φχ(|z| > R/2) 2 ≤ c ε φ 2 + R εR

I2 ≤ ε

(34)

where c is a constant independent of ε. Finally, since the derivative of u is bounded, we have the inequality |u(|y|/R) − u(|z|/R)|2 ≤ c|y − z|2 /R2 . This implies |φ(y)||φ(z)| |y − z|2 dydz I3 ≤ c |y − z|4 R2 R3 c ε|φ(y)|2 + (1/ε)|φ(z)|2 (35) ≤ 2 dydz R R3 |y − z|2 cε 1 φχ(|z| > R/2) 2 . ≤ φ 2 + R εR

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1115

4.1.2 Localization error for the operator Pf2 Lemma 4.3 There exists c < ∞ such that for all ε > 0 and all R large enough,

Pf U R ψ, Pf U R ψ + Pf V R ψ, Pf V R ψ − Pf2 ψ, ψ ≤

c

Nf ψ, ψ R2

(36)

holds for ψ ∈ Q(Hf ) ∩ Q(Nf ). Proof. The operator Pf maps each n-photon sector into itself. Therefore, it is (n) suﬃcient to restrict the proof to Fs . We have R R R 2

Pf uR n ψn , Pf un ψn + Pf vn ψn , Pf vn ψn − Pf ψn , ψn R R R =

∇i uR n ψn , ∇j un ψn + ∇i vn ψn , ∇j vn ψn − ∇i ∇j ψn , ψn i,j

=

R R R

uR n ∇i ψn , un ∇j ψn + vn ∇i ψn , vn ∇j ψn − ∇i ∇j ψn , ψn

i,j

+2

R

(∇i uR n )(∇j ψn ), un ψn

+

(37)

(∇i vnR )(∇j ψn ), vnR ψn

i,j

+

R R R

ψn ∇i uR n , ψn ∇j un + ψn ∇i vn , ψn ∇j vn .

i,j 2 R 2 Since (uR n ) + (vn ) = 1, the ﬁrst three terms on the right-hand side of the second equality sign in (37) add up to zero. Similarly, by rewriting the subsequent pair of terms as 2 R 2

(∇i (uR n ) )(∇j ψn ), ψn + (∇i (vn ) )(∇j ψn ), ψn , i,j R R R we again obtain zero. Next, we note that ∇i uR n and ∇j un (∇i vn and ∇j vn ) for i = j are bounded functions with supports overlapping only on a set of measure zero. This implies that in the last sum in (37), only the terms with i = j may be diﬀerent from zero. Thus, we obtain R R R

ψn ∇i uR n , ψn ∇j un + ψn ∇i vn , ψn ∇j vn i,j

=

2 R 2

ψn |∇i uR n | , ψn + ψn |∇i vn | , ψn

i

(38)

2 R 2 = n ψn |∇1 uR n | , ψn + ψn |∇1 vn | , ψn c ≤ n 2 ψn 2 , R 2 where in the last inequality, we used that for some constant c, we have |∇uR n| ≤ −2 R 2 −2 cR and |∇vn | ≤ cR .

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Ann. Henri Poincar´e

4.1.3 Localization error for Pf Af (0) Lemma 4.4 Let ψ ∈ Q(Pf Af (0)) ∩ D(Pf ) ∩ D(Nf ), and assume that for some p0 ∈ (6/5, 2], ∇k aλ (k)ψ F ∈ Lp0 (R3 ) + L2 (R2 ). Then, the inequality Pf Af (0)U R ψ, U R ψ + Pf Af (0)V R ψ, V R ψ − Pf Af (0)ψ, ψ ≤

c R1+δ

(39)

holds with δ = (p0 − 6/5)/2.

Proof. Throughout this proof, we will write dy for integration over the y variable, and summation over the polarization λ. Here and in the rest of the paper, we deﬁne Gλ (x) as the Fourier transform of the vector function ελ (k) 1

|k| 2

ζ(k).

In addition, everywhere where it does not lead to any misunderstanding, we will omit the photon polarization index λ. We have

Pf D(0)U R ψ, U R ψ + Pf D(0)V R ψ, V R ψ − Pf D(0)ψ, ψ n √ R R R =i n + 1 G(−yn+1 )ψn+1 (∇i ψn ) uR n+1 un +vn+1 vn −1 dy1 . . . dyn+1 n

+ =:

G(−yn+1 )ψn+1 ψn

i=1 n

R R R uR n+1 ∇i un + vn+1 ∇i vn

dy1 . . . dyn+1

i=1

(an + bn ) .

(40)

n R R R We ﬁrst estimate the term an . We denote F = uR n+1 un + vn+1 vn − 1. For |yn+1 | R R ≤ R/2, either Y ∞ = |yn+1 | and then un+1 (Y ) = un (y1 , . . . , yn ) = 1 and R vn+1 (Y ) = vnR (y1 , . . . , yn ) = 1, or Y ∞ = |yk |, for some k = n + 1, and then R R R un+1 (Y ) = uR n (y1 , . . . , yn ) and vn+1 (Y ) = vn (y1 , . . . , yn ). In both cases, we get F = 0. Thus for δ > 0 suﬃciently small, we have n √ G(−yn+1 )ψn+1 (∇i ψn )F dy1 . . . dyn+1 |an | = n + 1 |yn+1 |≥R/2 i=1 √ ≤ n+1 (1 + |yn+1 |)1−δ |G(−yn+1 )||ψn+1 |(1 + |yn+1 |)2δ |yn+1 |≥R/2

×

1 (Pf ψ)n dy1 . . . dyn+1 (1 + |yn+1 |)1+δ

Binding Conditions for Atomic N -electron Systems

Vol. 4, 2003

≤

1 R1+δ

1117

√ n + 1|ψn+1 |(1 + |yn+1 |)2δ (1 + |yn+1 |)1−δ

|G(−yn+1 )||(Pf ψ)n |dy1 . . . dyn+1 .

(41)

Applying the Schwarz inequality, we arrive at |an | ≤

21+δ √ n + 1ψn+1 (1 + |yn+1 |)2δ L2n+1 (1 + |yn+1 |)1−δ G L2 (dyn+1 ) R1+δ (Pf ψ)n L2n ,

(42)

where for brevity, L2k := L2 (dy1 , . . . , dyk ). According to Lemma 7.1 in the appendix, one ﬁnds that (1 + |yn+1 |)1−δ G L2 (dyn+1 ) is ﬁnite. Therefore, c √ n + 1ψn+1 (1 + |yn+1 |)2δ 2L2 + (Pf ψ)n 2L2n . (43) |an | ≤ 1+δ n+1 R n n We note that implies

n

∇k aλ (k)ψ F ∈ Lp0 (R3 , dk) + L2 (R3 , dk)

(n + 1) ψn+1 (y, .) 2L2n (1 + |y|)2 ∈ Lq0 /2 (R3 , dy) + L1 (R3 , dy) ,

1 p0

with + q10 = 1, by the Hausdorﬀ-Young inequality. Consequently, one can straightforwardly verify that for δ = (p0 − 6/5)/2, √ n + 1ψn+1 (1 + |yn+1 |)2δ 2L2 < c . (44) n+1

n

Moreover,

n

(Pf ψ)n 2L2n < c ,

(45)

since ψ ∈ D(Pf ). Inequalities (43)–(45) imply that c |an | ≤ 1+δ . R n

(46)

Let us turn to the estimate of bn . If maxi=1,...,n |yi | = |yn+1 |, then n R R R uR n+1 (y1 ,...,yn+1 )∇i un (y1 ,...,yn ) + vn+1 (y1 ,...,yn+1 )∇i vn (y1 ,...,yn ) =0 . i=1 R If max{y1 , . . . , yn+1 } = |yn+1 |, then ∇i uR n = ∇i vn = 0 for all (y1 , . . . , yn ), such that one ﬁnds maxk=1,...,n |yk | > |yi |. This means that except on a set of measure R R R zero in Rn , the functions uR n+1 ∇i un + vn+1 ∇i vn have disjoint supports. Therefore, n i=1

R R R uR n+1 ∇i un + vn+1 ∇i vn ≤

c . R

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Ann. Henri Poincar´e

R Moreover, ∇i uR N and ∇i vN have support in the set {|yi | ∈ [R/2, R]}, thus, since from the above, we only have to consider the region where |yn+1 | > maxi=1,...,n |yi |, we get |yn+1 | > R/2, hence c√ |bn | ≤ n |G(−yn+1 )| |ψn+1 | |ψn |dy1 . . . dyn+1 R |yn+1 |>R/2 c (47) ≤ (1 + |yn+1 |)−1/2 |G(−yn+1 )|(1 + |yn+1 |)1/2 R |yn+1 |>R/2 √ × |ψn | n|ψn+1 |dy1 . . . dyn+1 .

Applying the Schwarz inequality and Lemma 7.1, we obtain from (47) c |bn | ≤ 3/2 ψ 2 + Nf ψ 2 . R n

(48)

Inequalities (46) and (48) complete the proof of Lemma 4.4. 4.1.4 Localization error for Af (0)2 Lemma 4.5 Let ψ ∈ Q(Af (0)2 ) ∩ D(Nf ), and let for some p0 ∈ (6/5, 2] ∇k aλ (k)ψ ∈ Lp0 (R3 ) + L2 (R3 ). Then, the inequality

Af (0)2 U R ψ, U R ψ + Af (0)2 V R ψ, V R ψ − Af (0)2 ψ, ψ ≤

c R1+δ

(49)

holds with δ = (p0 − 6/5)/2. Proof. Using the canonical commutation relations, we have Af (0)2 = D(0)2 + D∗ (0)2 + 2ReD∗ (0)D(0) + cI, where the constant c depends on the ultraviolet cutoﬀ. Therefore, it is suﬃcient to compute the localization error for D(0)2 and D∗ (0)D(0). We have

D(0)2 U R ψ, U R ψ + D(0)2 V R ψ, V R ψ − D(0)2 ψ, ψ √ √ = n + 1 n + 2 G(yn+2 )G(yn+1 )ψn+2 ψn n

(50)

R R R × uR n+2 un + vn+2 vn − 1 dy1 . . . dyn+2 .

In the region where maxi=1,...,n+2 |yi | = max{|yn+1 |, |yn+2 |}, we ﬁnd R R 2 R R R 2 un+2 uR n + vn+2 vn − 1 (y1 , . . . , yn+2 ) = (un ) + (vn ) − 1 (y1 , . . . , yn ) = 0 . (51)

Vol. 4, 2003

Binding Conditions for Atomic N -electron Systems

1119

In the region where maxi=1,...,n+2 |yi | = |yn+2 | ≤ R/2, we have R uR n+2 (y1 , . . . , yn+2 ) = un (y1 , . . . , yn ) = 1

and

R vn+2 (y1 , . . . , yn+2 ) = vnR (y1 , . . . , yn ) = 0.

This yields (51) in that case. Similarly, in the region where maxi=1,...,n+2 |yi | = |yn+1 | ≤ R/2, equation (51) holds. Therefore, in (50), it suﬃces to carry out the integration in the region {(y1 , . . . , yn+2 ) | |yn+1 | ≥ R/2}∪{(y1 , . . . , yn+2 ) | |yn+2 | ≥ R/2}. Let us consider the integral in the ﬁrst region. The other will be treated the same way. We have √ √ n + 2 n + 1 G(yn+2 )G(yn+1 )ψn+2 ψn uR uR + v R v R − 1 dy1 . . . dyn+2 n+2 n n+2 n √ 21+δ ≤ 1+δ |G(yn+1 )|(1 + |yn+1 |)1−δ |G(yn+2 )| n + 1|ψn | R √ × n + 2|ψn+2 |(1 + |yn+1 |)2δ dy1 . . . dyn+2 . (52) Applying the Schwarz inequality and using (44) as in Lemma 4.4, we obtain the estimate √ √ n + 1 n + 2 G(yn+2 )G(yn+1 )ψn+2 ψn n

R 1 R R +v v −1 dy . . . dy un+2 uR ≤ c 1+δ . 1 n+2 n n+2 n R

We have

D∗ (0)D(0)U R ψ, U R ψ + D∗ (0)D(0)V R ψ, V R ψ − D∗ (0)D(0)ψ, ψ = (n + 1) G(yn+1 )G(zn+1 )ψn+1 (y1 , . . . , yn , yn+1 )ψn+1 (y1 , . . . , yn , zn+1 ) n

R × uR n+1 (y1 , . . . , yn , yn+1 )un+1 (y1 , . . . , yn , zn+1 )

R (y1 , . . . , yn , yn+1 )vnR (y1 , . . . , yn , zn+1 ) − 1 dy1 . . . dyn+1 dzn+1 . +vn+1

As before, in the region where both yn+1 and zn+1 are less than R/2, the expression inside the integral is zero. Without any loss of generality, we may assume that yn + 1 > R/2. In that case, the expression above is bounded by (n + 1)R1+δ ψn+1 (y1 , . . . , yn , yn + 1)χ(|yn + 1| ≥ R/2)G(−yn + 1) 2 +(n + 1)R−(1+δ) ψn+1 (y1 , . . . , yn , zn + 1)G(−zn + 1) 2 .

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Ann. Henri Poincar´e

Similarly to (52), we obtain ψn+1 (y1 , . . . , yn , yn + 1)χ(|yn + 1| ≥ R/2)G(−yn + 1) 2 ≤ R−2(1+δ) ψn+1 2 . Therefore,

D∗ (0)D(0)U R ψ, U R ψ + D∗ (0)D(0)V R ψ, V R ψ − D∗ (0)D(0)ψ, ψ ≤

c . R1+δ

This concludes the proof. 4.1.5 Localization error for the operator σ.Bf (0)

Lemma 4.6 Let ψ ∈ Q(σ.Bf (0)) ∩ D(Nf ), and assume that there exists p0 ∈ (6/5, 2], such that ∇k aλ (k)ψ ∈ Lp0 (R3 ) + L2 (R3 ). Then, the inequality

σ.Bf (0)U R ψ, U R ψ + σ.Bf (0)V R ψ, V R ψ − σ.Bf (0)ψ, ψ ≤

c R1+δ

(53)

holds with δ = (p0 − 6/5)/2. The proof of Lemma 4.6 is similar to the one of Lemma 4.5, with a large number of simpliﬁcations.

4.2

Proof of Theorem 4.1

We let

ΦR := U R Ω0 ,

(54)

where Ω0 is a normalized ground state eigenfunction of the operator T (0), and where U R is deﬁned in (28). We recall that we have T (0)Ω0 , Ω0 = Σ0 Ω0 2 . We would like to show that the value of the quadratic form associated to T (0) at ΦR is, for large R, close to the value of the quadratic form associated to T (0) at Ω0 . First, we notice that Ω0 fulﬁlls all the conditions of Lemmata 4.1-4.6 which implies that

T (0)Ω0 , Ω0 = T (0)U R Ω0 , U R Ω0 + T (0)V R Ω0 , V R Ω0 +

C(R) , R

where C(R) tends to zero as R tends to inﬁnity. Thus, since T (0)V R Ω0 , V R Ω0 ≥ Σ0 V R Ω0 2 , we obtain

T (0)ΦR , ΦR ≤ ≤

|C(R)| − Σ0 T (0)V R Ω0 , V R Ω0 R |C(R)| |C(R)| = Σ0 ΦR 2 + , Σ0 (1 − V R Ω0 2 ) + R R

Σ0 +

which proves ii) of Theorem 4.1.

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1121

To complete the proof of Theorem 4.1, it suﬃces to prove the two Inequalities (20) and (23). Let us start with (20). √ R | D(x)ΦR , ΦR | ≤ n + 1 |G(x − yn+1 )| |ΦR n+1 | |Φn |dy1 dyn+1 n

√ 2 = n+1 |G(x − yn+1 )|(1 + |x − yn+1 |)|ΦR n+1 | |x| |yn+1 |≤R n ×

(1 + |yn+1 |)2δ R |Φ |dy1 dyn+1 . (1 + |yn+1 |)2δ n

By applying the Schwarz inequality, we get √ R | D(x)ΦR , ΦR | ≤ n + 1 |G(x − yn+1 )| |ΦR n+1 | |Φn |dy1 dyn+1

(55)

n

=

2 G(x − yn+1 )(1 + |x − yn+1 |)(1 + |yn+1 |)−2δ ΦR n |x| n √ × n + 1(1 + |yn+1 |)2δ ΦR n+1 .

We recall that from Lemma 7.1 that |G(x − yn+1 )(1 + |x − yn+1 |)| ∈ Lr (R3 ) for all r > 2. Therefore, for p > 3/(3 − 2δ), and q given by 1/p + 1/q = 1, we have (1 + |yn+1 |)−2δ q < ∞. Thus, G(x − yn+1 )(1 + |x − yn+1 |)(1 + |yn+1 |)−2δ ΦR n ≤ G(x − yn+1 )(1 + |x − yn+1 |)χ(|yn+1 | ≤ R) p (1 + |yn+1 |)−2δ q ΦR n . Moreover, for |x| > 2R, the norm G(x − yn+1 )(1 + |x − yn+1 |)χ(|yn+1 | ≤ R) p tends to zero as R → ∞. This estimate together with (55) yields √ 2 2 2δ R 2 ε(x) ΦR . | D(x)ΦR , ΦR | ≤ n + n + 1(1 + |yn+1 |) Φn+1 |x| n Conditions C1 iii) and C1 iv) together with the above inequality conclude the proof of (20) if we pick δ = (p0 − 6/5)/2. The proofs of (21), (22), and (23) are similar.

5 Approximate ground state for a system with an external potential In the present section, we consider the Pauli-Fierz Hamiltonian for M electrons with an external potential HM =

M 2 √ √ −i∇x ⊗ If + αAf (x ) + ασ · Bf (x ) + V (x ) ⊗ If =1

+

1 2

1≤k,≤M

W (xk − x ) ⊗ If + Iel ⊗ Hf ,

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Ann. Henri Poincar´e

el acting on H = HM ⊗ F. The brackets · , · will from here on denote the scalar product on H. Furthermore, for the rest of this section, we will write operators of the form Iel ⊗ Af or Bel ⊗ If on H simply as Af or Bel , respectively, in order not to overburden the notation. The precise meaning will be clear from the context. We assume that the Condition C2 is fulﬁlled for this system, which implies, in particular, that the operator HM has a ground state. We will construct an approximation to the ground state which is spatially localized with respect to the electron and photon variables, and whose energy is close to the ground state energy.

5.1

Localization of the electrons

We start with localization in the electron conﬁguration space. To this end, we recall from ( 6) that Υ denotes the ground state of HM . For u given by (27), we R R el deﬁne ΥR = (ΥR 0 , Υ1 , . . . , Υn , . . .) ∈ H = HM ⊗ F by   ΥR n = u

2

M

i=1

|xi |2

R

  Υn ,

where Υn is the n-photon component of Υ. Notice that on the support of ΥR , we have |xi | ≤ R/2 for i = 1, . . . , M. Lemma 5.1 For all R > 1,

HM ΥR , ΥR ≤ EM + 1−

c , R2

c ≤ ΥR ≤ 1 . R2

(56) (57)

The proof of this Lemma follows immediately from standard localization error estimates for Schr¨ odinger operators [5], and the Condition C2 iii).

5.2

Localization of photons

Our next goal is to localize all photons in a ball of radius 2R centered at the origin. R R el For this purpose, we deﬁne the function ΨR = (ΨR 1 , Ψ2 , . . . , Ψn , . . .) ∈ HM ⊗ F as ΨR = U 2R ΥR . (58) el where U R straightforwardly extends the operator deﬁned on F in (28) to HM ⊗ F. We note here that the localization radius for photons is chosen to be four times larger than that for the electrons. The consequence is that the contribution of the “external” photons to the magnetic vector-potential will be negligible within the region where the electrons are localized.

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Similarly to Lemma 4.1, we ﬁnd that there exists c < ∞, such that for all ε > 0, and all R large enough,

Hf U 2R ΥR , U 2R ΥR + Hf V 2R ΥR , V 2R ΥR − Hf ΥR , ΥR

ε c V R ΥR 2 R R ≤ Nf Υ , Υ . + R εR ΥR 2

(59)

Obviously, it suﬃces to compute the localization error only for the operator √ √ (−i∇x1 + αAf (x1 ))2 + ασ · Bf (x1 ) + Hf . In the rest of this section, we will denote x = x1 . Lemma 5.2 The following estimate holds 2R R U Υ , i∇x Af (x)U 2R ΥR + V 2R ΥR , i∇x Af (x)V 2R ΥR − ΥR , i∇x Af (x)ΥR c ≤ 1+δ Nf ΥR 2 + ∇x ΥR 2 , R (60) where δ = (p0 − 6/5)/2 and p0 is given by C2 ii). Proof. The proof of this lemma is very similar to the one of Lemma 4.4. 2R R U Υ , i∇x D(x)U 2R ΥR + V 2R ΥR , i∇x D(x)V 2R ΥR − ΥR , i∇x D(x)ΥR √ R ≤ dx n + 1 |Gλ (x − yn+1 )| |ΥR (61) n+1 | |∇x Υn | |x|≤ R 2

n

2R 2R 2R ×(u2R n un+1 + vn vn+1 − 1)dy1 . . . dyn+1 . 2R 2R 2R Similarly to Lemma 4.4, we show that (u2R n un+1 + vn vn+1 − 1) is nonzero only if |yn+1 | ≥ R. This implies |x − yn+1 | ≥ |yn+1 |/2 ≥ R/2. Therefore, the integral in (61) can be estimated by √ 1 2δ dx n + 1 ΥR n+1 (1 + |yn+1 |) 1+δ R |x|≤ R 2 n (62) Gλ (x − yn+1 )(1 + |x − yn+1 |)1−δ ∇x ΥR × n λ

Since the term ∇x ΥR is ﬁnite, the rest of the proof is not diﬀerent from the one of Lemma 4.4. Similarly to Lemmata 4.5 and 4.6 and the above Lemma 5.2, one can prove that 2R R 2 U Υ , A (x)U 2R ΥR + V 2R ΥR , A2 (x)V 2R ΥR − ΥR , A2 (x)ΥR ≤ c R1+δ (63)

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and 2R R U Υ , σ · Bf (x)U 2R ΥR + V 2R ΥR , σ · Bf (x)V 2R ΥR − ΥR , σ · Bf (x)ΥR c ≤ 1+δ R (64) Theorem 5.1 (Energy of the approximate ground state) For arbitrarily ﬁxed ε > 0 and R large enough, the following statements hold. i) EM ΨR 2 ≤ HM ΨR , ΨR ≤ EM ΨR 2 +

ε ΨR 2 R

(65)

ii) Let z ∈ R3 be an external variable, i.e., the function ΨR does not depend on z. Then, for |z| > 4R | D(z)ΨR , ΨR | ≤

c(z) , |z|

(66)

| D(z)2 ΨR , ΨR | ≤

c(z) , |z|2

(67)

| D∗ (z)D(z)ΨR , ΨR | ≤

c(z) , |z|2

(68)

and | K(z)ΨR , ΨR | ≤

c(z) , |z|

(69)

where c(z) is a function independent of R that tends to zero as |z| tends to inﬁnity. Proof. Applying Lemma 5.1, and Inequalities (60), (63), (64), we obtain

HM U 2R ΥR , U 2R ΥR + HM V 2R ΥR , V 2R ΥR −

ε c ≤ EM + 2 . R R

(70)

ε c . + R R2

(71)

Using EM V 2R ΥR 2 ≤ HM V 2R ΥR , V 2R ΥR , we get

HM U 2R ΥR , U 2R ΥR ≤ EM U 2R ΥR 2 +

Since ΥR → 1 as R → ∞, we get (65). The proof of ii) is analogous to the proof of Lemma 5.2 and Theorem 4.1 iii).

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6 Proof of Theorem 2.1 The previous discussion enables us to construct a normalized trial function ΓR,b ∈ el ˜ R,N −1 as HN ⊗ F. For given N ∈ N, we deﬁne Ψ R ˜ R,N −1 = Ψ , Ψ ΨR

where ΨR is the function deﬁned in Section 5.2 for a system of M = N −1 electrons. As a natural candidate for a trial state for the proof of Theorem 2.1, one could consider the state ϕ = (ϕ0 , ϕ1 , . . .) deﬁned by ϕn =

n

ΘR,b j (y1 , . . . , yj , λ1 , . . . , λj , xN , sN )·

j=0

×

(72)

˜ R,N −1 (yj+1 , . . . , yn , λj+1 , . . . , λn , x1 , . . . , xN −1 , s1 , . . . , sN −1 ). Ψ n−j

However, since the components ϕn are neither symmetric in the photon, nor antisymmetric in the electron variables, our next goal is to symmetrize the function it in the electron variables. ϕn in the photon variables, and to antisymmetrize We denote by Sn,j the set of nj possible partitions g of the set of n indices {1, . . . , n} into two subsets C1 and C2 with j and n − j elements respectively. Let i1 (g), . . . , ij (g) be the indices in C1 and ij+1 (g), . . . , in (g) in C2 . We deﬁne the function ˜R,N −1 )(y1 , . . . , yn , λ1 , . . . , λn , x, s) (Πpn,j (g)ΘR,b j ψn−j : = ΘR,b j (yi1 , . . . , yij , λi1 , . . . , λij , xN , sN ) R,N −1 (yij+1 , . . . , yin , λij+1 , . . . , λin , x1 , . . . , xN −1 , s1 , . . . , sN −1 ) . × ψ˜n−j

Evidently, ˜ R,b := Γ n

n −1/2 n j=0

j

˜ R,N −1 Πpn,j (g)ΘR,b j Ψn−j

(73)

g∈Sn,j

is symmetric with respect to the permutation of photon variables. ˜ R,b To construct a combination of the functions Γ which is antisymmetric in n the electron variables, let us consider the set of all transpositions πi i = 1, . . . , N , which exchange a pair of electron variables (xi , si ) with (xN , sN ), including the trivial transposition (xN , sN ) ↔ (xN , sN ). For an arbitrary function ϕ(x1 , . . . , xN , s1 , . . . , sN ), let (Πel i ϕ)(x1 , . . . , xN , s1 , . . . , sN ) := ϕ(πi (x1 , . . . , xN , s1 , . . . , sN )). Then, we deﬁne ΓR,b n =

n j=0

N −1/2

−1/2 N n p R,b ˜ R,N −1 (−1)κ(i) Πel , i Πn,j (g)Θj Ψn−j j i=1 g∈Sn,j

(74)

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where κ(i) = 0 if i = N , and κ(i) = 1 otherwise. Obviously, R,b R,b el ΓR,b = (ΓR,b 0 , Γ1 , . . . , Γn , . . .) ∈ HN ⊗ F. el Notice that ΓR,b is a normalized function in HN ⊗ F, since if |b| > 5R, the summands in (74) have for diﬀerent i disjoint supports in electron variables, and thus

˜ R,b 2 = 1 ΓR,b 2 = Γ

(75)

and ˜ R,b , Γ ˜ R,b ). (HN ΓR,b , ΓR,b ) = (HN Γ ˜ R,b is not antisymmetric in all electron variables, the quadratic Although the state Γ ˜ R,b is well deﬁned. form of HN at Γ ˜ R,N −1 have a ﬁnite expectation numFurthermore, both functions ΘR,b and Ψ ber of photons, say, N1 and N2 , respectively. Evidently, this implies that ΓR,b has a ﬁnite expected photon number N1 + N2 . We remark that for |b| > 5R, and each of the terms in the sum Πel i

˜R,N −1 , Πpn,j (g)ΘR,b j ψn−j

g∈Sn,j R,N −1 and ψ˜n−j have disjoint supports, thus one ﬁnds ΘR,b j

˜ R,b , Γ ˜ R,b ), (HN ΓR,b , ΓR,b ) = (HN Γ ˜ R,b = (Γ ˜ R,b , Γ ˜ R,b , . . .) is the state prior to antisymwhere, as we recall from ( 73), Γ 0 1 metrization in the electron variables. Hence, instead of estimating the quadratic form of the operator HN with respect to the state ΓR,b , we may estimate it with ˜ R,b . Although this state is not antisymmetric in all electron variables, respect to Γ the quadratic form is well deﬁned. ˜ R,b , the variables (xN , sN ) are We recall that in our notation for the state Γ R,b the arguments of Θ , while (x1 , . . . , xN −1 , s1 , . . . , sN −1 ) are the arguments of ˜ R,b = 1. ψ˜R,N −1 , and furthermore, that Γ

6.1

Preliminary results

To prove Theorem 2.1, we ﬁrst need to state several lemmata. Lemma 6.1 For |b| > 8R, there exists c > 0 independent of R such that the following estimate holds ˜ R,N −1 , Ψ ˜ R,N −1 + c

Hf ΓR,b , ΓR,b ≤ Hf ΘR,b , ΘR,b + Hf Ψ

R3/2

Nf ΓR,b , ΓR,b . |b|5/2

Binding Conditions for Atomic N -electron Systems

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1127

Proof. We have ˜ R,b Hf Γ n = n|∇y1 |

n −1/2 n j=0

j

˜ R,N −1 . Πpn,j (g)ΘR,b j Ψn−j

(76)

g∈Sn,j

Let us start with one of the functions in the sum (76). We take for example ˜ R,N −1 (yj+1 , . . . , yn ). All other terms can the expression n|∇y1 |ΘR,b j (y1 , . . . , yj )Ψn−j be treated similarly. ˜ R,b , Γ ˜ R,b , this term appears twice, in In the quadratic form Hf Γ ˜ R,N −1 (yj+1 , . . . , yn ), n |∇y1 |ΘR,b j (y1 , . . . , yj )Ψn−j ˜ R,N −1 (yj+1 , . . . , yn ) , ΘR,b j (y1 , . . . , yj )Ψn−j and in

˜ R,N −1 (yj+1 , . . . , yn ), n |∇y1 |ΘR,b j (y1 , . . . , yj )Ψn−j

˜ R,N −1 ΘR,b j−1 (y2 , . . . , yj )Ψn−j+1 (y1 , yj+1 , . . . , yn ) .

