Operators for Parabolic Block Spin Transformations
Tadeusz Balaban, Joel Feldman, Horst Knörrer, Eugene Trubowitz
aa r X i v : . [ m a t h - ph ] S e p Operators for Parabolic Block SpinTransformations
Tadeusz Balaban , Joel Feldman ∗ , Horst Kn¨orrer , and EugeneTrubowitz Department of MathematicsRutgers, The State University of New [email protected] ∼ feldman/ ∼ knoerrer/ October 17, 2018
Abstract
This paper is a contribution to a program to see symmetry breaking in aweakly interacting many Boson system on a three dimensional lattice at lowtemperature. It is part of an analysis of the “small field” approximation to the“parabolic flow” which exhibits the formation of a “Mexican hat” potentialwell. Bounds on the fluctuation integral covariance, as well as on some otherlinear operators, are an important ingredient in the renormalization group stepanalysis of [7, 8]. These bounds are proven here. ∗ Research supported in part by the Natural Sciences and Engineering Research Council ofCanada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich. ontents Introduction
In [7, 8], we exhibit, for a many particle system of weakly interacting Bosons in threespace dimensions, the formation of a potential well of the type that typically leadsto symmetry breaking in the thermodynamic limit. To do so, we use the block spinrenormalization group approach. In previous papers [4, 1, 2, 3] (followed by a simplechange of variables) we have written the partition function of such a system on adiscrete torus in terms of a functional integral on a 1 + 3 dimensional space X = (cid:0) Z /L tp Z (cid:1) × (cid:0) Z /L sp Z (cid:1) with positive integers L tp , L sp . Up to corrections which are exponentially small inthe coupling constant, and up to a multiplicative normalization factor, this repre-sentation is of the form Z h Q x ∈X dψ ( x ) ∗ ∧ dψ ( x )2 πı i e A ( ψ ∗ ,ψ ) χ ( ψ ) (1.1)with an action A of the form A ( ψ ∗ , ψ ) = − h ψ ∗ , D ψ i − V ( ψ ∗ , ψ ) + µ h ψ ∗ , ψ i + E ′ ( ψ ∗ , ψ ) (1.2)Here • D = 1l − e − h − e − h ∂ , where ∂ the forward time derivative (see (2.9) below and h is – up to a scaling – the single particle Hamiltonian. • V ( ψ ∗ , ψ ) is a quartic monomial that describes the coupling between the particles • µ is related to the chemical potential of the system • E ′ ( ψ ∗ , ψ ) is perturbatively small • χ ( ψ ) is a “small field cut off function”.See [7, (1.3), (1.4)].For the block spin renormalization group action, we pick a “block rectangle” oflength L in the “time direction” and L in “space directions”, where L is a sufficientlylarge odd positive integer, and a corresponding nonnegative, compactly supportedfunction q ( x ) on Z × Z (the averaging profile). The choice of this kind of rectangleis characteristic of “parabolic scaling”. See [7, Definition 1.3, Remark 1.4, Definition1.11.d]. For simplicity we assume that L sp and L tp are powers of L .The block spin averaging operator, which we denote Q , maps functions on thelattice X to functions on the “coarse” lattice X (1) − = (cid:0) L Z /L tp Z (cid:1) × (cid:0) L Z /L sp Z (cid:1) . All bounds achieved so far are uniform in the volume of this torus. X (1)0 = (cid:0) Z / L L tp Z (cid:1) × (cid:0) Z / L L sp Z (cid:1) .The “scaled” block spin averaging operator maps functions on the lattice X = (cid:0) L Z / L L tp Z (cid:1) × (cid:0) L Z / L L sp Z (cid:1) to functions on the unit lattice X (1)0 .In the n th renormalization group step, we end up considering functions on thechain of lattices X ( n +1) − ⊂ X ( n )0 ⊂ X ( n − ⊂ · · · ⊂ X (1) n − ⊂ X (0) n where, for integers j ≥ − n ≥ X ( n ) j = (cid:0) ε j Z /ε n + j L tp Z (cid:1) × (cid:0) ε j Z /ε n + j L sp Z (cid:1) with ε j = L j The subscript in X ( n ) j determines the “coarseness” of the lattice — nearest neighbourpoints are a distance ε j = L j apart in the time direction and a distance ε j = L j apart in spatial directions. The superscript in X ( n ) j determines the number of pointsin the lattice — |X ( n ) j | = |X | /L n for all j . We usually write X (0) n = X n . See [7,Definition 1.5.a or Appendix A.1].The ( n + 1) st block spin transformation involves the passage from X ( n )0 to itssublattice X ( n +1) − . The averaging operations determine linear maps Q : H ( n )0
7→ H ( n +1) − and Q n : H n = H (0) n
7→ H ( n )0 where H ( n ) j = L (cid:0) X ( n ) j (cid:1) denotes the (finite dimensional) Hilbert space of functionson X ( n ) j with integral R X ( n ) j du = ε j P u ∈X ( n ) j and the real inner product h α , α i j = Z X ( n ) j α ( u ) α ( u ) du Again see [7, Definition 1.5.a or Appendix A.3]. In § Q , Q n , their Fourier transforms, and relatedoperators.Scaling is performed by the linear isomorphisms L : X ( n ) j → X ( n ) j − ( u , u ) ( L u , L u )For a function α ∈ H ( n ) j , define the function L ∗ ( α ) ∈ H ( n ) j − by L ∗ ( α )( L u ) = α ( u ) .See [7, Appendix A.2]. In particular, after rescaling and multiplication with the“scaling factor” L n , the differential operator D in (1.2) becomes the operator D n = L n L − n ∗ (cid:0) − e − h − e − h ∂ (cid:1) L n ∗ H n . This operator is discussed in § X ( n )0 to a functionalintegral on X ( n +1) − is an averaging procedure over, roughly speaking, a rectangle ofsize L in the time direction and size L in the spatial directions. This passage isanalyzed using stationary phase techniques that involve • the determination of critical fields on X ( n )0 (that are functions of external fields on X ( n +1) − ) for an appropriate action, and • a functional integral over “fluctuation fields” around the critical field.The covariance for the integral over the fluctuation fields has been identified in [7,(1.15)] and is bounded in § n renormalization group steps is – afterrescaling – a field on X n , called the “background field”, that is a function of anexternal field on X ( n )0 . It is crucial in our representation of the partition function. See[7, Theorem 1.17]. The “leading order” part of the background field is linear in theexternal field and has been identified in [7, Proposition 1.14]. It is the composition ofan operator, from H ( n )0 to H n determined by the averaging profile q , and an operator S n on H n which can be viewed as a Green’s function for the differential operator D n (plus a mass term). This operator, S n , is discussed in § n + 1, we usea well known algebraic relation between these critical fields and the background fieldsat step n + 1 given in [6, Proposition 9] and [7, Proposition 3.4.a]. The operators inthe linearization of this relation, and various other linearizations, are studied in § Definition 1.1.
For any operator A : H ( n − j ) j → H ( n − k ) k , with kernel A ( u, u ′ ), andfor any mass m ≥
0, we define the norm k A k m = max n sup u ∈X ( n − k ) k Z X ( n − j ) j du ′ e m | u − u ′ | | A ( u, u ′ ) | , sup u ′ ∈X ( n − j ) j Z X ( n − k ) k du e m | u − u ′ | | A ( u, u ′ ) | o In the special case that m = 0, this is just the usual ℓ – ℓ ∞ norm of the kernel.5s we point out in [5, Lemmas 12 and 13] this norm is related to the analyticityproperties of the Fourier transform. In this paper we use the following Fouriertransform conventions.The dual lattice of X ( n ) j isˆ X ( n ) j = (cid:0) πε n + j L tp Z / πε j Z (cid:1) × (cid:0) πε n + j L sp Z / πε j Z (cid:1) For a function α ∈ H ( n ) j ˆ α ( p ) = Z X ( n ) j α ( u ) e − ip · u du α ( u ) = Z ˆ X ( n ) j ˆ α ( p ) e iu · p dp (2 π ) where R ˆ X ( n ) j dp (2 π ) = ε n + j L tp L P p ∈ ˆ X ( n ) j . The maps L : ˆ X ( n ) j − → ˆ X ( n ) j ( q , q ) ( L q , L q )are again linear isomorphisms, and, for a function α ∈ H ( n ) j , \L ∗ ( α )( q ) = L ˆ α ( L q ) (1.3)The quotient map dual to the inclusion X ( n ) j ⊂ X ( n − k ) j + k isˆ π ( j + k,j ) n + j : ˆ X ( n − k ) j + k → ˆ X ( n ) j (1.4)When the indices are clear from the context we suppress them and write ˆ π .The estimates of this paper are used in [7, 8]. In particular, the constructionof the background fields and the critical fields in [9] uses a contraction mappingargument around the linearizations of § § Convention 1.2.
Most estimates in this paper are bounds on norms of operators asin Definition 1.1. The (finite number of) constants that appear in these bounds areconsecutively labelled Γ , Γ , · · · , γ , γ , · · · , m , m , · · · . All of these constants Γ j , γ j , m j are independent of L and the scale index n . We define Γ op to be the maximumof the Γ j ’s, and, in [7, 8, 9], refer to the estimates using only this constant Γ op .6 Block Spin Operators
In this chapter, we analyze the block spin “averaging operators” Q of [7, Definitions1.1.a and 1.11.d] and Q n of [7, Definition 1.11.d] as well as the operator Q n of [7,Definition 1.5.b]. Recall that Q : H ( n )0 → H ( n +1) − is defined by( Qψ )( y ) = P x ∈ Z × Z q ( x ) ψ ( y + [ x ]) (2.1)where [ x ] denotes the class of x ∈ Z × Z in the quotient space X ( n )0 . The averagingprofile q is the q –fold convolution of the characteristic function, 1 ( x ), of the rectangle (cid:2) − L − , L − (cid:3) × (cid:2) − L − , L − (cid:3) , normalized to have integral one. That is, q = L q q times z }| { ∗ ∗ · · · ∗ q ≥ The operator Q n = Q (1) · · · Q ( n ) = (cid:0) L − ∗ Q (cid:1) n L n ∗ : H n = H (0) n → H ( n )0 (2.2)where Q ( j ) = L − j ∗ Q L j ∗ : H ( n − j ) j → H ( n − j +1) j − . The operator Q n = a (cid:16)
1l + n − X j =1 1 L j Q j Q ∗ j (cid:17) − The Fourier transform of the characteristic function 1 is σ ( L k ) σ ( k ) with k ∈ ˆ X ( n )0 andwith σ ( k ) = sin (cid:0) k (cid:1) Y ν =1 sin (cid:0) k ν (cid:1) (2.3)Therefore ˆ q ( k ) = u + ( k ) q with u + ( k ) = σ ( L k ) L σ ( k ) (2.4)and, by [5, Lemma 9.a] [ ( Qψ )( k ) = X k ∈ ˆ X ( n )0ˆ π ( k )= k ˆ q ( k ) ˆ ψ ( k ) (2.5)for all ψ ∈ H ( n )0 and k ∈ ˆ X ( n +1) − . See Remark 2.7 for a discussion of the condition q >
2. The condition that q be even is imposedpurely for convenience. emark 2.1. (a) Since q is even, σ ( k ) q is an entire function of k ∈ C × C that is periodic withrespect to the lattice 2 π ( Z × Z ). Also σ ( p j ) q = σ (cid:0) ˆ π ( j, n ( p j ) (cid:1) q for all p j ∈ ˆ X ( n − j ) j (b) For all φ ∈ H n and k ∈ ˆ X ( n )0 , \ ( Q n φ )( k ) = X p ∈ ˆ X n ˆ π ( p )= k u n ( p ) q ˆ φ ( p ) with u n ( p ) = ε n σ ( p ) σ ( L − n p )(c) For all ψ ∈ H ( n )0 and k ∈ ˆ X ( n )0 , [ Q n ψ ( k ) = ˆ Q n ( k ) ˆ ψ ( k ) whereˆ Q n ( k ) = a (cid:20) n − X j =1 X pj ∈ ˆ X ( n − j ) j ˆ π ( pj )= k L j u j ( p j ) q (cid:21) − (d) The functions u n ( p ) and u + ( p ) are entire in p and are invariant under p ν → − p ν for each 0 ≤ ν ≤ p ν ↔ p ν ′ for all 1 ≤ ν, ν ′ ≤ B + = (cid:0) Z /L Z (cid:1) × (cid:0) Z /L Z (cid:1) ˆ B + = (cid:0) πL Z / π Z (cid:1) × (cid:0) πL Z / π Z (cid:1) = ker ˆ π (0 , − n − B j = (cid:0) ε j Z / Z (cid:1) × (cid:0) ε j Z / Z (cid:1) ˆ B j = (cid:0) π Z / πε j Z (cid:1) × (cid:0) π Z / πε j Z (cid:1) = ker ˆ π ( j, n for each integer j ≥
0. In this notation, the representations of Q , Q n and Q n of(2.5) and parts (b) and (c) are [ ( Qψ )( k ) = X ℓ ∈ ˆ B + u + ( k + ℓ ) q ˆ ψ ( k + ℓ ) \ ( Q n φ )( k ) = X ℓ ∈ ˆ B n u n ( k + ℓ ) q ˆ φ ( k + ℓ )ˆ Q n ( k ) = a (cid:20) n − X j =1 P ℓ j ∈ ˆ B j L j u j ( k + ℓ j ) q (cid:21) − [ ( Qψ )( k ) = P ℓ ∈ ˆ B + u + ( k + ℓ ) q ˆ ψ ( k + ℓ ), for example, k ∈ ˆ X ( n +1) − is rep-resented by the element of πε n L tp Z × πε n L sp Z having minimal components and ℓ is represented by the element of πε − Z × πε − Z having minimal components.Similarly (cid:0) d Q ∗ θ (cid:1) ( k + ℓ ) = u + ( k + ℓ ) q ˆ θ ( k ) (cid:0) d Q ∗ n ψ (cid:1) ( k + ℓ n ) = u n ( k + ℓ n ) q ˆ ψ ( k ) Proof. (a) Any two points of ˆ X ( n − j ) j with the same image in ˆ X ( n )0 under ˆ π ( j, n differby 2 π times an integer vector. The formula follows.(b) By (1.3) and (2.5), we have, for α ∈ H ( n − j ) j and p j − ∈ ˆ X ( n − j +1) j − \ ( Q ( j ) α )( p j − ) = L j \ ( QL j ∗ α )( L − j p j − ) = L j X k ∈ ˆ X ( n − j )0ˆ π ( k )= L − jpj − ˆ q ( k ) \ ( L j ∗ α )( k )= 1 L q X k ∈ ˆ X ( n − j )0ˆ π ( k )= L − jpj − σ ( L k ) q σ ( k ) q ˆ α (cid:0) L j k (cid:1) = 1 L q X pj ∈ ˆ X ( n − j ) j ˆ π ( pj )= pj − σ ( L − j +1 p j ) q σ ( L − j p j ) q ˆ α ( p j )so that, by part (a), \ ( Q n φ )( p ) = 1 L q n X pj ∈ ˆ X ( n − j ) j ˆ π ( pj )= pj − ≤ j ≤ n σ ( p ) q σ ( L − p ) q σ ( L − p ) q σ ( L − p ) q · · · σ ( L − n +1 p n ) q σ ( L − n p n ) q ˆ φ ( p n )= ε q n X pj ∈ ˆ X ( n − j ) j ˆ π ( pj )= pj − ≤ j ≤ n σ ( p n ) q σ ( L − p n ) q σ ( L − p n ) q σ ( L − p n ) q · · · σ ( L − n +1 p n ) q σ ( L − n p n ) q ˆ φ ( p n )= ε q n X pn ∈ ˆ X n ˆ π ( pn )= p σ ( p n ) q σ ( L − n p n ) q ˆ φ ( p n )(c) follows from part (b) and [5, Lemma 9.a].(d) is obvious since sin z sin zm is even and entire for any nonzero integer m .9n Lemmas 2.2 and 2.3 we derive a number of bounds on the kernels u n and u + that appear in the representations for Q n and Q of Remark 2.1.e. In Proposition 2.4we analyze the operator ˆ Q n . Then in Remark 2.5 and Lemma 2.6 we study how tomove derivatives past Q and Q n .When dealing with the asymmetry between “temporal” and “spatial” scaling weset, for convenience, L ν = ( L for ν = 0 L for ν = 1 , , ) ε n,ν = ( ε n = L n = L n for ν = 0 ε n = L nν = L n for ν = 1 , , ) (2.6) Lemma 2.2.
Let q ∈ N . Assume that | Re k ν | ≤ π , | Im k ν | ≤ for each ≤ ν ≤ .(a) (cid:12)(cid:12) u n ( k + ℓ ) (cid:12)(cid:12) ≤ Q ν =0 24 | ℓ ν | + π for all ℓ ∈ ˆ B n . We use | ℓ ν | to denote the magnitudeof the smallest representative of ℓ ν in its equivalence class, as an element of ˆ B n .There is a constant Γ , depending only on q , such that k Q n k m =1 ≤ Γ .(b) (cid:12)(cid:12) u n ( k + ℓ ) (cid:12)(cid:12) ≤ h Q ≤ ν ≤ ℓν =0 | k ν | i Q ν =0 24 | ℓ ν | + π if = ℓ ∈ ˆ B n .(c) (cid:12)(cid:12) u n ( k ) − (cid:12)(cid:12) ≤ | k | .(d) If ℓ ∈ ˆ B n and ℓ ˜ ν = 0 for some ≤ ˜ ν ≤ , then u n ( k + ℓ ) = sin (cid:0) k ˜ ν (cid:1) v n, ˜ ν ( k + ℓ ) with (cid:12)(cid:12) v n, ˜ ν ( k + ℓ ) (cid:12)(cid:12) ≤ Q ν =0 24 | ℓ ν | + π .(e) For all ℓ ∈ ˆ B n , (cid:12)(cid:12) Im u n ( k + ℓ ) (cid:12)(cid:12) ≤ | Im k | Q ν =0 24 | ℓ ν | + π (cid:12)(cid:12) Im u n ( k + ℓ ) q (cid:12)(cid:12) ≤ q | Im k | h Q ν =0 24 | ℓ ν | + π i q (f ) Recall that | Re k ν | ≤ π , | Im k ν | ≤ for each ≤ ν ≤ . We have π ≤ | u n ( k ) | ≤ π If, in addition, k is real (cid:0) π (cid:1) ≤ u n ( k ) ≤ (cid:0) π (cid:1) roof. Set s ( x ) = sin xx . By the definitions of u n ( p ) in Remark 2.1.b, σ ( k ) in (2.3)and ε n,ν in (2.6), u n ( p ) = sin p ε n sin ε n p Y ν =1 sin p ν ε n sin ε n p ν = Y ν =0 s ( p ν / s ( ε n,ν p ν /
2) (2.7)(a) We may assume without loss of generality that ℓ ν is bounded, as a real number,by πε n,ν − π . (Recall that ε n,ν is an odd natural number.) So (cid:12)(cid:12) Re k ν + ℓ ν (cid:12)(cid:12) is alwaysbounded by πε n,ν . Consequently, the hypotheses of Lemma A.1.c, with x + iy = k ν + ℓ ν and ε = ε n,ν , are satisfied and (cid:12)(cid:12)(cid:12)(cid:12) sin ( k ν + ℓ ν ) ε n,ν sin ε n,ν ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( ℓ ν = 0 | Re k ν + ℓ ν | if | ℓ ν | ≥ π ) ≤ | ℓ ν | + π since | Re k ν | ≤ π and ℓ ν ∈ π Z .When q > | u n ( k + ℓ ) | q is summable in ℓ and the bound on k Q n k m =1 followsfrom [5, Lemma 12.c]. When q = 1, we use that the action of Q n : H (0) n → H ( n )0 in position space is ( Q n φ )( x ) = P v ∈ ε n Z × ε n Z u n ( v ) φ ( x + [ v ]) (2.8)where [ v ] denotes the class of v ∈ ε n Z × ε n Z in the quotient space X (0) n = (cid:0) ε n Z /ε n L tp Z (cid:1) × (cid:0) ε n Z /ε n L sp Z (cid:1) and x runs over X ( n )0 = (cid:0) Z /ε n L tp Z (cid:1) × (cid:0) Z /ε n L sp Z (cid:1) When q = 1, the averaging profile u n is the characteristic function, 1 ( v ) (the depen-dence on n is suppressed in the notation), of the rectangle (cid:16)(cid:2) − , (cid:3) × (cid:2) − , (cid:3) (cid:17) ∩ (cid:0) ε n Z × ε n Z (cid:1) Note that u n = 1 is already normalized to have integral one. The L – L ∞ norm of u n (with mass zero) is exactly one. The L – L ∞ norm, with mass m = 1, of u n isbounded by exp (cid:8)(cid:12)(cid:12)(cid:0) , , , (cid:1)(cid:12)(cid:12)(cid:9) = e . 11b) If ℓ ν = 0, (cid:12)(cid:12) k ν + ℓ ν (cid:12)(cid:12) ≥ (cid:12)(cid:12) Re k ν + ℓ ν (cid:12)(cid:12) ≥ (cid:0) π + | ℓ ν | (cid:1) so that, by Lemma A.1.a, the denominator ε n,ν (cid:12)(cid:12) sin ε n,ν ( k ν + ℓ ν ) (cid:12)(cid:12) ≥ ε n,ν √ π ε n,ν (cid:0) π + | ℓ ν | (cid:1) = √ π (cid:0) π + | ℓ ν | (cid:1) On the other hand, the numerator, by Lemma A.1.b, (cid:12)(cid:12) sin ( k ν + ℓ ν ) (cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) k ν (cid:1)(cid:12)(cid:12) ≤ | k ν | As ℓ = 0, there is at least one ν with ℓ ν = 0. For each ν with ℓ ν = 0 bound thefactor (cid:12)(cid:12)(cid:12)(cid:12) sin ( k ν + ℓ ν ) ε n,ν sin ε n,ν ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ π | k ν || ℓ ν | + π ≤ | k ν || ℓ ν | + π Bound the remaining factors, with ν having ℓ ν = 0, by | ℓ ν | + π as in part (a). Alltogether Y ν =0 (cid:12)(cid:12)(cid:12)(cid:12) sin ( k ν + ℓ ν ) ε n,ν sin ε n,ν ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20) Y ≤ ν ≤ ℓν =0 | k ν | (cid:21) Y ν =0 | ℓ ν | + π (c) For both z = ε n,ν k ν and z = k ν , | z | ≤ π + 1 < (cid:12)(cid:12) sin zz − (cid:12)(cid:12) ≤ | z | Using ab − ( a − − ( b − b we have, by Lemma A.1.a, (cid:12)(cid:12)(cid:12)(cid:12) sin( k ν ) ε n,ν sin( ε n,ν k ν ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ | k ν | + | ε n,ν k ν | | sin( ε n,ν k ν ) | / | ε n,ν k ν | ≤ π √ (1 + ε n,ν ) | k ν | ≤ | k ν | Finally, using Y ν =0 A ν − A − Y ν =1 A ν + ( A − Y ν =2 A ν + ( A − A + ( A − (cid:12)(cid:12) u n ( k ) − (cid:12)(cid:12) ≤ | k | + 4 − | k | + 4 | k | + | k | ≤ | k | (cid:0) k ˜ ν (cid:1) in thenumerator sin ( k ˜ ν + ℓ ˜ ν ) = ( − ℓ ˜ ν π sin (cid:0) k ˜ ν (cid:1) is pulled out of u n , leaving v n, ˜ ν , rather than being bounded by | k ˜ ν | .(e) By Lemma A.1.c (cid:12)(cid:12)(cid:12)(cid:12) Im sin ( k ν + ℓ ν ) ε n,ν sin ε n,ν ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Im k ν | | ℓ ν | + π In general, for any complex numbers z j = r j e iθ j , 1 ≤ j ≤ J , (cid:12)(cid:12)(cid:12) Im J Q j =1 z j (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) sin (cid:16) J P j =1 θ j (cid:17)(cid:12)(cid:12)(cid:12) J Q j =1 r j Repeatedly using (cid:12)(cid:12) sin( θ + θ ′ ) (cid:12)(cid:12) = (cid:12)(cid:12) sin( θ ) cos( θ ′ ) + cos( θ ) sin( θ ′ ) (cid:12)(cid:12) ≤ | sin( θ ) | + | sin( θ ′ ) | we have (cid:12)(cid:12)(cid:12) Im J Q j =1 z j (cid:12)(cid:12)(cid:12) ≤ J X j =1 (cid:12)(cid:12) sin( θ j ) (cid:12)(cid:12) J Q j =1 r j = J X j =1 (cid:12)(cid:12) Im z j (cid:12)(cid:12) Q j ′ = j | z j ′ | So (cid:12)(cid:12) Im u n ( k + ℓ ) (cid:12)(cid:12) ≤ (cid:16) P ν =0 | Im k ν | (cid:17) Q ν =0 24 | ℓ ν | + π ≤ | Im k | Q ν =0 24 | ℓ ν | + π and (cid:12)(cid:12) Im u n ( k + ℓ ) q (cid:12)(cid:12) ≤ q | Im k | h Q ν =0 24 | ℓ ν | + π i q (f) Just apply Lemma A.1.a,b separately to all of the numerators and denominatorsin the right hand side of (2.7). When k is real, apply Lemma A.1.d instead.13 emma 2.3. Let q ∈ N . Assume that | Re k ν | ≤ πL ν and | Im k ν | ≤ L ν for each ≤ ν ≤ .(a) (cid:12)(cid:12) u + ( k + ℓ ) (cid:12)(cid:12) ≤ Q ν =0 24 L ν | ℓ ν | + π for all ℓ ∈ ˆ B + . We use | ℓ ν | to denote the magnitude ofthe smallest representative of ℓ ν in its equivalence class, as an element of ˆ B + .(b) (cid:12)(cid:12) u + ( k + ℓ ) (cid:12)(cid:12) ≤ h Q ν ∈I + L ν | k ν | ih Q ν =0 24 L ν | ℓ ν | + π i for all ℓ ∈ ˆ B + with ℓ = 0 . Here I + isany subset of (cid:8) ν (cid:12)(cid:12) ≤ ν ≤ , ℓ ν = 0 (cid:9) .(c) (cid:12)(cid:12) u + ( k ) − (cid:12)(cid:12) ≤ P ν =0 L ν | k ν | .(d) If ℓ ∈ ˆ B + and ℓ ˜ ν = 0 for some ≤ ˜ ν ≤ , then u + ( k + ℓ ) = sin (cid:0) L ˜ ν k ˜ ν (cid:1) v + , ˜ ν ( k + ℓ ) with (cid:12)(cid:12) v + , ˜ ν ( k + ℓ ) (cid:12)(cid:12) ≤ Q ν =0 24 L ν | ℓ ν | + π .(e) For all ℓ ∈ ˆ B + , (cid:12)(cid:12) Im u + ( k + ℓ ) (cid:12)(cid:12) ≤ | Im L k | Q ν =0 24 L ν | ℓ ν | + π (cid:12)(cid:12) Im u + ( k + ℓ ) q (cid:12)(cid:12) ≤ q | Im L k | h Q ν =0 24 L ν | ℓ ν | + π i q Proof. (a) We may assume without loss of generality that ℓ ν is bounded, as a realnumber, by π − πL ν . (Recall that L is an odd natural number.) So we will alwayshave (cid:12)(cid:12) Re k ν + ℓ ν (cid:12)(cid:12) ≤ π . So, by Lemma A.1.c with ε = L ν and x + iy = L ν ( k ν + ℓ ν ), (cid:12)(cid:12)(cid:12)(cid:12) sin L ν ( k ν + ℓ ν ) L ν sin ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( ℓ ν = 0 L ν | Re k ν + ℓ ν | if | L ν ℓ ν | ≥ π ) ≤ L ν | ℓ ν | + π since | Re k ν | ≤ πL ν , | Im k ν | ≤ L ν and ℓ ν ∈ πL ν Z .(b) If ℓ ν = 0, (cid:12)(cid:12) k ν + ℓ ν (cid:12)(cid:12) ≥ (cid:12)(cid:12) Re k ν + ℓ ν (cid:12)(cid:12) ≥ (cid:0) πL ν + | ℓ ν | (cid:1) and (cid:12)(cid:12) Re k ν + ℓ ν (cid:12)(cid:12) ≤ π so that, by Lemma A.1.a, the denominator L ν (cid:12)(cid:12) sin ( k ν + ℓ ν ) (cid:12)(cid:12) ≥ √ π (cid:0) π + L ν | ℓ ν | (cid:1)
14n the other hand, the numerator, by Lemma A.1.b, (cid:12)(cid:12) sin L ν ( k ν + ℓ ν ) (cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) L ν k ν (cid:1)(cid:12)(cid:12) ≤ L ν | k ν | = L ν | k ν | As ℓ = 0, there is at least one ν with ℓ ν = 0. Bound each factor with ν ∈ I + by (cid:12)(cid:12)(cid:12)(cid:12) sin L ν ( k ν + ℓ ν ) L ν sin ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π √ L ν | k ν | L ν | ℓ ν | + π Bound the remaining factors by L ν | ℓ ν | + π .(c) has the same proof as that of Lemma 2.2.c, with k ν replaced by L ν k ν and ε n,ν replaced by L ν .(d) The proof is similar to that for part (b), except that the factor sin (cid:0) L ˜ ν k ˜ ν (cid:1) in thenumerator sin L ˜ ν ( k ˜ ν + ℓ ˜ ν ) = ( − L ˜ νℓ ˜ ν π sin (cid:0) L ˜ ν k ˜ ν (cid:1) is pulled out of u + , leaving v + , ˜ ν , rather than being bounded by L ˜ ν | k ˜ ν | . Also, each (cid:12)(cid:12)(cid:12) sin Lν ( k ν + ℓ ν ) L ν sin ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12) with ν = ˜ ν is bounded by L ν | ℓ ν | + π .(e) By Lemma A.1.c (cid:12)(cid:12)(cid:12)(cid:12) Im sin L ν ( k ν + ℓ ν ) L ν sin ( k ν + ℓ ν ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Im L ν k ν | L ν | ℓ ν | + π The proof now continues as in Lemma 2.2.e, just by substituting k ν → L ν k ν , ℓ ν → L ν ℓ ν and ε n,ν = L ν . Proposition 2.4.