(77)

˜ R,b , Γ ˜ R,b that contain All other cross terms appearing in the quadratic form Hf Γ R,b R,N −1 ˜ the function n|∇y1 |Θj (y1 , . . . , yj )Ψ (yj+1 , . . . , yn ) are zero, because at least n−j for one variable, the supports of the functions in the scalar product are disjoint. Let us now estimate (77). The function ˜ R,N −1 (yj+1 , . . . , yn ) ΘR,b j (y1 , . . . , yj )Ψn−j is supported in the region {|y1 | ≥ |b| − 2R} whereas ˜ R,N −1 ΘR,b j−1 (y2 , . . . , yj )Ψn−j+1 (y1 , yj+1 , . . . , yn ) is supported in the region {|y1 | ≤ 2R}. Applying Lemma 7.2 with |b| > 8R, we arrive at ˜ R,N −1 (yj+1 , . . . , yn ), n |∇y1 |ΘR,b j (y1 , . . . , yj )Ψn−j ˜ R,N −1 (y1 , yj+1 , . . . , yn ) ΘR,b (y , . . . , y ) Ψ 2 j j−1 n−j+1 (78) 3/2 R R,N −1 2 ˜ ≤ c n 5/2 ΘR,b (y , . . . , y ) Ψ (y , . . . , y ) 1 j j+1 n j n−j |b| 2 ˜ R,N −1 , + ΘR,b j−1 (y2 , . . . , yj )Ψn−j+1 (y1 , yj+1 , . . . , yn ) which implies 3/2

˜ R,b , Γ ˜ R,b , Γ ˜ R,b ≤ c R Nf Γ ˜ R,b

Hf Γ |b|5/2 (79) n −1 n ˜ R,N −1 , Πp (g)ΘR,b Ψ ˜ R,N −1 . + Ψ n

|∇y1 |Πpn,j (g)ΘR,b j n−j n,j j n−j j n j=0 g∈Sn,j

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For ﬁxed n and j, in the sum ˜ R,N −1 , Πp (g)ΘR,b Ψ ˜ R,N −1 ,

|∇y1 |Πpn,j (g)ΘR,b j Ψn−j n,j j n−j g∈Sn,j

n−1 R,b ˜ R,N −1 . Theretimes in Ψ and n−j−1 the variable y1 appears n−1 n−j j−1 times in Θj fore, the second term on the right-hand side of (79) can be rewritten as −1

n n n−1 R,b ˜ R,N −1 2 n

|∇y1 |ΘR,b j (y1 , . . . , yj ), Θj (y1 , . . . , yj ) Ψn−j j j − 1 n j=1 −1

n n n−1 2 + n ΘR,b j j n − j − 1 n j=1 ˜ R,N −1 (y1 , . . . , yn−j ), Ψ ˜ R,N −1 (y1 , . . . , yn−j )

|∇y1 |Ψ n−j n−j =

n

R,b ˜ R,N −1 2 j |∇y1 |ΘR,b j (y1 , . . . , yj ), Θj (y1 , . . . , yj ) Ψn−j

n j=1

+

n

2 ˜ R,N −1 (y1 , . . . , yn−j ), Ψ ˜ R,N −1 (y1 , . . . , yn−j ) (n − j) ΘR,b j |∇y1 |Ψn−j n−j

n j=1

˜ R,N −1 , Ψ ˜ R,N −1 . = Hf ΘR,b , ΘR,b + Hf Ψ (80) The relations (79) and (80) imply the statement of the Lemma.

Lemma 6.2 For any ε > 0 and |b| large enough, N ˜ R,b , Γ ˜ R,b

i∇x A(x )Γ =1

≤ i∇xN A(xN )ΘR,b , ΘR,b +

N −1

˜ R,N −1 , Ψ ˜ R,N −1

i∇x A(x )Ψ

=1

ε ∇xN ΘR,b 2 + ΘR,b 2 + 2(|b| − 2R) N −1 ε ˜ R,N −1 2 + Ψ ˜ R,N −1 2 . + ∇x Ψ 2(|b| − 2R)

(81)

=1

Furthermore, N N −1 R,b ˜ R,b R,b R,b ˜ ˜ R,N −1 , Ψ ˜ R,N −1

σ · B(x )Γ , Γ ≤ σ · B(xN )Θ ,Θ +

σ · B(x )Ψ =1

=1

+

ε ΘR,b 2 + (|b| − 2R)

N −1 =1

ε ˜ R,N −1 2 , Ψ (|b| − 2R)

(82)

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and N N −1 2 R,b R,b 2 R,b R,b 2 R,N −1 R,N −1 ˜ ,Γ ˜ − D (xN )Θ , Θ − ˜ ˜

D (x )Ψ ,Ψ D (x )Γ =1

=1

≤

ε ΘR,b 2 + (|b| − 2R)

N −1 =1

ε ˜ R,N −1 2 . Ψ (|b| − 2R)

(83)

Moreover, N ∗ D (x )D(x )Γ ˜ R,b , Γ ˜ R,b − D∗ (xN )D(xN )ΘR,b , ΘR,b =1

−

=1

≤

R,N −1 ˜ R,N −1 ˜

D (x )D(x )Ψ ,Ψ

N −1

∗

(84)

N −1 ε ε ˜ R,N −1 2 . ΘR,b 2 + Ψ (|b| − 2R) (|b| − 2R) =1

˜ R,N −1 , the variable x1 appears only in the function Proof. We recall that in ΘR,b j Ψn−j R,N −1 ˜ Ψ , and the variable xN only in ΘR,b n−j j . Permutations of photon variables do not change this fact. We have, for k = 1, . . . , N , ˜ R,b D(xk )Γ n−1 n −1/2 (85) √ n ˜ R,N −1 L2 (R3 ⊗C2 ,dy ) , Ψ = n G(xk − yn ), Πpn,j (g)ΘR,b n j n−j j j=0 g∈Sn,j

where as before, dyn means integration with respect to yn and summation over the associated polarization λn . ˜ R,N −1 in the sum (73). Let us start with one of the functions Πpn,j (g)ΘR,b j Ψn−j For ﬁxed g, two variants are possible. Either the index n is in C1 , and the function ˜ R,N −1 depends on yn . ΘR,b depends on the photon variable yn , or the function Ψ j n−j n−1 For ﬁxed n and j, the ﬁrst variant occurs j−1 times, whereas the second one n−1 times. Let us consider the function occurs n−j−1 −1/2 n ˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn ) . ΘR,b j (xN , y1 , . . . , yj )Ψn−j j ˜ R,b , Γ ˜ R,b , it appears only once in the In the quadratic form N k=1 i∇xk D(xk )Γ scalar product with

−1/2 √ n−1 n∇xk G(xk − yn ) ΘR,b j (xN , y1 , . . . , yj ) j−1 ˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn−1 ) ×Ψ n−j

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Ann. Henri Poincar´e

which, in the case k = N , is equal to n−j

−1 n−1 ˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn ), Ψ n−j n−j−1

˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn−1 ) ΘR,b 2 , ∇xk G(xk − yn )Ψ n−j−1 j

(86)

and in the case k = N , n−j

n−1 n−j−1

−1

˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn )∇xN ΘR,b (xN , y1 , . . . , yj ), Ψ n−j j

˜ R,N −1 (x1 , . . . , xN −1 , yj+1 , . . . , yn−1 )ΘR,b (xN , y1 , . . . , yj ) . G(xN − yn )Ψ n−j−1 j (87) ˜ R,N −1 , give the same All other terms in (73) with the same j, and with yn in Ψ n−j N ˜ R,b , Γ ˜ R,b . Summing up these n−1 contricontribution to k=1 i∇xk D(xk )Γ n−j−1 butions in the case k = N yields ˜ R,N −1 , ∇x G(xk − yn )Ψ ˜ R,N −1 ΘR,b 2 , n−j Ψ (88) k n−j n−j−1 j and in the case k = N , ˜ R,N −1 ∇x ΘR,b , G(xN − yn )Ψ ˜ R,N −1 ΘR,b . n−j Ψ N n−j j n−j−1 j

(89)

If we sum ﬁrst over m = n − j, and then the terms (88) over j, we get ˜ R,N −1 , ∇x D(xk )Ψ ˜ R,N −1 ΘR,b 2 .

Ψ k

(90)

Let us compute ﬁrst the sum over n − j of the terms (89), and estimate them according to (66). We obtain for ε > 0, and |b| suﬃciently large,

|∇xN ΘR,b j |,

c(xN ) R,b ε R,b 2 2 , |Θj | ≤ ∇xN ΘR,b j + Θj |xN | 2(|b| − 2R)

(91)

where we used that |xN | ≥ |b| − 2R, and c(xN ) tends to zero, as |xN | tends to inﬁnity. Therefore, ˜ R,N −1 ∇xN ΘR,b , G(xN − yn )Ψ ˜ R,N −1 ΘR,b n−j Ψ n−j j n−j−1 j n j (92) ε R,b 2 R,b 2 ≤ ∇xN Θ + Θ . 2(|b| − 2R)

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˜ R,b , Γ ˜ R,b of In analogy to (90) and (92), the contribution to N k=1 i∇xk D(xk )Γ R,b the terms for which the variable yn is in Θ , is, for k = N , equal to

i∇xN D(xN )ΘR,b , ΘR,b

(93)

and for k = N , it can be estimated by ε ˜ R,N −1 2 + Ψ ˜ R,N −1 2 . ∇xk Ψ 2(|b| − 2R)

(94)

This completes the proof of (81). Let us next prove Inequality (83). The operator D2 (xk ) acts as n −1/2 √ √ n 2 R,b ˜ D (xk )Γ = n n−1 j n−2 j=0

(95)

g∈Sn,j

× G(xk − yn )G(xk −

˜ R,N −1 L2 (X,dy )⊗L2 (X,dy ) yn−1 ), Πpn,j (g)ΘR,b n n−1 j Ψn−j

,

where X := R3 ⊗ C2 . Assume that in the decomposition g, we have the indices n ∈ C2 and (n − 1) ∈ C2 . Then, both variables yn and yn−1 appear in the function ˜ R,N −1 . For ﬁxed n and j, we have n−2 such cases. Similar to (88) in the case Ψ n−j

n−j−2

k = N , and to (89) in the case k = N , we obtain, respectively, ˜ R,N −1 , G(xk − yn ) G(xk − yn−1 )Ψ ˜ R,N −1 ΘR,b 2 , (96) n−j n−1−j Ψ n−j n−j−2 j and ˜ R,N −1 ΘR,b , G(xN − yn ) G(xN − yn−1 )Ψ ˜ R,N −1 ΘR,b . n−j n−j −1 Ψ n−j j n−j−1 j (97) Now, summing each of these expressions over m = n − j and j, and applying (67), we arrive at ˜ R,N −1 , Ψ ˜ R,N −1 ΘR,b 2

D2 (xk )Ψ (98) for k = N , and

ε ΘR,b 2 (|b| − 2R)2

(99)

for k = N . Let us now consider g with n ∈ C1 and (n − 1) ∈ C1 , which implies that the variables yn and yn−1 are in ΘR,b j . We get

for k = N , and for k = N .

˜ R,N −1 2

D2 (xN )ΘR,b , ΘR,b Ψ

(100)

ε ˜ R,N −1 2 Ψ (|b| − 2R)2

(101)

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Finally, let us address the case where one of the indices n, n − 1 belongs to C1 and the other one to C2 . In this case, one of thevariables yn and yn−1 appears ˜ R,N −1 , and the other one in ΘR,b . We have 2 n−2 such cases. Note that in in Ψ n−j j j−1 each such case, either |G(xk − yn )| or |G(xk − yn−1 )| is small, and the contribution of the sum of these terms can be estimated as ε1 ˜ R,N −1 , Ψ ˜ R,N −1

Nf ΘR,b , ΘR,b + Nf Ψ (102) |b| − 2R ε ˜ R,N −1 2 . ΘR,b 2 + Ψ ≤ |b| − 2R The estimates (98)–(102) imply (83) The proof of (82) is very similar to the one of (81).

6.2

Proof

To prove Theorem 2.1, we will show that for suitably chosen parameters R and |b|, the trial function ΓR,b satisﬁes

HN ΓR,b , ΓR,b < EN −1 + Σ0 .

(103)

We recall that HN =

N

−i∇x +

2 √ √ αAf (x ) + ασ · Bf (x ) + V (x )

=1

1 + 2

W (|xk − x |) + Hf ,

(104)

1≤k,≤N

and that, as was shown in the previous section, Inequality (103) is equivalent to ˜ R,b , Γ ˜ R,b < EN −1 + Σ0 .

HN Γ For M ∈ N, we deﬁne IM (x1 , . . . , xM ) =

M

V (x ) +

=1

1 2

W (xk − x ).

1≤k,≤M

Obviously, we have N

˜ R,b , Γ ˜ R,b + IN (x1 , . . . , xn )Γ ˜ R,b , Γ ˜ R,b

−∆ Γ

=1

=

N −1

˜ R,N −1 , Ψ ˜ R,N −1

−∆ Ψ

(105)

=1

˜ R,N −1

˜ R,N −1

R,b

R,b

+ IN −1 (x1 , . . . , xN −1 )Ψ ,Ψ + −∆Θ , Θ $ # N −1 ˜ R,N −1 , Ψ ˜ R,N −1 ΘR,b , ΘR,b W (xi − xN )Ψ + V (xN ) +

i=1

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Vol. 4, 2003

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˜ R,N −1 are normalized. On the support of the where we used that ΘR,b and Ψ R,b ˜ R,N −1 , function Θ , we have |xN | ≤ |b| + R and on the support of the function Ψ −1 |xi − xN | ≥ |b| − 2R. This implies, for |b|R suﬃciently large, that on the support ˜ R,b (deﬁned in (73)), of Γ V (xN ) +

N −1

W (xi − xN ) < −

i=1

γ1 (N − 1) ν γ0 + <− , |b| + R |b| − 2R 2|b|

(106)

for ν = γ0 − γ1 (N − 1) > 0. Thus, (105) and (106) yield N

˜ R,b , Γ ˜ R,b + IN (x1 , . . . , xn )Γ ˜ R,b , Γ ˜ R,b

−∆ Γ

=1

≤

N −1

˜ R,N −1 , Ψ ˜ R,N −1 + IN −1 (x1 , . . . , xN −1 )Ψ ˜ R,N −1 , Ψ ˜ R,N −1

−∆ Ψ

(107)

=1

+ −∆ΘR,b , ΘR,b −

ν . 2|b|

˜ R,N −1 ≤ c Ψ ˜ R,N −1 ( = 1, . . . , N − 1), and Taking into account that ∇x Ψ R,b R,b that ∇xN Θ ≤ c Θ , with a constant c independent of R, we derive from (81) N N −1 R,b ˜ R,b ˜ ˜ R,N −1 , Ψ ˜ R,N −1

∇ A(x ) Γ , Γ −

∇x A(x )Ψ x =1 =1 ε R,b R,b − ∇xN A(xN )Θ , Θ ≤ . |b| − 2R

(108)

Similarly to (108), and using (21), (22), (67), and (68), we have N

˜ R,b , Γ ˜ R,b

A2 (x )Γ

=1

≤

N −1 =1

(109)

ε ˜ R,N −1 , Ψ ˜ R,N −1 + A2 (xN )ΘR,b , ΘR,b +

A2 (x )Ψ |b| − 2R

Along the same lines, we have for the magnetic term, using (23) and (69), N

˜ R,b , Γ ˜ R,b

σ · B(x )Γ

=1

≤

N −1 =1

ε ˜ R,N −1 , Ψ ˜ R,N −1 + σ · B(xN )ΘR,b , ΘR,b + .

σ · B(x )Ψ |b| − 2R

(110)

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According to Lemma 6.1 we have ˜ R,b ≤ Hf ΘR,b , ΘR,b + Hf Ψ ˜ R,N −1 ˜ R,b , Γ ˜ R,N −1 , Ψ

Hf Γ R3/2 ˜ R,b , Γ ˜ R,b .

Nf Γ |b|5/2 ˜ R,b , Γ ˜ R,b ≤ c Ψ ˜ R,N −1 2 + ΘR,b 2 . Equality (75) implies that Nf Γ +c

Collecting the estimates (107)–(111) we obtain for any ε > 0 and suﬃciently large R, 6ε cR3/2 ˜ R,b , Γ ˜ R,b ≤ EN −1 + Σ0 − ν +

HN Γ + 5/2 . 2|b| |b| − 2R |b|

(111)

To complete the proof of the Theorem, we pick ﬁrst R large enough to have ε < 48−1 ν, and then pick |b| suﬃciently large to satisfy the inequality (R|b|−1 )3/2 < δ(4c)−1 , which implies ˜ R,b , Γ ˜ R,b < EN −1 + Σ0 .

HN Γ

7 Appendix Lemma 7.1 We deﬁne Gλ as

Gλ (y) = F

ελ (k) 1

|k| 2

ζ(k)

where F denotes the Fourier transform. Then, for λ = 1, 2 and arbitrary ε > 0, |Gλ (y)(1 + |y|)| ∈ L2+ε (R3 ). Proof. The statement of the Lemma follows from the Hausdorﬀ-Young inequality, ελ,i (k) 2−ε and the fact that for arbitrarily ε > 0, ∇k (R3 ), for i = 1, 2, 3, 1 ζ(k) is in L |k| 2

which can be checked directly.

Lemma 7.2 Let ϕ1 (x) ∈ H 1/2 (R3 ) with support in the ball of radius aR centered at the origin, and ϕ2 (x) ∈ H 1/2 (R3 ) with support outside the ball of radius bR centered at the origin. Then for b > 2a, | |∇|ϕ1 , ϕ2 | ≤

a3/2 ϕ1 2 + ϕ2 2 . 1/2 5/2 3 π R(b − a) 1

(112)

Proof. Consider the function u deﬁned in (27). Then, for χ1 (x) = u(|x|/(bR)) and χ2 (x) = 1 − χ21 (x), we have, according to [10, Theorem9]

|∇|(ϕ1 + ϕ2 ), ϕ1 + ϕ2 − |∇|ϕ1 , ϕ1 − |∇|ϕ2 , ϕ2 1 |ϕ1 (x) + ϕ2 (x)| |ϕ1 (y) + ϕ2 (y)| 2 ≤ |χi (x) − χ2i (y)|dydy . 2 2π |x − y|4 i=1,2

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Since χ1 = 1 on the support of ϕ1 , χ1 = 0 on the support of ϕ2 , we obtain

|∇|(ϕ1 + ϕ2 ), ϕ1 + ϕ2 − |∇|ϕ1 , ϕ1 − |∇|ϕ2 , ϕ2 = ≤ ≤

2Re |∇|ϕ1 , ϕ2 1 |ϕ1 (x)| |ϕ2 (y)| dydy π2 |x − y|4 2 a3/2 ϕ1 2 + ϕ2 2 . π31/2 R(b − a)5/2

Acknowledgments Jean-Marie Barbaroux and Semjon Vugalter were ﬁnancially supported by the Bayerisch-Franz¨ osisches Hochschulzentrum, and by the European Union through the IHP network of the EU No. HPRN-CT-2002-00277. Thomas Chen was supported by a Courant Instructorship.

References [1] V. Bach, T. Chen, J. Fr¨ ohlich, I.M. Sigal, Smooth Feshbach map and operatortheoretic renormalization group methods, preprint mp-arc 02-214, 2002. To appear in J. Funct. Anal. [2] V. Bach, J. Fr¨ ohlich, I.M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation ﬁeld, Comm. Math. Phys. 207 (2), 249–290 (1999). [3] T. Chen, Operator-theoretic infrared renormalization and construction of dressed 1-particle states in non-relativistic QED, preprint arXiv: mathph/0108021, 2001. [4] T. Chen, V. Vougalter, S.A. Vugalter, The increase of binding energy and enhanced binding in non-relativistic QED, J. Math. Phys. 44 (5), (2003). [5] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry, Text and Monographs in Physics, Springer Verlag, Berlin, 1st edition, 1987. [6] J. Fr¨ ohlich, Existence of dressed one electron states in a class of persistent models, Fortschritte der Physik 22, 159–189 (1974). [7] C. G´erard, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri Poincar´e 1 (3), 443–459 (2000). [8] M. Griesemer, E.H. Lieb, M. Loss, Ground states in non-relativistic quantum electrodynamics, Inv. Math 145, 557–595 (2001).

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[9] T. Kato, On the existence of solutions of the Helium wave equation, Trans. Amer. Math. Soc. 70, 212–218 (1951). [10] E.H. Lieb, H.-T. Yau, The stability and instability of relativistic matter, Comm. Math. Phys. 118, 177–213 (1988). [11] M. Reed, B. Simon, Methods of modern mathematical physics, volume 4: Analysis of operators. Academic Press, New York, 1st edition, 1978. [12] G.M. Zhislin, A study of the spectrum of the Schr¨ odinger operator for a system of several particles, Trudy Moskov. Mat. Obsc. 9, 81–120 (1960). Jean-Marie Barbaroux Centre de Physique Th´eorique Luminy Case 907 F-13288 Marseille Cedex 9, France email: [email protected] Thomas Chen Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012-1185, USA email: [email protected] Semjon Vugalter Mathematik, Universit¨ at M¨ unchen Theresienstrasse 39 D-80333 M¨ unchen, Germany email: [email protected] Communicated by Bernard Helﬀer submitted 14/07/03, accepted 18/08/03

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 1137 – 1167 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061137-31 DOI 10.1007/s00023-003-0158-0

Annales Henri Poincar´ e

Charge Superselection Sectors for Scalar QED on the Lattice ´ J. Kijowski, G. Rudolph and C. Sliwa Abstract. The lattice model of scalar quantum electrodynamics (Maxwell ﬁeld coupled to a complex scalar ﬁeld) in the Hamiltonian framework is discussed. It is shown that the algebra of observables O(Λ) of this model is a C ∗ -algebra, generated by a set of gauge-invariant elements satisfying the Gauss law and some additional relations. Next, the faithful, irreducible and non-degenerate representations of O(Λ) are found. They are labeled by the value of the total electric charge, leading to a decomposition of the physical Hilbert space into charge superselection sectors. In the appendices we give a uniﬁed description of spinorial and scalar quantum electrodynamics and, as a byproduct, we present an interesting example of weakly commuting operators, which do not commute strongly.

1 Introduction The ideas of axiomatic and algebraic quantum ﬁeld theory in the sense of Wightman, Haag and Kastler [1] have played an important role in clarifying basic nonperturbative structures of quantum physics. In particular, there has been developed a general scheme for superselection rules [2], which, however, does not apply to theories with massless particles. Thus, an extension of these ideas to realistic gauge theories is still a big challenge. Some partial results in this direction already exist, see a series of papers by Strocchi and Wightman ([3], [4] and [5]). In particular, in [3] Quantum Electrodynamics was considered. It was shown that if one insists in locality and Lorentz covariance, one is rather naturally led to a theory with indeﬁnite metric. Within this scheme, the charge superselection rule for QED was proven, but a decomposition of the physical Hilbert space into a direct sum of subspaces carrying deﬁnite total charge was not obtained. For a deep discussion of charged states in QED we refer to [6] and further references therein. Studying simple toy models, e.g., a Z2 -gauge theory with Z2 -matter ﬁelds [7], one can realize the full programme, which one would like to implement for realistic theories. In [7], the authors were able to determine the ground state and charged states explicitly. Using methods of Euclidean quantum ﬁeld theory, it was possible to show that – for some regions in the space of coupling constants – the thermodynamic limit for charged states can be controlled. In this paper we also discuss a simpliﬁed model, we put scalar quantum electrodynamics on a ﬁnite lattice. In the context of lattice approximation complicated operator theoretic problems arising in (continuum) quantum ﬁeld theory become simpler, whereas problems typical for gauge theories remain and can be, there-

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fore, discussed separately. We consider scalar QED in the hamiltonian approach on a ﬁnite cubic (3-dimensional) lattice and work in the non-compact formulation, where the gauge potential remains Lie-algebra-valued on the lattice level. Our starting point are the commutation relations for canonically conjugate pairs of gauge-dependent lattice ﬁelds. Then, in a ﬁrst step we construct the algebra of observables as the algebra of gauge invariant operators fulﬁlling the Gauss law and show that the charge superselection rule holds. Algebraically, the observable algebra is generated by electric and magnetic ﬂux operators, together with gauge invariant operators bilinear in the matter ﬁelds. These gauge invariant generators fulﬁl a number of algebraic identities, which by using the technical tool of a lattice tree can be reduced essentially. It turns out that the observable algebra is the tensor product of an associative algebra generated by canonical commutation relations (electromagnetic part) and an associative algebra generated by a certain Lie algebra (matter ﬁeld part). Using Woronowicz’s theory of C ∗ -algebras generated by unbounded elements (see [17]), we can endow the observable algebra with a C ∗ -structure. Within this setting, we are able to classify all faithful, irreducible and nondegenerate representations of the observable algebra and to obtain the physical Hilbert space as a direct sum of representation spaces labeled by the total charge. We stress that the restriction to non-degenerate representations of C ∗ -algebras (or, equivalently, to integrable representations of the underlying Lie algebras) is of fundamental importance, as is shown in Appendix A. We have obtained a similar result for the case of spinorial QED earlier, see [10] and [11]. However, from the mathematical point of view, the problems occurring in this paper are much more complicated due to the fact that the matter ﬁeld part of the observable algebra is generated by a non-compact Lie algebra. Consequently, the observable algebra is inﬁnite-dimensional and its representations are more diﬃcult to control. For some earlier work on scalar lattice QED we refer to [9] and for basic notions concerning lattice gauge theories we refer to [19] and references therein. Recently, we have started to investigate lattice QCD in the above spirit, see [12]. Our paper is organized as follows: In Section 2 we discuss the standard second quantization procedure on the lattice. We deﬁne the ﬁeld algebra, discuss the Gauss law and introduce the notion of boundary data. In Section 3 we construct observables in terms of ﬁeld operators and give an abstract deﬁnition of the observable algebra in terms of generators and relations. Finally, we endow it with a C ∗ -structure. In Section 4 we ﬁnd all faithful, irreducible and non-degenerate representations of the observable algebra and prove that they are labeled by the eigenvalues of the total charge operator. This yields the superselection structure. Finally, we discuss some perspectives of our approach. In Appendix A we give an example of a non-integrable representation carrying non-integer charge and in Appendix B we present a uniﬁed description for both bosonic and fermionic matter.

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2 Scalar QED on the lattice Continuum scalar quantum electrodynamics (QED) is the theory of a complexvalued scalar ﬁeld φ interacting with the electromagnetic ﬁeld Aµ . The classical Lagrangian of this model is deﬁned as follows 1 1 L = − Fµν F µν − Dµ φDµ φ − V (|φ|2 ), 4 2

(2.1)

where Dµ φ = ∂µ φ + igAµ φ, Fµν = ∂µ Aν − ∂ν Aµ , g = e/. Local gauge transformations are given by ˜ φ(x) = e−igλ(x) φ,

A˜µ (x) = Aµ (x) + ∂µ λ(x).