Let q ∈ N . There are constants Γ , depending only on q , and Γ ,depending only on a , such that the following hold for all L > Γ .(a) On the domain (cid:8) k ∈ C × C (cid:12)(cid:12) | Im k ν | < for each ≤ ν ≤ (cid:9) ˆ Q n ( k ) isanalytic, and invariant under k ν → − k ν for each ≤ ν ≤ and under k ν ↔ k ν ′ for all ≤ ν, ν ′ ≤ , and obeys a ≤ | ˆ Q n ( k ) | ≤ a Re ˆ Q n ( k ) ≥ a If k is real a ≤ ˆ Q n ( k ) ≤ a .(b) If | Re k ν | ≤ π and | Im k ν | ≤ for each ≤ ν ≤ , then (cid:12)(cid:12) ˆ Q n ( k ) − a n (cid:12)(cid:12) ≤ a | k | where a n = a − L − − L − n c) k Q n k m =1 ≤ Γ Proof. (a) Recall, from Remark 2.1.e, that ˆ Q n ( k ) = a (cid:20) n − P j =1 P ℓ j ∈ ˆ B j L j u j ( k + ℓ j ) q (cid:21) − .By Lemma 2.2.a, n − X j =1 P ℓ j ∈ ˆ B j L j | u j ( k + ℓ j ) | q ≤ ∞ X j =1 1 L j P ℓ ∈ π Z × π Z Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q = c q L − where c q = h P j ∈ Z (cid:0) /π [2 | j | +1 (cid:1) q i . Just pick L large enough that c q L − < and usethat, if | z | ≤ Re z = Re (1+ z ) | z | ≥ / / = (b) Using O ( | k | ) to denote any function that is bounded by a constant, dependingonly on q ,1 + n − X j =1 P ℓ j ∈ ˆ B j L j u j ( k + ℓ j ) q = n − X j =0 1 L j + n − X j =1 1 L j [ u j ( k ) q − n − X j =1 P ℓj ∈ ˆ B jℓ =0 L j u j ( k + ℓ j ) q ≤ − L − n − L − + n − X j =1 1 L j O (cid:0) | k | (cid:1) by Lemma 2.2.b,c ≤ − L − n − L − + L O (cid:0) | k | (cid:1) So ˆ Q n ( k ) = a (cid:2) − L − n − L − + L O (cid:0) | k | (cid:1)(cid:3) − = a n (cid:2) L − L − − L − n O (cid:0) | k | (cid:1)(cid:3) − and it suffices to choose Γ large enough that ≤ − L − − L − n ≤ (cid:12)(cid:12)(cid:12) L − L − − L − n O (cid:0) | k | (cid:1)(cid:12)(cid:12)(cid:12) ≤ | k | (cid:12)(cid:12)(cid:12) L − L − − L − n O (cid:0) | k | (cid:1)(cid:12)(cid:12)(cid:12) ≤ for all allowed k ’s and L ’s.(c) follows immediately from part (a) and [5, Lemma 12.b] with X fin = X crs = X ( n )0 . 16e now define operators Q ( ± ) n,ν and Q ( ± )+ ,ν so that the next remark holds. Remark 2.5.
Let 0 ≤ ν ≤
3. We have ∂ ν Q ∗ n = Q (+) n,ν ∂ ν ∂ ν Q n = Q ( − ) n,ν ∂ ν ∂ ν Q ∗ = Q (+)+ ,ν ∂ ν ∂ ν Q = Q ( − )+ ,ν ∂ ν If S : H n → H n and T : H ( n )0 → H ( n )0 are linear operators that are translationinvariant with respect to X n and X ( n )0 , respectively, then Q ( − ) n,ν SQ (+) n,ν = Q n SQ ∗ n Q ( − )+ ,ν T Q (+)+ ,ν = QT Q ∗ To prepare for the definitions, recall that the forward derivatives of α ∈ H ( n ) j aredefined by ( ∂ ν α )( x ) = ε j,ν (cid:2) α ( x + ε j,ν e ν ) − α ( x ) (cid:3) (2.9)where e ν is a unit vector in the ν th direction. The Fourier transforms (cid:0)d ∂ ν φ (cid:1) ( p ) = 2 ie iε n,ν p ν / ε n,ν p ν / ε n,ν ˆ φ ( p ) for all φ ∈ H n and p ∈ ˆ X n (cid:0) d ∂ ν ψ (cid:1) ( k ) = 2 ie ik ν / sin( k ν /
2) ˆ ψ ( k ) for all ψ ∈ H ( n )0 and k ∈ ˆ X ( n )0 (2.10) (cid:0) c ∂ ν θ (cid:1) ( k ) = 2 ie iL ν k ν / L ν k ν / L ν ˆ θ ( k ) for all θ ∈ H ( n +1) − and k ∈ ˆ X ( n +1) − Set u (+) n,ν ( p ) = Y ≤ ν ′ ≤ ν ′ = ν sin p ν ′ ε n,ν ′ sin ε n,ν ′ p ν ′ u (+)+ ,ν ( k ) = Y ≤ ν ′ ≤ ν ′ = ν sin L ν ′ k ν ′ L ν ′ sin k ν ′ u ( − ) n,ν ( p ) = sin p ν ε n,ν sin ε n,ν p ν Y ν ′ =0 sin p ν ′ ε n,ν ′ sin ε n,ν ′ p ν ′ u ( − )+ ,ν ( k ) = sin L ν k ν L ν sin k ν Y ν ′ =0 sin L ν ′ k ν ′ L ν ′ sin k ν ′ and ζ (+) n,ν ( k, ℓ n ) = e iε n,ν ( k + ℓ n ) ν / e − ik ν / cos ℓ n,ν ζ (+)+ ,ν ( k , ℓ ) = e i ( k + ℓ ) ν / e − iL ν k ν / cos L ν ℓ ν ζ ( − ) n,ν ( k, ℓ n ) = e ik ν / e − iε n,ν ( k + ℓ n ) ν / cos ℓ n,ν ζ ( − )+ ,ν ( k , ℓ ) = e iL ν k ν / e − i ( k + ℓ ) ν / cos L ν ℓ ν Define the operators Q (+) n,ν : H ( n )0 → H n and Q ( − ) n,ν : H n → H ( n )0 by (cid:0) \ Q (+) n,ν ψ (cid:1) ( k + ℓ n ) = ζ (+) n,ν ( k, ℓ n ) u (+) n,ν ( k + ℓ n ) u n ( k + ℓ n ) q − ˆ ψ ( k ) (cid:0) \ Q ( − ) n,ν φ (cid:1) ( k ) = X ℓ n ∈ ˆ B n ζ ( − ) n,ν ( k, ℓ n ) u ( − ) n,ν ( k + ℓ n ) u n ( k + ℓ n ) q − ˆ φ ( k + ℓ n ) (2.11)17nd the operators Q (+)+ ,ν : H ( n +1) − → H ( n )0 and Q ( − )+ ,ν : H ( n )0 → H ( n +1) − by (cid:0) \ Q (+)+ ,ν θ (cid:1) ( k + ℓ ) = ζ (+)+ ,ν ( k , ℓ ) u (+)+ ,ν ( k + ℓ ) u + ( k + ℓ ) q − ˆ θ ( k ) (cid:0) \ Q ( − )+ ,ν ψ (cid:1) ( k ) = X ℓ ∈ ˆ B + ζ ( − )+ ,ν ( k , ℓ ) u ( − )+ ,ν ( k + ℓ ) u + ( k + ℓ ) q − ˆ ψ ( k + ℓ ) (2.12) Proof of Remark 2.5.
For the “ Q ∗ n ” and “ Q n ” cases, it suffices to observe that (cid:0) ie iε n,ν ( k + ℓ ) ν / ε n,ν ( k + ℓ ) ν / ε n,ν (cid:1) u n ( k + ℓ ) = ζ (+) n,ν ( k, ℓ ) u (+) n,ν ( k + ℓ ) (cid:0) ie ik ν / sin( k ν / (cid:1) and (cid:0) ie ik ν / sin( k ν / (cid:1) u n ( k + ℓ ) = ζ ( − ) n,ν ( k, ℓ ) u ( − ) n,ν ( k + ℓ ) (cid:0) ie iε n,ν ( k + ℓ ) ν / ε n,ν ( k + ℓ ) ν / ε n,ν (cid:1) and ζ ( − ) n,ν ( k, ℓ ) u ( − ) n,ν ( k + ℓ ) ζ (+) n,ν ( k, ℓ ) u (+) n,ν ( k + ℓ ) = u n ( k + ℓ ) for all k , ℓ , ν . We remark that “ Q ( − ) n,ν SQ (+) n,ν = Q n SQ ∗ n ” should not be surprising since ∂ ν Q n SQ ∗ n = Q ( − ) n,ν SQ (+) n,ν ∂ ν and Q n SQ ∗ n is translation invariant on the unit scale and socommutes with ∂ ν . The proof for the “ Q ∗ ” and “ Q ” cases are virtually identical. Lemma 2.6.
Let ≤ ν ≤ , ℓ ∈ ˆ B + and ℓ n ∈ ˆ B n .(a) ζ (+) n,ν ( k, ℓ n ) u (+) n,ν ( k + ℓ n ) and ζ ( − ) n,ν ( k, ℓ n ) u ( − ) n,ν ( k + ℓ n ) are entire in k and ζ (+)+ ,ν ( k , ℓ ) u (+)+ ,ν ( k + ℓ ) and ζ ( − )+ ,ν ( k , ℓ ) u ( − )+ ,ν ( k + ℓ ) are entire in k (b) Assume that | Re k ν ′ | ≤ π , | Im k ν ′ | ≤ , | Re k ν ′ | ≤ πL ν ′ and | Im k ν ′ | ≤ L ν ′ for each ≤ ν ′ ≤ . Then (cid:12)(cid:12) ζ (+) n,ν ( k, ℓ n ) u (+) n,ν ( k + ℓ n ) (cid:12)(cid:12) ≤ e Y ≤ ν ′ ≤ ν ′ = ν | ℓ n,ν ′ | + π (cid:12)(cid:12) ζ (+)+ ,ν ( k , ℓ ) u (+)+ ,ν ( k + ℓ ) (cid:12)(cid:12) ≤ e Y ≤ ν ′ ≤ ν ′ = ν L ν | ℓ ν ′ | + π (cid:12)(cid:12) ζ ( − ) n,ν ( k, ℓ n ) u ( − ) n,ν ( k + ℓ n ) (cid:12)(cid:12) ≤ e | ℓ n,ν | + π Q ν ′ =0 24 | ℓ n,ν ′ | + π (cid:12)(cid:12) ζ ( − )+ ,ν ( k , ℓ ) u ( − )+ ,ν ( k + ℓ ) (cid:12)(cid:12) ≤ eL ν | ℓ ν | + π Q ν ′ =0 24 L ν ′ | ℓ ν ′ | + π c) There is a constant Γ , depending only on q , such that (cid:13)(cid:13) Q ( ± ) n,ν (cid:13)(cid:13) m =1 ≤ Γ .Proof. (a) The proof is virtually identical to that of Remark 2.1.d.(b) The proof is virtually identical to that of Lemmas 2.2.a and 2.3.a.(c) By (2.11), the Fourier transform of Q ( ± ) n,ν is ζ ( ± ) n,ν ( k, ℓ n ) u ( ± ) n,ν ( k + ℓ n ) u n ( k + ℓ n ) q − ,which by part (b) and Lemma 2.2.a, is bounded in magnitude by e Y ≤ ν ′ ≤ ν ′ = ν | ℓ n,ν ′ | + π Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q − As q >
2, the claim now follows by [5, Lemma 12.c].
Remark 2.7.
The principle obstruction to allowing q = 1 arises when a differentialoperator ∂ ν is intertwined with the block spin averaging operator Q n , as happens inRemark 2.5. See, for example, the proof of Lemma 2.6.c. We use the condition q > § §
5, 6. (See Lemma 5.5.) We use the condition q > Differential Operators
In [7, Definition 1.5.a] we associated to an operator h on L (cid:0) Z /L sp Z (cid:1) the operators D n = L n L − n ∗ (cid:0) − e − h − e − h ∂ (cid:1) L n ∗ (3.1)Here ∂ is the forward time derivative of (2.9). In this chapter we assume that h isthe periodization (see [5, § h on L (cid:0) Z (cid:1) whoseFourier transform ˆ h ( p ) • is entire in p and invariant under p ν → − p ν for each 1 ≤ ν ≤ • is nonnegative when p is real and is strictly positive when p ∈ R \ π Z • obeys ˆ h ( ) = ∂ ˆ h ∂ p ν ( ) = 0 for 1 ≤ ν ≤ H = h ∂ ˆ h ∂ p µ ∂ p ν ( ) i ≤ µ,ν ≤ . Remark 3.1. (a) The operator D n is the periodization of a translation invariant operator D n ,acting on L (cid:0) ε n Z × ε n Z (cid:1) , whose Fourier transform isˆ D n ( p ) = ε n p e − ˆ h ( ε n p ) (cid:20) sin ε n p ε n p (cid:21) + p − e − ˆ h ( ε n p ) ε n p − ip e − ˆ h ( ε n p ) sin ε n p ε n p with p = ( p , p ) ∈ C × C .(b) ˆ D n ( p ) is entire in p and invariant under p ν → − p ν for each 1 ≤ ν ≤ D n ( p ) has nonnegative real part when p is real. Proof. (a) follows from (1.3) and the observation, by (2.10), that the Fourier trans-form of ∂ , on Z , is 2 ie ik / sin( k /
2) = − ( k /
2) + i sin( k )(b) and (c) are obvious. Lemma 3.2.
There are constants γ , Γ and a function ¯ m ( c ) > that depend onlyon ˆ h and in particular are independent of n and L , such that the following hold. a) For all p ∈ R × R , (cid:12)(cid:12) ˆ D n ( p ) (cid:12)(cid:12) ≥ γ (cid:0) | p | + P ν =1 | p ν | (cid:1) We use | p | , | p ν | and | p | to refer to the magnitudes of the smallest representa-tives of p ∈ C , p ν ∈ C and p ∈ C in C / πε n Z , C / πε n Z and C / πε n Z , respec-tively.(b) For all p ∈ C × C with ε n p , ε n p having modulus less than one, ˆ D n ( p ) = − ip + ε n p +
12 3 P ν,ν ′ =1 H ν,ν ′ p ν p ν ′ + O (cid:16) ε n | p | + ε n | p | (cid:17) The higher order part O ( · ) is uniform in n and L .(c) We have, for all p ∈ C × C with ε n | Im p | ≤ and ε n | Im p | ≤ , (cid:12)(cid:12) ˆ D n ( p ) (cid:12)(cid:12) ≤ Γ (cid:0) | p | + P ν =1 | p ν | (cid:1) and (cid:12)(cid:12) ∂ ℓi ∂p ℓiν ˆ D n ( p ) (cid:12)(cid:12) ≤ Γ | p | + | p | ] ℓi − if ν = 0 , ℓ i = 1 , | p | + | p | ] ℓi/ − if ≤ ν ≤ , ≤ ℓ i ≤ (d) For all c > and p ∈ C × C , with | p | + | p | ≥ c and | Im p | ≤ ¯ m ( c ) , (cid:12)(cid:12) ˆ D n ( p ) (cid:12)(cid:12) ≥ γ (cid:0) | p | + P ν =1 | p ν | (cid:1) (e) For all c > and all p in the set (cid:8) p ∈ C × C (cid:12)(cid:12) | Im p | ≤ ¯ m ( c ) , | p | ≥ c (cid:9) ∪ (cid:8) p ∈ C × C (cid:12)(cid:12) | Im p | ≤ ¯ m ( c ) , | ε n p | ≥ c (cid:9) we have Re ˆ D n ( p ) ≥ γ (cid:0) ε n | p | + P ν =1 | p ν | (cid:1) roof. (a) By the hypotheses on ˆ h ,1 − e − ˆ h ( ε n p ) ε n p = ˆ h ( ε n p ) + O (ˆ h ( ε n p ) ) ε n p = ε n p · H p + O ( | ε n p | ) ε n p for ε n p is a real neighbourhood of . This is strictly positive and bounded awayfrom 0 on some real neighbourhood of 0, uniformly in ε n . Since ˆ h is continuous andstrictly positive on R \ π Z , there is a constant γ ′ >
0, independent of ε n , suchthat 1 − e − ˆ h ( ε n p ) ε n | p | ≥ γ ′ and e − ˆ h ( ε n p ) ≥ γ ′ for all p ∈ R . The claim now follows from Lemma A.1.a.(b) Expanding sin zz = 1 + O ( | z | ), for | z | ≤
1, and e − ˆ h ( ε n p ) = 1 − ε n P ν,ν ′ =1 H ν,ν ′ p ν p ν ′ + O (cid:16)(cid:0) ε n | p | (cid:1) (cid:17) for ε n | p | ≤ D n ( p ) = h − ip + ε n p ih O (cid:16)(cid:0) ε n | p | (cid:1) + (cid:0) ε n p (cid:1) (cid:17)i +
12 3 P ν,ν ′ =1 H ν,ν ′ p ν p ν ′ + O (cid:16) ε n | p | (cid:17) = − ip + ε n p +
12 3 P ν,ν ′ =1 H ν,ν ′ p ν p ν ′ + O (cid:16) ε n | p | + ε n | p || p | + ε n | p | (cid:17) The claim now follows from ε n | p || p | ≤ (cid:0) ε n | p | (cid:1) / (cid:0) ε n | p | (cid:1) / (c) Since ˆ D n ( p ) is periodic with respect to πε n Z × πε n Z , we may assume that ε n Re p and each ε n Re p ν , 1 ≤ ν ≤ π . By Lemma A.1.b, (cid:12)(cid:12) sin zz (cid:12)(cid:12) ≤ z ∈ C with | Im z | ≤
1. Since ε n p runs over a compact set (indepen-dently of n and L ), e − ˆ h ( ε n p ) is bounded. So (cid:12)(cid:12)(cid:12)(cid:12) ε n p e − ˆ h ( ε n p ) (cid:20) sin ε n p ε n p (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ip e − ˆ h ( ε n p ) sin ε n p ε n p (cid:12)(cid:12)(cid:12)(cid:12) ≤ const | p | and (cid:12)(cid:12)(cid:12)(cid:12) − e − ˆ h ( ε n p ) ε n (cid:12)(cid:12)(cid:12)(cid:12) ≤ const | ε n p | ε n ≤ const | p | (cid:12)(cid:12) ˆ D n ( p ) (cid:12)(cid:12) .The bounds ∂∂p ε n h sin( ε n p ) ε n i = 2 sin( ε n p ) = O (1) ∂∂p h sin( ε n p ) ε n i = cos( ε n p ) = O (1) ∂ ∂p ε n h sin( ε n p ) ε n i = 2 ε n cos( ε n p ) = O ( ε n ) ∂ ∂p h sin( ε n p ) ε n i = − ε n sin( ε n p ) = O ( ε n ) ∂∂ p ν h ε n e − ˆ h ( ε n p ) i = O ( | p | ) ∂ ∂ p ν h ε n e − ˆ h ( ε n p ) i = O (1 + ε n | p | ) = O (1) ∂ ∂ p ν h ε n e − ˆ h ( ε n p ) i = O ( ε n | p | + ε n | p | ) ∂ ∂ p ν h ε n e − ˆ h ( ε n p ) i = O ( ε n + ε n | p | + ε n | p | )= O ( ε n ) = O ( ε n )together with ε n ≤ const1+ | p | + | p | ε n | p | ≤ constyield the bounds on the derivatives.(d) Write p = P + i Q with P = ( P , P ) , Q = ( Q , Q ) ∈ R × R . We may choose ¯ m ( c )sufficiently small that, if | P + i Q | ≥ c and | Q | ≤ ¯ m ( c ), then | P + i Q | + P ν =1 | P ν + i Q ν | ≤ (cid:16) | P | + P ν =1 | P ν | (cid:17) + (cid:16) | Q | + P ν =1 | Q ν | (cid:17) ≤ (cid:16) | P | + P ν =1 | P ν | (cid:17) If γ > is chosen large enough, then, for all such P , Q , we have, by parts (a) and (c), (cid:12)(cid:12) ˆ D n ( P ) (cid:12)(cid:12) ≥ γ (cid:0) | P | + P ν =1 | P ν | (cid:1) ≥ γ (cid:0) | P + i Q | + P ν =1 | P ν + i Q ν | (cid:1)(cid:12)(cid:12)(cid:12) ˆ D n ( P + i Q ) − ˆ D n ( P ) (cid:12)(cid:12)(cid:12) ≤ (cid:16) | P + i Q | + P ν =1 | P ν + i Q ν | (cid:17) | Q | Recalling that | P + i Q | + P ν =1 | P ν + i Q ν | ≥ min n c , c o , it now suffices to choose¯ m ( c ) small enough that2 Γ ¯ m ( c ) ≤ γ min n c , c o and 2 Γ ¯ m ( c ) ≤ γ p = P + i Q with P = ( P , P ) , Q = ( Q , Q ) ∈ R × R . Since ˆ D n ( P + i Q )is periodic with respect to P ∈ πε n Z × πε n Z , we may assume that (cid:12)(cid:12) ε nP (cid:12)(cid:12) ≤ π and (cid:12)(cid:12) ε n P ν (cid:12)(cid:12) ≤ π , for each 1 ≤ ν ≤
3. If the constant γ was chosen small enough, then,as in part (a),Re ˆ D n ( P ) = ε nP e − ˆ h ( ε n P ) (cid:20) sin ε nP ε nP (cid:21) + P − e − ˆ h ( ε n P ) ε n P ≥ γ (cid:0) ε nP + | P | (cid:1) Hence it suffices to prove that it is possible to choose ¯ m = ¯ m ( c ) so that (cid:12)(cid:12)(cid:12) Re ˆ D n ( P + i Q ) − Re ˆ D n ( P ) (cid:12)(cid:12)(cid:12) ≤ γ | P | when | P | ≥ c and | Q | ≤ ¯ m (3.2)and that (cid:12)(cid:12)(cid:12) Re ˆ D n ( P + i Q ) − Re ˆ D n ( P ) (cid:12)(cid:12)(cid:12) ≤ γ c when | P | ≤ c , | ε nP | ≥ c and | Q | ≤ ¯ m (3.3)This is a consequence of the following bounds on the derivatives of the real parts ofthe three terms making up ˆ D n ( P + i Q ) in Remark 3.1.a. For the first term, (cid:12)(cid:12)(cid:12)(cid:12) ddt ε n ( P + it Q ) e − ˆ h ( ε n P ) (cid:20) sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const (cid:2) ε n | P + i Q | | Q | + ε n | P + i Q | ε n | Q | (cid:3) ≤ const ¯ m ddt Re ε n ( P + it Q ) h e − ˆ h ( ε n P + itε n Q ) − e − ˆ h ( ε n P ) i(cid:20) sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 (cid:12)(cid:12)(cid:12)(cid:12) d dt ε n ( P + it Q ) h e − ˆ h ( ε n P + itε n Q ) − e − ˆ h ( ε n P ) i(cid:20) sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const (cid:2) ε n | Q | + ε n | P + i Q | | Q | (cid:0) ε n | Q | + ε n | Q | (cid:1) + ε n | P + i Q | (cid:0) ε n | Q | + ε n | Q | (cid:1) (cid:3) ≤ const ¯ m For the second term, ddt Re − e − ˆ h εn P + itεn Q ) ε n (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 (cid:12)(cid:12)(cid:12)(cid:12) d dt − e − ˆ h εn P + itεn Q ) ε n (cid:12)(cid:12)(cid:12)(cid:12) ≤ const ε n (cid:0) ε n | Q | (cid:1) ≤ const ¯ m (cid:12)(cid:12)(cid:12)(cid:12) ddt Re i ( P + it Q ) e − ˆ h ( ε n P ) sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const | Q | + const | P + i Q | ε n | Q |≤ const ¯ m (cid:12)(cid:12)(cid:12)(cid:12) ddt Re i ( P + it Q ) h e − ˆ h ( ε n P + itε n Q ) − e − ˆ h ( ε n P ) i sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Re i P h ddt ˆ h ( ε n P + itε n Q ) i t =0 e − ˆ h ( ε n P ) sin ε n ( P ) ε nP (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ε nP (cid:0) Q · ˆ h ′ ( ε n P ) (cid:1) e − ˆ h ( ε n P ) sin ε n ( P ) ε nP (cid:12)(cid:12)(cid:12)(cid:12) ≤ const ε n | P | | P | ¯ m ≤ const | P | ¯ m (cid:12)(cid:12)(cid:12)(cid:12) d dt i ( P + it Q ) h e − ˆ h ( ε n P + itε n Q ) − e − ˆ h ( ε n P ) i sin ε n ( P + it Q ) ε n ( P + it Q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const (cid:2) | Q | (cid:0) ε n | Q | + ε n | Q | (cid:1) + | P + i Q | (cid:0) ε n | Q | + ε n | Q | (cid:1) (cid:3) ≤ const ¯ m Now choose ¯ m = ¯ m ( c ) small enough that (3.2) and (3.3) are satisfied.25 The Covariance
The covariance for the fluctuation integral in [8] is C ( n ) = ( aL Q ∗ Q + ∆ ( n ) ) − where ∆ ( n ) = ((cid:0)
1l + Q n Q n D − n Q ∗ n (cid:1) − Q n if n ≥ D if n = 0 ) : H ( n )0 → H ( n )0 See [7, (1.15) and (1.14)]. In Lemma 4.2 we study the properties of ∆ ( n ) and inCorollary 4.5 we study the properties of C ( n ) and its square root. Remark 4.1.
Let n ≥ ( n ) is the periodization of a translation invariant operator ∆∆∆ ( n ) ,acting on L (cid:0) Z × Z (cid:1) , whose Fourier transform isˆ∆∆∆ ( n ) ( k ) = ˆ Q n ( k ) (cid:16) Q n ( k ) X ℓ ∈ ˆ B n u n ( k + ℓ ) q ˆ D − n ( k + ℓ ) (cid:17) − = ˆ Q n ( k ) ˆ D n ( k )ˆ D n ( k ) + ˆ Q n ( k ) P ℓ u n ( k + ℓ ) q ˆ D − n ( k + ℓ ) ˆ D n ( k )with k ∈ C × C , where u n ( p ) and ˆ B n were defined in parts (b) and (e) ofRemark 2.1, respectively.(b) ˆ∆∆∆ ( n ) ( k ) is invariant under k ν → − k ν for each 1 ≤ ν ≤ ( n ) ( k ) has nonnegative real part when k is real. Lemma 4.2.