(2.2)

For a given Cauchy hyperplane Σ = {t = const} in Minkowski space, the above Lagrangian gives rise to an inﬁnite-dimensional Hamiltonian system in variables (Ak , E k , φ, π) with the Hamiltonian H=

1 1 1 2 + V (|φ|2 ), Ek E k + Bk B k + |π|2 + |Dφ| 2 2 2

(2.3)

where B = curl A is the magnetic ﬁeld, E is the electric ﬁeld (the momentum canonically conjugate to A) and π denotes the momentum canonically conjugate to φ . Let us take a ﬁnite, regular, cubic lattice Λ contained in Σ, with lattice spacing a, and let us denote the set of n-dimensional lattice elements by Λn , n = 0, 1, 2, 3. Such elements are (in increasing order of n) called sites, links, plaquettes and cubes. We approximate every continuous conﬁguration (Ak , E k , φ, π) in the following way: Λ0 x −→ φx

:=

φ(x) ∈ C ,

(2.4)

Λ x −→ πx

:=

(2.5)

ˆ −→ A Λ1 (x, x + k) ˆ x,x+k

:=

ˆ −→ E Λ1 (x, x + k) ˆ x,x+k

:=

π(x) ∈ C , Ak dl ∈ R , ˆ (x,x+k) E k dσk ∈ R .

0

ˆ σ(x,x+k)

(2.6) (2.7)

ˆ denotes a plaquette of the dual lattice, dual to the link (x, x+ k) ˆ ∈ Here σ(x, x+ k) 1 Λ . A local gauge transformation of a lattice conﬁguration is given by: φ˜x

=

exp(−igλx ) φx ,

π ˜x A˜x,x+kˆ

= =

exp(−igλx ) πx , Ax,x+kˆ + λx+kˆ − λx ,

(2.8) (2.9) (2.10)

where Λ0 x −→ λx ∈ R. The electric ﬁeld E is gauge invariant. Note that we have chosen the non-compact lattice approximation, where the potential and the ﬁeld strength remain Lie-algebra-valued on the lattice level.

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We deﬁne second quantization of the lattice theory by postulating the folˆ π ˆ E, ˆ φ, lowing canonical commutation relations for the lattice quantum ﬁelds (A, ˆ) corresponding to the classical lattice ﬁelds given by (2.4)–(2.7): ˆ = φˆx , π = 2iδxy 1 (2.11) ˆy ˆy∗ , φˆ∗x , π ˆ ˆ Aˆx,x+kˆ , E (2.12) = iδ(x,x+k),(y,y+ ˆ ˆ y,y+ˆ l l) 1 . ˆ The remaining commutators have to vanish. Here, δ(x,x+k),(y,y+ ˆ ˆ l) = 0 if (x, x + k) and (y, y + ˆl) are diﬀerent links, δ(x,x+k),(y,y+ ˆ ˆ l) = 1 if they coincide and have = −1 if they coincide and have opposite the same orientation and δ(x,x+k),(y,y+ ˆ ˆ l) orientations. All irreducible representations in the strong (Weyl) sense of the above algebra are equivalent to the Schr¨odinger representation, see [8], of wave functions Ψ ∈ H0 ≡ L2 (A, φ) . We denote the ﬁeld algebra of bounded operators on H0 , generated ˆ π ˆ E, ˆ φ, by (A, ˆ ) and fulﬁlling (2.11) and (2.12), by F (Λ) . It is endowed with a ˆ are self-adjoint. Obviously, F (Λ) contains natural ∗-operator, such that Aˆ and E a lot of unphysical (gauge-dependent) elements. Moreover, the above electric ﬁeld ˆ does not automatically satisfy the Gauss law. In what follows we will present an E explicit construction of the algebra O(Λ) of observables (gauge invariant operators satisfying the Gauss law), together with a complete classiﬁcation of its irreducible representations. The group of local gauge transformations acts on F (Λ) by automorphisms, whose generators are given by i ˆ Ex,x+kˆ − qˆx ) ∈ End(F (Λ)) , Λ0 x −→ Gˆx := − (

(2.13)

ˆ k

e ˆ Im(φˆ∗x π ˆx ) − 1 (2.14) denoting the operator of electric charge at x . Thus, the corresponding (local) Gauss law constraint, which has to be imposed on observables, has the following form ˆ E ˆx . (2.15) ˆ = q x,x+k with

qˆx =

ˆ k

ˆ of total electric charge putting We deﬁne the operator Q ˆ := Q qˆx .

(2.16)

x∈Λ0

Summing up the local Gauss laws over all lattice sites, we see that non-trivial ˆ can only arise from non-trivial boundary data, which values of the total charge Q we are now going to introduce. For this purpose we consider also external links

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of Λ, connecting lattice sites belonging to the boundary ∂Λ with “the rest of the world”. We denote these external links by (x, ∞) and consider their electric ﬂuxes Ex,∞ . Then we obtain from the Gauss law ˆx,∞ , ˆ= (2.17) E Q x∈∂Λ0

where we denote ∂Λn := ∂Λ ∩ Λn . In this paper we assume that the ﬂuxes Ex,∞ are constant in time. For purposes of the construction of the complete ﬁeld theory via a limit of Then the lattice approximations, we may treat Λ as a piece of a bigger lattice Λ. ˆ boundary ﬂux operators Ex,∞ belong to F (Λ) and – due to locality of the theory – must commute with all elements of the ﬁeld algebra F (Λ). Physically, these external ﬂuxes measure the “violation of the local Gauss law” on the boundary ∂Λ , ˆx,∞ := qˆx − E Eˆx,x+kˆ . (2.18) ˆ k

This is due to the fact, that the “world outside of Λ” has been discarded on this level of approximation. According to the above discussion, we assume that the above elements belong to the center of the algebra F (Λ). Mathematically, admitting non-vanishing elements of this type is equivalent to admitting gauge dependence of quantum states under the action of boundary gauges ∂Λ0 x → ξ(x) ∈ U (1) . ˆ deﬁnes a superselection rule, giving As will be shown, the charge operator Q ˆ = Q1 ˆ on every superselection sector. Consequently, the only consistent choice Q ˆx,∞ = Ex,∞ 1 ˆ on every superselection sector, where for the external ﬂuxes is E Ex,∞ are c-numbers fulﬁlling Q= Ex,∞ . (2.19) x∈∂Λ0

Therefore, we treat external ﬂuxes as prescribed, classical boundary conditions and show that representations characterized by the same value Q, but corresponding to diﬀerent external ﬂux distributions fulﬁlling (2.17) are equivalent. For a more detailed discussion of this point we refer to [12]. we must also prescribe When considering Λ as a piece of a bigger lattice Λ, the interaction of the magnetic degrees of freedom on Λ with the rest of the world: in continuum theory, an additional condition for B or B ⊥ on the boundary is necessary. In lattice theory, these quantities live on external plaquettes of lattice cubes, adjacent to ∂Λ. In what follows, we simply assume the boundary condition B = 0 over the whole boundary ∂Λ. Due to the Maxwell equation E˙ ⊥ = curl2 B , ˆx,∞ are time-independent. We this condition is compatible with the fact that E stress that other boundary conditions could be considered as well.

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3 The observable algebra The observable algebra O(Λ) will be deﬁned by imposing the local Gauss law and gauge invariance. To implement gauge invariance we have to take those elements of F (Λ) , which commute with all generators Gˆx . In Subsection 3.1 we give a complete list of gauge invariant generators, built from elements of the ﬁeld algebra. These generators are not independent, they have to fulﬁl a number of relations. Moreover, as an additional relation we impose the Gauss law. We deﬁne O(Λ) as a C ∗ -algebra generated by unbounded elements, fulﬁlling these relations.

3.1

Generators and defining relations

We start with giving a complete list of generators of O(Λ) on a purely algebraic level. Obviously, the electric ﬁeld ˆ →E ˆ Λ1 (x, x + k) ˆ ∈ R, x,x+k ˆ ˆl) at x , as well as the magnetic ﬂux through the rectangular plaquette (x; k, ˆ ˆ spanned by the vectors k and l , ˆ ˆl) → B ˆ ˆ ˆ∈ R, Λ2 (x; k, x;k,l deﬁned by ˆ ˆ ˆ = Aˆ ˆ ˆ ˆ ˆ + Aˆ ˆ ˆ ˆ + Aˆ ˆ B ˆ + Ax+k,x+ x;k,l x,x+k k+l x+k+l,x+l x+l,x

(3.1)

are gauge invariant. With every lattice path γ, starting at x and ending at y, we associate the following set of generators: ∗ ˆ ˆ ˆ φˆy , Lγ = φx exp(ig A) (3.2) γ ˆ γ = φˆ∗ exp(ig A)ˆ ˆ πy , M (3.3) x γ ˆ φˆy , Nˆγ = π ˆx∗ exp(ig A) (3.4) γ ˆγ = π ˆ πy . R ˆx∗ exp(ig A)ˆ (3.5) γ

ˆ γ, N ˆγ or R ˆ γ . It is We will use the symbol Pˆγ as a place holder for any of Lˆγ , M ˆ ˆ ˆ clear that the set (B, E, Pγ ) generates the observable algebra. Gauge invariance of the theory, together with canonical commutation relaˆ π ˆ E, ˆ φ, tions fulﬁlled by elements of ﬁeld algebra generators (A, ˆ ), impose many ˆ ˆ ˆ relations between generators (B, E, Pγ ) of the observable algebra. The proposition below contains a set of relations, which is minimal in the following sense:

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Taking them, together with the Gauss law, as deﬁning relations of O(Λ), we show that its faithful irreducible representations are unique, for a given value of the total charge. Proposition 3.1 The following relations between generators of the observable algebra hold: 1. For any lattice site x we have the Gauss law: ˆ k

e ˆ xx ) − e 1 ˆ, Eˆx,x+kˆ = Im(M

(3.6)

where the index “xx” on the right-hand side stands for the trivial path at x. ˆ ˆl, n 2. For any lattice site x and three independent vectors (k, ˆ ) , spanning a lattice cube at x, we have the Bianchi identity: ˆ ˆ ˆ+ B ˆ ˆ +B ˆ ˆ +B ˆ ˆ ˆ ˆ ˆ= 0 . B ˆ + Bx+ˆ ˆ n + Bx+k;ˆ x;k,l x;ˆ n,k x;l,ˆ n x+ˆ n;ˆ l,k l;k,ˆ n,l

(3.7)

3. For two diﬀerent lattice paths γ1 and γ2 , both from x to y , we have ˆ −1 )Pˆγ1 , Pˆγ2 = exp(ig B γ γ2 1

(3.8)

where γ1−1 γ2 is the loop composed of γ1−1 and γ2 (γ2 is adjoint to the end

ˆ γ = Aˆ . of γ1−1 ), and B γ 4. The following commutation relations between the electromagnetic and the matter ﬁeld generators hold: ˆ ˆˆ Pˆγ , B x;k,l ˆ Pˆγ , E ˆ x,x+k ˆy,y+ˆn ˆ ˆ ˆ, E B x;k,l

= 0,

(3.9)

ˆ = eδγ,(x,x+k) ˆ Pγ ,

(3.10)

= iδ∂(x;k, ˆˆ l),(y,y+ˆ n) .

(3.11)

ˆ ˆ Here δγ,(x,x+k) ˆ = 0 if (x, x + k) ∈ γ , δγ,(x,x+k) ˆ = 1 if (x, x + k) ∈ γ and has the same orientation as γ and δγ,(x,x+k) ˆ = −1 otherwise. In the last formula, ˆ ˆ ˆ ˆl). ∂(x; k, l) denotes the boundary of the oriented plaquette (x; k, 5. The following commmutation relations between the matter ﬁeld generators hold: If γ is a path from x to y, and γ is a path from x to y , then we have [Lˆγ , Lˆγ ] = ˆ γ ] = [Lˆγ , M

0

(3.12)

2iδxy Lˆγ γ

(3.13)

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ˆγ ] = [Lˆγ , N ˆ γ ] = [Lˆγ , R ˆ γ, M ˆ γ ] = [M ˆγ ] = ˆ γ, N [M ˆ γ, R ˆ γ ] = [M ˆγ ] = ˆγ , N [N ˆγ , R ˆ γ ] = [N ˆγ, R ˆ γ ] = [R

Ann. Henri Poincar´e

2iδx y Lˆγγ ˆ γ γ + δx y M ˆ γγ ) 2i(δxy N ˆ γ γ − δx y M ˆ γγ ) 2i(δxy M

(3.14)

0

(3.17)

ˆ γγ 2iδxy R ˆγγ − δxy N ˆγ γ ) 2i(δx y N ˆ γγ 2iδx y R

(3.18)

0.

(3.21)

(3.15) (3.16)

(3.19) (3.20)

Thus, the invariant ﬁelds Pˆγ generate a Lie algebra. 6. The ∗-operator acts on the matter ﬁeld generators as follows: ˆ γ )∗ = N ˆγ −1 , (Nˆγ )∗ = M ˆ γ −1 , (R ˆ γ )∗ = R ˆ γ −1 . (Lˆγ )∗ = Lˆγ −1 , (M

(3.22)

7. For any path γ from x to y and any path γ from y to z , the following identity holds: (3.23) Lˆγ Lˆγ = Lˆyy Lˆγγ , Proof. By a number of lengthy, but simple calculations, which we leave to the reader. In particular, equation (3.6) follows from equation (2.14), with the right-hand ˆ side expressed in terms of the generators M, qˆx =

e ˆ xx ) − e 1 ˆ. Im(M

(3.24)

Summing over all lattice points yields the expression for the global charge: ˆ= e ˆ xx ) − N e 1 ˆ, Q Im(M (3.25) 0 x∈Λ

with N being the number of lattice points. Now we are able to give an abstract deﬁnition of the observable algebra, which does not refer any more to the ﬁeld algebra we started with. Definition 3.2 The observable algebra O(Λ) is a C ∗ -algebra generated by abstract ˆ E, ˆ Pˆγ ) . The generators satisfy the following axioms: elements (B, 1. The Gauss law (3.6). 2. The Bianchi identities (3.7). 3. Identities (3.8), relating generators Pˆγ for two diﬀerent paths with common end points.

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4. The commutation relations (3.9)–(3.11) between electromagnetic and matter ﬁeld generators. 5. The Lie algebra commutation relations (3.12)–(3.21) among the matter ﬁeld generators. ˆ and B ˆ∗ = B ˆ , whereas the ˆ and B ˆ are self-adjoint, Eˆ ∗ = E 6. The generators E matter ﬁeld generators fullﬁl (3.22). 7. There is a single pair (γ0 , γ0 ) of non-trivial paths with a common intermediate point y0 , γ0 connecting x0 with y0 = x0 and γ0 connecting y0 with z0 = y0 , such that the following identity holds: Lˆγ0 Lˆγ0 = Lˆy0 y0 Lˆγ0 γ0 .

(3.26)

The notion “C ∗ -algebra generated by abstract elements” is meant in the sense of Woronowicz [17] and will be explained in Subsection 3.3. From the above axioms we obtain the following basic Lemma 3.3 1. All generators Pˆγ are normal: [Pˆγ∗ , Pˆγ ] = 0 . 2. For any path γ from x to y and any path γ from y to z, the following identities hold: Lˆγ Lˆγ ˆ γ ˆγR R

= =

Lˆyy Lˆγγ , ˆ γγ , ˆ yy R R

(3.27) (3.28)

If, moreover, the end z of γ diﬀers from y, then the following identity holds: ˆ γ Lˆγ M

=

ˆ γγ . Lˆyy M

(3.29)

If, moreover, also the beginning x of γ diﬀers from y, then the following identity holds: ˆ γ Nˆγ M

ˆ γγ = R ˆ γγ Lˆyy . = Lˆyy R

3. The element Zˆ :=

1

x∈Λ0

ˆ xx ) = Im(M

1 2i

ˆ xx − Nˆxx M

(3.30) (3.31)

x∈Λ0

commutes with all elements Pˆ and, therefore, belongs to the center of the observable algebra.

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Proof. Points 1 and 3. are easily proved by direct inspection. To prove point 2 we have to consider two cases: the generic one, when all the three points x, y and z are diﬀerent and the non-generic one, when two of them coincide. We begin with the generic case. Suppose that x is diﬀerent from x0 , y0 and z0 . Choose any path ˆ α to the right on both sides of (3.26), (i.e., taking α from x to x0 . Acting with adM 1 ˆ the commutator with 2i Mα ) we obtain: 1 2i

ˆ α = Lˆαγ0 Lˆγ − Lˆy0 y0 Lˆαγ γ = 0 . Lˆγ0 Lˆγ0 − Lˆy0 y0 Lˆγ0 γ0 , M 0 0 0

(3.32)

Using identity (3.8), we can replace the path αγ0 by any other path from x to y0 and the path γ0 by any other path from y0 to z0 . Acting successively with ˆ or adN ˆ , we may shift in the same way the endpoints appropriate operators adM y0 and z0 to any other generic positions y and z. We stress that the above procedure holds for both generic and non-generic (i.e., x0 = z0 ) initial position. The cases x = y or y = z are trivial. Finally, the only non-trivial non-generic case of equation (3.27), namely x = z, may be obtained in a similar way from the generic case, by ˆ β , where β is a path from z to x. This operation shifts point z to acting with adM x. This ends the proof of formula (3.27). ˆ zz on (3.27), we directly get a generic Acting, in the generic case, with adR ˆ xx , we get a generic case of (3.30). case of (3.29). Acting on it once more with adR ˆ zz on the latter identity, we get a Finally, acting successively two times with adR generic case of (3.28). Non generic cases x = z of (3.28), (3.29) and (3.30) are also easily obtained ˆ β , where β is a path from z to by acting on corresponding generic cases with adM x. This operation shifts point x to z. Remarks 1. As will be seen later, the real and the imaginary parts of generators of the observable algebra are, in any physical representation, unbounded self-adjoint operators. Writing bilinear relations for unbounded operators is, in general, meaningless. However, as will be seen in the sequel, part of the observable algebra O(Λ) is generated from a certain Lie subalgebra of the Lie algebra deﬁned by formulae (3.12)–(3.21). We will show that non-degenerate representations of O(Λ) are given by unitary representations of the corresponding Lie group (or, equivalently, from integrable representations of the Lie algebra). Commutation relations (3.12)–(3.21), together with integrability of the representation, imply that in equations (3.8), (3.26), (3.27), (3.28) and (3.30) we always multiply strongly commuting observables. Therefore, the products used in these equations will be always unambiguously deﬁned. The same argument applies also to the right-hand side of equation (3.29) and to its left-hand side, provided x = z. The only problem could come from the leftˆ γ do not hand side of (3.29), for x = z, because the two elements Lˆγ and M commute.

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Let us discuss this point in more detail: Without any loss of generality, we may limit ourselves to the case γ = γ −1 . Observe that we have ˆ γ = ˆ γ + iImM ˆ γ Lˆγ M ReLˆγ + iImLˆγ · ReM = +

ˆ γ − ImLˆγ ImM ˆ γ ReLˆγ ReM ˆ γ + ImLˆγ ReM ˆ γ . i ReLˆγ ImM

(3.33)

It follows immediately from the commutation relations that ReLˆγ commutes ˆ γ . Similarly, ReM ˆ γ commutes with ImLˆγ . This means, that the with ImM imaginary part of the above expression is unambiguously deﬁned as a product of commuting observables. To give a meaning also to its real part we take the unambiguously deﬁned identity Lˆγ Lˆγ = Lˆyy Lˆγγ ,

(3.34)

ˆ β , where β is a path from z to x. As and act on both of its sides with adM a result we get precisely the real part of expression (3.33), which is, therefore, unambiguously deﬁned by the adjoint action of self-adjoint operators on self-adjoint operators. This shows that the above relations of the observable algebra can be meaningfully formulated on the level of unbounded operators. 2. Applying successively operators adLˆ to identity (3.26), as we did in the above proof, we may produce a lot of new identities. Each of them could also have been used as a deﬁning relation instead of (3.26). In the Lemma, we have listed only those identities, which will be used in the sequel. 3. By point 3 of the above Lemma, the observable Zˆ deﬁnes a superselection rule. Therefore, the physical Hilbert space is a direct sum of charge superselection sectors, H = ⊕α Hα , ˆ . Due to deﬁnition (3.25), the same is true for the on which Zˆ acts as Zα 1 ˆ global charge Q and, moreover, 1e Qα = Zα − N . As will be shown later, Zα may assume only integer values which proves that also 1e Qα is integer.

3.2

Generating the observable algebra from the tree data

A convenient way to solve relations between generators is to choose a tree, i.e., to choose a unique path connecting any pair of lattice sites. More precisely, a tree is a pair (x0 , T ), where x0 is a distinguished lattice site (called root) and T is a set of lattice links such that for any lattice site x there is exactly one path from x0 to x, with links belonging to T . Suppose, we have chosen a tree. Then, for any pair (x, y) of lattice sites, there is a unique along tree path from x to y. Denote by Pˆx,y the generator Pˆγ corresponding to this path. Due to equation (3.8), the remaining generators Pˆ may be expressed in terms of those, provided we know the magnetic

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ˆ To choose independent quantities among the electromagnetic generators, ﬁeld B. ˆ and denote by Bˆ take any oﬀ tree link (x, x + k), ˆ the magnetic ﬂux through x,x+k ˆ and the the surface spanned by the closed path composed of the link (x, x + k) ˆ unique on-tree path from x + k to x. It is easily seen that, solving the Bianchi identities (3.7), starting from the root and moving outside, we can reconstruct ˆ ˆ ˆ from Bˆ all the magnetic ﬂuxes B ˆ . Among electric ﬁelds we may also x;k,l x,x+k choose only those which correspond to oﬀ tree links as independent quantities. ˆ starting from the boundary and moving Indeed, solving the Gauss law (3.6) for E, towards the root, with local charges qˆx expressed in terms of generators Pˆx,y and ˆx,∞ given, we can reconstruct all the remaining elements the boundary data E ˆ Ex,x+kˆ . This way we get the following Proposition 3.4 For a given tree T , the set of tree data ˆ /T , ˆ (Pˆx,y , Bˆx,x+kˆ , E ˆ ) , (x, x + k) ∈ x,x+k constitute a complete set of generators of O(Λ) . For further details of the proof, we refer the reader to the (completely analogous) proof for the case of QED with fermions, contained in [10]. Obviously, the tree data inherit commutation relations from Deﬁnition 3.2. In particular, from point 4 of this deﬁnition, we read oﬀ the following commutation relations: ˆ ˆ (3.35) [Bˆx,x+kˆ , E ˆˆ y,y+ˆ l ] = iδ(x,y) δ(k, l) 1, ˆ ∈ where (x, x + k) / T , (y, y + ˆl) ∈ / T . Hence, the independent electromagnetic ﬁelds fulﬁll the canonical commutation relations. The corresponding associative algebra, ˆ ∈ generated by these ﬁelds, will be denoted by OTem . Moreover, for (z, z + k) / T we have: [Pˆxy , Bˆz,z+kˆ ] = 0 ,

(3.36)

ˆ [Pˆxy , E ˆ] = 0 , z,z+k

(3.37)

which means that OTem commutes with the subalgebra OTmat ⊂ O(Λ) , spanned by the matter ﬁeld generators Pˆxy . This fact, together with Proposition 3.4, implies that the observable algebra decomposes as follows: O(Λ) = OTem ⊗ OTmat .

(3.38)

We know already from point 5 of Proposition 3.1, respectively from point 5 of Deﬁnition 3.2, that the algebra OTmat is generated by a Lie algebra. The tree data inherit, of course, a Lie algebra structure, which we are now going to describe. For this purpose, let us label all the lattice sites by integers k = 1, . . . , N . (In what follows, it does not matter, which one among the points xk coincides with the previously chosen tree root x0 .) To simplify notation we shall write Pˆkl instead of

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Pˆxk ,xl . Then, from point 5 of Deﬁnition 3.2, we have the following commutation relations: 1 ˆ ˆ i [Lkl , Lmn ] 1 ˆ ˆ i [Lkl , Mmn ] 1 ˆ ˆ i [Lkl , Nmn ] 1 ˆ ˆ i [Lkl , Rmn ] 1 ˆ ˆ i [Mkl , Mmn ] 1 ˆ ˆ i [Mkl , Nmn ] 1 ˆ ˆ i [Mkl , Rmn ] 1 ˆ ˆ i [Nkl , Nmn ] 1 ˆ ˆ i [Nkl , Rmn ] 1 ˆ ˆ i [Rkl , Rmn ]

= 0,

(3.39)

= 2δkn Lˆml , = 2δml Lˆkn , ˆml + δml M ˆ kn ) , = 2(δkn N ˆ ml − δml M ˆ kn ) , = 2(δkn M

(3.40)

= 0,

(3.44)

ˆ ml , = 2δkn R ˆkn − δkn Nˆml ) , = 2(δml N ˆ kn , = 2δml R

(3.45)

= 0.

(3.48)

(3.41) (3.42) (3.43)

(3.46) (3.47)

Moreover, from point 6 of this deﬁnition, we have: ˆ kl )∗ = Nˆlk , (N ˆkl )∗ = M ˆ lk , (R ˆ kl )∗ = R ˆ lk . (Lˆkl )∗ = Lˆlk , (M

(3.49)

These relations deﬁne a complex Lie algebra, denoted by gmat T , with Lie bracket 1 [·, ·] and with conjugation “∗”, given by (3.49). Consider now the Lie algebra i gl(2N, C) (with ordinary Lie bracket [A, B] = AB − BA), and with conjugation “∗” deﬁned as follows: A∗ := −1(N,N ) A† 1(N,N ) , A ∈ gl(2N, C) .

(3.50)

Here “†” denotes Hermitian conjugation and 1(N,N ) := 10N −10N , with 1N being the unit (N × N )-matrix. Lemma 3.5 The mapping F : gmat −→ gl(2N, C) , T deﬁned by F (Lˆkl ) :=

1 i

F (Nˆkl ) :=

i

−Ekl

iEkl iEkl Ekl

iEkl Ekl −Ekl iEkl

ˆ kl ) := F (M

i

ˆ kl ) := F (R

2 i

−iEkl

Ekl −Ekl −iEkl

−Ekl −iEkl −iEkl Ekl

(3.51) ,

(3.52)

with (Ekl ) being the canonical basis of gl(N, C), is an isomorphism of complex Lie algebras with conjugation, F ( 1i [X , Y]) ∗

=

F (X ) = for X , Y ∈ gmat . T

[F (X ), F (Y)] , F (X )∗ ,

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Proof. By a lengthy, but simple calculation, which we leave to the reader. ˆ ˆ In what follows, we shall often omit writing F and identify F (P) with P. mat We denote the real form of gmat T , corresponding to “∗” by gT . The elements of this real Lie algebra are physical observables, spanned by the self-adjoint eleis identiﬁed with the ments (RePˆkl , ImPˆkl ). Under the above isomorphism, gmat T real form u(N, N ) := {A ∈ gl(2N, C) : A∗ = A} of gl(2N, C) , corresponding to the conjugation deﬁned by (3.50). Exponentiating u(N, N ) we obtain the corresponding connected Lie group U (N, N ) , consisting of those linear transformations of C2N which preserve the hermitian form deﬁned by 1(N,N ) : U (N, N ) := {U ∈ Mat2N ×2N (C) : U † 1(N,N )U = 1(N,N )} . As we will see, non-degenerate representations of OTmat will be given by intregrable ∼ representations of gmat = u(N, N ), this means unitary representations of the group T U (N, N ). Observe that due to identiﬁcation (3.51) we have Zˆ =

1

ˆ kk ) = i1(2N ) . Im(M

(3.53)

k

This element generates the center u(1) of u(N, N ). The restriction of any nondegenerate representation of the observable algebra to the center can be integrated to a representation of the subgroup {exp(iτ 1(2N ) ) , τ ∈ R1 } ∼ = U (1) . This implies that the spectrum of Zˆ must be integer. We conclude that the specˆ trum of the charge operator Q e , deﬁned by formula (3.25) must be integer, too. Observe that equations (3.27)–(3.30) (implied by axiom (3.26)) may be rewritten in the following form: For any triple xi , xj , xk of lattice points we have: Lˆij Lˆjk ˆ ij R ˆ jk R

= Lˆjj Lˆik , ˆ ik . ˆ jj R = R

(3.55)

ˆ ik . = Lˆjj M

(3.56)

ˆ ik = R ˆ ik Lˆjj . = Lˆjj R

(3.57)

(3.54)

If, moreover, j = k, then we have: ˆ jk Lˆij M If, moreover, i = j, then we have: ˆij M ˆ jk N

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Functional analytic structure

Now we are able to endow the observable algebra with a functional analytic structure. Recall that O(Λ) is the tensor product of OTem and OTmat , see (3.38). It turns out that both components can be endowed with the structure of a C ∗ -algebra generated by unbounded elements in the sense of Woronowicz, see [17]. Since OTmat is generated by the Lie algebra u(N, N ), we take the C ∗ -algebra ∗ C (U (N, N )) and factorize it with respect to the ideal generated by relations (3.54)–(3.57). (In fact, it is suﬃcient to impose only one of these relations: the proof of Lemma 3.3 shows that one of these identities implies the remaining ones.) The C ∗ -algebra C ∗ (G) of any locally compact group may be obtained as the C ∗ -completion of the group algebra of G. By deﬁnition, the latter is the space L1 (G) of integrable functions on the group, with convolution providing the product structure. The completion is taken with respect to the following norm:

f := supπ π(f ) ,

(3.58)

where f ∈ L1 (G), the supremum is taken over all representations π of the group and π(f ) denotes the operator obtained by smearing the representation over the group with the function f , see [14], [18]. We note that C ∗ (G) is one of the classes of examples considered by Woronowicz. It is generated by any basis of the Lie algebra of G, see [16] and [17]. We stress that these generators are not elements of C ∗ (G), they are only aﬃliated in the C ∗ -sense. Thus, take the algebra C ∗ (U (N, N )). To impose relations (3.54)–(3.57) on generators (more precisely, their real and imaginary parts) we multiply them from both sides by elements of C ∗ (U (N, N )) belonging to the common dense domain of the generators (e.g., the so-called smooth elements, corresponding to functions of the class C0∞ (U (N, N )) ⊂ L1 (U (N, N ))). This way we generate a double-sided ideal J ⊂ C ∗ (U (N, N )). We deﬁne the matter algebra OTmat as the quotient: OTmat ∼ = C ∗ (U (N, N ))/J .