There are constants m > , γ , Γ and Γ , such that, for L > Γ , thefollowing hold.(a) ˆ∆∆∆ ( n ) ( k ) is analytic on | Im k | < m .(b) For all k ∈ C × C with | Im k | ≤ m . ˆ∆∆∆ ( n ) ( k ) = − ik + (cid:0) a n + ε n (cid:1) k +
12 3 P ν,ν ′ =1 H ν,ν ′ k ν k ν ′ + O (cid:0) | k | (cid:1) ˆ∆∆∆ ( n ) ( k ) ˆ D − n ( k ) = 1 + ik a n + O (cid:0) | k | (cid:1) The higher order part O ( · ) is uniform in n and L . c) (cid:12)(cid:12) ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) ≤ a and (cid:12)(cid:12) ∂∂k ν ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ ∂k ν ∂k ν ′ ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) ≤ Γ for all ≤ ν, ν ′ ≤ and k ∈ C × C with | Im k | < m .(d) There is a function ρ ( c ) > , which is defined for all c > and which dependsonly on m , q , ˆ h and a and, in particular, is independent of n and L , such that Re ˆ∆∆∆ ( n ) ( k ) ≥ ρ ( c ) for all k ∈ C × C with | k | ≥ c and | Im k | ≤ m .(e) For all k ∈ C × C with | Im k | ≤ m and | Re k ν | ≤ π for all ≤ ν ≤ , (cid:12)(cid:12) ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) ≥ γ (cid:12)(cid:12) ˆ D n ( k ) (cid:12)(cid:12) (f ) ˆ D − n ( p ) ˆ∆∆∆ ( n ) ( p ) is analytic on | Im p | ≤ m . Furthermore, for all p ∈ C × C with | Im p | ≤ m , (cid:12)(cid:12) ˆ D − n ( p ) ˆ∆∆∆ ( n ) ( p ) (cid:12)(cid:12) ≤ Γ | p | + P ν =1 | p ν | Here, as usual, | p | and | p ν | refer to the magnitudes of the smallest representa-tives of p ∈ C and p ν ∈ C in C / πε n Z and C / πε n Z respectively.(g) (cid:12)(cid:12) ∂∂ k ν ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) ≤ Γ | k ν | for all ≤ ν ≤ and k ∈ C × C with | Im k | < m .Proof. We first prove part (b). Using that • u n ( k ) q = 1 + O (cid:0) | k | (cid:1) by Lemma 2.2.b • ˆ Q n ( k ) = a n + O (cid:0) | k | (cid:1) by Proposition 2.4.b • | ˆ D n ( k ) | ≤ const (cid:0) | k | + | k | (cid:1) by Lemma 3.2.cand that, for ℓ = 0, • | u n ( k + ℓ ) | q ≤ const | k | q Q ν =0 1( | ℓ | ν | + π ) q by Lemma 2.2.a • | ˆ D − n ( k + ℓ ) | ≤ const by Lemma 3.2.dwe obtain, by Remark 4.1.a and Lemma 3.2.b,ˆ∆∆∆ ( n ) ( k ) = ˆ Q n ( k ) ˆ D n ( k )ˆ Q n ( k ) u n ( k ) q + ˆ D n ( k ) + O (cid:0) | k | (cid:1) = ˆ D n ( k ) a n + O (cid:0) | k | (cid:1) a n + ˆ D n ( k ) + O (cid:0) | k | (cid:1)
27 ˆ D n ( k ) n − a n ˆ D n ( k ) + O (cid:0) | k | (cid:1)o = − ik + ε n k +
12 3 P ν,ν ′ =1 H ν,ν ′ k ν k ν ′ − a n ˆ D n ( k ) + O (cid:0) | k | (cid:1) = − ik + (cid:0) a n + ε n (cid:1) k +
12 3 P ν,ν ′ =1 H ν,ν ′ k ν k ν ′ + O (cid:0) | k | (cid:1) This also shows that, in a neighbourhood of the origin, ˆ∆∆∆ ( n ) ( k ) is analytic andbounded in magnitude by 2 a . For the second expansion,ˆ∆∆∆ ( n ) ( k ) ˆ D − n ( k ) = a n + O (cid:0) | k | (cid:1) a n + ˆ D n ( k ) + O (cid:0) | k | (cid:1) = 1 − a n ˆ D n ( k ) + O (cid:0) | k | (cid:1) = 1 + ik a n + O (cid:0) | k | (cid:1) (a), (c) We first prove the analyticity and the bound (cid:12)(cid:12) ˆ∆∆∆ ( n ) ( k ) (cid:12)(cid:12) ≤ a of part (c). Wehave already done so for a neighbourhood of the origin. So it suffices to consider | k | > c , for some suitably small c . Since ˆ∆∆∆ ( n ) is periodic with respect to 2 π Z × π Z ,it suffices to consider k in the set M ( m ) = (cid:8) k ∈ C × C (cid:12)(cid:12) | Re k ν | ≤ π for all 0 ≤ ν ≤ , | Im k | ≤ m , | k | > c (cid:9) Recall, from Remarks 2.1.d and 3.1.b, that u n ( p ) and ˆ D n ( p ) are entire. If m issmall enough, then, by Lemma 3.2.d, the functions k ˆ D n ( k + ℓ ), ℓ ∈ ˆ B n , n ≥ M ( m ). Hence each term in the infinite sum1 + ˆ Q n ( k ) X ℓ ∈ ˆ B n u n ( k + ℓ ) q ˆ D − n ( k + ℓ )is analytic and it suffices to prove that, for k ∈ M ( m ), the sum converges uniformlyand (cid:12)(cid:12) Q n ( k ) P ℓ ∈ ˆ B n u n ( k + ℓ ) q ˆ D − n ( k + ℓ ) (cid:12)(cid:12) ≥ .By Proposition 2.4.a, Remark 2.1.d and Lemma 3.2.d, there is an l > | ˆ Q n ( k ) | X ℓ ∈ ˆ B n | ℓ |≥ l (cid:12)(cid:12) u n ( k + ℓ ) q ˆ D − n ( k + ℓ ) (cid:12)(cid:12) ≤ a X ℓ ∈ ˆ B n | ℓ |≥ l γ π Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q ≤ const l ≤ (4.1)Hence we have uniform convergence and (cid:12)(cid:12)(cid:12) Q n ( k ) X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) Q n ( k ) X ℓ ∈ ˆ B n | ℓ |
3. Observe that ∂ ˆ∆∆∆ ( n ) ∂k ν ( k ) = k ν Z ∂ ˆ∆∆∆ ( n ) ∂k ν (cid:0) k ( t ) (cid:1) dt with k ( t ) ν ′ = ( k ν ′ if ν ′ = νtk ν , if ν = ν ′ since ∂ ˆ∆∆∆ ( n ) ∂k ν (cid:0) k (0) (cid:1) = 0, by Remark 4.1.b. Now apply the bound on the second deriva-tive from part (c).We next consider the “resolvents” R ( n ) ζ = (cid:0) ζ − aL Q ∗ Q − ∆ ( n ) (cid:1) − in preparation for studying C n and √ C n . We shall use [5, Lemma 14], with ε T = 1 ε X = 1 L T = L L X = L L T = L tp L X = L sp and X fin = X ( n )0 Z fin = Z × Z X crs = X ( n +1) − Z crs = L Z × L Z B = B + We wish to apply [5, Lemma 14], with A → D n = aL Q ∗ Q + ∆ ( n ) (scaled). By [5,Lemmas 9.b and 5.b], for each ℓ, ℓ ′ ∈ ˆ B = ˆ B + , a k ( ℓ, ℓ ′ ) → d n, k ( ℓ, ℓ ′ ) = aL u + ( k + ℓ ) q u + ( k + ℓ ′ ) q + δ ℓ,ℓ ′ ˆ∆ ( n ) ( k + ℓ ) (4.3)Here we are denoting • momenta dual to the L –lattice by k ∈ (cid:0) R / πL Z (cid:1) × (cid:0) R / πL Z (cid:1) and • momenta dual to the unit lattice Z × Z by k ∈ (cid:0) R / π Z (cid:1) × (cid:0) R / π Z (cid:1) anddecompose k = k + ℓ or k = k + ℓ ′ with k in a fundamental cell for (cid:0) R / πL Z (cid:1) × (cid:0) R / πL Z (cid:1) and ℓ, ℓ ′ ∈ (cid:0) πL Z / π Z (cid:1) × (cid:0) πL Z / π Z (cid:1) = ˆ B + .We also use d n, k to denote the ˆ B + × ˆ B + matrix (cid:2) d n, k ( ℓ, ℓ ′ ) (cid:3) ℓ,ℓ ′ ∈ ˆ B + . Observe, by Remark2.1.d and Lemma 4.2.a, that d n, k (cid:0) ℓ, ℓ ′ (cid:1) is analytic in the strip | Im k | < m .Let (cid:2) v ℓ (cid:3) ℓ ∈ ˆ B + and (cid:2) w ℓ (cid:3) ℓ ∈ ˆ B + be any vectors in L ( ˆ B + ). Then, if k = L − ( k ), (cid:10) ¯ v, d n, k w (cid:11) = aL h P ℓ ∈ ˆ B + u + ( L − ( k )+ ℓ ) q v ℓ ih P ℓ ∈ ˆ B + u + ( L − ( k )+ ℓ ) q w ℓ i + X ℓ ∈ ˆ B + ˆ∆ ( n ) ( L − ( k )+ ℓ ) v ℓ w ℓ = X ℓ,ℓ ′ ∈ ˆ B v L ( ℓ ) d ( s ) n,k ( ℓ, ℓ ′ ) w L ( ℓ ′ ) d ( s ) n,k ( ℓ, ℓ ′ ) = aL u + (cid:0) L − ( k + ℓ ) (cid:1) q u + (cid:0) L − ( k + ℓ ′ ) (cid:1) q + δ ℓ,ℓ ′ ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) Lemma 4.3.
There are constants m , λ , Γ > , such that, for all L > Γ and k ∈ C × C with | Im k | < m , the following hold.(a) Write k = L − ( k ) . For both the operator and (matrix) ℓ – ℓ ∞ norms k d n, k k ≤ Γ k d ( s ) n,k k ≤ Γ (b) Let λ ∈ C be within a distance λ L of the negative real axis. Then the resolvent (cid:13)(cid:13)(cid:0) λ − d ( s ) n,k (cid:1) − (cid:13)(cid:13) ≤ Γ L This is true for both the operator and (matrix) ℓ – ℓ ∞ norms.Proof. Since d n, k + p ( ℓ, ℓ ′ ) = d n, k ( ℓ + p, ℓ ′ + p ), for all p, ℓ, ℓ ′ ∈ ˆ B + , we may alwaysassume that | Re k | ≤ πL and | Re k ν | ≤ πL for 1 ≤ ν ≤ (cid:2) aL u + ( k + ℓ ) q u + ( k + ℓ ′ ) q (cid:3) ℓ,ℓ ′ ∈ ˆ B + and (cid:2) aL u + (cid:0) L − ( k + ℓ ) (cid:1) q u + (cid:0) L − ( k + ℓ ′ ) (cid:1) q (cid:3) ℓ,ℓ ′ ∈ ˆ B have finite L – L ∞ norms and the matrix elements of (cid:2) δ ℓ,ℓ ′ ˆ∆ ( n ) ( k + ℓ ) (cid:3) ℓ,ℓ ′ ∈ ˆ B + and (cid:2) δ ℓ,ℓ ′ ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1)(cid:3) ℓ,ℓ ′ ∈ ˆ B are all bounded.(b) Case | k | ≤ c , with c being chosen later in this case: We first consider thosediagonal matrix elements of d ( s ) n,k ( ℓ, ℓ ′ ) having k, ℓ such that | L − ( k + ℓ ) | < ˜ c , where˜ c > L . Then, by Lemma 4.2.b,we have the following. • If at least one of ℓ ν , 1 ≤ ν ≤ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) ≥ c L , provided m and ˜ c are chosen small enough. Here c and the constraints on m and ˜ c depend only on the largest and smallest eigenvalues of (cid:2) H ν,ν ′ (cid:3) , a n and the O ( | k | ).To see this, denote by k and ℓℓℓ the spatial parts of k and ℓ and observe that ◦ (cid:12)(cid:12) Re L − ( k + ℓℓℓ ) (cid:12)(cid:12) ≥ L max (cid:8) π, | ℓℓℓ | (cid:9) , 32 (cid:12)(cid:12) Im L − ( k + ℓℓℓ ) (cid:12)(cid:12) = L (cid:12)(cid:12) Im k (cid:12)(cid:12) ≤ m L and ◦ (cid:12)(cid:12) Im L − ( k + ℓ ) (cid:12)(cid:12) = L (cid:12)(cid:12) Im k (cid:12)(cid:12) ≤ m L .In controlling the contribution from O (cid:16)(cid:12)(cid:12) L − ( k + ℓ ) (cid:12)(cid:12) (cid:17) when | ℓ | L is larger than | ℓℓℓ | L ,we have to use that, in this case, the real part of (cid:0) a n + ε n (cid:1)(cid:0) L − ( k + ℓ ) (cid:1) is at leasta strictly positive constant times ℓ L . • If ℓ ν = 0 for all 1 ≤ ν ≤ ℓ = 0, then − sgn ℓ Im ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) ≥ π L ,provided m and ˜ c is chosen small enough. To see this observe that ◦ (cid:12)(cid:12) Re L − ( k + ℓℓℓ ) (cid:12)(cid:12) = L | Re k | ≤ √ πL , ◦ (cid:12)(cid:12) Im L − ( k + ℓℓℓ ) (cid:12)(cid:12) = L (cid:12)(cid:12) Im k (cid:12)(cid:12) ≤ m L , ◦ sgn ℓ Re L − ( k + ℓ ) ≥ L max (cid:8) π, | ℓ | (cid:9) , ◦ (cid:12)(cid:12) Re L − ( k + ℓ ) (cid:12)(cid:12) ≤ L | ℓ | and ◦ (cid:12)(cid:12) Im L − ( k + ℓ ) (cid:12)(cid:12) = L (cid:12)(cid:12) Im k (cid:12)(cid:12) ≤ m L . • If ℓ = 0, then, by parts (a) and (b) of Lemma 2.3, aL (cid:12)(cid:12) u + (cid:0) L − ( k + ℓ ) (cid:1) q (cid:12)(cid:12) ≤ aL | k | h Q ν =0 24 | ℓ ν | + π i q • If ℓ = 0 then Re (cid:8) aL u + (cid:0) L − ( k ) (cid:1) q + ˆ∆ ( n ) (cid:0) L − ( k ) (cid:1)(cid:9) ≥ a L , provided m and ˜ c arechosen small enough. To see this observe that ◦ (cid:12)(cid:12) u + (cid:0) L − ( k ) (cid:1) − (cid:12)(cid:12) ≤ | k | , by Lemma 2.3.c and (cid:12)(cid:12) u + (cid:0) L − ( k ) (cid:1)(cid:12)(cid:12) ≤ (cid:0) π (cid:1) by Lemma2.3.a. ◦ Re ˆ∆ ( n ) (cid:0) L − ( k ) (cid:1) ≥ − c (cid:0) m L + ˜ c (cid:0) | k | L (cid:1) (cid:1) where c depends only on a n , the largesteigenvalue of (cid:2) H ν,ν ′ (cid:3) , and the O ( | k | ).Note that we have now fixed ˜ c . Now we consider the remaining matrix elements. • For the remaining diagonal matrix elements we have | L − ( k + ℓ ) | ≥ ˜ c and then,by Lemma 4.2.d, Re ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) ≥ ρ (˜ c ) • Finally, the off–diagonal matrix elements of d ( s ) n,k obey, by parts (a) and (b) ofLemma 2.3, aL (cid:12)(cid:12) u + (cid:0) L − ( k + ℓ ) (cid:1) q u + (cid:0) L − ( k + ℓ ′ ) (cid:1) q (cid:12)(cid:12) ≤ aL | k | h Q ν =0 24 | ℓ ν | + π i q h Q ν =0 24 | ℓ ′ ν | + π i q Hence the off–diagonal part of d ( s ) n,k has Hilbert-Schmidt, matrix and, as q > L – L ∞ norms all bounded by a universal constant times aL | k | .Thus λ − d ( s ) n,k has diagonal matrix elements of magnitude at leastmin (cid:8) c L , π L , a L , ρ (˜ c ) (cid:9) − aL | k | h Q ν =0 24 | ℓ ν | + π i q − λ L L – L ∞ norm bounded by a universal constant times aL | k | .It now suffices to choose c and λ small enough that every diagonal matrix elementhas magnitude at least L min (cid:8) c , π , a , ρ (˜ c ) (cid:9) and the off diagonal part has L – L ∞ norm bounded by L min (cid:8) c , π , a , ρ (˜ c ) (cid:9) and then do a Neumann expansion.(b) Case | k | ≥ c , with the c just chosen: We may assume that | Re k ν | ≤ π foreach 0 ≤ ν ≤ • If | L − ( k + ℓ ) | < ˜ c and ℓ ν = 0 for at least one 1 ≤ ν ≤ | L − ( k + ℓ ) | ≥ ˜ c ,then Re ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) ≥ min (cid:8) ρ (˜ c ) , c L (cid:9) . The proof of this given in the case | k | ≤ c applies now too. • If | L − ( k + ℓ ) | < ˜ c , ℓ ν = 0 for all ν ≥ ( n ) (cid:0) L − (Re k + ℓ ) (cid:1) ≥ (cid:12)(cid:12) ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) − ˆ∆ ( n ) (cid:0) L − (Re k + ℓ ) (cid:1)(cid:12)(cid:12) ≤ π Γ L | Im k | The first bound follows immediately from Remark 4.1.c. The second bound followsfrom Lemma 4.2.c (for the L Im k = Im ( L − ( k + ℓ ) contribution) and Lemma4.2.g (for the L Im k ν contribution, with 1 ≤ ν ≤