(3.59)

It is worthwhile to notice that the same ideal may be obtained as the space of elements whose norm vanishes, if we replace (3.58) by the supremum over only those representations, which fulﬁll additional identities (3.54)–(3.57). Thus, OTmat could be deﬁned also as the completion of L1 (U (N, N )) with respect to the latter norm (cf. [16], [17]). Next, we endow the electromagnetic component OTem with a C ∗ -structure. Again, the theory of Woronowicz can be applied. By the von Neumann theorem, all irreducible representations of the (electromagnetic) canonical commutation relations (3.35) are isomorphic to the Sch¨ odinger representation. We take as OTem ∗ the C -algebra CB(H) of compact operators acting on the Hilbert space H of this representation. Here, no additional identities have to be imposed. Again, the ˆ generators Bˆx,x+kˆ and E y,y+ˆ l are aﬃliated with the algebra.

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OTmat OTem ,

Ann. Henri Poincar´e

Finally, the C ∗ -algebra O(Λ) is, by deﬁnition, the minimal tensor product of with OTem . Then, for elements A, aﬃliated with OTmat , and B, aﬃliated with A ⊗ B is aﬃliated with O(Λ).

Remarks 1. We stress that O(Λ) is a C ∗ -algebra without unit. On the abstract level, the ˆ appearing in formulae (3.24) and (3.25) has to be understood as element 1 the identity in the multiplier algebra M (O(Λ)), which is a subalgebra of the set of aﬃliated elements. 2. As noticed by Woronowicz [15], there exists an abstract deﬁnition of C ∗ (G), which applies to any topological group G, not necessarily being a Lie group. More precisely, a C ∗ -algebra A and a homomorphism α : G → M (A) of G into unitary elements of the multiplier algebra M (A) is called a C ∗ -algebra of G and denoted C ∗ (G) if the pair (A, α) is universal in the following sense: for any other such pair (B, β), there exists a morphism ϕ ∈ M or(A, B), such that β = ϕ ◦ α (for the deﬁnition of M or(A, B) see [17]). An eﬀective construction in case of a locally compact group consists in taking the above-mentioned C ∗ -completion of the group algebra of G.

4 Representations of the observable algebra and the charge superselection structure We are going to construct all faithful, irreducible and non-degenerate representations of the observable algebra O(Λ) . Concerning the electromagnetic part, the von Neumann theorem guarantees uniqueness of representations. As for the matter part, there is a one-to-one correspondence (cf. [14], [18]) between non-degenerate representations of the algebra C ∗ (U (N, N )) with unitary representations of the Lie group U (N, N ). Thus, by Deﬁnition 3.59, the non-degenerate representations of OTmat are given by those faithful, irreducible and integrable representations of the Lie algebra gmat T , which respect the additional relations (3.54)–(3.57). To ﬁnd to a certain Lie subalgebra hmat ⊂ gmat , such them, we shall further reduce gmat T T T mat mat that it also generates OT and that the enveloping algebra of hT does not inherit any identities from the relations deﬁning the ideal J . We will show that, of the above type, relations (3.54)–(3.57) enable us given a representation of hmat T to construct a unique representation of OTmat , for a given value of total charge. Moreover, every representation of OTmat is obtained this way. , we ﬁx one lattice point, say xN , and take To deﬁne the Lie subalgebra hmat T ˆ and M’s, ˆ which start at this point. More precisely, we denote: only those L’s (4.1) (4.2)

R

:= LˆN k , ˆ Nk , := M := LˆN N ,

K

ˆ NN ) , := Re (M

(4.4)

Qk Pk

(4.3)

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for k = 1, . . . , N − 1. Observe that P and Q are normal, [P, P ∗ ] = [Q, Q∗ ] = 0, whereas R and K are Hermitean, R∗ = R , K ∗ = K. Denoting Qk = qk1 + iqk2 and Pk = p1k + ip2k , with qka and pak being self-adjoint, we read oﬀ from relations (3.39)–(3.48) the following commutation relations:

a 1 i qk , 1 i R, a 1 i qk , a 1 i pk ,

1 b pl 1 K 1 K 1 K

R δkl δ ab ,

=

(4.5)

=

R,

(4.6)

=

qka pak

,

(4.7)

,

(4.8)

=

whereas the remaining commutators vanish. Observe that formulae (4.5) are the commutation relations of the real Heisenberg Lie algebra H2(N −1) , generated by 2(N − 1) canonically conjugate pairs (qja , 1 paj ) and with center generated by R. The 1-dimensional Lie algebra generated by 1 K acts by (4.6)–(4.8) on H2(N −1) , endowing hmat with the structure of T a semidirect sum, = R1 ⊕s H2(N −1) . hmat T

(4.9)

does not inherit any identiObserve that the enveloping algebra spanned by hmat T ties from the deﬁning relations (3.54)–(3.57), indeed.

4.1

Representations of the Lie algebra hmat T

Theorem 4.1 There are exactly two faithful, irreducible and integrable representations of the Lie algebra hmat T . They are both deﬁned on the Hilbert space H := L2 (CN −1 × R1 ), and are given by the following formulae: (Qk Ψ)(z1 , . . . , zN −1 , λ)

=

(Pk Ψ)(z1 , . . . , zN −1 , λ)

=

(RΨ)(z1 , . . . , zN −1 , λ)

=

(KΨ)(z1 , . . . , zN −1 , λ)

=

±eλ zk Ψ(z1 , . . . , zN −1 , λ) , 2 ∂ eλ Ψ(z1 , . . . , zN −1 , λ) , i ∂ z¯k ±e2λ Ψ(z1 , . . . , zN −1 , λ) , ∂ Ψ(z1 , . . . , zN −1 , λ) , i ∂λ

(4.10) (4.11) (4.12) (4.13)

where Ψ ∈ H and (z1 , . . . , zN −1 , λ) ∈ CN −1 × R1 . Proof. We use the following matrix representation of hmat T : 

k X(k, r, p, q) :=  0 0

 p r 0 q , 0 −k

(4.14)

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where qT , p ∈ R2(N −1) , k, r ∈ R1 , and the generators are identiﬁed as follows: 1 K

=

R = =

1 1 pj 1 2 pj qk1 qk2

X(1, 0, 0, 0) , X(0, 1, 0, 0) , X(0, 0, δji , 0) ,

=

i X(0, 0, δ2(N −1)+j , 0) ,

=

X(0, 0, 0, δji ) ,

=

i X(0, 0, 0, δ2(N −1)+j ) .

Exponentiating representation (4.14), we obtain the following matrix representation of the connected, simply connected Lie group HTmat , corresponding to hmat T ,   u a c b , g(u, c, a, b) =  0 1 (4.15) 0 0 u−1 where aT , b ∈ R2(N −1) , c ∈ R1 and u ∈ R+ . For u = 1, this formula yields the standard matrix representation of the (4(N − 1) + 1)-dimensional real Heisenberg group H2(N −1) . On the other hand, for a = 0 = b and c = 0, it yields the real (multiplicative) group in 1 dimension. Thus, HTmat coincides with the following semidirect product: (4.16) HTmat = R+ ×s H2(N −1) . Since H2(N −1) is a closed normal subgroup of HTmat , we can apply the method of induced representations, see [18]. The faithful, unitary, irreducible representations of H2(N −1) are labeled by t ∈ R∗ = R1 \ 0 , (Ut (g(1, c, a, b))f )(x) = eit(bx+c) f (x + a) ,

(4.17)

with f ∈ L2 (R2(N −1) ), see [18]. Any two representations labeled by t and t , with t = t , are unitarily inequivalent. We have to ﬁnd the orbits of the action of HTmat ˆ 2(N −1) of equivalence classes of unitary irreducible representations on the space H of H2(N −1) . This action reduces to the action of R+ , ˆ 2(N −1) × R+ (Ut , g(u, 0, 0, 0)) → Ut ◦ Adg(u, 0, 0, 0) ∈ H ˆ 2(N −1) . H Using formula (4.17) it can be shown that – up to unitary equivalence – we have Ut ◦ Adg(u, 0, 0, 0) = Uu2 t . Thus, there are two orbits, O+ = t ∈ R1 : t > 0 , and

O− = t ∈ R1 : t < 0 .

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For both orbits, the stabilizer of each point on the orbit is conjugate to H2(N −1) , and R+ acts transitively on each orbit. Consequently, there are two equivalence classes of faithful, unitary, irreducible representations U± of HTmat , induced from the representations Ut , deﬁned on the spaces H± = L2 O± , L2 (R2(N −1) ) of functions on O± with values in the representation space L2 (R2(N −1) ) of H2(N −1) . Identifying both orbits with R1 , by putting t = ±e2λ , we have H± ∼ = 2 2(N −1)+1 ) , and a simple calculation yields the following induced representaL (R tions: 2λ

(U± (g(u, c, a, b))f )(x, λ) = e±ie

u(e−λ bx+c)

f (x + eλ a, λ + ln u) .

(4.18)

Diﬀerentiating these representations along the one-parameter subgroups generated by the above-deﬁned basis elements of HTmat and identifying R2(N −1) with C(N −1) , we obtain exactly formulae (4.11)–(4.13). Observe that the transformation ˆ xy , = −R I Lˆxy (4.19) ˆ xy I R (4.20) = −Lˆxy , ˆxy , ˆ xy = N (4.21) I M ˆ xy , ˆxy = M I N (4.22) preserves the deﬁning relations (3.12)–(3.28) and, therefore, generates an automorphism of OTmat . This automorphism intertwines the two representations given by Theorem 4.1. Thus, it is suﬃcient to consider the positive representation only, because the other one is equivalent to one obtained from the positive representation by relabeling of the elements of OTmat . Such a relabeling changes, however, the physical meaning of some observables. This is, in particular, true for the electric ˆ xx ) changes under the transformation I. Consecharge. Indeed, the sign of Im(M quently, the deﬁnition (3.24) of the electric charge, would have to be supplemented by the sign of the representation. Therefore, we choose the positive representation ˆ xx . once for ever. This choice implies the positivity of all the elements Lˆxx and R It is interesting to observe that we could postulate the positivity of only one of them, say Lˆx0 x0 , as an additional axiom for the observable algebra. Indeed, identity (3.26) implies ∗ Lˆx0 x0 · Lˆyy = Lˆx0 y · Lˆyx0 = Lˆx0 y · Lˆx0 y > 0 . Similarily, we get for x = y

∗ ˆ xx · Lˆyy = Nˆxy · M ˆ yx = N ˆxy · Nˆxy > 0 . R

Hence, choosing the positive sign of Lˆx0 x0 we obtain positivity for all of them.

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Ann. Henri Poincar´e

Local charge operators

Lemma 4.2 In the positive representation we have (ˆ qxk Ψ)(z1 , . . . , zN −1 , λ) =

∂ e Ψ(z1 , . . . , zN −1 , λ) , i ∂ argzk

(4.23)

for k = 1, . . . , N − 1. Proof. Due to identity ˆ Nk = M ˆ kk LˆN N = RM ˆ kk , Q∗k Pk = LˆkN M and to formulae (4.10)–(4.13), we have: 2 ∂ ˆ kk = R¯ RM zk . i ∂ z¯k

(4.24)

The operator R is strictly positive and, hence, we may skip it on both sides of the equation. This way we obtain: ˆ kk = z¯k 2 ∂ . M i ∂ z¯k

(4.25)

The imaginary part of this operator gives us (4.23).

Remark. Deriving (4.25) from (4.24), we have, in fact, divided both sides of the equation by R = LˆN N . Such an operation is, of course, meaningless on the level of the abstract algebra OTmat . On the level of representations, however, we have proved that R cannot be a divisor of zero. Hence, this operation is fully justiﬁed. Corollary 4.3 In every representation of OTmat , the local charge operators qˆxk take integer eigenvalues (in units of the elementary charge) only. Thus, the spectrum of each qˆxk is given by Sp(ˆ qxk ) = eN .

4.3

Constructing representations of OTmat from representations of hmat T

We will show that each irreducible, faithful and non-degenerate representation of OTmat , assigned to a given tree, is uniquely generated by a corresponding representation of its Lie subalgebra hmat T , provided the total electric charge Q carried by the matter ﬁeld is ﬁxed. Suppose that a representation of OTmat is given. Choosing instead of xN any (xi ) of Lie subalgebras. The reother reference point xi , we obtain a family hmat T strictions of the above representation to these subalgebras are all given by Theorem 4.1. In particular, for x1 , the corresponding Lie subalgebra hmat (x1 ) is generated T

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by the following observables: k Q P k

:=

:= R := := K

Lˆ1k , ˆ 1k , M

(4.27)

Lˆ11 ,

(4.28)

ˆ 11 ) , Re (M

(4.29)

(4.26)

for k = 2, . . . , N . The corresponding positive representation is equivalent to the following one: k Ψ)(µ, w2 , . . . , wN ) = (Q w2 , . . . , wN ) = (P k Ψ)(µ, Ψ)(µ, (R w2 , . . . , wN ) = Ψ)(µ, (K w2 , . . . , wN ) =

w2 , . . . , wN ) , eµ wk Ψ(µ, ∂ 2 Ψ(µ, w2 , . . . , wN ) , eµ i ∂w ¯k w2 , . . . , wN ) , e2µ Ψ(µ, ∂ Ψ(µ, w2 , . . . , wN ) , i ∂µ

(4.30) (4.31) (4.32) (4.33)

∈H ∼ with Ψ = L2 (R1 × CN −1 ) . We stress that if we dealt with the restrictions of to the two subalgebras under an arbitrary representation of the Lie algebra gmat T consideration, then these restrictions would be completely independent. Here, the additional constraints (3.27)–(3.28) imply a relation between the two Lie algebra in terms of the representations, which enables us to express the wave function Ψ wave function Ψ uniquely. For this purpose, take the polar decomposition of wN , wN = eλ ξ , where |ξ| = 1 and eλ = |wN | . For any Ψ ∈ H , deﬁne the following function Φ ∈ H: Φ(µ, w2 , . . . , wN ) := eµ−λ Ψ(eµ ξ −1 , w2 ξ −1 , . . . , wN −1 ξ −1 , λ) .

(4.34)

It is easy to check that this formula deﬁnes an isometric isomorphism from H to . (The factor in front of Ψ is necessary to convert the radial measure |z1 |d|z1 | H z1 into the measure dµ and the coming from the two-dimensional measure dz1 d¯ measure dλ into the radial measure |wN |d|wN |, coming from the two-dimensional must be unitary, too. measure dwN dw ¯N .) Hence, the transformation from Φ to Ψ We are going to prove that this transformation is of a special type, consisting in multiplication by a phase factor only. Lemma 4.4 Given a faithful, irreducible, positive and non-degenerate representa and Φ diﬀer by a phase factor depending only tion of OTmat , the wave functions Ψ upon the phase ξ of wN . More precisely, for any quantum state Ψ we have: = ξ Qe Φ , Ψ

(4.35)

where Q is the total charge carried by the representation under consideration.

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Proof. For k = 2, . . . , N − 1 , we have in the ﬁrst representation: k , R¯ z1 zk = Q∗1 Qk = Lˆ1N LˆN k = LˆN N Lˆ1k = RQ

(4.36)

k = z¯1 zk . Thus, Q k acts on Φ in the same which, due to positivity of R implies Q way as on Ψ : k = z¯1 zk = eµ · ξ · wk · ξ −1 = eµ wk . Q For k = N , this also holds true: ∗ N = Lˆ1N = LˆN 1 = Q∗1 = z¯1 eλ = eµ ξeλ = eµ wN . Q Moreover, in the ﬁrst representation we have : ∗ = RLˆ11 = LˆN N Lˆ11 = LˆN 1 LˆN 1 = Q∗1 Q1 = e2λ z¯1 z1 = Re2µ , RR = e2µ . We conclude that the “position operators” (R, Q k ) of and, consequently, R This implies that Φ the second representation act in the same way on Φ as on Ψ. may diﬀer only by a phase factor, and Ψ = exp(if )Φ , Ψ

(4.37)

where f is real. Next, we prove that the phase factor f does not depend upon variables µ , |wN | and wk , for k = 2, . . . , N − 1 . For this purpose, observe that in the ﬁrst representation we have ∗ 2 ∂ ˆ 11 = LˆN 1 M ˆ N 1 = Q∗1 P1 = R¯ ˆ 11 = LˆN N M z1 · , RM i ∂ z¯1 and, consequently ˆ 11 = z¯1 · 2 ∂ . M i ∂ z¯1

(4.38)

The real part of this operator applied to Φ gives = ∂ , K i ∂µ

(4.39)

acting on Ψ. We conclude that the phase factor cannot which coincides with K depend upon µ. To prove that it does not depend upon wk , for k = 2, . . . , N − 1, neither, observe that in the ﬁrst representation we have ∗ 2 ∂ ˆ 1k = LˆN 1 M ˆ N k = Q∗ Pk = R¯ z1 · , RP k = LˆN N M 1 i ∂ z¯k and, consequently, 2 ∂ P k = z¯1 · . i ∂ z¯k

(4.40)

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When applied to Φ it yields 2 ∂ 2 ∂ = eµ · , P k = eµ ξ · i ∂ z¯k i ∂w ¯k

(4.41)

exactly as in the Ψ-representation. This implies that f does not depend upon wk . Finally, to prove the independence of f from |wN | observe that in the second representation holds ∗ 2 ∂ M ˆ N N = Lˆ11 M ˆ N N = Lˆ1N M ˆ 1N = Q ∗ P N = R w R ¯N · , N i ∂w ¯k and, consequently, 2 ∂ ˆ NN = w ¯N · . M i ∂w ¯N yields The real part of this operator applied to Ψ 2 ∂ ∂ ˆ N N ) = Re w K = Re (M |wN | ¯N · = i ∂w ¯N i ∂ |wN | ∂ 1 |wN | . = |wN | i ∂ ln |wN |

(4.42)

(4.43)

∂ . Transformed On the other hand, in the ﬁrst representation K acts on Ψ as i ∂λ to the representation in terms of Φ it gives precisely (4.43). This implies that the phase factor f may depend only upon the phase of wN , f = f (ξ). To prove the speciﬁc form of f in equation (4.35), we diﬀerentiate Φ in formula (4.34) and get N −1 ∂ e ∂ e ∂ e Φ=− Φ= Ψ. i ∂ argwN i ∂ argξ i ∂ argzk

(4.44)

k=1

Thus, by Lemma 4.2 we have: N −1 ∂ e Φ=− qˆxk Φ = (ˆ qxN − Q)Φ . i ∂ argwN

(4.45)

k=1

On the other hand, in the second representation we have: = qˆxN Ψ =

e ∂ = e ∂ (exp(if )Φ) Ψ i ∂ argwN i ∂ argξ ∂f exp(if ) e Φ + (ˆ qxN − Q)Φ , ∂ argξ

(4.46) (4.47)

which implies Q ∂f = . ∂ argξ e

(4.48)

As a consequence we get the following

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Theorem 4.5 Given a value Q ∈ eN of the total electric charge carried by the matter ﬁeld, there exists exactly one faithful, irreducible, positive and non-degenerate representation of the algebra OTmat , assigned to a given tree. This representation is uniquely generated by its restriction to the Lie subalgebra hmat T . For diﬀerent values of Q these representations are inequivalent. There is no non-degenerate representation of OTmat corresponding to Q ∈ / eN. Proof. Choosing diﬀerent points xk , together with the corresponding Lie subalgebras hmat we may T (xk ), and transforming between diﬀerent representations, ∗ ˆ kl and Nˆkl = M ˆ lk , for k = l, in a single, uniquely calculate elements Lˆkl , M ﬁxed representation, say the ﬁrst one, described by the wave function Ψ. Then, we ∗ ˆ ˆ ˆ ˆ can represent also all the “ultralocal” observables Lkk , Mkk and Nkk = Mkk in this ﬁxed representation. Indeed, in the representation based on xk we have Lˆkk = R, whereas qˆxk ˆ Mkk = K + i +1 (4.49) e according to formulae (4.4) and (3.24). Finally, we ﬁnd the representation of the ˆ kl by choosing any xn , diﬀerent from both xk and xl , and using observables R formula ˆ kl = Nˆkn M ˆ nl . Lˆnn R (4.50) We denote the irreducible representation space corresponding to eigenvalue ˆ by Hmat (Z) . Z of the total charge 1e Q T

4.4

Charge superselection structure

Now we use the fact that, for a given tree, the observable algebra decomposes, O(Λ) = OTem ⊗ OTmat ,

(4.51)

see (3.38). But the electromagnetic part OTem is generated by a ﬁnite number of canonically conjugate pairs. Thus, its integrable representations are (up to unitary equivalence) unique, due to the von Neumann theorem [8]. Thus, the faithful, irreducible and non-degenerate representations of O(Λ) are labeled by the irreducible charge sectors of the matter ﬁeld part OTmat , as described in the previous subsection. For a given tree, we get the physical Hilbert space as a direct sum of charge superselection sectors, (4.52) H(Λ) = HTem ⊗ HTmat (Z) . Z

Finally, it can be easily shown that a diﬀerent choice of the tree induces a similar decomposition of O(Λ) as above and the two decompositions are related to each other via an isomorphism of the corresponding electromagnetic observable algebras.

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5 Discussion As we have seen, the fact that the observable algebra is generated by a certain Lie algebra, is extremely helpful for the classiﬁcation of its irreducible representations. It should be also helpful for constructing the thermodynamic limit, because for taking the limit N → ∞ in the generating Lie algebra there seems to exist appropriate mathematical tools for studying the resulting representations, see [20]. However, it is doubtful, whether the classiﬁcation of charge superselection sectors obtained here will carry over to the thermodynamic limit in a straightforward way. One should rather expect that these considerations will have to be supplemented by a discussion of ﬁeld dynamics. For this purpose we deﬁne the lattice version of the ﬁeld Hamiltonian (2.3) by the following procedure. In formulae (3.2)–(3.5) we ˆ and use the substitution put γ = (x, x + k)

ˆ x+k

ˆ 1 ˆ + iagAk , A)

exp(ig x

φˆx+kˆ

φˆx + a∂k φˆx .

(5.53) (5.54)

Consequently, we obtain the following approximation: ∗ 1 2 ˆ ˆ |Dφ(x)| Lˆ(x,x+k) Lˆ(x,x+k) . 2 (Lˆxx )−1 ˆ − Lxx ˆ − Lxx a ˆ k

This leads to the lattice version of the Hamiltonian (2.3): ˆ =H ˆ el−mag + H ˆ matter + H ˆ int , H

(5.55)

where the electromagnetic, matter and interaction parts of the Hamiltonian are given by: 2 a 2 ˆ ˆ ˆˆ , ˆ el−mag = a E B H + (5.56) ˆ x,x+k x;k,l 2 2 ˆ ˆˆ (x,x+k) x;k, l 1 3 ˆ ˆ ˆ Rxx + V (Lxx ) , Hmatter = a (5.57) 2 x ˆ int = a ˆ 2 H (Lˆxx )−1 |Lˆ(x,x+k) (5.58) ˆ − Lxx | . 2 ˆ x,k

Given a ﬁnite lattice, we have thus approximated the ﬁeld by a quantummechanical system whose dynamics is governed by the positively deﬁned Hamiltonian1 (5.55). Its ground state will be treated as a ﬁnite-lattice-approximation of the vacuum. Suppose that this vacuum state has been found. Then, applying the 1 When considering diﬀerent boundary conditions, an additional surface term in the Hamiltonian will be necessary.

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strategy outlined in [10], one should ﬁnd the vacuum in the thermodynamic limit as a projective limit of the above vacuum states corresponding to ﬁnite lattices. Using this vacuum, the physically admissible representations of the observable algebra (in the thermodynamic limit) could be singled out via the GNS-construction. To ﬁnd the “true” vacuum will be, of course, extremely diﬃcult, but may be one can ﬁnd an approximation, which is “much better” than the perturbative vacuum. Then one can hope to get deeper insight into nonperturbative aspects of this model. The same remark applies to one- or multi-particle states. Of course, the ultra-violet limit of the theory (obtained by sending the lattice spacing a to zero) is, probably, much more diﬃcult to investigate. But, in principle, the strategy outlined in [10] applies also here.

Acknowledgments The authors are very much indebted to L. S. Woronowicz for helpful discussions and remarks. One of the authors (J. K.) is grateful to Professor E. Zeidler for his hospitality at the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

Appendix A

A non-integrable representation carrying non-integer charge

Quantization of charge is due to integrability of the representations of the Lie algebra generating the matter ﬁeld part of the observable algebra. Below, we construct a weak (non-integrable) representation, which carries non-integer charge. At the same time we construct an interesting example of weakly commuting operators, which do not commute strongly (cf. [13]). We limit ourselves to a single lattice point (N = 1) and multiply the wave function by the phase factor ξ c = exp(ic · argz), c ∈ N , as in formula (4.35). After this operation the spectrum of the charge operator ˆj := qˆ = 1 ∂ , e i ∂ argz

(A.1)

gets shifted by the value c. It is interesting that a similar “shift” can be also deﬁned for any real number c ∈ R. Of course, the result of such a “shift” cannot be unitarily equivalent to the original operator ˆj. As will be seen in the sequel, this leads to a simple example of canonical commutation relations, which are fulﬁlled only in the weak sense: momenta pˆ1 and pˆ2 do not commute strongly or, in other words, Pˆ = pˆ1 + iˆ p2 is not normal.

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Consider, therefore, the Hilbert space H = L2 (C) and deﬁne the following two groups of unitary transformations: ˆ1 (t)Ψ)(z) = (U ˆ2 (s)Ψ)(z) = (U

1 (ϕ1 (z + t)Ψ(z + t)) , ϕ1 (z) 1 (ϕ2 (z + is)Ψ(z + is)) , ϕ2 (z)

(A.2) (A.3)

where ϕ1 (respectively ϕ2 ) is a phase factor obtained from the multi-valued function ϕ(z) := exp(ic · argz) , by the cut along the real (respectively the imaginary) positive half-axis. Denote by pˆa , a = 1, 2, the self-adjoint generators of these groups: ˆa (t) = exp(itpa ) . U These operators may be easily calculated on their common, dense domain D, consisting of those functions, which are smooth and vanish identically in a neighbourhood of the two cuts. Indeed, for Ψ ∈ D, a straightforward calculation gives: (ˆ p1 Ψ)(z) = (ˆ p2 Ψ)(z) =

1 ϕ1 (z) i ∂ 1 ϕ2 (z) i ∂

∂ Re ∂ Im

z z

(ϕ1 (z)Ψ(z)) ,

(A.4)

(ϕ2 (z)Ψ(z)) .

(A.5)

This immediately implies weak commutation: pˆ1 pˆ2 Ψ = pˆ2 pˆ1 Ψ . However, the commutation relations are not satisﬁed in the strong sense, because ˆ2 do not commute. Indeed, take any smooth function Ψ with ˆ1 and U the groups U support contained in the square S = {−2 ≤ Re z, Im z ≤ −1}. It is easy to check that we have ˆ2 (3)U ˆ1 (3)Ψ = exp(2πic)U ˆ1 (3)U ˆ2 (3)Ψ , U (A.6) and, whence, we obtain the strong commutation only for c ∈ N. A simple geometric interpretation of this result consists in interpreting the multi-valued function Φ = ϕ · Ψ as a function deﬁned on the Riemann surface R of the logarithm, R = {(|z|, argz) : |z| ∈ R+ , argz ∈ R1 } , and satisfying the condition: Φ(|z|, argz + 2π) = exp(2πic)Φ(|z|, argz) . with scalar product deﬁned by integration These functions form a Hilbert space H over any closed set D ⊂ R, which covers almost the whole space C only once.