3, — note that on the linesegment from L − (Re k + ℓ ) to L − ( k + ℓ ), (cid:12)(cid:12) ∂ ˆ∆ ( n ) ∂k ν (cid:12)(cid:12) is bounded by Γ | L − ( k + ℓ ) ν | = Γ L | k ν | ≤ Γ π +3 m L ). Furthermore − sgn ℓ Im ˆ∆ ( n ) (cid:0) L − (Re k + ℓ ) (cid:1) ≥ π L if ℓ = 0 (cid:12)(cid:12) ˆ∆ ( n ) (cid:0) L − (Re k ) (cid:1)(cid:12)(cid:12) ≥ γ γ min (cid:8) c √ , c (cid:9) L if ℓ = 0In the case ℓ = 0, the proof of the bound given in the case | k | ≤ c applies nowtoo. (Just apply it to Re k .) The bound for the case ℓ = 0 follows from Lemma4.2.e and Lemma 3.2.a. • For all ℓ , by parts (a) and (e) of Lemma 2.3, (cid:12)(cid:12) u + (cid:0) L − ( k + ℓ ) (cid:1) q (cid:12)(cid:12) ≤ h Q ν =0 24 | ℓ ν | + π i q (cid:12)(cid:12) Im u + (cid:0) L − ( k + ℓ ) (cid:1) q (cid:12)(cid:12) ≤ q | Im k | h Q ν =0 24 | ℓ ν | + π i q We split λ − d ( s ) n,k into three pieces (cid:0) λ − d ( s ) n,k (cid:1) ( ℓ, ℓ ′ ) = D ( ℓ, ℓ ′ ) − P ( ℓ, ℓ ′ ) + I ( ℓ, ℓ ′ )34here D ( ℓ, ℓ ′ ) = δ ℓ,ℓ ′ d ℓ with d ℓ = λ − − ˆ∆ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) ℓ ν = 0 for some ν ≥ ( n ) (cid:0) L − ( k + ℓ ) (cid:1) | L − ( k + ℓ ) | ≥ ˜ c ˆ∆ ( n ) (cid:0) L − (Re k + ℓ ) (cid:1) otherwise λ − = min { Re λ, } + i Im λP ( ℓ, ℓ ′ ) = aL v ( ℓ ) v ( ℓ ′ ) with v ( ℓ ) = Re u + (cid:0) L − ( k + ℓ ) (cid:1) q We have chosen λ − so that Re λ − ≤ | λ − λ − | ≤ λ L . As P is a rank one operator( D − P ) − ( ℓ, ℓ ′ ) = d ℓ δ ℓ,ℓ ′ + − κ aL v ( ℓ ) d ℓ v ( ℓ ′ ) d ℓ ′ with κ = X ℓ ′′ aL d ℓ ′′ v ( ℓ ′′ ) We have shown above that | d ℓ | ≥ λ L Re d ℓ ≤ ( | L − ( k + ℓ ) | < ˜ c , ℓ ν = 0 for all ν ≥ − λ L otherwise | v ( ℓ ) | ≤ h Q ν =0 24 | ℓ ν | + π i q | I ( ℓ, ℓ ′ ) | ≤ (cid:0) λ L + 4 π Γ L | Im k | (cid:1) δ ℓ,ℓ ′ + 32 q | Im k | aL h Q ν =0 24 | ℓ ν | + π i q h Q ν =0 24 | ℓ ′ ν | + π i q provided we choose 0 < λ ≤ min (cid:8) c , ρ (˜ c ) , π , γ γ c √ , γ γ c (cid:9) .Since Re z ≤ ⇒ Re z ≤
0, and v ( ℓ ) ∈ R for all ℓ , we have that Re κ ≤ (cid:12)(cid:12) − κ (cid:12)(cid:12) ≤ k ( D − P ) − k ℓ − ℓ ∞ ≤ L λ k I k ℓ − ℓ ∞ ≤ Γ ′ L (cid:0) λ + | Im k | (cid:1) with the constant λ > a and λ and the constant Γ ′ dependingonly on Γ , a and q . It now suffices to choose λ and m smaller than λ ′ and usea Neumann expansion to give (cid:13)(cid:13)(cid:0) λ − d ( s ) n,k (cid:1) − (cid:13)(cid:13) ℓ − ℓ ∞ ≤ L λ roposition 4.4. Let m , λ , Γ be as in Lemma 4.3 and use R − to denote thenegative real axis in C . Set O C = (cid:8) z ∈ C (cid:12)(cid:12) dist( z, R − ) > λ L , | z | < Γ + 1 (cid:9) C Γ λ L O = (cid:8) z ∈ C (cid:12)(cid:12) dist( z, R − ) > λ L , | z | < Γ + 2 (cid:9) and let • C = ∂ O C , oriented counterclockwise • f : O → C be analytic.Then f (cid:0) ( aL Q ∗ Q + ∆ ( n ) ) ( s ) (cid:1) , defined by [5, (12) and Lemma 15.a], exists and thereis a constant Γ such that k f (cid:0) ( aL Q ∗ Q + ∆ ( n ) ) ( s ) (cid:1) k m ≤ Γ L sup ζ ∈ C | f ( ζ ) | Proof.
Apply [5, Lemma 14] with ˆ a k ( ℓ, ℓ ′ ) = d ( s ) n,k ( ℓ, ℓ ′ ) and X fin = X ( n )1 Z fin = ε Z × ε Z X crs = X ( n +1)0 Z crs = Z × Z B = B and m = 3 m , m ′ = 2 m and m ′′ = m . Then vol c = 1, B = L . Observe inparticular that, by Lemma 4.3, the spectrum of d ( s ) n,k is contained in (cid:8) z ∈ C (cid:12)(cid:12) dist( z, R − ) > λ L , | z | ≤ Γ (cid:9) which is the shaded region in the figure above. So the lemma gives (cid:13)(cid:13) f (cid:0) ( aL Q ∗ Q + ∆ ( n ) ) ( s ) (cid:1)(cid:13)(cid:13) m ′′ ≤ C m ′− m ′′ π vol c | C | sup ζ ∈ C | f ( ζ ) | sup | Im k | = m ′ ζ ∈ C X ℓ,ℓ ′ ∈ ˆ B (cid:12)(cid:12) ( ζ − ˆ d ( s ) n, k ) − ( ℓ, ℓ ′ ) |≤ C m ′− m ′′ π (Γ + 1)(3 π + 2) sup ζ ∈ C | f ( ζ ) | L sup | Im k | = m ′ ζ ∈ C sup ℓ ∈ ˆ B X ℓ ′ ∈ ˆ B (cid:12)(cid:12) ( ζ − ˆ d ( s ) n, k ) − ( ℓ, ℓ ′ ) |≤ C m ′− m ′′ π (Γ + 1)(3 π + 2) sup ζ ∈ C | f ( ζ ) | L Γ L by Lemma 4.3.b. Recall Convention 1.2. f ( z ) = z , f ( z ) = √ z and f ( z ) = √ z , where √ z isthe principal value of the square root gives Corollary 4.5.
The operators C ( n ) , √ C ( n ) and (cid:0) √ C ( n ) (cid:1) − all exist. There is aconstant Γ such that (cid:13)(cid:13) L − ∗ C ( n ) L ∗ (cid:13)(cid:13) m , (cid:13)(cid:13)p L − ∗ C ( n ) L ∗ (cid:13)(cid:13) m , (cid:13)(cid:13)(cid:0)p L − ∗ C ( n ) L ∗ (cid:1) − (cid:13)(cid:13) m ≤ Γ L The Green’s Functions
In this chapter, we discuss the inverses of the operators D n + Q ∗ n Q n Q n These inverses, and variations thereof, are constituents of the leading part of thepower series expansion of the background fields of [7, 8, 9]. See [7, Proposition 1.14]and [9, Proposition 2.1]. In Proposition 5.1, below, we show that for sufficientlysmall µ , the operators D n + Q ∗ n Q n Q n − µ are invertible, and we estimate the decayof the kernels S n ( µ )( x, y ) of their inverses S n ( µ ) = (cid:2) D n + Q ∗ n Q n Q n − µ (cid:3) − By Remark 2.5 ∂ ν (cid:0) S n ( µ ) ∗ (cid:1) − = (cid:0) S (+) n,ν ( µ ) (cid:1) − ∂ ν ∂ ν S n ( µ ) − = (cid:0) S ( − ) n,ν ( µ ) (cid:1) − ∂ ν (5.1)where S (+) n,ν ( µ ) = (cid:2) D ∗ n + Q (+) n,ν Q n Q ( − ) n,ν − µ (cid:3) − S ( − ) n,ν ( µ ) = (cid:2) D n + Q (+) n,ν Q n Q ( − ) n,ν − µ (cid:3) − (5.2)and Q (+) n,ν , Q ( − ) n,ν were defined in (2.11). We shall write S n = S n (0) S ( ± ) n,ν = S ( ± ) n,ν (0) (5.3)The main result extends the statement of [7, Theorem 1.13]. It is Proposition 5.1.
There are constants µ up , m > and Γ , depending only on q , h and a , and in particular independent of n and L > Γ , such that, for | µ | ≤ µ up ,the operators D n + Q ∗ n Q n Q n − µ and D ∗ n + Q (+) n,ν Q n Q ( − ) n,ν − µ , D n + Q (+) n,ν Q n Q ( − ) n,ν − µ are invertible, and their inverses S n ( µ ) and S (+) n,ν ( µ ) , S ( − ) n,ν ( µ ) , respectively, fulfill k S n ( µ ) k m , k S ( ± ) n,ν ( µ ) k m ≤ Γ k S n ( µ ) − S n k m , k S ( ± ) n,ν ( µ ) − S ( ± ) n,ν k m ≤ | µ | Γ This Proposition is proven following the proof of Lemma 5.5.
Example 5.2.
As a model computation, we evaluate the inverse transform s ( x ) = Z R × R e ip · x − ip + m + p + p + p dp d p (2 π )
38f ˆ s ( p ) = − ip + m + p . It is designed to mimic the behaviour of S n in the limit n → ∞ . Write x = ( t, x ) ∈ R × R . We first compute the p integral. Observe thatthe integrand has exactly one pole, which is at p = − i ( m + p ), and that the e ip t in the integrand forces us to close the contour in the upper half plane when t > t <
0. Thus Z ∞−∞ e ip t + i p · x − ip + m + p dp (2 π ) = ( t > e ( m + p ) t + i p · x π ) if t < s ( x ) = 0 for t > t < s ( x ) = Z R e − ( m + p ) | t | e i p · x d p (2 π ) = e − m | t | Y j =1 Z ∞−∞ e −| t | p j e i p j x j d p j π = ( π | t | ) / e − m | t | e − x | t | Here are some observations about s ( x ). • Since m | t | + x | t | ≥ m | x | (the minimum is at | t | = | x | m ), s ( x ) decays exponentiallyfor large | x | in all directions. • For x = 0, lim t ր e − x | t | = 0, so s ( x ) is continuous everywhere except at x = 0. • s ( x ) has an integrable singularity at x = 0. There are a number of ways to see this.For example, the inequality e − x | t | ≤ const (cid:0) | t || x | (cid:1) κ/ implies that | s ( x ) | is boundednear x = 0 by a constant times | t | (3 − κ ) / | x | κ . This is integrable if 1 < κ < • If we send x → x = − γ | t | ln | t | , s ( x ) ≈ const | t | / − γ . • We see, using x j e − x j | t | ≤ const | t | , that (cid:0) t + P j =1 x j (cid:1) | s ( x ) | is bounded and expo-nentially decaying. Note that t +Σ j x j has an integrable singularity at the origin,since R ∞−∞ t +Σ j x j dt = const √ Σ j x j .As preparation for and in addition to the position space estimates of Proposition5.1, we also derive bounds on the Fourier transforms of these and related operators.To convert bounds in momentum space into bounds in position space, we shall use[5, Lemma 12], with X fin = X n Z fin = ε n Z × ε n Z X crs = X ( n )0 Z crs = Z × Z B = B n (5.4)We shall routinely use | p | , | p ν | and | p | to refer to the magnitudes of the smallestrepresentatives of p ∈ C , p ν ∈ C and p ∈ C in C / πε n Z , C / πε n Z and C / πε n Z ,respectively. 39he operators S n ( µ ) act on functions on the lattice X n , but they are only transla-tion invariant with respect to the sublattice X ( n )0 . An exponentially decaying operatorwhich is fully translation invariant, and has the same local singularity as S n , is theoperator S ′ n = (cid:2) D n + a n exp {− ∆ n } (cid:3) − where∆ n = ∂ ∗ ∂ + (cid:0) ∂ ∗ ∂ + ∂ ∗ ∂ + ∂ ∗ ∂ (cid:1) (5.5)and a n = a − L − − L − n as in Proposition 2.4.b, and the forward derivatives ∂ ν are definedin (2.9). Obviously S ′ n has Fourier transform b S ′ n ( p ) = (cid:2) ˆ D n ( p ) + a n exp {− ∆ n ( p ) } (cid:3) − where ∆ n ( p ) = (cid:2) sin ε n p ε n (cid:3) + P ν =1 (cid:2) sin ε n p ν ε n (cid:3) Before we discuss the properties of S ′ n and of the difference δS = S n − S ′ n we note Remark 5.3.
The Fourier transform ∆ n ( p ) of the four dimensional Laplacian ∆ n isentire. For p ∈ R × R , ∆ n ( p ) ≥ π (cid:2) | p | + | p | (cid:3) For p ∈ C × C with ε n | Im p | ≤ ε n | Im p | ≤ | ∆ n ( p ) | ≤ (cid:2) | p | + | p | (cid:3) | ∂∂p ν ∆ n ( p ) | ≤ | p ν | | ∂ ℓ ∂p ℓν ∆ n ( p ) | ≤ ε ℓ − n,ν if ℓ ≥ ε n,ν of (2.6). For p ∈ C × C with | Im p | ≤ n ( p ) ≥ − π + π (cid:2) | p | + | p | (cid:3) Proof.
For the first two claims, just apply parts (a) and (b) of Lemma A.1. For thederivatives, use ddθ (cid:2) sin( ηθ ) η (cid:3) = sin(2 ηθ ) η = ⇒ d ℓ dθ ℓ (cid:2) sin( ηθ ) η (cid:3) = ± (2 η ) ℓ − η ( sin(2 ηθ ) for ℓ oddcos(2 ηθ ) for ℓ evenFor the final claim, write p = P + i Q with P , Q ∈ R × R . ThenRe ∆ n ( P + i Q ) ≥ ∆ n ( P ) − (cid:12)(cid:12) Re ∆ n ( P + i Q ) − ∆ n ( P ) (cid:12)(cid:12) ≥ π (cid:2) | P | + | P | (cid:3) − | Q || P + i Q |≥ π | P + i Q | − (cid:0) π (cid:1) | Q | | P + i Q | − π | Q | ≥ π | P + i Q | − (cid:8) π (cid:0) π (cid:1) + π (cid:9) | Q |
40y the resolvent identity δS = S n − S ′ n = − S ′ n (cid:2) Q ∗ n Q n Q n − a n exp {− ∆ n } (cid:3) S n S n and δS are translation invariant with respect to the sublattice X ( n )0 of X n . By“Floquet theory” (see [5, Lemma 1]), their Fourier transforms b S n ( p, p ′ ) , c δS ( p, p ′ ) , p, p ′ ∈ ˆ X n vanish unless π ( n, n ( p ) = π ( n, n ( p ′ ), i.e. unless there are k ∈ ˆ X ( n )0 and ℓ, ℓ ′ ∈ ˆ B n such that p = k + ℓ , p ′ = k + ℓ ′ . The blocks b S − n,k ( ℓ, ℓ ′ ) = b S − n ( k + ℓ, k + ℓ ′ )and c δS k ( ℓ, ℓ ′ ) = c δS ( k + ℓ, k + ℓ ′ ) are given by b S − n,k ( ℓ, ℓ ′ ) = ˆ D n ( k + ℓ ) δ ℓ,ℓ ′ + u n ( k + ℓ ) q ˆ Q n ( k ) u n ( k + ℓ ′ ) q c δS k ( ℓ, ℓ ′ ) = − X ℓ ′′ ∈ ˆ B n b S ′ n ( k + ℓ ) (cid:2) u n ( k + ℓ ) q ˆ Q n ( k ) u n ( k + ℓ ′′ ) q − a n e − ∆ n ( k + ℓ ) δ ℓ,ℓ ′′ (cid:3) b S n,k ( ℓ ′′ , ℓ ′ )(5.6)where u n and ˆ Q n are given in parts (b) and (e) of Remark 2.1. Lemma 5.4.
There are constants m > and Γ , such that the following hold forall L > Γ .(a) b S ′ n ( p ) is analytic in | Im p | < m and obeys (cid:12)(cid:12) b S ′ n ( p ) (cid:12)(cid:12) ≤ Γ | p | + | p | and (cid:12)(cid:12) ∂ ∂p b S ′ n ( p ) (cid:12)(cid:12) , (cid:12)(cid:12) ∂ ∂ p ν b S ′ n ( p ) (cid:12)(cid:12) ≤ Γ (1+ | p | + | p | ) for ≤ ν ≤ there.(b) For all ℓ, ℓ ′ ∈ ˆ B n , b S n,k ( ℓ, ℓ ′ ) is analytic in | Im k | < m and obeys (cid:12)(cid:12) b S n,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ Γ | ℓ | +Σ ν =1 | ℓ ν | n δ ℓ,ℓ ′ + | ℓ ′ | +Σ ν =1 | ℓ ′ ν | Q ν =0 1( | ℓ ν | +1) q Q ν =0 1( | ℓ ′ ν | +1) q o there.(c) For all ℓ, ℓ ′ ∈ ˆ B n , c δS k ( ℓ, ℓ ′ ) is analytic in | Im k | < m and obeys (cid:12)(cid:12)c δS k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ Γ exp (cid:8) − Σ ν =0 | ℓ ν | (cid:9) δ ℓ,ℓ ′ + Γ | ℓ | +Σ ν =1 | ℓ ν | n Q ν =0 1( | ℓ ν | +1) q Q ν =0 1( | ℓ ′ ν | +1) q o | ℓ ′ | +Σ ν =1 | ℓ ′ ν | there. Recall Convention 1.2. d) For all u, u ′ ∈ X n , (cid:12)(cid:12) S n ( u, u ′ ) − S ′ n ( u, u ′ ) (cid:12)(cid:12) ≤ Γ e − m | u − u ′ | (cid:12)(cid:12) S ′ n ( u, u ′ ) (cid:12)(cid:12) ≤ Γ min n e − m | u − u ′| | u − u ′ | + | u − u ′ | , L n o Proof. (a) Obviously b S ′ n ( p ) − is entire. For real p (cid:12)(cid:12) b S ′ n ( p ) − (cid:12)(cid:12) ≥ a n exp {− ∆ n ( p ) } + const (cid:8) | p | + | p | (cid:9) ≥ const (cid:8) | p | + | p | (cid:9) (5.7)by Remark 5.3, Lemma 3.2.a, and the fact that ∆ n ( p ) , Re ˆ D n ( p ) ≥ p . Thebound on (cid:12)(cid:12) ∂∂p ν ˆ D n ( p ) (cid:12)(cid:12) of Lemma 3.2.c and Remark 5.3 shows that (5.7) is valid forall | Im p | < m , if m is chosen sufficiently small.We now bound the derivatives. For any 0 ≤ ν ≤ ℓ ∈ N , ∂ ℓ ∂p ℓν c S ′ n ( p ) is a finitelinear combination of terms of the form c S ′ n ( p ) j j Y i =1 ∂ ℓi ∂p ℓiν (cid:2) ˆ D n ( p ) + a n exp {− ∆ n ( p ) } (cid:3) with each ℓ i ≥ P ji =1 ℓ i = ℓ . By Remark 5.3, all derivatives of exp {− ∆ n ( p ) } oforder ℓ i ∈ N are bounded by const ℓi [1+ | p | + | p | ] ℓi . Hence, by Lemma 3.2.c, (cid:12)(cid:12) ∂ ℓi ∂p ℓiν (cid:2) ˆ D n ( p ) + a n exp {− ∆ n ( p ) } (cid:3)(cid:12)(cid:12) ≤ const ( | p | + | p | ] ℓi − if ν = 0, ℓ i = 1 , | p | + | p | ] ℓi/ − if ν ≥
1, 1 ≤ ℓ i ≤ (cid:12)(cid:12) b S ′ n ( p ) (cid:12)(cid:12) ≤ const1+ | p | + | p | , (cid:12)(cid:12)(cid:12)c S ′ n ( p ) j j Y i =1 ∂ ℓi ∂p ℓiν (cid:2) ˆ D n ( p ) + a n exp {− ∆ n ( p ) } (cid:3)(cid:12)(cid:12)(cid:12) ≤ const1+ | p | + | p | j Y i =1 ( | p | + | p | ] ℓi if ν = 0, ℓ i = 1 , | p | + | p | ] ℓi/ if ν ≥
1, 1 ≤ ℓ i ≤ c ′′ > | k | ≥ c ′′ and | Im k | < m , analyticity and thebound (cid:12)(cid:12) b S n,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ Γ ′ | ℓ | +Σ ν =1 | ℓ ν | δ ℓ,ℓ ′ + Γ ′ | ℓ | +Σ ν =1 | ℓ ν | Q ν =0 1( | ℓ ν | +1) q Q ν =0 1( | ℓ ′ ν | +1) q | ℓ ′ | +Σ ν =1 | ℓ ′ ν | ′ depending on c ′′ ) which follows from the representation S n = D − n − D − n Q ∗ n ∆ ( n ) Q n D − n (see [6, Remark 10.b]) and Lemmas 3.2.d, 2.2.a and 4.2.c.For | k | < c ′ , with c ′ to be shortly chosen sufficiently small, and | Im k | < m weuse the representationˆ S − n,k ( ℓ, ℓ ′ ) = ˆ D n ( k + ℓ ) δ ℓ,ℓ ′ + u n ( k + ℓ ) q ˆ Q n ( k ) u n ( k + ℓ ′ ) q = D ℓ,ℓ ′ + B ℓ,ℓ ′ (5.8)with D ℓ,ℓ ′ = ˆ D n ( k + ℓ ) δ ℓ,ℓ ′ + ( a n if ℓ, ℓ ′ = 00 otherwise B ℓ,ℓ ′ = ( ˆ Q n ( k ) u n ( k ) q − a n if ℓ = ℓ ′ = 0 u n ( k + ℓ ) q ˆ Q n ( k ) u n ( k + ℓ ′ ) q otherwiseBy parts (c) and (d) of Lemma 3.2, assuming that | k | < c ′ with c ′ small enough, D is invertible and the inverse is a diagonal matrix with every diagonal matrix elementobeying (cid:12)(cid:12) D − ℓ,ℓ (cid:12)(cid:12) ≤ Γ ′′ | ℓ | +Σ ν =1 | ℓ ν | for some Γ ′′ which is independent of c ′ . By parts (b) and (c) of Lemma 2.2 andparts (a) and (b) of Proposition 2.4, (cid:12)(cid:12) B ℓ,ℓ ′ (cid:12)(cid:12) ≤ const a, q | k | Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q Q ν =0 (cid:0) | ℓ ′ ν | + π (cid:1) q So if | k | < c ′ with c ′ small enough, D + B is invertible with the inverse given bythe Neumann expansion D − + P ∞ p =1 ( − p D − (cid:0) BD − (cid:1) p . Since D and B are bothanalytic on | Im k | < X ℓ ∈ π Z const a, q | k | (cid:16) Q ν =0 24 | ℓ ν | + π (cid:17) q Γ ′′ | ℓ | +Σ ν =1 | ℓ ν | (cid:16) Q ν =0 24 | ℓ ν | + π (cid:17) q < if c ′ is small enough, we again get the desired analyticity and bound on (cid:12)(cid:12) b S n,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) .(c) Just apply Remark 5.3 and parts (a) and (b) of this lemma, Lemma 2.2.a andProposition 2.4.a and the fact that X ℓ ∈ π Z (cid:16) Q ν =0 1 | ℓ ν | + π (cid:17) q | ℓ | +Σ ν =1 | ℓ ν | (cid:16) Q ν =0 1 | ℓ ν | + π (cid:17) q
43s bounded uniformly in n and L to (5.6).(d) The bound on (cid:12)(cid:12) S n ( u, u ′ ) − S ′ n ( u, u ′ ) (cid:12)(cid:12) follows from part (c) by [5, Lemma 12.b]with the replacements (5.4). The bound on (cid:12)(cid:12) S ′ n ( u, u ′ ) (cid:12)(cid:12) follows from part (a), notingin particular that Γ (1+ | p | + | p | ) ∈ L ( ˆ Z fin ), and | u ν − u ′ ν | j S ′ n ( u, u ′ ) = Z ˆ Z fin ∂ j b S ′ n ∂p jν ( p ) e − ip · ( u − u ′ ) d p (2 π ) and (cid:12)(cid:12) S ′ n ( u, u ′ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z ˆ Z fin b S ′ n ( p ) e − ip · ( u − u ′ ) d p (2 π ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Γ Z ˆ Z fin d p (2 π ) = Γ L n We now prove the analog of Lemma 5.4 for the operators S (+) n,ν and S ( − ) n,ν of (5.3).As in (5.5)–(5.6), we decompose S (+) n,ν = S ′ n ∗ + δS (+) ν S ( − ) n,ν = S ′ n + δS ( − ) ν with δS (+) ν = S (+) n,ν − S ′ n ∗ = − S ′ n ∗ (cid:2) Q (+) n,ν Q n Q ( − ) n,ν − a n exp {− ∆ n } (cid:3) S (+) n,ν δS ( − ) ν = S ( − ) n,ν − S ′ n = − S ′ n (cid:2) Q (+) n,ν Q n Q ( − ) n,ν − a n exp {− ∆ n } (cid:3) S ( − ) n,ν They have Fourier representations [ δS (+) ν,k ( ℓ, ℓ ′ ) = − X ℓ ′′ ∈ ˆ B n b S ′ n ( k + ℓ ) (cid:2) U (+) n,ν ( k, ℓ ) ˆ Q n ( k ) U ( − ) n,ν ( k, ℓ ′′ ) − a n e − ∆ n ( k + ℓ ) δ ℓ,ℓ ′′ (cid:3) b S (+) n,ν,k ( ℓ ′′ , ℓ ′ ) [ δS ( − ) ν,k ( ℓ, ℓ ′ ) = − X ℓ ′′ ∈ ˆ B n b S ′ n ( k + ℓ ) (cid:2) U (+) n,ν ( k, ℓ ) ˆ Q n ( k ) U ( − ) n,ν ( k, ℓ ′′ ) − a n e − ∆ n ( k + ℓ ) δ ℓ,ℓ ′′ (cid:3) b S ( − ) n,ν,k ( ℓ ′′ , ℓ ′ )(5.9)where, by (2.11), U (+) n,ν ( k, ℓ ) = ζ (+) n,ν ( k, ℓ ) u (+) n,ν ( k + ℓ ) u n ( k + ℓ ) q − U ( − ) n,ν ( k, ℓ ′′ ) = ζ ( − ) n,ν ( k, ℓ ′′ ) u ( − ) n,ν ( k + ℓ ′′ ) u n ( k + ℓ ′′ ) q − Lemma 5.5.
There are constants m > and Γ such that the following hold forall L > Γ . a) For all ℓ, ℓ ′ ∈ ˆ B n , b S ( ± ) n,ν,k ( ℓ, ℓ ′ ) is analytic in | Im k | < m and obeys (cid:12)(cid:12) b S ( ± ) n,ν,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ Γ | ℓ | +Σ ν =1 | ℓ ν | n δ ℓ,ℓ ′ + | ℓ ′ | +Σ ν =1 | ℓ ′ ν | Q ν =0 1( | ℓ ν | +1) q − Q ν =0 1( | ℓ ′ ν | +1) q o there.(b) For all ℓ, ℓ ′ ∈ ˆ B n , c δS k ( ℓ, ℓ ′ ) is analytic in | Im k | < m and obeys (cid:12)(cid:12) [ δS ( ± ) ν,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ Γ exp (cid:8) − Σ ν =0 | ℓ ν | (cid:9) δ ℓ,ℓ ′ + Γ | ℓ | +Σ ν =1 | ℓ ν | Q ν =0 1( | ℓ ν | +1) q − Q ν =0 1( | ℓ ′ ν | +1) q | ℓ ′ | +Σ ν =1 | ℓ ′ ν | there.(c) For all u, u ′ ∈ X n , (cid:12)(cid:12) S ( ± ) n,ν ( u, u ′ ) − S ′ n ( u, u ′ ) (cid:12)(cid:12) ≤ Γ e − m | u − u ′ | .Proof. (a) For any c ′′ > | k | ≥ c ′′ and | Im k | < m , analyticity andthe desired bound follows from the representations (apply [6, Remark 10.b] with R = Q ( − ) n,ν , R ∗ = Q (+) n,ν and use Remark 2.5 to give R D − R ∗ = Q − D − Q ∗− ) S (+) n,ν = D ∗ n − − D ∗ n − Q (+) n,ν ∆ ( n ) Q ( − ) n,ν D ∗ n − S ( − ) n,ν = D − n − D − n Q (+) n,ν ∆ ( n ) Q ( − ) n,ν D − n and Lemmas 3.2.d, 2.2.a, 2.6.b and 4.2.c.For | k | < c ′ , with c ′ to be shortly chosen sufficiently small, and | Im k | < m weuse the representation (cid:0) ˆ S ( − ) n,ν,k (cid:1) − ( ℓ, ℓ ′ ) = ˆ D n ( k + ℓ ) δ ℓ,ℓ ′ + U (+) n,ν ( k, ℓ ) ˆ Q n ( k ) U ( − ) n,ν ( k, ℓ ′ ) = D ℓ,ℓ ′ + B ℓ,ℓ ′ (5.10)with D ℓ,ℓ ′ = ˆ D n ( k + ℓ ) δ ℓ,ℓ ′ + ( a n if ℓ, ℓ ′ = 00 otherwise B ℓ,ℓ ′ = ( ˆ Q n ( k ) u n ( k ) q − a n if ℓ = ℓ ′ = 0 U (+) n,ν ( k, ℓ ) ˆ Q n ( k ) U ( − ) n,ν ( k, ℓ ′ ) otherwiseand the obvious analog for (cid:0) ˆ S (+) n,ν,k (cid:1) − — just replace ˆ D n ( k + ℓ ) with its complexconjugate. As in the proof of Lemma 5.4.b, D is invertible and the inverse is adiagonal matrix with every diagonal matrix element obeying (cid:12)(cid:12) D − ℓ,ℓ (cid:12)(cid:12) ≤ Γ ′ | ℓ | +Σ ν =1 | ℓ ν | ′ which is independent of c ′ . By parts (b) and (c) of Lemma 2.2, parts(a) and (b) of Proposition 2.4, and part (b) of Lemma 2.6, (cid:12)(cid:12) B ℓ,ℓ ′ (cid:12)(cid:12) ≤ const a, q | k | Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q − Q ν =0 (cid:0) | ℓ ′ ν | + π (cid:1) q So if | k | < c ′ with c ′ small enough, D + B is invertible with the inverse given bythe Neumann expansion D − + P ∞ p =1 ( − p D − (cid:0) BD − (cid:1) p . As q >
1, the desiredanalyticity and the desired bound on (cid:12)(cid:12) b S ( ± ) n,ν,k ( ℓ, ℓ ′ ) (cid:12)(cid:12) follow as in the proof of Lemma5.4.b(b) is proven just as Lemma 5.4.c.(c) follows from part (b) by [5, Lemma 12.b]. Proof of Proposition 5.1.