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is an isomorphism of Hilbert spaces. The mapping H Ψ → Φ := ϕΨ ∈ H ˆ2 (3)U ˆ1 (3) and U ˆ1 (3)U ˆ2 (3) describe When interpreted in terms of H, the operators U dragging Φ along the spiral surface R with respect to the two opposite helicities. ˆ =x x2 , where x ˆ1 (resp. x ˆ2 ) is the operator of Now, consider the operator X ˆ1 +iˆ multiplication by the real (resp. the imaginary) part of z. Formulae (A.2) and (A.3) ˆ1 satisfy the canonical commutation relations in the strong imply that pˆ1 and x ˆ2 . Hence, formally Pˆ = pˆ1 + iˆ p2 sense. The same is true for pˆ2 and the operator x ˆ and X = xˆ1 + iˆ x2 satisfy the commutation relations (2.11) for quantum mechanics of two degrees of freedom in the weak sense. But, these relations are not satisﬁed strongly, because the real and the imaginary parts of Pˆ , although being self-adjoint, do not commute strongly. It is easy to show that the formal “charge operator” built from them, ˆ ∗ Pˆ , qˆ := Im X is self-adjoint and has spectrum shifted by c with respect to the ordinary spectrum: Sp qˆ = {n + c : n ∈ N} . ˆ ∗ X, ˆ M ˆ =X ˆ ∗ Pˆ , Nˆ = Pˆ ∗ X ˆ and R ˆ = Pˆ ∗ Pˆ We conclude that the operators Lˆ = X satisfy the axioms of the observable algebra weakly, but do not provide its strong (integrable) realization.

B A unified description of the bosonic and the fermionic case Consider at each lattice point xk the following bosonic annihilation and creation operators: ak := √12 Reφˆk + i Reˆ πk , (B.1) (B.2) πk , bk := √12 Imφˆk + i Imˆ a∗k := √12 Reφˆk − i Reˆ (B.3) πk , b∗k := √12 Imφˆk − i Imˆ πk . (B.4) Then, take their combinations: χk

:=

√1 2

(ak + ibk ) =

1 2

ϕ∗k

:=

√1 2

(a∗k + ib∗k ) =

1 2

χ∗k

:=

√1 2

(a∗k − ib∗k ) =

1 2

ϕk

:=

√1 2

(ak − ibk ) =

1 2

φˆk + i π ˆk φˆk − i π ˆk φˆ∗k − i π ˆk∗ φˆ∗k + i π ˆk∗

,

(B.5)

,

(B.6)

,

(B.7)

.

(B.8)

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The entire information about the ﬁeld algebra may be encoded in the following objects: ∗ χk χk ∗ ψk = = , ψ . (B.9) k ϕ∗k ϕk In [10] and [11] we considered spinoral QED, where the matter ﬁeld was described by a similar structure. The only diﬀerence was, that there both χ and ϕ carried an additional spinorial index K = 1, 2, (but this only multiplies the number of degrees of freedom). These objects fulﬁll the canonical (anti)-commutation relations: ˆ, [χk , χ∗l ]∓ = [ϕk , ϕ∗l ]∓ = δkl 1

(B.10)

where the upper sign always applies to the bosonic and the lower to the fermionic case. The remaining (anti)-commutators vanish. Due to (2.2), under a gauge transformation the ﬁeld ψ is multiplied by e−igλ(x) , whereas ψ ∗ is multiplied by eigλ(x) . Hence, as the observable algebra generators we may use the following gauge invariant combinations: ˆ ϕ∗j , (B.11) lij := ±ϕi exp(ig A) γ ˆ χj , (B.12) rij := −χ∗i exp(ig A) γ ˆ χj , (B.13) mij := i ϕi exp(ig A) γ ˆ ϕ∗ . (B.14) nij := i χ∗i exp(ig A) j γ

1∓1 2 ,

which equals 0 for the bosonic and 1 Here, by we denote the number := for the fermionic case. It is easy to check that these generators fulﬁll the following universal commutation relations, the same for bosons as for fermions: [lij , lkl ] =

−δkj lil + δil lkj ,

(B.15)

[lij , mkl ] =

−δkj mil ,

(B.16)

[lij , nkl ] = [lij , rkl ] =

δil nkj , 0,

(B.17) (B.18)

0, δkj lil − δil rkj ,

(B.19) (B.20)

[mij , rkl ] = [nij , nkl ] =

−δkj mil , 0,

(B.21) (B.22)

[nij , rkl ] = [rij , rkl ] =

δil nkj , −δkj ril + δil rkj .

(B.23) (B.24)

[mij , mkl ] = [mij , nkl ] =

But the conjugation is diﬀerent in both cases: ∗ lij = lji ,

∗ rij = rji ,

m∗ij = ±nji .

(B.25)

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It is easy to check that, under the matrix presentation (3.51) and (3.52), we have: iE 0 kl mkl := i 00 E0kl (B.26) lkl := 0 0 0 0 0 0 rkl := 0 iEkl . (B.27) nkl := i Ekl 0 In the bosonic case, the conjugation (B.25) implies (3.50) and, therefore, the algebra of self-adjoint observables is generated by u(N, N ). In the fermionic case, (B.25) implies A∗ = −A† and the algebra of self-adjoint observables is generated by two copies of u(2N ), corresponding to two values of the spinorial index K. In both cases, the following formula for the total charge holds: 1 Q = l + rii − 1 . (B.28) ii e

References [1] F. Streater and A. Wightman, PCT, Spin and Statiscics and All That, W.A. Benjamin, New York 1964 R. Haag and D. Kastler, J. Math. Phys. 5, 848 (1964). [2] S. Doplicher, R. Haag and J. Roberts, Commun. Math. Phys. 23, 199 (1971). [3] F. Strocchi and A. Wightman, J. Math. Phys. 15, 2198 (1974). [4] F. Strocchi, Commun. Math. Phys. 56, 57 (1977). [5] F. Strocchi, Phys. Rev. D17, 2010 (1978). [6] D. Buchholz, Commun. Math. Phys. 85, 49 (1982) D. Buchholz, Phys. Lett. B174, 331 (1986). [7] K. Fredenhagen and M. Marcu, Commun. Math. Phys. 92, 81 (1983). [8] J. v.Neumann, Math. Ann. 104, 570 (1931). [9] J. Kijowski, A. Thielmann, J. of Geom. and Phys. 19, 173 (1996). [10] J. Kijowski, G. Rudolph and A. Thielmann, Commun. Math. Phys. 188, 535 (1997). ´ [11] J. Kijowski, G. Rudolph and C. Sliwa, Lett. Math. Phys. 43, 299 (1998). [12] J. Kijowski, G. Rudolph, J. Math. Phys. 43 No 4, 1796–1808 (2002). [13] E. Nelson, Annals of Mathematics 70, 572 (1959). [14] J. Dixmier, Les C ∗ -alg`ebres at leurs repr´esentations, Gauthier-Villars, 1967. [15] S.L. Woronowicz, privat communication.

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[16] S.L. Woronowicz, K. Napi´ orkowski, Rep. Math. Phys. 31, 353 (1992). [17] S.L. Woronowicz, Reviews in Mathematical Physics Vol. 7, No. 3, 481 (1995). [18] A.A. Kirillov, Elementy teorii predstavlenii, “Nauka”, Moscow 1972. [19] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Phys., vol. 159, Springer (1982) E. Seiler, “Constructive Quantum Field Theory: Fermions”, in Gauge Theories: Fundamental Interactions and Rigorous Results, eds. P. Dita, V. Georgescu, R. Purice. [20] V.G. Kac and A.K. Raina, Bombay Lectures On Highest Weight Representations of Inﬁnite Dimensional Lie Algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientiﬁc 1987. ´ J. Kijowski and C. Sliwa Center for Theoretical Physics Polish Academy of Sciences al. Lotnik´ ow 32/46 PL-02-668 Warsaw, Poland email: [email protected] email: [email protected] G. Rudolph Institut f¨ ur Theoretische Physik Universit¨ at Leipzig Augustusplatz 10/11 D-04109 Leipzig, Germany email: [email protected] Communicated by Klaus Fredenhagen submitted 10/10/02, accepted 17/07/03

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 4 (2003) 1169 – 1218 c Birkh¨ auser Verlag, Basel, 2003 1424-0637/03/061169-50 DOI 10.1007/s00023-003-0159-z

Annales Henri Poincar´ e

A Converse Hawking-Unruh Eﬀect and dS2/CFT Correspondence Daniele Guido and Roberto Longo Abstract. Given a local quantum ﬁeld theory net A on the de Sitter spacetime dS d , where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e., particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh eﬀect. Such positive energy evolutions always exist as noncommutative ﬂows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS d . Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets A on dS 2 and local conformal non-isotonic families (pseudonets) B on S 1 . The pseudonet B gives rise to two local conformal nets B± on S 1 , that correspond to the H± horizon components of A, and to the chiral components of the maximal conformal subnet of A. In particular, A is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iﬀ the translations on H± have positive energy and the translations on H∓ are trivial. This is the case iﬀ the one-parameter unitary group implementing rotations on dS 2 has positive/negative generator.

1 Introduction The thermalization eﬀects discovered by Hawking [31], resp. by Unruh [54], have shown the concept of particle to be gravity, resp. observer, dependent; the two eﬀects being related by Einstein equivalence principle. Unruh eﬀect deals in particular with a quantum ﬁeld theory on Minkowski spacetime: an observer O with uniform acceleration a feels, in its proper Rindler a . As noticed in [50], this can spacetime W , the Hawking temperature TH = 2π be also explained by the Bisognano-Wichmann theorem [2]: the one-parameter automorphism group describing the evolution of O in its proper observable algebra A(W ) satisﬁes the KMS thermal equilibrium condition at inverse temperature βH = TH−1 , see [30]. The Gibbons-Hawking eﬀect [22] occurs in the de Sitter spacetime with radius 1 . Again we may ρ. Here every inertial observer O feels the temperature TGH = 2πρ express this fact by saying that the one-parameter automorphism group describing the evolution of O in its proper observable algebra A(W ) satisﬁes the KMS thermal

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−1 equilibrium condition at inverse temperature βGH = TGH , where W is here the static de Sitter spacetime ([20] for the two-dimensional case). In the Minkowski spacetime, the KMS property for a uniformly accelerated observer O can be taken as a ﬁrst principle, then the basic structure follows, in particular the Poincar´e symmetries with positive energy are then derived [24, 11, 25, 29] and [16, 13] for a related approach. See also [29, 40] for weaker thermal conditions. In the de Sitter space, the KMS property for the geodesic observer O can be taken as a ﬁrst principle [6, 9, 44]; in particular the value of TGH is then ﬁxed automatically [6]. We mention at this point that actual observations in cosmology indicates that, on a large scale, our universe is isotropic, homogeneous and repulsively expanding. The de Sitter spacetime thus provides a good approximation model, at least asymptotically. Hence de Sitter spacetime and, more generally, RobertsonWalker spacetimes with positive scalar curvature are basic objects to be studied. The ﬁrst aim of this paper is to study a sort of converse to the abovementioned thermalization eﬀects. Starting with the curved de Sitter space, where a geodesic observer is thermalized, we wish to ﬁnd a diﬀerent observer whose quantum evolution has positive generator, namely feels to be in a ground state. In other words, we want to keep the same state, but choose a time evolution w.r.t. which the state becomes a ground state. Now an observer in dS d whose world line is an orbit of a boost experiences a temperature 1 1 T = + a2 ≥ TGH , 2π ρ2

[44], with a the modulus of the intrinsic uniform acceleration, contrary to our aim. The dethermalization eﬀect to take place is indeed a non-trivial matter. To understand this point notice that we are looking for a particle whose acceleration compensates the curvature of the underlying space so that, at least locally, the particle’s picture of the spacetime is ﬂat. Yet the particle’s acceleration is a vector, but the curvature is a tensor so that, even in the constant scalar curvature case, there is no obvious way to fulﬁll the above requirement. Indeed, it turns out that this cooling down eﬀect is linked to the conformal invariance and, in two spacetime dimensions, to a holography in a sense similar to the one studied in the anti-de Sitter spacetime [43], as we shall explain. As is known, Minkowski spacetime M d is conformal to a double cone in the Einstein static universe E d . On the other hand dS d is conformal to a rectangular strip of E d . Using this fact one can directly set up up a bijective correspondence between local conformal nets on M d and on dS d . Less obviously, this sets up a correspondence between positive energy-momentum local conformal nets on M d and local conformal nets on dS d with the KMS property for geodesic observers. At this point it is immediate that, given any local, conformal, KMS geodesic net on dS d , there exists a timelike conformal geodesic ﬂow µ on dS d that gives rise

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to a quantum evolution with positive generators: they are simply the ones that correspond to timelike translations on M d . Let us remark that µ has only a local action on dS d , namely in general µt x “goes outside” dS d for large t. We may ask whether the ﬂow µ promotes to a ground state quantum evolution for a general local net on dS d . In a sense we need to proceed similarly to mechanics when one passes from a passive description (in terms of coordinates) to an active description (in terms of tensors). The answer is yes, but the evolution is only partially geometric. We shall show that there exists a one-parameter unitary group V with positive generator such that, in particular, V (t)A(O)V (−t) = A(µt O) for certain regions O and for all t ∈ R such that µt O is still in dS d , and ˜t ), V (t)A(O)V (−t) ⊂ A(O for all double cones O contained in the steady-state universe subregion N of dS ˜t depend(or contained in the complement of N ), for a suitable double cone O ing on O and t, cf. Remark 4.11. The unitary group V is constructed by the Borchers-Wiesbrock methods [4, 5, 55] and, in the conformal case, coincides with the previously considered one where the geometric meaning is complete. Our analysis then proceeds to determine the maximal subnet of A where the geometric meaning is complete. For any net A we show that there exists a unique maximal expected conformal subnet, and this net has the required property. Finally we consider more speciﬁcally the case of a two-dimensional de Sitter spacetime, with the aim of describing a local net on dS 2 via holographic reconstruction, namely in terms of a suitable conformal theory on S 1 . For diﬀerent approaches to dS/CF T correspondence in the two-dimensional or in the higherdimensional case, see, e.g., [51]. To this end we introduce the notion of pseudonet on S 1 . This is a family of local von Neumann algebras associated with intervals of S 1 where isotony is not assumed. Moreover, we assume M¨obius covariance, commutativity between the algebra of an interval and that of its complement, the existence of an invariant cyclic (vacuum) vector, and the geometric meaning of the modular groups. We shall show that a local conformal pseudonet B encodes exactly the same information of a SO0 (2, 1)-covariant local net A on dS 2 with the geodesic KMS property, namely we have a bijective correspondence, holography, SO0 (2, 1) − covariant local nets on dS 2 ↔ local conformal pseudonets on S 1 . The pseudonet naturally lives on one component H+ or H− of the cosmological horizon (choosing the other horizon component would amount to pass to the conjugate pseudonet), and the holographic reconstruction is based on a 1 : 1 geometric correspondence between wedges in dS 2 and their projections on H± . Conformal invariance and chirality may then be described in terms of the pseudonet.

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A net A on dS 2 gives, by restriction, two nets A± on H± , that turn out to be conformal, hence A± extend to conformal nets on S 1 . Then Aχ := A+ ⊗ A− is a two-dimensional chiral conformal net. It turns out that Aχ is naturally identiﬁed with a conformal subnet of A, indeed it is the chiral subnet of the maximal conformal subnet of A. From a diﬀerent point of view, the pseudonet B gives naturally rise to a pair of local conformal (i.e., M¨ obius covariant) nets B± on S 1 that correspond to A± . Finally we address the question of when a net on dS 2 is holographically reconstructed by a conformal net, namely when the associated pseudonet is indeed isotonic. Let τ be the Killing ﬂow (2.2) which restricts to the translations on H+ , and U is the associated one-parameter unitary subgroup of the de Sitter group representation. Then U has positive generator if and only if B is isotonic. This is perhaps a point where the relation between the dethermalization eﬀect, conformal invariance and holography is more manifest. Indeed the two-dimensional case is the only case where the de Sitter group admits positive energy representations. This means that a massless particle on H+ may evolve according with the ﬂow τ (cf. Subsection 2.1.4) hence feels a dethermalized vacuum if the representation is positive energy. However this is exactly the case where the net is conformal and “lives on H+ ”, namely the restriction A− is trivial. An analogous result holds replacing H+ with H− .

2 General structure 2.1

Geometrical preliminaries

We begin to recall some basic structure, mainly geometrical aspects, that will undergo our analysis. 2.1.1 Expanding universes and Gibbons-Hawking eﬀect As is known [22], a spacetime M with repulsive (i.e., positive) cosmological constant has certain similarities with a black hole spacetime. M is expanding so rapidly that, if γ is a freely falling observer in M, there are regions of M that are inaccessible to γ, even if he waits indeﬁnitely long; in other words the past of the world line of γ is a proper subregion N of M. The boundary H of N is a cosmological event horizon for γ. As in the black hole case, one argues that γ detects a temperature related to the surface gravity of H. This is a quantum eﬀect described by quantum ﬁelds on M (see below); heuristically: spontaneous particle pairs creation happens on H, negative energy particles may tunnel into the inaccessible region, the others contribute to the thermal radiations. 2.1.2 de Sitter spacetime The spherically symmetric, complete vacuum solution of Einstein equation with cosmological constant Λ > 0 is dS d , the d-dimensional de Sitter spacetime. By

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Hopf theorem, if d > 2, dS d is the unique complete simply connected spacetime with constant curvature R = 2dΛ/(d − 2) (if d = 2 this characterizes the universal covering of dS 2 ). dS d may be deﬁned as a pseudosphere, namely the submanifold of the ambient Minkowski spacetime Rd+1 x20 − x21 − · · · − x2d = −ρ2 (d−1)(d−2) where the de Sitter radius is ρ = . dS d is maximally symmetric, 2Λ isotropic and homogeneous; the de Sitter group SO(d, 1) acts transitively by isometries of dS d . The geodesics of dS d are obtained by intersecting dS d with twodimensional planes through the origin of Rd+1 , see, e.g., [46, 45]. In particular the world line of a material freely falling observer is a boost ﬂow line, say  x0 (t) = ρ sinh t        x1 (t) = ρ cosh t x2 (t) = 0 (2.1)    · · ·     xd (t) = 0 whose past is the steady-state universe, the part of dS d lying in the region N = {x : x1 > x0 } and the cosmological horizon H is the intersection of dS d with the plane {x : x0 = x1 }. The orbits of uniformly accelerated observers are obtained by intersecting dS d with arbitrary planes of Rd+1 [47], of course only timelike and lightlike sections describe material and light particles, the others have constant imaginary acceleration. 2.1.3 Killing ﬂows We brieﬂy recall a few facts about the proper spacetime and the corresponding evolution of an observer. Let M be a Lorentzian manifold and γ : R → M a timelike or lightlike geodesic. The proper spacetime of the observer associated with γ is the causal completion W of γ. The relative acceleration of nearby particles is measured by the second derivative of the variation vector ﬁeld V on γ; by deﬁnition, if x : R × (−δ, δ) → M is a smooth map with γ(u) = x(u, 0) then V (u) ≡ ∂v x(u, v)|v=0 . If x is geodesic, namely every map u → x(u, v) is a geodesic, then V is a Jacobi vector ﬁeld, namely V = RV γ γ where R is the curvature tensor, showing that in general there is a non-zero tidal force RV γ γ (we use proper time parametrization in the timelike case). On the other hand, if all maps u → x(u, v) are ﬂow lines of a Killing ﬂow τ , and x(u, v) = τu (x(0, v)), then the tidal forces vanish. Indeed V (u) is the image of V (0) under the diﬀerential of τu , thus V (u) parallel to V (0) because γ is geodesic. Therefore the relative velocity, hence the relative acceleration, is 0.

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In other words a Killing ﬂow having the geodesic γ as a ﬂow line describes an evolution which is static with respect to the freely falling particle associated with γ. We shall consider, in particular, the case where M is dS d and γ is a boost line; then W is, by deﬁnition, a wedge and the evolution associated with γ is described by the same one-parameter subgroup of the de Sitter group. 2.1.4 The two Killing ﬂows of a lightlike particle We consider now a null geodesic γ in dS d . It lies in a section by a two-dimensional plane of Rd+1 through the origin that contains a lightlike vector of dS d . By the transitivity of the SO0 (d, 1)-action, we may assume the plane is {x : x0 = x1 , x2 = · · · = xn = 0}. Contrary to the timelike geodesic situation (2.1), which is the ﬂow of a unique Killing ﬂow (boosts), there are here two possible Killing ﬂows with an orbit in this section. As γ is lightlike, there is no proper time associated with γ. We may parametrize γ, for example, as γ1 (s) = x(s) with  x0 (s) = s      x   1 (s) = s x2 (s) = 0 (2.2)    ···     xd (s) = 0, s ∈ R, or γ2 (t) = x(t) with

 x0 (t) = et     t    x1 (t) = e x2 (t) = 0     ···    xd (t) = 0,

(2.3) t ∈ R,

namely γ2 (t) = γ1 (et ). The supports of the two curves are of course diﬀerent, one is properly contained in the other: in the ﬁrst case it is the entire line, while in the second case it is only a half-line. Now s → γ2 (s) is a ﬂow line of the boosts (2.1), the observable algebra is A(W ) with W the wedge as above, and we are in the Hawking-Unruh situation. The boundary of W is a “black hole” horizon for the boosts: the observer associated with γ2 cannot send a signal out of W and get it back. Also t → γ1 (t) is the ﬂow line of a Killing ﬂow τ . If d = 2 we may use the usual x1 2 identiﬁcation of a point (x0 , x1 , x2 ) ∈ R3 with the matrix x ˜ = x0x+x x0 −x2 , so 1 that the determinant of x ˜ is the square of the Lorentz length of x. Now P SL(2, R) acts on R3 by the adjoint map A ∈ P SL(2, R) → Ad A ∈ SO(2, 1) where Ad A : x ˜ → A˜ xAT .

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The map Ad is an isomorphism of P SL(2, R) with SO0 (2, 1), the connected component of the identity of SO(2, 1), (we often identify P SL(2, R) and SO0 (2, 1)) and the ﬂow τ is given by

1 t 1 t 1 0 τt = Ad :x ˜→ x˜ . (2.4) 0 1 0 1 t 1 (If d > 2, τ is given by the same formula, but acts trivially on the xk coordinate, k > 2.) Proposition 2.1 The above ﬂow τ is the unique Killing ﬂow having the curve (2.2) as a ﬂow line. τ is lightlike on H and otherwise spacelike. Proof. The statement is proved by elementary computations.

In this case the observable algebra of γ2 is A(N ) = A(dS d ) (= B(H) in the irreducible case), and τ acts on the cosmological event horizon H = {x : x1 = x0 }, the boundary of N = {x : x1 > x0 }.

2.2

Quantum ﬁelds and local algebras

So far we have mainly discussed geometrical aspects of dS d . We now consider a quantum ﬁeld on dS d , but we assume that back reactions are negligible, namely the geometry of dS d is not aﬀected by the ﬁeld. Let us denote by K the set of double cones of dS d , namely K is the set of non-empty open regions of dS d with compact closure that are the intersection of the future of x and the past of y, where x, y ∈ dS d and y belongs to the future of x. A wedge is the limit case where x and y go to inﬁnity. We shall denote by ˜ the set of double cones, possibly with one or two W the set of wedges and by K ˜ ⊃ K ∪ W. Elements of K ˜ are obtained by intersecting a vertex at inﬁnity, thus K family of wedges. The ﬁeld is described by a (local) net A with the following properties. a) Isotony and locality. A is an inclusion preserving map O ∈ K → A(O)

(2.5)

from double cones O ⊂ dS d to von Neumann algebras A(O) on a ﬁxed Hilbert space H. A(O) is to be interpreted as the algebra generated by all observables which can be measured in O. For a more general region D ⊂ dS d the algebra A(D) is deﬁned as the von Neumann algebra generated by the local algebras A(O) with O ⊂ D, O ∈ K. The local algebras are supposed to satisfy the condition of locality, i.e., A(O1 ) ⊂ A(O2 ) if O1 ⊂ O2 ,

(2.6)

where O denotes the spacelike complement of O in dS and A(O) the commutant of A(O) in B(H).

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b) Covariance. There is a continuous unitary representation U of the de Sitter group SO0 (d, 1) on H such that for each region O ⊂ dS d U (g)A(O)U (g)−1 = A(gO),

g ∈ SO0 (d, 1).

(2.7)

c) Vacuum with geodesic KMS property. There is a unit vector Ω ∈ H, the vacuum vector, which is U -invariant and cyclic for the global algebra A(dS d ). The corresponding vector state ω given by ω(A) = (Ω, AΩ),

(2.8)

has the following geodesic KMS-property (see [6]): For every wedge W the restriction ω A(W ) satisﬁes the KMS-condition at some inverse temperature β > 0 with respect to the time evolution (boosts) ΛW (t), t ∈ R, associated with W . In other words, for any pair of operators A, B ∈ A(W ) there exists an analytic function F in the strip D ≡ {z ∈ C : 0 < Imz < β}, bounded and continuous on the closure ¯ of D, such that D F (t) = ω(Aαt (B)), F (t + iβ) = ω(αt (B)A),

t ∈ R,

where αt = Ad U (ΛW (t)). d) Weak additivity. For each open region O ⊂ dS we have A(gO) = A(dS d ) ,

(2.9)

(2.10)

g∈SO0 (d,1)

where the lattice symbol ∨ denotes the generated von Neumann algebra. Proposition 2.2 ([6, Borchers-Buchholz]) The following hold: • Reeh-Schlieder property: Ω is cyclic for A(O) for each ﬁxed open non-empty region O (hence it is separating for A(O) if the interior of O is non-empty). • Wedge duality: For each wedge W we have A(W ) = A(W ). • Gibbons-Hawking temperature: The inverse temperature is β = 2πρ. • PCT symmetry: The representation U of SO0 (d, 1) extends to a (anti-) unitary representation of SO(d, 1) acting covariantly on A. The Reeh-Schlieder property is obtained by using the KMS property in place of the analyticity due to the positivity of the energy in the usual argument in the Minkowski space. Wedge duality then follows as usual by the geometric action of the modular group due to Takesaki theorem; that is to say, if D = W is a wedge and L ⊂ A(W ) is a von Neumann algebra cyclic on Ω and globally stable under AdΛW , then L = A(W ); this is a known fact in Minkowski spacetime, see, e.g., [11]. Note that this argument also shows that the deﬁnition of the von Neumann algebra A(W ) is univocal if W is a wedge. Concerning the construction of the PCT anti-unitary, a corresponding result in the Minkowski space is contained in [25].

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Lemma 2.3 Let A satisfy a), b) and c). Then A is weakly additive iﬀ the ReehSchlieder property holds. Proof. Because of Proposition 2.2 it is suﬃcient to show that if O ∈ K and A(O)Ω = H then the von Neumann algebra L generated by the union of A(gO), g ∈ SO0 (d, 1), is equal to A(dS d ). Now L ⊃ ∨t A(ΛW (t)O) and the latter is equal A(W ) by Takesaki theorem. Thus L ⊃ A(W1 ) for every wedge W1 because SO0 (d, 1) acts transitively on W and we conclude L = A(dS d ) because every double cone is contained in a wedge. It follows as in [25, Proposition 3.1] (see also [6]) that the center Z of A(W ) coincides with the center of A(dS d ) and A has a canonical disintegration, along Z, into (almost everywhere) irreducible nets. Moreover A is irreducible, i.e., A(dS d ) = B(H), if and only if Ω is the unique U -invariant vector (see also [26]). We shall say that A satisﬁes Haag duality if A(O) = A(O ) for all double cones O ∈ K. Now it is elementary to check that every double cone is an intersection of wedges, indeed ˜ O= WO , O ∈ K, where WO denotes the set of wedges containing O. We then deﬁne the dual net Aˆ as ˆ A(O) ≡ A(W ) , W ∈WO

ˆ ) = A(W ) if W is a wedge, hence A(D) ˆ Note that A(W = A(D) if every double cone O ⊂ D is contained in a wedge W ⊂ D (this is the case if D is union of wedges). By wedge duality the net Aˆ is local (two spacelike separated double cones are contained in two spacelike separated wedges) and satisﬁes all properties a)–d). The following proposition is the version of a known fact in Minkowski space, cf. [48]. Proposition 2.4 Aˆ is Haag dual: ˆ ) (= A(O )), O ∈ K , ˆ = A(O A(O)

and A = Aˆ iﬀ A satisﬁes Haag duality. ˆ i ), hence ˆ = ∩i A(W Proof. Let {Wi } be the set of wedges in WO . Then A(O) ˆ ˆ ) ⊂ A(O) ˆ A(O) = (∩i A(Wi )) = ∨i A(Wi ) ⊂ A(O . To check the last part, it suﬃcient to assume that A satisﬁes Haag duality ˆ Indeed in this case A(O) = A(O ) = (∨i A(W )) = and show that A = A. i ˆ ∩i A(Wi ) = A(O).

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If τ is a ﬂow in dS d , in general we may expect a corresponding quantum evolution only if τ static, namely if τ is a Killing ﬂow (see Subsect. 2.1.4). In this case there is a one-parameter unitary group U on H implementing τ covariantly: U (t)A(O)U (t)∗ = A(τt O). Indeed U is a one-parameter subgroup of the unitary representation of SO0 (d, 1), the connected component of the identity in SO(d, 1). In particular, if γ : u ∈ R → γ(u) ∈ dS d is a timelike or lightlike geodesic, the evolution for the observer associated with γ is given by a Killing ﬂow having γ in one orbit. Now the observable algebra associated with γ is A(W ), where W is the causal envelope of γ, which is globally invariant with respect to τ . If γ describes a material particle, namely τ is a boost, then W is the wedge region globally invariant with respect to such boosts. By the geodesic KMS property, AdU is a one-parameter automorphism group of A(W ) that satisﬁes the KMS 1 [20, 9] and this corresponds, as thermal equilibrium condition at temperature 2πρ is known, to the Hawking-Unruh eﬀect [31, 54]. 2.2.1 Subnets Given a net A on dS d on a Hilbert space H, namely A satisﬁes properties a), b), c), d), we shall say the B is a subnet of A if B : O ∈ K → B(O) ⊂ A(O) is a isotonic map from double cones to von Neumann algebras such that U (g)B(O)U (g)−1 = B(gO),

g ∈ SO0 (1, d),

where U is the representation of the de Sitter group associated with A. B is extended to any region as above. Clearly B satisﬁes the properties a), b) and c), but for the cyclicity of Ω. By the Reeh-Schlieder theorem argument (∨g∈SO0 (d,1) B(gO))Ω = B(O)Ω

˜ , O∈K

(2.11)

where the bar denotes the closure, thus we have B(W )Ω = HB ≡ B(dS d )Ω

(2.12)

for every W ∈ W, because the de Sitter group acts transitively on W and every double cone is contained in a wedge. Thus B(D)Ω is independent of the region D ⊂ dS 2 if D contains a wedge. Clearly B acts on HB and we denote by B0 its restriction to HB . Note that B0 is net satisfying all properties a), b), c), but not necessarily d).

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We shall say that a subnet B is expected (in A) if for every O ∈ K there exists a vacuum preserving conditional expectation of A(O) onto B(O) εO : A(O) → B(O) such that εO A(O0 ) = εO0

˜ . O0 ⊂ O , O0 , O ∈ K

It is easily seen that εO is given by εO (X)Ω = EO XΩ,

X ∈ A(O) ,

where EO is the orthogonal projection onto B(O)Ω. Lemma 2.5 If B is expected, then B0 is weakly additive. Proof. Let O be a double cone and W ⊃ O a wedge. If X ∈ A(O) we have εO (X)Ω = EO XΩ = εW (X)Ω = EW XΩ = EXΩ where E is the projection on HB , hence B(O)Ω = εO (A(O))Ω = EA(O)Ω = EH = HB , namely B0 (O) is cyclic on Ω. The statement then follows by Lemma 2.3.

The following Lemma is elementary, but emphasizes a property that holds on dS d but not on M d and makes a qualitative diﬀerence in the subnet analysis in the two cases. Lemma 2.6 Let O be a double cone in dS d . Then the union of all wedges in WO ≡ {W ∈ W : W ⊃ O} has non-empty causal complement (it is the double cone antipodal to O). ¯ denote its antipodal point. If W is a wedge, then W is Proof. If x ∈ dS d , let x ¯. If {Wi } is a family of the antipodal of W , hence W contains x iﬀ W contains x wedges, then

x∈ Wi ⇔ x ¯∈ Wi = ( Wi ) . i

i

i

Thus if ∩i Wi = O the spacelike complement of ∪i Wi is the antipodal of O.

Proposition 2.7 Let A be a local net on dS d on a Hilbert space H, and B a subnet. ˆ Setting B(O) = ∩W ∈WO B(W ), the following hold: ˆ 0 on HB (the dual net of B0 ). (i) B restricts to B ˆ (ii) B0 is Haag dual iﬀ B = B. ˆ ˆ (iii) B is an expected subnet of A. ˆ (iv) B is expected in A iﬀ B0 is Haag dual.

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Proof. (i): By Lemma 2.7 D = ∅ where D ≡ ∪{W : W ∈ WO }. Hence Ω is separating for A(D), so the map X ∈ B(D) → X HB is an isomorphism between B(D) on H and B(D) on HB . It follows that the operation ∩W ∈WO B(W ) of taking intersection commutes with the restriction map. 0 = B0 , thus iﬀ B = Bˆ by the (ii): By Proposition 2.4 B is Haag dual iﬀ B previous point. (iii): If W is a wedge then by the geodesic KMS property and Takesaki theorem there exists a vacuum preserving conditional expectation εW : A(W ) → B(W ) such that εW (X)E = EXE, X ∈ A(W ), where E is the orthogonal projection onto B(W )Ω = HB , (cf. [18, 41]). ˆ Let X ∈ A(O) and W ∈ WO . Since X ∈ A(W ), we have εW (X) ∈ B(W ), so there exists YW ∈ B(W ) such that YW E = EXE. If W1 is another wedge in WO then YW1 Ω = EXΩ = YW Ω, thus YW1 = YW because Ω is separating for B(W1 ) ∨ B(W ) by Lemma 2.6 and Reeh-Schlieder theorem. Thus the operator Y ≡ YW is independent of W ∈ WO and belongs to B(W ) for all wedges in WO , ˆ namely Y ∈ B(O). The map X → Y is clearly a vacuum preserving conditional ˆ ˆ expectation from A(O) onto B(O). (iv): If B0 , then B = Bˆ by (i) and B is expected in Aˆ by (iii). ˆ We have Conversely, assume that B is expected in A. ˆ ˆ B(O) = B(W ) ⊂ A(W ) = A(O) , W ∈WO

W ∈WO

ˆ ˆ so, if X ∈ B(O), then X ∈ A(O) and EXE = XE, namely εO (X) = X, so X ∈ B(O). Thus Bˆ = B.

3 Conformal ﬁelds 3.1

Basics on the conformal structure

It is a known fact that several interesting spacetimes can be conformally embedded in the Einstein static universe, see [32, 1]. We shall recall here some embeddings and we begin with a discussion about conformal transformations. 3.1.1 The conformal group and the conformal completion Two metrics on a manifold are said to belong to the same conformal class if one is a multiple of the other by a strictly positive function. Given two semi-Riemannian manifolds M1 , M2 , a local conformal map is a triple (D1 , D2 , T ) where D1 ⊂ M1 , D2 ⊂ M2 are open, non-empty sets and T : D1 → D2 is a diﬀeomorphism which pulls back the metric on M2 to a metric in the same conformal class as the original metric on M1 .

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With M a d-dimensional semi-Riemannian manifold, a conformal vector ﬁeld is a vector ﬁeld Z on M that satisﬁes the conformal Killing-Cartan equation: there exists a function f such that ∇Z X, Y + X, ∇Z Y = f X, Y ,

(3.1)

for all vector ﬁelds X, Y, Z. Conformal vector ﬁelds form a Lie algebra (they exponentiate to local oneparameter groups of local conformal maps, see below). We shall now assume d ≥ 3 (our discussion will motivate deﬁnitions also in the d = 2 case). The dimension of the Lie algebra of the conformal Killing vector ﬁelds is then ﬁnite and indeed lower or equal than (d + 1)(d + 2)/2, the equality holding if and only if the manifold is conformally ﬂat, namely the metric tensor is equal to the ﬂat one up to a nonvanishing function [53]. Such Lie algebra is called the conformal Lie algebra of M, and is denoted by conf(M). Let us observe that such Lie algebra does not really depend on the metric on M, but only on the conformal class, namely two metrics on M in the same conformal class give rise to the same Lie algebra conf(M). Let us recall now that a Lie group G acts locally on a manifold M if there exists an open set W ⊂ G × M and a C ∞ map T :W →M (g, x) → Tg x

(3.2) (3.3)

with the following properties: (i) ∀x ∈ M, Vx ≡ {g ∈ G : (g, x) ∈ W } is an open connected neighborhood of the identity e ∈ G; (ii) Te x = x, ∀x ∈ M; (iii) If (g, x) ∈ W , then VTg x = Vx g −1 and moreover for any h ∈ G such that hg ∈ Vx Th Tg x = Thg x. In general, a vector ﬁeld satisfying equation (3.1) gives rise to a one-parameter family of (non-globally deﬁned) transformations that are local conformal mappings, namely to a local action of R on M by means of local conformal maps, therefore conf(M) exponentiates to a (connected, simply connected) group acting on M by local conformal mappings. We shall call this Lie group the local conformal group of M, and denote it by CONFloc (M). A manifold M is conformally complete if the elements of CONFloc (M) are everywhere deﬁned maps, i.e., CONFloc (M) is contained in CONF(M), the group of global conformal transformations of M. Obviously, in this case CONFloc (M) is contained in CONF0 (M), the connected component of the identity in CONF(M). Lemma 3.1 The stabilizer H of a point x under the action of the group CONFloc (M) is a closed subgroup.

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Proof. Let us prove the group property. Indeed if g ∈ Vx stabilizes x then Vx is g-invariant, by (iii) above. Then, if g, h ∈ Vx stabilize x, h ∈ Vx = Vx g −1 , namely hg ∈ Vx , and clearly Thg x = x. Now assume gn → g, gn ∈ Vx and gn x = x. Then there exists n0 such that, for n > n0 , gn−1 g ∈ Vx , therefore g = gn · gn−1 g ∈ Vx , and gx = x follows by continuity. Let us assume that CONFloc (M) acts transitively on M. We may therefore = CONFloc identify M with an open subspace of the homogeneous space M (M)/H. Clearly, the Lie algebra conf(M) coincides with the Lie algebra conf(M), Therefore the local conformal group of M and CONFloc (M) acts globally on M. acts globally on M, namely M is conformally (= the local conformal group of M) complete. Let us note that in general the action of CONFloc (M) may be non-eﬀective on M, namely there may be non-identity elements of CONFloc (M) acting trivially. is a quotient of CONFloc (M). Therefore in general CONF0 (M) Now we come back to the case of a non-conformally complete manifold M on which CONFloc (M) acts transitively, and suppose that there exists a discrete such that M is a fundamental domain for Γ, central subgroup Γ of CONF(M) is conformally namely ∪Γ γM is dense in M and the γM’s are disjoint. Then M/Γ is denoted by M and is complete and M embeds densely in it. In this case, M/Γ called the conformal completion of M. (In the cases we shall consider, and possibly in all cases, the choice of Γ is unique, thus the deﬁnition of M does not depend on Γ.) We now summarize the construction of the conformal completion in the following diagram: conf. Killing v. ﬁelds

M −−−−−−−−−−−−−→ conf(M)   completion ˜ M/Γ

M

exponential

−−−−−−−→

CONFloc (M)   (transitive case)H stabilizer

Γ discrete central ˜ = CONFloc (M)/H ←−−−−−−−−−−− M

Clearly CONF(M) acts on M by restriction. Such action is indeed quasi global [10], namely the open set {x ∈ M : (g, x) ∈ W } is the complement of a meager set Sg , and the following equation holds: lim Tg x = ∞,

x→x0

g ∈ G,

x0 ∈ Sg ,

(3.4)

where x approaches x0 out of Sg and a point goes to inﬁnity when it is eventually out of any compact subset of M. It has been proved in [10] that any quasi-global action of a Lie group G on a manifold M gives rise to a unique G-completion,

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namely to a unique manifold M in which M embeds densely and on which the action of G is global. Since CONF(M) acts quasi-globally on M, we shall follow the standard usage in physics and call it the conformal group of M, denoting it by Conf(M). Of course when M is conformally complete CONF(M) = Conf(M). In the d = 2 case, the Lie algebra of conformal Killing vector ﬁelds is inﬁnitedimensional. The above discussion goes through by considering a ﬁnite-dimensional Lie subgroup. In the Minkowski spacetime (and in conformally related spacetimes, see Section 3.2) this is the Lie algebra of the group generated by the Poincar´e group and the ray inversion map. Analogous considerations can be made for isometries, namely by replacing conformal Killing vector ﬁelds by Killing vector ﬁelds which are obtained setting f = 0 in equation (3.1). These gives rise to the local one-parameter groups with values in Isoloc (M), the local isometry group. If a Lorentzian manifold is geodesically complete then Isoloc (M) acts globally on it (cf., e.g., [36]). 3.1.2 The embedding of M d Einstein static universe E d = R × S d may be deﬁned as the cylinder with radius 1 around the time axis in the d+1-dimensional Minkowski spacetime M d+1 , equipped with the induced metric. Denoting the coordinates in M d+1 by (tE , xE , wE ) and the coordinates in d M by (tM , xM ), we consider the embedding  2 −1/2 xM   xE = (η + rM ) 2 −1/2 wE = sgn(η)(η + rM )   tE = arctan(tM + rM ) + arctan(tM − rM )

(3.5)

which maps the d-dimensional Minkowski space into the Einstein universe, where 2 we have set rM = |xM |, η = 12 (1 + t2M − rM ). If we now use the cylindrical coordinates (tE , θE , ψE ) in M d+1 to describe d E , and the cylindrical coordinates (tM , rM , θM ) in M d , we get 2 • the metric tensor of E d is ds2E = dt2E −dψE −sin2 ψE dΩ(θE )2 , where dΩ(θE )2 denotes the metric tensor of the (d − 2)-dimensional unit sphere; 2 2 − rM dΩ(θ M )2 ; • the metric tensor for M 4 is ds2M = dt2M − drM • the embedding above can be written as

   θE = θM ψE = arctan(tM + rM ) − arctan(tM − rM )   tE = arctan(tM + rM ) + arctan(tM − rM ).

(3.6)

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A simple calculation shows that the metric tensor on M d is pulled back to the following metric on E d :

1 tE + ψE tE − ψE 2 2 2 ds = sec sec ds2E , 4 2 2 showing that the embedding is conformal and that the image of M d is the “double cone” of E d given by (3.7) −π < tE ± ψE < π. Remark 3.2 In the two-dimensional case ψ is the only angle coordinate, hence it ranges from −π to π, and the previous inequality is indeed drawn as a double cone. In higher dimension, ψ ∈ [0, π], however the inequality still describes a double cone in E d with center (t0 , v0 ): {(t, v) ∈ R × S d−1 : |t − t0 | + d(v, v0 ) < π},

(3.8)

where t0 = 0, v0 is the point xE = 0, wE = 1, and d(·, ·) denotes the geodesic distance in E d . The conformal Lie algebra of M d is o(d, 2). If d ≥ 3, the quotient of the universal covering of SO0 (d, 2) by the stabilizer of a point is E d . However the action of the universal covering of SO0 (d, 2) is not eﬀective, since there is a Z2 component in the center acting trivially on E d . The corresponding quotient is (the identity component of) the conformal group of E d . Indeed let us now consider the map γ in E d = R×S d−1 given by γ : (tE , v) → (tE + π, −v), where v → −v is the antipodal map. It is easy to see that γ belongs to Conf(E d ) and the “double cone” above is a fundamental domain for the corresponding action of Z on E d . Therefore the quotient is the conformal completion M d of M d , which is usually called the Dirac-Weyl compactiﬁcation of M d . Since γ 2 is central in Conf 0 (E d ), the quotient SO0 (d, 2) = Conf(E d )/2Z acts on M d . If d is even, such action is not eﬀective on M d , and the (quasi-global) conformal group of M d is P SO0 (d, 2). If d is odd, the action is eﬀective, and Conf(M d ) = SO0 (d, 2). If d = 2, the conformal group is inﬁnite-dimensional, however we shall still set conf(M 2 ) = o(2, 2). Moreover, E 2 is not simply connected, indeed the procedure 2 the universal covering of E 2 . However, E 2 is outlined above would give as M 2 the only globally hyperbolic covering of M where the image of M 2 has empty space-like complement. As we shall see below, this condition is necessary in order d = E d to lift a conformal net on E 2 to a local net, therefore we shall write M when d = 2 too. 3.1.3 The embedding of dS d The de Sitter space dS d (of radius ρ) may be described in terms of the coordinates (τ, θ S , ψS ), where τ varies in (0, π), (θS are spherical coordinates in S d−2 and

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ψS varies in [0, π], such that (θ S , ψS ) are spherical coordinates in S d−1 . Then the embedding of dS d in M d+1 is t = −ρ cot τ (3.9) x = ρ · (sin τ )−1 v(θ S , ψS ), where v(θ S , ψS ) denotes a point in S d−1 expressed in terms of spherical coordinates. In terms of these coordinates the metric tensor is ds2 =

ρ2 (dτ 2 − dψS2 − sin2 (ψS )dΩ(θ S )2 ). sin2 τ

Therefore the embedding of dS d in E d  ϕE    θ E  ψ E    tE

= ϕS = θS = ψS =τ

(3.10)

is conformal and maps dS d to the “rectangle” of E d 2 {(tE , xE , wE ) : |xE |2 + wE = 1, 0 < tE < π}.

(3.11)

Again, the rectangle is a fundamental domain for the action of Z on E d induced by Γ. Therefore Conf(M d ) = Conf(dS d ) and M d = dS d , Conf(dS d ) acting quasiglobally on dS d . When d = 2 we deﬁne conf(dS 2 ) = o(2, 2). Let us note that, opposite to the M 2 case, the homogeneous space given by the quotient of CONFloc (dS 2 ) by the stabilizer of a point is exactly E 2 , not its covering. 3.1.4 The conformal steady-state universe The intersection of the conformal images of M d and dS d in E d is the steady-state universe. Composing the previous maps we may therefore obtain a conformal map from the subspace {tM > 0} in the Minkowski space to the steady-state subspace of the de Sitter space. The map can be written as a map from {tM > 0} in M d to M d+1 , with range {(t, x, w) ∈ M d+1 : −t2 + |x|2 + w2 = ρ2 , w > t}:  t2 − |xM |2 − 1   t = −ρ · M   2tM    xM x=ρ· tM      t2 − |xM |2 + 1   w = −ρ · M . 2tM

(3.12)

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The image of the steady-state universe in E d is not a fundamental domain for some Γ, therefore there is no conformally complete manifold in which it embeds densely. Let us note for further reference that time translations for tM > 0 are mapped to endomorphisms of the steady-state universe, and that the (incomplete) time-like geodesic {xM = 0, tM > 0} is mapped to the de Sitter-complete geodesic  t2 − 1   t = −ρ · M    2tM  x=0 (3.13)   2   t +1   w = −ρ · M . 2tM

Fig. 3. The embeddings of Minkowski space, de Sitter space, and steady-state universe in Einstein universe.

Minkowski space. de Sitter space. Steady-state universe.

3.2

Conformal nets on de Sitter and Minkowski spacetimes

Let M be a spacetime on which the local action of the conformal group is transitive. A net A of local algebras on M is conformal if there exists a unitary local representations U of CONFloc (M) acting covariantly: for each ﬁxed double cone O there exists a neighborhood U of the identity in CONFloc (M) such that gO ⊂ M for all g ∈ U and U (g)A(O)U (g)−1 = A(gO),

∀g ∈ U .

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and the local Proposition 3.3 A conformal net A on M lifts to a net A on M representation U lifts to a true representation U of CONFloc (M) under which A is covariant. O ⊂ M, Proof. The results follow as in [10]: for any region O1 ≡ gO ⊂ M, −1 g ∈ CONFloc (M), we set A(gO) = U (g)A(O)U (g ), and observe that A(O1 ) is well deﬁned since CONFloc (M) is simply connected. Then we extend the net A by additivity. on all M Let us notice that we did not assume A to be local, namely that commutativity at spacelike distance is satisﬁed. In particular we did not prove that A is local. This was proved in [10] for local nets on the Minkowski space, and the proof easily extends to a spacetime M where the conformal group acts quasi globally and whose conformal completion is the Dirac-Weyl space, as is the case for the de Sitter space. We shall prove a more general result here. We say that a (local) unitary representation of Conf(E d ) has positive energy if the generator of the one-parameter group of time translations on E d is positive. Let us denote by K the set of double cones of E d (the deﬁnition is analogous as in the de Sitter case), and by ΛO the one-parameter group of conformal transformation of E d , that can be deﬁned by requiring that ΛW is the boost oneparameter group associated with W if W is a wedge of M d embedded in E d , and ΛO (t) = gΛW (t)g −1 if O ∈ K and g is a conformal transformation such that gW = O. We shall say that a net on E d satisﬁes the double cone KMS property if, for any O ∈ K, (Ω, · Ω) is a KMS state on the algebra associated with O w.r.t. the evolution implemented by U (ΛO (·)). = E d . Then a local conformal net A Theorem 3.4 Let M be a spacetime s.t. M on M with positive energy lifts to a local net A˜ on E d which is covariant under the (orientation-preserving) conformal group CONF+ (E d ). A˜ satisﬁes Haag duality and the double cone KMS property. which is covariant under Proof. By the above proposition we get a net on M the universal covering of SO(d, 2). Then modular unitaries associated with double cones act geometrically, as in [10] Lemma 2.1. Now we ﬁx two causally disjoint double cones O, O1 ⊂ M. Then if ΛO is the one-parameter group of conformal transformations corresponding to the (rescaled) modular group of A(O), we have O (t)O1 ) for any t. Let us assume for the moment that A(O) commutes with A(Λ that d > 2. Then ΛO leaves globally invariant the spacelike complement O of O indeed its action is implemented by the modular group of A(O ) at −t. in M, Therefore the algebra O (t)O1 ) A(Λ t∈R

), is globally stable under the action of ∆it , commutes is a subalgebra of A(O O with A(O) and is cyclic for the vacuum. By the Takesaki theorem the subalgebra

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). This implies that the net is local by covariance. indeed coincides with A(O One then proves the geometric action of J, thus extending the representation to is a conformal transformations which does not preserve the time orientation. U d representation of the conformal group of E rather than of its simply connected two-covering by a spin and statistics argument (cf. [25, 38, 27]). The last properties are proved as in [10]. In the low-dimensional case we may assume that O1 is “on the right” of O and ∪t∈R ΛO (t)O1 = OR , where OR is the closest smallest region on the right of O which is globally invariant under ΛO (t). As before, we prove that A(OR ) = A(O) . Now there exists a suitable conformal rotation whose lift R(t) to E d satisﬁes R(π)O = OR , R(π)OR = (OR )R , and so on. Therefore, A(R(2π)O) = A(O) = A(O), namely the net actually lives on E d . The rest of the proof goes on as before. Remark 3.5 In the proof above, we proved in particular that, when d ≤ 2, the extension of the net satisfying locality necessarily lives on E d , and not on its universal covering. Besides the Minkowski space and the de Sitter space, Theorem 3.4 applies to the Robertson-Walker space RW d , to the Rindler wedge and many others. In particular, there is a bijection between isomorphism classes: local conformal nets on M d local conformal nets on dS d In the following theorem we describe what the positive energy condition on M d becomes on dS d under this correspondence. Theorem 3.6 There is a natural correspondence between isomorphism classes of (i) Local conformal nets on M d with positive energy; (ii) Local conformal nets on dS d with the KMS property for geodesic observers; (iii) Local conformal nets on E d with positive energy. Here positive energy on E d = R × S d−1 means that the one-parameter group of time translations (on R) is implemented by a unitary group with positive generator. 2) of SO0 (d, 2) is Proof. (i) ⇔ (iii): Let us note that the universal covering SO(d, also the universal covering of SL(2, R). Since the covariance unitary representation U of SO(d, 2) is the same, it suﬃces to show that the two one-parameter unitary subgroup of U in question both have or not have positive generators. Let us consider the group generated by time translation, dilations and ray inversion in M d . This group is isomorphic to P SL(2, R) and acts on time axis of M d . U restricts to a unitary representation of P SL(2, R) thus, by a well-known fact (see, e.g., [37], positivity of time translations on M d is equivalent to positivity of conformal rotations (the generator corresponding to the rotation subgroup of P SL(2, R) is positive). Now the above rotation group provides the time translations on E d , hence the positivity of the corresponding one-parameter subgroup of

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2) is a consequence of the mentioned equivalence of positive energy condiSO(d, tions for unitary representations of (the universal covering of) P SL(2, R). (iii) ⇒ (ii) has been proved above. (ii) ⇔ (i): It is known that a Poincar´e covariant net for which the vacuum is KMS for the algebra of a wedge acted upon by the boosts satisﬁes the spectrum condition, see, e.g., [16, 11]. Since geodesic KMS property for a net on dS d is equivalent to the KMS property for any wedge for the corresponding net on M d , we get the thesis. If O is a double cone with vertices x and y, call B a basis of O if B is the part of a Cauchy surface contained in O and the closure of B contains the points lightlike to both x and y. We shall say that a net A satisﬁes the local time-slice property if for any double cone O A(O) = A(O) with O ∈ K and O ⊂ O an open slice around a basis B of O, namely a tubular neighborhood of B contained in O (thus O = O ). Note that, by an iteration/compactness argument, for the local timelike slice property to hold it is enough to assume A(O) = A(O) where O is obtained by O by removing arbitrarily small neighborhoods of the vertices of O (by using additivity). Corollary 3.7 Let A be a conformal net on dS d . A is Haag dual iﬀ it satisﬁes the local time-slice property. Proof. With dS d is embedded in E d as above and O a double cone in dS d , we have A(O) = A(OE ) where OE is the causal complement of O in E d . Thus A is Haag d ), where O = OE ∩ dS d is the causal complement dual on dS iﬀ A(O ) = A(OE d d of O in dS . Now OE is a double cone in E and O is a timelike slice for O , so Haag duality on dS d is satisﬁed iﬀ the time-slice holds for OE . We can now map, by a conformal transformation, OE to any other double cone contained in dS d , thus the time-slice property holds on dS d iﬀ it holds for OE . Thus, under a general assumption (local time-slice property), all conformal nets on dS 2 are Haag dual. One should compare this with the Minkowski spacetime case, where Haag duality for conformal nets is equivalent to a strong additivity requirement: removing a point from the basis B of O we have A(O) = A(B \{pt}) [33]. As a consequence, if two conformal nets on M d and dS d are conformally related as above, then Haag duality on M d =⇒ Haag duality on dS d but the converse is not true.

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Modular covariance and the maximal conformal subnet

We have shown that on spacetimes that can be conformally embedded in E d , a local, locally conformal net can be lifted to a local, globally conformal net on E d with the double cone KMS property, namely to a net for which the modular groups of double cones have a geometric action. Indeed a converse is true. Assume we have = E d . We shall say that a subregion O of M is a a spacetime M such that M double cone if it can be conformally identiﬁed with a double cone in E d . Given a net A on the double cones of M acting on a Hilbert space with a given vector Ω, such that, for any double cone O ⊆ M, Ω is cyclic and separating for A(O), we shall consider the following property for the algebra A(O): ˜ ⊂ O, we have • Local modular covariance: for every double cone O ˜ it ˜ ∆it O A(O)∆O = A(ΛO (−2πt)O). Local modular covariance was introduced in [23] under the name of weak modular covariance, where it was proven that weak modular covariance for wedges plus essential duality is equivalent to modular covariance, hence reconstructs the Poincar´e covariant representation, for nets on the Minkowski space. Theorem 3.8 Let O0 be a spacetime which can be conformally identiﬁed with a double cone in E d , and assume we are given a net O → A(O) of local algebras, O ⊂ O0 , acting on a Hilbert space with a given vector Ω, such that, for any double cone O ⊆ O0 , Ω is cyclic and separating for A(O) and the local modular covariance property holds. Then the net extends to a conformal net on (the universal covering of ) E d . If A is local, then the extended net is indeed a local conformal net on E d . The proof requires some steps. We ﬁrst construct “half-sided modular translations”. Let us identify O0 with a future cone in M d , and denote by v → τ + (v) the subgroup of the conformal group isomorphic to Rd consisting of M d translations, in such a way that when v is a causal future-pointing vector τ + (v) implements endomorphisms of O0 . These transformations can be seen as conformal translations which ﬁx the upper vertex of O0 . In the same way we get a family v → τ − (v) of conformal translations ﬁxing the lower vertex of O0 , and such that τ − (v) implements endomorphisms of O0 when v is a causal past-pointing vector. For any causal future-pointing vector v, we may implement the translation a la t → τ + (tv) by a one-parameter unitary group T + (tv) with positive generator ` −it + + −2πt Wiesbrock. Borchers relations are satisﬁed: ∆it T (v)∆ = T (e v). O0 O0 Translations T − (v), for causal (past-pointing) vector v, are constructed analogously. Lemma 3.9 The T ± translations associated with O0 act geometrically on subregions, whenever it makes sense: Ad T ± (v)A(O) = A(τ ± (v)O),

if τ ± (v)O ⊂ O0 .

(3.14)

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Proof. First assume O is compactly contained in O0 and the translation “goes inside”, namely it is of the form τ + (v) with v a causal future-pointing vector or τ − (v) with v a causal past-pointing vector. Then there exists an ε > 0 such that τ + (εv)O ⊂ O0 . Therefore, by Borchers relations, T + (ε(e−2πt − 1)v) = ∆it O0 ∆−it τ + (εv)O0 , and the thesis follows. Then, by additivity, one can remove the hypothesis that O is compactly contained in O0 . Indeed, by local modular covariance, for any O ⊂ O0 , the von Neumann algebra generated by the local algebras associated with compactly contained subregions of O is globally stable for ∆it O , therefore, by Takesaki Theorem, it coincides with A(O). We have proved that (3.14) holds for τ + (v) whenever v is a causal future-pointing vector, hence, applying Ad T + (−v), one gets Ad T + (−v)A(O) = A(τ + (−v)O) whenever τ + (−v)O ⊂ O0 . The thesis follows. Lemma 3.10 T + is indeed a representation of Rd , and the same holds for T − . They act geometrically on subregions, whenever it makes sense. Proof. First we prove, as in [23], that [T + (v), T + (w)] = 0. By the previous point, the multiplicative commutator c(s, t) := T + (−sv)T + (−tw)T + (sv)T + (tw)

(3.15)

has a geometric action, hence stabilizes, the algebras A(O), for s, t ≥ 0. Therefore it commutes with ∆it O and with the translations themselves. With simple manipulations we get c(s, t) = c(−s, −t) = c(−s, t)∗ = c(s, −t)∗ , namely c(s, t) commutes with translations for any s, t, hence T + (sv), T + (tw) generate a central extension of R2 . By positivity of the generators the commutator has to vanish. In an analogous way one shows that c(t) := T + (−t(v + w))T + (tv)T + (tw) is central, hence is a one-parameter group, and by Borchers relations c(λt) = c(t) for any positive λ, namely c(t) = 1. The relations for T − and the geometric action follows as before. Now we construct the group G. For any O ⊆ O0 , deﬁne G(O) as the group generated by ˜ {∆it ˜ : O ⊆ O}. O Lemma 3.11 G(O) is independent of O. Proof. Let us note that G(τ ± (v)O) is a subgroup of G(O) and clearly contains T ± (v), hence coincides with G(O). Repeating this argument we get that G(O) does not depend on O. We shall denote this group simply by G. Let us note that G is generated by a ﬁnite number of one-parameter groups: setting Ok = τ + (vk )O0 , k = 1, . . . , d, Ok+d = τ − (vk )O0 , k = 1, . . . , d, the one-parameter groups ∆it Ok , k = 0, . . . , 2d generates all translations T ± (v), hence G by covariance.

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Then we construct the central extension. The one-parameter groups Λk k = 1, . . . , d, generate the conformal group SO0 (d, 2). Pick (d + 1)(d + 2)/2 functions gi (t) with values in SO0 (d, 2), each given by a product of Λk ’s , such that the Lie algebra elements gi (0) form a basis for so(d, 2). Since (d + 1)(d + 2)/2 ≥ 2d + 1 one may assume that gi (t) = ΛOk−1 (−2πt), i = 1, . . . , 2d + 1. Then the map F (t) = g1 (t1 ) · · · gn (tn ) n = (d + 1)(d + 2)/2, is a local diﬀeomorphism from Rn to the conformal group. Now use the identiﬁcation ΛO (−2πt) ↔ ∆it O to get a map G from Rn to the group G = G(O0 . . . O2d ) generated by the ∆it , Ok k = 0, . . . , 2d, and ﬁnally obtain a map H = G · F −1 from a neighborhood V of the identity in SO0 (d, 2), to G. Observe that Ad H(g)A(O) = A(gO) whenever each step makes sense. Lemma 3.12 The inverse of the map H gives rise to a homomorphism from G to SO0 (d, 2) which is indeed a central extension. Proof. First we show that H is a local homomorphism to G/Z, Z denoting the center of G. ˜ compactly contained in O0 . Now, possibly restricting V, Choose a region O ˜ ⊂ O for any g ∈ V. As a consequence, if g, h, gh ∈ V, one may assume that g O ˜ namely then H(gh)∗ H(g)H(h) implements an automorphism of A(O), for O ⊂ O, commutes with the corresponding modular groups, hence is in the center of G. Now we extend the map H to a homomorphism from the universal covering 2) of SO0 (d, 2) to G/Z, and observe that since all normal subgroups of SO(d, 2) are central, we get an isomorphism from a suitable covering C˜ of SO0 (d, 2) SO(d, to G/Z. The inverse gives rise to a homomorphism from G to SO0 (d, 2) which is indeed a central extension. Proof of Theorem 3.8. The arguments in [11] show that the extension is weak Lie type, hence gives rise to a strongly continuous representation U of SO(d, 2). Such representation acts geometrically on the algebras A(O) whenever it makes sense, therefore, by Proposition 3.3 we get a CFT on (the universal covering of) E d . If A is local, the extension is indeed a local net on dS d by Theorem 3.4. In the following corollary we characterize conformal theories in terms of local modular covariance. = E d . Then there is a natural Corollary 3.13 Let M be a spacetime for which M correspondence between • Local conformal nets on E d with positive energy; • Local nets on M with local modular covariance for double cones. Proof. Assume we are given a local net A on M satisfying local modular covariance for double cones. For any double cone O ⊂ M, Theorem 3.8 gives a local conformal net AO on E d , based on the restriction of A to O. Now embed M in E d , and observe that, by Lemma 3.11, if O1 ⊂ O2 ⊂ dS d the two nets AOi , i = 1, 2, on E d coincide. From this we easily get that all nets

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AO based on A|O coincide, hence their restriction to M coincides with A. The converse implication follows by Theorem 3.4. Note that the unitary representation of the conformal group is unique [10] because it is generated by the unitary modular groups associated with double cones (cf. Thm. 3.8). A further consequence of Corollary 3.13 is additivity for a conformal net A: if O, Oi are double cones and O ⊂ ∪i Oi , then A(O) ⊂ ∨i A(Oi ). This can be proved by the argument in [21]. We now return to the de Sitter spacetime dS d , with any dimension d. Let A be a local net on dS d and B be a subnet of A. We shall say that B is a conformal subnet if its restriction B0 to HB is a conformal net. Now, given any local net A and O ∈ K, we set −it C(O) = {X ∈ A(O) : ∆it O0 X∆O0 ∈ A(ΛO0 (−2πρt)O), ∀O0 ∈ K, O0 ⊃ O}. (3.16) It is immediate to check that C(O) is a von Neumann subalgebra of A(O). Moreover C is covariant w.r.t. the unitary representation of SO0 (d, 1) because if X ∈ C(O), then Ad U (g)X ∈ A(gO) and, for any O0 ⊃ gO, it Ad ∆it O0 U (g)X = Ad U (g)∆g−1 O0 X

∈ Ad U (g)A(Λg−1 O0 (−2πρt)O) = A(ΛO0 (−2πρt)gO),

(3.17)

namely U (g)XU (g)−1 ∈ C(gO). Finally C is isotonic, thus C is a subnet of A. Theorem 3.14 A local net A on dS d has a unique maximal conformal expected subnet C. It is given by Equation (3.16). Proof. Let B be a conformal expected subnet of A. Then B0 is weakly additive by Lemma 2.5, hence satisﬁes the Reeh-Schlieder property by Lemma 2.3. So the projection E onto HB implements all the expectations εO and commutes with all ∆O by Takesaki theorem. By Corollary 3.13, local modular covariance is satisﬁed for B(O), hence if X is an element of the algebra B(O) it belongs to C(O). Thus we have only to show that the subnet C is conformal and expected. −it Clearly ∆it O C(O)∆O = C(O), thus C is expected by Takesaki theorem. Also, by construction, local modular covariance holds true, so C is conformal by Corollary 3.13.

4 The dethermalization eﬀect In the ﬂat Minkowski spacetime, the world line of an inertial particle is a causal line. The corresponding evolution on a quantum ﬁeld in the vacuum state is implemented by a one-parameter translation unitary group whose inﬁnitesimal generator, “energy”, is positive. A uniformly accelerated observer feels a thermalization

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(Unruh eﬀect): its orbit is the orbit of a one-parameter group of pure Lorentz transformation that, on the quantum ﬁeld, is implemented by a one-parameter automorphism group of the von Neumann algebra of the corresponding wedge that satisﬁes the KMS thermal equilibrium condition at Hawking temperature. On the other hand, if we consider an inertial observer on the de Sitter spacetime, its world line is the orbit of a boost and it is already thermalized, in the vacuum quantum ﬁeld state, at Gibbons-Hawking temperature. Our aim is to seek for a diﬀerent evolution on the de Sitter spacetime with respect to which the vacuum is dethermalized, namely becomes a ground state, an eﬀect opposite to the Unruh thermalization.

4.1

General evolutions

Let us recall a recent proposal of (quasi-)covariant dynamics for not necessarily inertial observers [15]. Our presentation, though strictly paralleling the one in [15], will diﬀer in some respects. Our description is in fact strictly local, therefore local conformal transformations will play the central role. The dynamics will consist of propagators describing the time evolution as seen by the observer, the main requirement being that the rest frame for the observer is irrotational. Let us consider a (not necessarily parametrized by proper time) observer in a given spacetime M, namely a timelike, future pointing C 1 curve γ : t ∈ (−a, a) → γt ∈ M. Then we look for a local evolution for the observer γ, namely a family of maps λt from M to M that satisfy the following physical requirements • λt γ0 = γt , t ∈ (−a, a). • Given x0 ∈ M, for each y0 in some neighborhood of x0 , the events λt (y0 ), t ∈ (−a, a), describe, potentially, the worldline of some material particle. This worldline is either disjoint from the observer’s worldline or coincides with it. • For a suitable y0 spacelike to x0 , the axis of a gyroscope carried by the observer at the space-time point λt (x0 ) points towards the point λt (y0 ) at all times t. As observed in [15], the previous conditions only depend on the conformal structure of the manifold. Therefore we will specify λt to be a local conformal transformation of M, or, more precisely, λ to be a curve in Conf(M). In this way the notion of local evolution only depends on the conformal class, namely if two metrics belong to the same conformal class they give rise to the same notion of local evolution. From the mathematical point of view, the above requests mean that the range of γ is an orbit of λ, and that for any t ∈ (−a, a), (λt )∗ (the diﬀerential of the transformation λt : M → M) maps orthogonal frames in Tγ0 M to orthogonal frames in Tγt M, in such a way that a tangent vector to the curve γ at t = 0 is mapped to a tangent vector to the curve γ at the point t, and that every orthogonal vector v to γ at t = 0 evolves without rotating to vectors orthogonal to γ, as we will explain.

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If we now ﬁx a metric g in the conformal class, we can choose the proper time parametrization, and then look for a curve λ ∈ Iso(M, g), the isometry group of M, namely for a local isometric evolution on γ. In this way orthonormal frames evolve to orthonormal frames. Recalling the notion of Fermi-Walker transport (cf. [49]), we may reformulate the conditions for a local isometric evolution on γ as follows: E1 : λ is a curve in Iso(M, g). E2 : λt γ0 = γt , t ∈ (−a, a). E3 : (λt )∗ is the Fermi-Walker transport along the curve γ. Clearly a local isometric evolution on γ does not exist in general; however, if it exists, it is unique. Proposition 4.1 Assume λt , λt , t ∈ (−a, a), satisfy properties E1 , E2 , E3 for a given observer γ. Then λt coincides with λt on a suitable neighborhood of γ0 . Proof. By assumption, λ−1 t · λt is a local isometry ﬁxing the point x0 whose diﬀer ential is the identity on Tx0 M. Then λ−1 t · λt acts identically on any geodesic at x0 , hence coincides with the identity on the injectivity radius neighborhood.

Concerning the existence problem, let us ﬁrst consider a geodesic observer. In this case the Fermi-Walker transport coincides with the parallel transport. Let us recall that a (semi-) Riemannian manifold is symmetric if for any p ∈ M there exists an involutive isometry σp such that p is an isolated ﬁxed point. It is easy to see that de Sitter, Minkowski, and Einstein spacetimes are symmetric. Proposition 4.2 (i) If γ is a geodesic observer, a local isometric evolution is indeed a one-parameter group of isometries. (ii) If M is symmetric, a local isometric evolution exists for any geodesic. Proof. (i) First we show that a local isometric evolution for a geodesic γ satisﬁes λt · λt = λ2t . Indeed, since λt is an isometry, λt γs , 0 ≤ s ≤ t describes a geodesic, and since (λt )∗ γ0 = γt , it describes the geodesic γt+s , 0 ≤ s ≤ t. As a consequence, λ2t implements the parallel transport on γ from Tγ0 M to Tγt M. By the uniqueness proved in Proposition 4.1 we get the statement. Now we observe that the previous property implies λt · λs = λt+s whenever s/t is rational, hence, by continuity, for any s and t. (ii) Since γ is geodesic, the Fermi-Walker transport coincides with the parallel transport (cf. [49]). On a symmetric manifold, the existence of isometries implementing the parallel transport is a known fact, see, e.g., [3], Thm 8.7. We now study the case of a generic observer. Assume λ is a C 1 one-parameter family of local diﬀeomorphisms of M and denote by Lt the vector ﬁeld given by d Lt (λt (x)) = ds λs (x)|s=t . Assume then that the x-derivatives of Lt (x) are jointly

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continuous, namely that the map (t, x, v) ∈ R × T M → (∇v Lt )(x)

(4.1)

is continuous. Lemma 4.3 Let γ be an orbit of λ: λt γ0 = γt , and let X be a λ-invariant vector ﬁeld on γ: (λt )∗ X(γ0 ) = X(γt ). Then, at the point γt , the covariant derivative of X on the curve γ satisﬁes (4.2) ∇γ ∗ X = ∇X Lt . Proof. Since X is invariant under λ, the commutator [X, Lt ] vanishes at the point γt . This fact can be proved via a simple computation, where two derivatives should be exchanged. Condition (4.1) ensures that Schwartz Lemma applies. Then the symmetry of the Levi-Civita connection implies ∇Lt X = ∇X Lt at the point γt . Since by deﬁnition Lt (γt ) = thesis.

d ds λs (γ0 )|s=t

=

d ds γs |s=t ,

we get the

The existence of a local isometric evolution for any observer has been proved in [15] for the de Sitter metric. Property (iii) of the following theorem gives an extension of this fact. Theorem 4.4 Let γ be an observer in M. The following hold: (i) There exists a local isometric evolution λ on γ satisfying condition (4.1) iﬀ for every t, γt extends locally to a Killing vector ﬁeld Lt satisfying (4.1) and (∇v Lt (γt ), w) = 0 for every vectors v, w in the rest space of γt . (ii) The existence of a local isometric evolution for any geodesic observer is equivalent to the existence of a local isometric evolution for any observer. (iii) If M is symmetric, a local isometric evolution exists for every observer. Proof. (i) A curve λ in Iso(M) satisfying E2 on γ gives rise, by derivation, to a oned parameter family of Killing vector ﬁelds Lt deﬁned by: Lt (λt (x)) = ds λs (x)|s=t . Clearly Lt satisﬁes Lt (γt ) = γt . Conversely a curve Lt of Killing vector ﬁelds verifying Lt (γt ) = γt gives rise to a curve of local isometries via the equations λ0 (x) = x dλs (x) = Lt (λt (x)). ds s=t

d Clearly ds λs (γt )|s=t = Lt (γt ) = γt , hence λt (γ0 ) = γt , namely condition E2 . By condition E2 , (λt )∗ maps vectors tangent to γ to vectors tangent to γ, hence, being isometric, preserves the rest frame for γ. Therefore it implements the

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Fermi-Walker transport if and only if tangent rest vectors evolve irrotationally, namely iﬀ the Fermi derivative Fγ ∗ X = 0 on γt for any λ-invariant vector ﬁeld X in the rest space of γ. According to [49], Proposition 2.2.1, if P denotes the projection on the rest space, this is equivalent to P ∇γ ∗ X = 0. By equation (4.2), this means that P ∇v Lt (γt ) = 0, v ∈ P Tγt M, ∀t, (4.3) which is our thesis. (ii) Assume the existence of a local isometric evolution for any geodesic observer. By Proposition 4.2, Lt does not depend on t, hence condition (4.1) is trivially satisﬁed. Then, reasoning as in (i) and taking into account that the Fermiderivative for a geodesic observer is indeed the Levi-Civita connection, we get ∇w L(x) = 0 for any x in the geodesic, w ∈ Tx M. Namely, the existence of a local isometric evolution for any geodesic observer is equivalent to the following: for any (x, v) ∈ T M, there exists a vector ﬁeld H = Hx,v deﬁned in a neighborhood U of x, such that, if γ is the geodesic determined by (x, v), H satisﬁes (V, ∇H W )(x) = (∇H V, W )(x), ∇w L(x) = 0, w ∈ Tx M, L(γs ) = γs , |s| < ε

x ∈ U,

x = γ(s),

|s| < ε,

where γ(s) ⊂ U for any |s| < ε. Since any such Hx,v would determine a local isometric evolution for γ, Proposition 4.1 implies uniqueness. Hence the existence of a local isometric evolution for any geodesic observer is equivalent to the existence and uniqueness of a local solution for the system above. Let us remember that the solutions of the ﬁrst equation (the Killing equation) form a ﬁnite-dimensional space V, therefore existence and uniqueness can be reformulated as the existence and uniqueness for the ﬁnite-dimensional linear system given by the last two equations, with L ∈ V. Clearly, both the linear operator and the coeﬃcients depend smoothly on (x, v) if the manifold (and the Riemannian metric) is smooth. Therefore, for any (continuous) curve γ, the one-parameter family of Killing ﬁelds Lt = Hγt ,γt satisﬁes conditions (4.1) and (4.3), namely, by point (i), the existence of a local isometric evolution for any observer. (iii) Immediately follows by Proposition 4.2 and point (ii).

4.2

Dethermalization for conformal ﬁelds

Besides the geometric question of existence of the curve t → λt ∈ Conf(M), there is a second existence problem if we want to describe the local dynamics in quantum ﬁeld theory. Indeed, it is not obvious that the local maps λt are unitarily implemented, or give rise to automorphisms of the net. This is clearly the case of a conformally covariant theory, but not the general case. The previous discussion on local evolutions shows that the evolution may change if we replace the original metric with another metric in the same conformal class. We shall show that, with a suitable choice of the new metric, the original

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observer will become an inertial observer in a (locally) ﬂat spacetime. Therefore, in a conformal quantum ﬁeld theory, the local evolution will be implemented by a one-parameter group with positive generator w.r.t. which the vacuum state is a ground state. We mention the analysis contained in [17], where the authors classify the global conformal vacua on a conformally ﬂat spacetime in terms of the global timelike Killing vector ﬁelds. In the Minkowski spacetime there is only one global timelike Killing vector ﬁeld, while for other “small” spacetimes one may have two nonequivalent Killing vector ﬁelds, as is the case of the Rindler wedge subregion where the boost ﬂow is also timelike. The two Killing vector ﬁelds give rise to diﬀerent vacua, and the vacuum for the Minkowskian Killing vector ﬁeld is thermalized w.r.t. the second Killing evolution. Our construction represents a converse to this procedure: starting with dS d , where the global Killing vector ﬁeld is unique and the de Sitter vacuum is thermalized, we restrict to a smaller spacetime where a global dethermalizing conformal Killing ﬂow exists. From a classical point of view then, the dethermalization is realized by replacing the original dynamics with a new ‘conformal’ dynamics. Let us note that such a change of the dynamics implies in particular a change in the time parametrization. Of course the absence of a preferred proper time parametrization occurs if the conformal structure alone is considered. As we shall see in the next sections, the evolutions λ will give rise only to a quasi-covariant dynamics in the sense of [15] for general (non-conformal) quantum ﬁelds. As seen in Subsection 3.1, there exists a conformal diﬀeomorphism Ψ between the steady-state universe subspace N of dS d containing a given complete causal d = {(x, t) ∈ M d , t > 0} in the Minkowski space, geodesics γ and the semispace M+ mapping γ to a causal geodesics γ˜ . However γ˜ is not complete, and can be identiﬁed with the half-line {x = 0, t > 0} in the timelike case, and with the half-line {x1 = t, xi = 0, i > 1, t > 0} in the lightlike case. Therefore we get the following. Proposition 4.5 If we replace the metric on N with the pull back via Ψ of the ﬂat d , there exists a local evolution µt , t > 0, from N into itself, given by metric on M+ the pull back of the time translations. Theorem 4.6 Let A be a conformal net on dS d and W a wedge causally generated by a geodesic observer γ. Then: (a) The local isometric evolution λ corresponding to the de Sitter metric is indeed global, there exists one-parameter unitary group U on the Hilbert space implementing λ and the vacuum is a thermal state at the Gibbons-Hawking temperature w.r.t. U . (b) The local isometric evolution µ corresponding to the ﬂat metric is unitarily implemented, namely there exists a one-parameter unitary group V on the Hilbert space such V (t) implements µt for t > 0, and the vacuum is a ground state w.r.t. V . If we extend the net A to a conformal net A˜ on the static Einstein universe, then V (t) acts covariantly on A˜ for every t ∈ R.

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Proof. The ﬁrst statement is simply a reformulation of assumption c) in Section 2. Concerning the second statement, note that µ extends to a global ﬂow on E d which is implemented by a one-parameter group V with positive generator. The thesis is then immediate.

Fig. 2. The ﬂow lines of the dethermalizing evolution µ in the steady-state universe.

Fig. 3. The ﬂow lines of the isometric evolution λ in the wedge contained in the steady-state universe.

Remark 4.7 In the conformal case, the dethermalizing evolution is not unique. In fact we may identify dS d with a rectangle in the Einstein universe (cf. Equation (3.11)), and then consider the corresponding metric on it. Again, the new evolution, which is given by time translations in E d , is dethermalized.

4.3

Dethermalization with noncommutative ﬂows

As anticipated, we construct here a quasi-covariant dynamics corresponding to the geometric dynamics described above, showing that the vacuum vector becomes a ground state w.r.t. this dynamics. Our ﬂow will be noncommutative in the sense it gives a noncommutative dynamical system, indeed it is a ﬂow on a quantum algebra of observables, although it will retain a partial geometric action. We begin with a no-go result. Proposition 4.8 Let U be a non-trivial unitary representation of SO0 (d, 1), d ≥ 2, and u the associated inﬁnitesimal representation of the Lie algebra so(d, 1). The following are equivalent:

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D. Guido, R. Longo

Ann. Henri Poincar´e

(i) There exists a non-zero L ∈ so(d, 1) such that u(L) is a positive or negative operator; (ii) d = 2 and U is the direct sum of irreducible representations that are either the identity or belong to the discrete series of SO0 (2, 1) ( P SL(2, R)). In this case L belongs to the cone generated by the translation generators. Proof. The set P of L ∈ so(d, 1) such that u(L) is a positive operator is a convex cone of so(d, 1), which is globally stable under the adjoint action of SO0 (d, 1), and P ∩ −P = 0 because so(d, 1) is a simple Lie algebra. Now every element L ∈ so(d, 1) can be written as a sum L = R + K, where R ∈ so(d) and K is a the generator of a boost one-parameter subgroup. Let then L belong to P and assume d > 2. We can then choose a rotation r ∈ SO(d) such that Adr(K) = −K. Set R ≡ Ad r(R) ∈ so(d). Since L ≡ Ad r(L) ∈ P, the element L + L = R + K + Ad r(R) + Ad r(K) = R + R belongs to P and to so(d), so it is enough to show that P ∩ so(d) = {0}. Indeed if R ∈ so(d), d > 2, we can choose a rotation r such that Ad r(R ) = −R , thus R = 0 if R ∈ P. We thus conclude that d = 2. Now every non-zero L ∈ SO0 (2, 1) P SL(2, R) is (conjugate to) the generator of either a boost, or translation, or rotation oneparameter group. If L is a boost generator, then L is conjugate to −L as above, thus L ∈ / P. The positivity of u(L), L a translation generator, is equivalent to the positivity of u(L), L a rotation generator (see, e.g., [37]) and is equivalent to U to be a direct sum of representations in the discrete series of U and, possibly, to the identity [42]. The next corollary states that the existence of a dethermalized covariant oneparameter dynamics is possible only if d = 2 and implies conformal covariance. Corollary 4.9 Given a local net A on the de Sitter space, assume there is a oneparameter group in SO0 (d, 1) which has positive generator in the covariance representation. Then A is conformally covariant. Proof. Assume the net is not conformally covariant. By the Proposition above, this implies d = 2. Then, Corollary 5.16 shows that positive energy representations in the two-dimensional case imply conformal covariance. Now we turn to a geodesic observer γ, and denote by W the wedge generated by the complete geodesic, by N the steady-state universe containing W , by λ the Killing ﬂow corresponding to the geodesic γ, by µt , t > 0, the conformal evolution of N described above. Let us observe that the time is reparametrized, namely γ˜t = γlog t = µt−1 γ0 . We also denote by R the spacetime reﬂection mapping W to its spacelike complement W .

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Theorem 4.10 Let A be a net of local algebras on the de Sitter spacetime. Then there exists a unique one-parameter unitary group V with the following properties: (i) Ω is a ground state w.r.t. V ; (ii) V implements a quasi covariant dynamics for the regions µt (W ), t ≥ 0, namely V (t)A(W )V (−t) = A(µt W ), t ≥ 0; (iii) Partial localization for negative times: V (−t)A(W )V (t) = A(Rµt W ) , t ≥ 0. Proof. By the geodesic KMS property, we get that A(µ1 (W )) ⊂ A(W ) is a half-sided modular inclusion. Therefore the theorem of Wiesbrock [55] gives a one-parameter group V with positive generator such that V (1)A(W )V (−1) = A(µ1 (W )) and satisfying the Borchers commutation relations U (λt )V (s)U (λ−t ) = V (et s) U (R)V (s)U (R) = V (−s). Then (i) is obvious, and the above relations give V (t)A(W )V (−t) = U (λlog t )V (1)U (λ− log t )A(W )U (λlog t )V (−1)U (λ− log t ) = A(µt W ), t ≥ 0, namely (ii). Property (iii) follows in an analogous way. The uniqueness now follows by the uniqueness for one-parameter groups with Borchers property [4] in the following lemma. Remark 4.11 By the above theorem we have the following localization properties for the noncommutative ﬂow Ad V : (i) If L is a region contained in the steady-state universe subregion N of dS and L = µs W for some s ≥ 0, then µt L ⊂ dS d if and only if t ∈ [−s, +∞) and, for such t, Ad V (t)A(L) = A(µt L). Analogously, considering µt as a global transformation on E d acting on dS d by restriction, using the geometric action of J, and Borchers commutation relations JV (t)J = V (−t), if L is a region in the complement of N and L = µ−s W for some s ≥ 0, then µt L ⊂ dS if and only if t ∈ (−∞, s] and, for such t, Ad V (t)A(L) = A(µt L). (ii) If O is a double cone contained in N , there exists s > 0 such that, for any s ≥ s, O ⊂ (µ−s W ) . Therefore, for any t ∈ R, A(µt−s W ) if t − s ≤ 0 Ad V (t)A(O) ⊂ (4.4) A(µt−s W ) if t − s ≥ 0. Assuming Haag duality on dS d we then get Ad V (t)A(O) ⊂ A(µt−s W ∩ dS d )

(4.5)

for any double cone O ⊂ N ; note that µt−s W ∩ dS d has non-empty spacelike complement in dS d . Analogous localization properties hold if O is contained in dS d \ N .

1202

D. Guido, R. Longo

Ann. Henri Poincar´e

Localization results for any double cone O ⊂ dS d would then follow by a form of strong additivity. Let us remark that more stringent localization properties would indeed imply a complete geometrical action [39]. Lemma 4.12 Let P be a von Neumann algebra on a Hilbert space H with cyclic and separating vector Ω ∈ H. Let V1 and V2 be Ω-ﬁxing one-parameter unitary groups on H such that Vk (t)PVk (−t) ⊂ P, t ≥ 0, (k = 1, 2) and V1 (1)PV1 (−1) = V2 (1)PV2 (−1). Suppose that the generators of V1 and V2 are positive. Then V1 = V2 . Proof. By Borchers theorem [4] we have ∆is Vk (t)∆−is = Vk (e−2πs t), t, s ∈ R, where ∆ is the modular operator associated with (P, Ω). We then have Ad V1 (t)(P) = Ad V2 (t)(P), t ≥ 0, because Ad Vk (e−2πs )(P) = Ad ∆is Vk (1)∆−is (P) = Ad ∆is Vk (1)(P) = Ad ∆is (P1 ), s ∈ R. Then Z(t) ≡ V2 (−t)V1 (t), t ≥ 0, is Ω-ﬁxing and implements an automorphism of P, thus commutes with ∆is . On the other hand ∆is Z(t)∆−is = Z(e−2πs t), due to the above commutation relations, so Z(t) = Z(e−2πs t) for all t ≥ 0 and all s ∈ R. Letting s → ∞ we conclude that Z(t) = 1, that is V1 (t) = V2 (t), for t ≥ 0 and thus for all t ∈ R because Vk (−t) = Vk (t)∗ . The following table summarizes the basic structure in the above discussion. space Minkowski de Sitter

orbit geodesic geodesic

ﬂow translations boosts

ω ground KMS

orbit hyperbola geodesic

ﬂow boosts µ

ω KMS ground

5 Two-dimensional de Sitter spacetime 5.1

Geometric preliminaries

Let us assume that dS 2 is oriented and time-oriented. Following Borchers [5], a wedge W at the origin (namely a wedge whose edge contains the origin) in the Minkowski space M d is determined by an ordered pair of linearly independent future-pointing lightlike vectors 1 , 2 ; W is the open cone spanned by 1 , −2 and vectors orthogonal to 1 , 2 (in the Minkowski metric). In order to make this correspondence 1 : 1 one can normalize the vectors in such a way that their timecomponent is 1. Clearly such a pair determines and is determined by the (d − 2) oriented hyperplane which is orthogonal to 1 and 2 w.r.t. the Minkowski metric (the edge of the wedge). In particular, when d = 3, it is determined by an oriented line ζ through the origin, e.g., by requiring that 1 , −2 , v determine the orientation in M 3 when v is an oriented vector in ζ. Denoting by x, x ˜ the intersection points

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of ζ with dS 2 , with x preceding x ˜ according to the orientation, it is clear that x ˜ is the symmetric of x w.r.t the origin, therefore x determines the wedge, so the map x → W (x) is a bijection between points of dS 2 and wedges. Any point x in dS 2 determines two lightlike lines given by the intersection of 2 dS with the tangent plane at x (the ruled lines through x of the hyperboloid). Let us denote them by hr (x), hl (x) in such a way that, if vr , vl are future pointing vectors in hr (x), hl (x) respectively, the pair (vr , vl ) determines the given orientation of dS 2 . Now let us consider an observer generating the wedge W (x). Then, the sets hr (x) = hr (x) ∪ hr (˜ x), hl (x) = hl (x) ∪ hl (˜ x) form a bifurcated Killing horizon for dS 2 [35], see also [27]), the Killing ﬂow being the one-parameter group of pure Lorentz transformations associated to the wedge W (x) and the set H = hr (x) ∪ hl (˜ x) is the event horizon for W (x), which splits in the two components x). H+ = hr (x), H− = hl (˜ Clearly any point x ∈ dS 2 determines a partition of the space into 6 disjoint regions: W (x) (the right of x), W (˜ x) (the left of x), V+ (x) (the closed future cone x) (the closed future cone at x ˜), V− (˜ x) at x), V− (x) (the closed past cone at x), V+ (˜ (the closed past cone at x ˜). Lemma 5.1 Two wedges W (x), W (y) have non-empty intersection if and only if y belongs to W (x) ∪ W (˜ x) ∪ V+ (x) ∪ V− (x). Proof. The “if” part is obvious. Concerning the “only if” part, assume that y is in the future of x ˜. Then W (y) is contained in the future of W (˜ x). Since the latter is the region in the future of the hA (x) horizon, while W (x) is contained in the past of hA (x), the thesis follows.

Fig. 4. Two-dimensional de Sitter space. The whole marked area is the steadystate universe, whose boundary is the event horizon. The striped area is the wedge region (static de Sitter spacetime), whose boundary is the black-hole horizon.

Now we may characterize the sets that are intersections of wedges.

1204

D. Guido, R. Longo

Ann. Henri Poincar´e

Lemma 5.2 In dS 2 , every non-empty open region O given by an intersection of wedges, is indeed an intersection of two (canonically determined) wedges, or, equiv˜ alently, O ∈ K. Proof. Let O be an open region given by intersection of wedges, and let X(O) the set of points x such that W (x) ⊃ O. Endow X(O) with the partial order relation of being “to the right”, namely x > y if x ∈ W (y). If x, y ∈ X(O) are not comparable, since O ⊂ W (x) ∩ W (y), Lemma 5.1 implies that one is in the future of the other. If x is in the future of y, deﬁne x ∨ y as the intersection of hl (x) with hr (y). Clearly, if O ⊂ W (x) ∩ W (y), then O ⊂ W (x ∨ y). Therefore X(O) is directed. Since O is open, the supremum of any ordered subset in X(O) belongs to X(O), hence there exists a maximal element. Directedness implies that such maximal element is indeed a maximum L(O) (the leftmost point of the closure of O). Analogously we get a minimum R(O) among the points y such that W (˜ y) ⊃ O (the rightmost point of the closure of O). Clearly O = W (L(O)) ∩ W (R(O)). Such generated by the points a set is the double cone (possibly degenerate, i.e., O ∈ K) F (O) = hr (L(O)) ∩ hl (R(O)), P (O) = hl (L(O)) ∩ hr (R(O)). We shall call L(O), R(O) the spacelike endpoints of O, and P (O), F (O) the timelike endpoints of O.

5.2

Geometric holography

Now we ﬁx the event horizon as the intersection of the plane x0 = y0 with the de Sitter hyperboloid, the two components being H± = {(t, t, ±ρ) : t ∈ R}. In the two-dimensional case, the orientation preserving isometry group of the de Sitter spacetime is isomorphic to SO0 (2, 1). On the other hand SO0 (2, 1) is isomorphic to P SL(2, R) and acts on (the one-point compactiﬁcation of) H+ or H− . We shall construct holographies based on this equality. The M¨ obius group is the semidirect product of P SL(2, R) with Z2 . Let us chose the following generators for its Lie algebra sl(2, R):

1 1 0 1 0 1 1 0 0 D= , T = , A= . (5.1) 2 0 −1 2 0 0 2 1 0 The following commutation relations hold: [D, T ] = T,

[D, A] = −A,

[T, A] = D.

(5.2)

We consider also the following orientation reversing element of the M¨obius group:

−1 0 r= . (5.3) 0 1 Let us observe that the following relations hold: rDr = D,

rT r = −T,

rAr = −A.

(5.4)

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We shall denote by β the usual action of the M¨ obius group on R ∪ {∞} as fractional linear transformations. Then β(r) implements the reﬂection x → −x, β(exp(tD)) implements the dilations, β(exp(tT )) implements the translations, and β(exp(tA)) implements the anti-translations (see [26]). Now we consider the two immersions ψ± : R → H± ⊂ dS 2 t → (t, t, ±ρ) of the real line in dS 2 as ±-horizon, and will look for actions α± of the M¨ obius group on dS 2 with the following property: whenever α± (g) preserves H± , then α± (g)ψ± (t) = ψ± (β(g)t).

(5.5)

Lemma 5.3 The previous requirement determines α± uniquely, in particular we have   0 1 0 α+ (D) = α− (D) = 1 0 0 0 0 0   0 0 1 1 0 0 1 α+ (T ) = −α− (T ) = 2 1 −1 0   0 0 1 1 0 0 −1 α+ (A) = −α− (A) = 2 1 1 0   −1 0 0 α+ (r) = α− (r) =  0 −1 0 0 0 1 where α± also denote the associated actions of sl(2, R). Moreover, the following relation holds: (5.6) α− (g) = α+ (rgr). Proof. It is easy to see that the subgroup (globally) stabilizing H+ coincides with the subgroup stabilizing H− and is generated by α+ (exp(tD)), α+ (exp(tT )), and α+ (r), as they are deﬁned in the statement, therefore the identiﬁcation is forced by Equation (5.5) for these elements. Equation (5.2) implies then the formula for α+ (A). The proof for α− is analogous. Relation (5.6) immediately follows from the previous equations and relations (5.4). Remark 5.4 By Lemma 5.3, it follows that α+ (T ) and α− (T ) have opposite signs, thus, given a unitary representation U of SO0 (2, 1), the generator of U (α+ (exp(tT )) is positive if and only if the generator of U (α− (exp(tT )) is negative.

1206

D. Guido, R. Longo

Ann. Henri Poincar´e

Now we deﬁne two maps Φ± from the set W of wedges in dS 2 to the set I of open intervals in (the one-point compactiﬁcation of) R such that, for any element g in the M¨ obius group and any wedge W , one has Φ± (α± (g)W ) = β(g)Φ± (W ).

(5.7)

Proposition 5.5 Let W ∈ W. Then ∂W ∩ H+ = ∅ ⇔ ∂W ∩ H− = ∅. The maps Φ± are uniquely determined by the further requirement that, for any such wedge, −1 (∂W ). Φ± (W ) = ψ±

(5.8)

Moreover they satisfy Φ± (W ) = Φ± (W ) Φ+ (W ) = β(r)Φ− (W ),

(5.9) (5.10)

where the prime denotes the spacelike complement in dS 2 and the interior of the complement in S 1 . Proof. Let us construct Φ+ , the construction of Φ− being analogous. For notational simplicity we shall drop the subscript + in the rest of the proof. Let W0 be the wedge W (0, 1, 0), according to the previous description. Since the Lorentz group acts transitively on wedges, property (5.7) may be equivalently asked for W0 only. Now Equation (5.8) implies Φ(W0 ) = I0 , where I0 denotes the positive half-line, hence we only have to test that equation Φ(α(g)W0 ) = β(g)I0 makes Φ welldeﬁned. This is equivalent to show that if α(g)W0 = W0 , then β(g)I0 = I0 . The stabilizer of r ), where

W0 is easily seen to be generated by α(exp(tD)) and α(ˆ 0 1 rˆ = , since 1 0   −1 0 0 α(ˆ r) =  0 1 0  . 0 0 −1 A direct computation shows that β(exp(tD)) and β(ˆ r ) stabilize I0 . Now we show that Equation (5.8) is always satisﬁed. Indeed, let ∂W ∩ H = ∅, namely either W = W (x) or W = W (˜ x), with x ∈ H. Then there exists g stabilizing H, either of the form exp(sT ), or of the form exp(sT )r, such that W = α(g)W0 . Equation (5.5) then implies the thesis. Now we prove the (5.9). Indeed, by (5.7), it is enough to prove it for only one wedge, e.g., W0 , where it follows immediately by (5.8). Concerning (5.10), we have Φ+ (α+ (g)W0 ) = β(g)Φ+ (W0 ) = β(g)Φ− (W0 ) = Φ− (α− (g)W0 ) = Φ− (α− (r)α+ (g)α− (r)W0 ) = β(r)Φ− (α+ (g)W0 ).

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Let us observe that the above-mentioned map trivially preserves inclusions, indeed no wedge is properly contained in another wedge of dS d , while the inverse map does not. Now we may pass to points. Indeed any point in dS 2 corresponds to a wedge: x → W (x). Also, any interval in the one-point compactiﬁcation of R determines its leftmost extreme: I → (I). Then we may deﬁne point maps as follows: ϕ± (x) = (Φ± (W (x))).

(5.11)

Lemma 5.6 The point maps ϕ± are equivariant, namely ϕ± (α± (g)x) = β(g)ϕ± (x).

(5.12)

Proof. Assume g to be orientation preserving. Then α± (g)W (x) = W (α± (g)x) and (β(I)) = β((I)). Therefore the result follows from (5.7). Assume now g to be orientation reversing. Given x ∈ dS 2 , we may write g as h1 rh2 , where α+ (h2 )x = x0 ≡ (0, 0, ρ). Then Equation (5.12) reduces to ϕ+ (α+ (r)x0 ) = β(r)ϕ+ (x0 ), which is obvious. The proof in the − case is analogous. Theorem 5.7 The wedge maps Φ± are induced by the point maps ϕ± , namely Φ± (W ) = {ϕ± (x) : x ∈ W }. The point maps ϕ± are given by the holographic projections x ∈ dS 2 → h∓ (x) ∩ H± , where H± are identiﬁed with R as before. Proof. We prove the second statement ﬁrst. Indeed, it is suﬃcient to show that the preimage under ϕ± of a point t in R is the ruled line h∓ (ψ± (t)). Equation (5.12) implies that this is simply the α± -orbit of the β-stabilizer of t, and that we may check the property for one point only, say t = 0. The elements of SO(2, 1) β-stabilizing 0 but not α± -stabilizing ψ± (0) are of the form exp(sT ), and the orbit of α± (exp(sT )) at ψ± (0) is exactly h∓ (ψ± (0)). Now we prove the ﬁrst statement in the + case. Let x ∈ W . By equivariance, we can move x and W in such a way that W = W (ψ+ t), and x ∈ h− (ψ+ (0)). Then the statement becomes ϕ+ (x) ∈ Φ+ (W ), i.e., t < ϕ+ (x), but this is obvious since x ∈ W . The proof for the − case is analogous. The maps ϕ± may be considered as geometric holographies, namely projection maps from the de Sitter space to (some part of) the horizon preserving the causal structure and intertwining the symmetry group actions. Of course one can construct holography maps onto the conformal boundary as well, simply associating with any x the intersection of h± (x) with the conformal boundary.

1208

5.3

D. Guido, R. Longo

Ann. Henri Poincar´e

Pseudonets

By a local conformal pseudonet B on a Hilbert space H (or simply a local pseudonet) we shall mean here a map B from the (proper, open, non-empty) intervals I of S 1 to von Neumann algebras on H with the following properties: • M¨ obius covariance. There exists a unitary representation U of P SL(2, R) on H such that U (g)B(I)U (g)−1 = B(gI), g ∈ P SL(2, R), I ∈ I. • Vacuum with Reeh-Schlieder property. There exists a unit, U -invariant vector Ω, cyclic for each B(I). • Interval KMS property. ∆it I = U (ΛI (−2πt)), I ∈ I, where ∆I is the modular operator associated with (B(I), Ω) and ΛI is the one-parameter subgroup of P SL(2, R) of special conformal transformations associated with I, see [10]. • Locality. B(I) and B(I ) commute elementwise for every I ∈ I (with I the interior of S 1 \ I). Note that we do not assume positivity of the energy (or negativity of the energy) nor isotony (or anti-isotony). Given a local pseudonet B on the Hilbert space H, let J be the canonical antiunitary from H to conjugate Hilbert space H. We deﬁne the conjugate pseudonet B on H by B(I) = JB(I )J, U (g) = JU (g)J, Ω = JΩ . We may deﬁne B directly on H with the same vacuum vector by choosing a reﬂection r on S 1 associated with any given interval I0 (say r : z → −z) and putting B(I) = B(rI ) with the covariance unitary representation U given by U(g) = U (rgr), g ∈ P SL(2, R). In this case B depends on the choice of r, but is well deﬁned up to unitary equivalence. The second conjugate of B is equivalent to B. B is isotonic iﬀ B is antiisotonic, and B has positive energy iﬀ B has negative energy. Note that B¯ is deﬁned also if Ω is not cyclic. Theorem 5.8 Let B be a local pseudonet. (i) Haag duality holds: B(I) = B(I ), I ∈ I. (ii) If Ω is unique U -invariant, then each B(I) is a type III1 factor. (iii) B is isotonic (resp. anti-isotonic) iﬀ it has positive energy (resp. negative energy). Proof. (i) By locality, B(I ) is a von Neumann subalgebra of B(I) , globally invariant with respect to the modular group Ad∆−it of B(I) , hence B(I ) = B(I) I by Takesaki theorem due to the Reeh-Schlieder property of Ω. (ii) If Ω is unique U -invariant, then, as in [26], Ad∆it I is ergodic on B(I), and this entails the III1 -factor property.

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(iii) If B is isotonic, then positivity of the energy follows from the interval KMS property, see, e.g., [27]. Conversely, if U has positive energy, let us prove isotony. Clearly it is enough to prove isotony for pairs I˜ ⊂ I having one extreme point in common, and, by SO(2, 1) covariance, we need only one pair, say I = (0, ∞), I˜ = (1, ∞), namely it is enough to show that translations T (t) implement endomorphisms of B(I) for positive t. By a classical argument, positivity is equivalent to the positivity of the self-adjoint generator of the translations T (t), therefore we have the four ingredients of the Borchers theorem: a vector Ω, the vacuum, which is invariant for the representation, hence for the modular group ∆it I of B(I) and for U (T (t)), the commutation relations between these one-parameter groups, the positivity of the generator of translations, and an (expected) implementation of B(I)-endomorphisms by U (T (t)) for positive t. Davidson [19] proved that the last property follows from the ﬁrst three ones if the following holds: there exists an ε > 0 such that the vacuum is cyclic for the set B(ε) consisting of all the x ∈ B(I) such that, for all t ∈ (0, ε), U (T (t))xU (T (−t)) is in B(I) (Theorem ˜ ⊂ B(1), hence the cyclicity follows. 3 ibid.). Now B(I) The equivalence between anti-isotony and negative energy is obtained by considering the conjugate pseudonet. Let us deﬁne the “isotonized” nets associated with B, resp. B: B+ (I0 ) =

B(I),

B− (I0 ) =

I⊃I0

B(I).

I⊃I0

Then B± is isotonic, thus it has positive energy (on the vacuum cyclic subspace). Moreover B+ (I) is globally invariant w.r.t. Ad∆it I thus, by Takesaki theorem, there is a vacuum preserving normal conditional expectation from B(I) onto B+ (I). It is easy to check that B− (I0 ) = B(I), I⊂I0

hence B− is expected in B and B+ (I) ∨ B− (I) ⊂ B(I). Proposition 5.9 If Ω is unique U -invariant, we have the von Neumann tensor product splitting B+ (I1 ) ∨ B− (I2 ) = B+ (I1 ) ⊗ B− (I2 ). Proof. First we show that B+ (I1 ) and B− (I2 ) commute for any I1 , I2 ∈ I. As B+ is a net, it is additive and we may assume that rI1 ∪ I2 has non-empty complement. We may then enlarge I2 in such a way that rI1 ⊂ I2 . Then B+ (I1 ) ⊂ B(I1 ), and B− (I2 ) ⊂ B(I2 ) = B(rI2 ) ⊂ B(I1 ) , namely they commute. Then, as in Theorem 5.8 (ii), the von Neumann algebras B± (I) are factors, hence they generate a von Neumann tensor product by Takesaki’s theorem [52].

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5.4

D. Guido, R. Longo

Ann. Henri Poincar´e

Holography and chirality

Let B be a local pseudonet on S 1 . Then we may associate with it a local net A on the wedges of dS 2 as follows: A(W ) ≡ B(Φ+ (W )).

(5.13)

Clearly, given a pseudonet on S 1 , we may obtain a net on the double cones of dS 2 by intersection, and such net will satisfy properties a), b), c) on a suitable cyclic subspace. Conversely, given a net on dS 2 , equation (5.13) gives rise to a pseudonet on S 1 . Let us consider the following property: • Intersection cyclicity. For any pair of intervals I1 ⊂ I2 , the vacuum vector is cyclic for the algebra B(I1 , I2 ) ≡ B(I) . (5.14) I1 ⊂I⊂I2

Theorem 5.10 The map (5.13) gives rise to a natural bijective correspondence between: • Haag dual nets A on dS 2 (satisfying properties a), b), c), d) in Section 2.2) • Local pseudonets B on S 1 satisfying intersection cyclicity. Proof. We only have to check that, setting A(O) = A(W ) W ⊃O

for any double cone O, the intersection cyclicity is equivalent to the Reeh-Schlieder property for double cones. We shall show that A(O) = B(ϕ+ (O), β(r)ϕ− (O) ).

(5.15)

Indeed, any double cone O can be described as a Cartesian product: O = I+ × I− , where I± = ϕ± (O). Therefore, W ⊃ O is equivalent to Φ± (W ) ⊃ I± . Setting I = Φ+ (W ) and making use of (5.10), this is in turn equivalent to I+ ⊆ I ⊆ β(r)I− . In particular, since any double cone is contained in some wedge, I+ , I− give rise to a double cone O = I+ × I− iﬀ I+ ⊆ β(r)I− . The thesis follows. We showed that any Haag dual net on dS 2 can be holographically reconstructed from a pseudonet on S 1 . Now we address the question of when such a net is conformal. Assuming intersection cyclicity, let us denote by ∆I1 ,I2 the modular operator associated with (B(I1 , I2 ), Ω) for a pair of intervals I1 ⊂ I2 . Theorem 5.11 Let B be a local pseudonet on S 1 satisfying intersection cyclicity, A the corresponding Haag dual net on dS 2 . Then A is conformal if and only if, for any I1 ⊂ L1 ⊂ L2 ⊂ I2 , −it ∆it I1 ,I2 B(L1 , L2 )∆I1 ,I2 = B(ΛI1 (−2πt)(L1 ), ΛI2 (2πt)(L2 ))

(5.16)

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˜ ⊂ O) can Proof. Let us note that local modular covariance (for the inclusion O be rephrased, in view of equation 5.15, as −it ˜ ˜ ∆it I1 ,I2 B(L1 , L2 )∆I1 ,I2 = B(ϕ+ (ΛO (t)O), β(r)ϕ− (ΛO (t)O) ),

(5.17)

˜ are determined by ϕ+ (O) = I1 , ϕ− (O) = β(r)I2 , ϕ+ (O) ˜ = L1 , where O and O ˜ = β(r)L . ϕ− (O) 2 We want to show that for any double cone O, and any x ∈ O, ϕ+ (ΛO (t)x) = ΛI (t)ϕ+ (x) where I = ϕ+ (O). It is enough to show the property when ∂O ∩ H+ is non-empty, since any other double cone can be reached via a transformation in the de Sitter group. In this case, I is identiﬁed with ∂O ∩ H+ . First we observe that ϕ+ (ΛO (t)x) = ΛO (t)ϕ+ (x), since, identifying de Sitter with Minkowski, ΛO splits as the product of the action on the chiral components. Since both are M¨ obius transformations on H+ leaving I globally invariant, they should coincide, possibly up a reparametrization. Finally, we ﬁnd a conformal transformation leaving H+ globally stable and mapping O onto a wedge W , therefore it is enough to check the equality on a wedge, where it follows by equivariance (5.12). ˜ we get ˜ ⊂ O, I˜ = ϕ+ (O), As a consequence, whenever O ˜ = ΛI (t)I. ˜ ϕ+ (ΛO (t)O) In an analogous way we get ˜ = Λϕ (O) (t)ϕ− (O). ˜ ϕ− (ΛO (t)O) − These equations show that relations (5.16) and (5.17) are equivalent, therefore the thesis follows by Theorem 3.6. Now we study the geometric interpretation of the isotonized nets B± . We have seen that any double cone O in dS 2 can be represented as O = I+ × I− , where I± = ϕ± (O). Then we may deﬁne the horizon components of a net A on dS 2 as the nets on S 1 given by A+ (I) = A(O) , A− (I) = A(O). (5.18) O:ϕ+ (O)⊃I

O:ϕ− (O)⊃I

Theorem 5.12 Let A be a Haag dual net on dS 2 , B the corresponding pseudonet on S 1 . Then horizon components correspond to isotonized nets: A± (I) = B± (I). As a consequence the horizon components are conformal nets.

(5.19)

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Proof. Since A is Haag dual, the chiral components may be equivalently deﬁned as A+ (I) = A(W ) , A− (I) = A(W ), (5.20) W :ϕ+ (W )⊃I

W :ϕ− (W )⊃I

and the equality (5.19) follows by Equation (5.13).

Remark 5.13 We could have also deﬁned the horizon restriction net for any component H± of the cosmological horizon, simply setting ∩W ⊃ψ± (I) A(W ), for any interval I ⊂ R. In general it is a larger subnet than the horizon component A± . Then we consider the conformal net on dS 2 given by Aχ (I+ × I− ) = A+ (I+ ) ∨ A− (I− ).

(5.21)

Theorem 5.14 Aχ is a conformal expected subnet of A, satisfying Aχ (I+ × I− ) = A+ (I+ ) ⊗ A− (I− ).

(5.22)

Indeed it is the chiral subnet of the maximal conformal expected subnet of A. Proof. The tensor product splitting follows by Proposition 5.9. As Aχ is chiral conformal, it is immediate that it satisﬁes the local time-slice property, hence it is a Haag dual conformal net. Thus it is expected by Proposition 2.7. The subnets A± may be considered as the chiral components of A. Indeed, they correspond to the two chiral nets on the lightlike rays for a conformal net on the two-dimensional Minkowski space. Therefore we shall say that A is a chiral net if it coincides with Aχ . The following table summarizes the chirality structure. Net on dS 2   restrictionto horizons

max. conf. subnet

−−−−−−−−−−−→ Conformal net on dS 2 Theorem 3.14   Theorem 3.6dS d −M d conf. equiv. chiral components

Two conf. nets on R ←−−−−−−−−−−− Conformal net on M 2 Theorem 5.14

We conclude this section characterizing the chiral nets on dS 2 with only one horizon component. Theorem 5.15 Let A be a local net of von Neumann algebras on the de Sitter spacetime such that H is positive, resp. negative, where H is the generator of the rotation subgroup. Then the associated pseudonet B, resp B is indeed a local net, which holographically reconstructs A: A(O) = B(ϕ± (O)). In particular A is conformal. Proof. If H is positive, the pseudonet is isotonic, by 5.8 (iii). Analogously, if H is negative, the pseudonet is anti-isotonic, hence B is isotonic. In both cases A is chiral, hence conformal.

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Corollary 5.16 The following are equivalent: • The representation U has positive (resp. negative) energy • B− (I) (resp. B+ (I)) is trivial and Aχ (O) = A(O) • A(I+ × I− ) = A+ (I+ ) (resp. = A− (I− )) • A is conformal and the translations on H− (resp. on H+ ) are trivial

Proof. Immediate by the above discussion. We end up with a “holographic” dictionary: dS 2 wedge W double cone O Haag dual nets A on dS 2 SO(2, 1) covariance horizon (chiral) components A± Reeh-Schlieder property for O conformal invariance positive (negative) energy for A chirality

S1 interval I pair of intervals I1 ⊂ I2 local pseudonets B on S 1 M¨ obius covariance isotonized nets B± intersection cyclicity property (5.16) isotony (anti-isotony) for B B = B+ ⊗ B−

6 Final comments Equivalence principle and dethermalization. As is well known, Einstein equivalence principle is a fundamental guiding principle in General Relativity, although it is valid only at the inﬁnitesimal (i.e., local) level. However, if one considers quantum eﬀects, one may notice a certain asymmetry, yet between inertial observers in diﬀerent spacetimes: the one in de Sitter spacetime feels the Gibbons-Hawking temperature, while the one in Minkowski spacetime is in a ground state. One way to describe the dethermalization eﬀect is to say that it “restores” the symmetry: being a quantum eﬀect, it needs a quantum (i.e., noncommutative) description. Only in the limit case where QFT becomes conformal (a situation closer to general covariance in classical general relativity) the dethermalization eﬀect is described by classical ﬂows. In the general case the noncommutative geometry is encoded in the net of local algebras (that takes the place of function algebras) and the dynamics is expressed in terms of this net. Other spacetimes. Although this paper has dealt essentially with de Sitter spacetime, a good part of our description obviously holds in more general spacetimes. As mentioned, several spacetimes are conformal to subregions of Einstein static universe. For a d-dimensional spacetime M in this class one can obviously extend the analysis made in the dS d case: one can set up a correspondence between local conformal nets on M and on M d , hence providing a KMS characterization of the conformal vacuum on M, and ﬁnding the evolutions corresponding to dethermalized observers. However, the partial geometric property of the noncommutative

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ﬂow with positive energy is established only in dS d case by using the large group of isometries of dS d . In particular we may consider a Robertson-Walker spacetime RW d . In the positive curvature case, RW d is R × S d−1 with metric ds2 = dt2 − f (t)2 dσ 2 , where dσ 2 is the metric on the unit sphere S d−1 and f (t) > 0 (in the general case S d−1 is a manifold of constant curvature K = 1, −1, 0). In this case we may also use the method of transplantation given in [14]. Classiﬁcation. Recently [34], diﬀeomorphism covariant local nets on the twodimensional Minkowski spacetime, with central charge less than one, have been completely classiﬁed. By the conformal equivalence Theorem 3.6 one immediately translates this result on dS 2 , namely one has a classiﬁcation of the two-dimensional diﬀeomorphism covariant local nets on dS 2 with central charge less than one. Models, modular localization. The methods in [12] provide a construction of (free) local nets on dS 2 associated with unitary representations of the de Sitter group SO0 (d, 1), and conformal nets on S 1 associated with unitary representations of P SL(2.R). The isomorphism between SO0 (2, 1) and P SL(2, R) gives the holography in these models and is at the basis of our general analysis.

Acknowledgments One of the authors (D.G.) wishes to thank the organizers and the participants to the E. Schr¨ odinger Institute program “QFT on CST”, Vienna 2002, for the kind invitation and many helpful discussions.

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