Set m = min { m , m } . As | u | + | u | is locally integrablein R , the pointwise bounds on (cid:12)(cid:12) S n ( u, u ′ ) − S ′ n ( u, u ′ ) (cid:12)(cid:12) and (cid:12)(cid:12) S ′ n ( u, u ′ ) (cid:12)(cid:12) , given in Lemma5.4.d, and on (cid:12)(cid:12) S ( ± ) n,ν ( u, u ′ ) − S ′ n ( u, u ′ ) (cid:12)(cid:12) , given in Lemma 5.5.c, imply k S n k m , k S ( ± ) n,ν k m ≤ ˜Γ with ˜Γ a constant, depending only on m , times max { Γ , Γ } . Setting µ up = and Γ to be the maximum of 2˜Γ (for k S n ( µ ) k m and k S ( ± ) n,ν ( µ ) k m ) and 2˜Γ (for k S n ( µ ) − S n k m and k S ( ± ) n,ν ( µ ) − S ( ± ) n,ν k m ) a Neumann expansion gives the specifiedbounds.We now formulate and prove two more technical lemmas that will be used else-where. Lemma 5.6.
There are constants m > and Γ such that, for all L > Γ , (cid:12)(cid:12)(cid:0) S n Q ∗ n (cid:1) ( y, x ) (cid:12)(cid:12) ≤ Γ e − m | x − y | k S n Q ∗ n k m ≤ Γ Proof.
From the definitions of S n , in (5.3), and ∆ ( n ) , at the beginning of §
4, one seesdirectly that S ( ∗ ) n Q ∗ n = D ( ∗ ) n − Q ∗ n ∆ ( n )( ∗ ) Q − n : L ( X crs ) → L ( X fin ) (5.11)The Fourier transform of the kernel, b ( y, x ), of the operator S n Q ∗ n isˆ b k ( ℓ ) = ˆ D − n (k + ℓ ) u n (k + ℓ ) q ˆ∆ ( n ) (k) Q n (k)
46y Lemma 4.2.a,c,f, Remark 3.1.b and Lemma 3.2.d, Proposition 2.4.a, Remark2.1.d and Lemma 2.2.a, ˆ b k ( ℓ ) is analytic in | Im k | < m and (cid:12)(cid:12) ˆ b k ( ℓ ) (cid:12)(cid:12) ≤ a Γ | k + ℓ | + P ν =1 | k ν + ℓ ν | Y ν =0 (cid:16) | ℓ ν | + π (cid:17) q The bound is uniform in n and L and is summable in ℓ , so the claims follow from [5,Lemma 12.c]. Lemma 5.7.
There are constants m > and Γ such that, for all L > Γ , theoperators (cid:0) S n ( µ ) Q ∗ n Q n (cid:1) ∗ (cid:0) S n ( µ ) Q ∗ n Q n (cid:1) , (cid:0) S n ( µ ) ∗ Q ∗ n Q n (cid:1) ∗ (cid:0) S n ( µ ) ∗ Q ∗ n Q n (cid:1)(cid:0) S ( ± ) n,ν ( µ ) Q (+) n,ν Q n (cid:1) ∗ (cid:0) S ( ± ) n,ν ( µ ) Q (+) n,ν Q n (cid:1) all have bounded inverses. The k · k m norms of the inverses are all bounded by Γ .Proof. We first consider the case that µ = 0. By (5.11), the operator S n Q ∗ n Q n = D − n Q ∗ n ∆ ( n ) : L ( X crs ) → L ( X fin )has Fourier transform ˜ b k ( ℓ ) = ˆ D − n ( k + ℓ ) u n ( k + ℓ ) q ˆ∆ ( n ) ( k )The operator ( S n Q ∗ n Q n ) ∗ ( S n Q ∗ n Q n ) maps L ( X crs ) to L ( X crs ) and has Fourier trans-form X ℓ ∈ ˆ B ˜ b − k ( − ℓ )˜ b k ( ℓ ) = X ℓ ∈ ˆ B ˆ D − n ( − k − ℓ ) ˆ D − n ( k + ℓ ) u n ( k + ℓ ) q ˆ∆ ( n ) ( − k ) ˆ∆ ( n ) ( k )For k real, X ℓ ∈ ˆ B ˜ b − k ( − ℓ )˜ b k ( ℓ ) = X ℓ ∈ ˆ B (cid:12)(cid:12) ˆ D − n ( k + ℓ ) u n ( k + ℓ ) q ˆ∆ ( n ) ( k ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) ˆ D − n ( k ) u n ( k ) q ˆ∆ ( n ) ( k ) (cid:12)(cid:12) ≥ γ inf | k ν |≤ π | u n ( k ) | q ≥ γ (cid:0) π (cid:1) q by Lemmas 4.2.e and 2.2.f. To show that half this the lower bound extends into astrip along the real axis that has width independent of n and L , we observe that47 all first order derivatives of u n ( k + ℓ ) are uniformly bounded by 2 Q ν =0 24 | ℓ ν | + π onsuch a strip by the Cauchy integral formula and Remark 2.1.d and Lemma 2.2.aand • u n ( k + ℓ ) itself is uniformly bounded by Q ν =0 24 | ℓ ν | + π on such a strip by Lemma2.2.a and • all first order derivatives of ˆ∆ ( n ) ( k ) are uniformly bounded on such a strip byLemma 4.2.c and • for ℓ = 0, all first order derivatives of ˆ D − n ( k + ℓ ) are uniformly bounded on sucha strip by parts (c) and (d) of Lemma 3.2 and • for ℓ = 0, all first order derivatives of ˆ D − n ( k ) ˆ∆ ( n ) ( k ) are uniformly bounded onsuch a strip by Lemma 4.2.f.The operator S ∗ n Q ∗ n Q n = D − n ∗ Q ∗ n ∆ ( n ) ∗ has Fourier transform ˜ b − k ( − ℓ ). So( S n Q ∗ n Q n ) ∗ ( S n Q ∗ n Q n ) = ( S ∗ n Q ∗ n Q n ) ∗ ( S ∗ n Q ∗ n Q n )The operators S ( − ) n,ν Q (+) n,ν Q n = D − n Q (+) n,ν ∆ ( n ) and S (+) n,ν Q (+) n,ν Q n = D − n ∗ Q (+) n,ν ∆ ( n ) ∗ map L ( X crs ) to L ( X fin ) and have Fourier transforms˜ c k ( ℓ ) = ˆ D − n ( ± k ± ℓ ) ζ (+) n,ν ( k, ℓ ) u (+) n,ν ( k + ℓ ) u n ( k + ℓ ) q − ˆ∆ ( n ) ( ± k )So the operators ( S ( ± ) n,ν Q (+) n,ν Q n ) ∗ ( S ( ± ) n,ν Q (+) n,ν Q n ) both map L ( X crs ) to L ( X crs ) and haveFourier transform X ℓ ∈ ˆ B ˜ c − k ( − ℓ )˜ c k ( ℓ )= X ℓ ∈ ˆ B ˆ D − n ( − k − ℓ ) ˆ D − n ( k + ℓ ) u (+) n,ν ( k + ℓ ) u n ( k + ℓ ) q − ˆ∆ ( n ) ( − k ) ˆ∆ ( n ) ( k )This is bounded just as P ℓ ∈ ˆ B ˜ b − k ( − ℓ )˜ b k ( ℓ ) was. The specified bounds, in the specialcase that µ = 0 follow.By Proposition 5.1, a Neumann expansion gives the desired bounds when µ isnonzero. 48 The Degree One Part of the Critical Field
In [9, Proposition 5.1 and (5.3)] we derive an expansion for the critical fields of theform ψ ( ∗ ) n ( θ ∗ , θ, µ, V ) = aL C ( n ) ( µ ) ( ∗ ) Q ∗ θ ( ∗ ) + ψ ( ≥ ∗ ) n ( θ ∗ , θ, µ, V )with the C ( n ) ( µ ) of [7, Proposition 1.15] and with ψ ( ≥ ∗ ) n being of degree at least 3in θ ( ∗ ) . In this section we derive bounds on a scaled version of C ( n ) ( µ ) ( ∗ ) Q ∗ θ ( ∗ ) andsome related operators. To do so we use the representation aL C ( n ) ( µ ) ( ∗ ) Q ∗ = (cid:0) aL Q ∗ Q + Q n (cid:1) − (cid:8) aL Q ∗ + Q n Q n ˇ S n +1 ( µ ) ( ∗ ) ˇ Q ∗ n +1 ˇ Q n +1 (cid:9) (6.1)of [9, (5.2)]. Here, as in [9, Lemma 2.4 and (5.1)],ˇ Q n +1 = L ∗ Q n +1 L − ∗ = QQ n : H n → H ( n +1) − ˇ Q − n +1 = L L ∗ Q − n +1 L − ∗ = aL −
1l + Q Q − n Q ∗ : H ( n +1) − → H ( n +1) − ˇ S n +1 ( µ ) = L L ∗ S n +1 ( L µ ) L − ∗ = (cid:8) D n − µ + ˇ Q ∗ n +1 ˇ Q n +1 ˇ Q n +1 (cid:9) − : H n → H n (6.2)These operators are all translation invariant with respect to X ( n +1) − . Asˇ S n +1 ( µ ) = ˇ S n +1 + µ ˇ S n +1 ˇ S n +1 ( µ ) with ˇ S n +1 = ˇ S n +1 (0)we have aL C ( n ) ( µ ) ( ∗ ) Q ∗ = aL C ( n )( ∗ ) Q ∗ + µA ψ,φ ˇ S ( ∗ ) n +1 ˇ S n +1 ( µ ) ( ∗ ) ˇ Q ∗ n +1 ˇ Q n +1 (6.3)where A ψ,φ = ( aL − Q ∗ Q + Q n ) − Q n Q n : H n → H ( n )0 (6.4)The operator A ψ,φ is also used in the course of bounding ψ ( ≥ ∗ ) n in [9, Proposition 5.1].The main results of this section are Proposition 6.1.
There are constants m > and Γ , Γ such that the followingholds, for each L > Γ and each µ obeying | L µ | ≤ µ up .(a) (cid:13)(cid:13) L − ∗ A ψ,φ L ∗ (cid:13)(cid:13) m =1 ≤ Γ and (cid:13)(cid:13) L − ∗ aL C ( n ) ( µ ) ( ∗ ) Q ∗ L ∗ (cid:13)(cid:13) m ≤ Γ Recall Convention 1.2. b) Let ≤ ν ≤ . There are operators A ψ,φ,ν and A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) such that ∂ ν A ψ,φ = A ψ,φ,ν ∂ ν ∂ ν aL C ( n ) ( µ ) ( ∗ ) Q ∗ = A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) ∂ ν and k L − ∗ A ψ,φ,ν L ∗ k m =1 ≤ Γ (cid:13)(cid:13) L − ∗ A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) L ∗ (cid:13)(cid:13) m ≤ Γ This proposition is proven at the end of this section, after Lemma 6.6. In this proofwe write aL C ( n )( ∗ ) Q ∗ = A ψ ( ∗ ) ,θ ( ∗ ) so that A ψ ( ∗ ) ,θ ( ∗ ) = ( aL − Q ∗ Q + Q n ) − (cid:8) aL − Q ∗ + Q n Q n ˇ S ( ∗ ) n +1 ˇ Q ∗ n +1 ˇ Q n +1 (cid:9) : H ( n +1) − → H ( n )0 Remark 6.2. (a) A ψ ( ∗ ) θ ( ∗ ) = Q − n Q ∗ ˇ Q n +1 + A ψ,φ D − n ( ∗ ) ˇ Q ∗ n +1 ˇ∆ ( n +1)( ∗ ) withˇ∆ ( n +1)( ∗ ) = L L ∗ ∆ ( n +1)( ∗ ) L − ∗ = (cid:8)
1l + ˇ Q n +1 ˇ Q n +1 D − n ( ∗ ) ˇ Q ∗ n +1 (cid:9) − ˇ Q n +1 being a fully translation invariant operator on H ( n +1) − .(b) Let 0 ≤ ν ≤
3. We have ∂ ν A ψ,φ = A ψ,φ,ν ∂ ν and ∂ ν A ψ ( ∗ ) θ ( ∗ ) ( µ ) = A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) ∂ ν where A ψ,φ,ν = (cid:2) − Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν (cid:3) Q − n,ν A ψ ( ∗ ) θ ( ∗ ) ν = Q − n Q (+)+ ,ν ˇ Q n +1 + A ψ,φ,ν L ∗ D − ∗ ) n +1 Q (+) n +1 ,ν ∆ ( n +1)( ∗ ) L − ∗ A ψ ∗ θ ∗ ν ( µ ) = A ψ ∗ θ ∗ ν + L µA ψ,φ,ν L ∗ S (+) n +1 ,ν S (+) n +1 ,ν ( L µ ) Q (+) n +1 ,ν Q n +1 L − ∗ A ψθν ( µ ) = A ψθν + L µA ψ,φ,ν L ∗ S ( − ) n +1 ,ν S ( − ) n +1 ,ν ( L µ ) Q (+) n +1 ,ν Q n +1 L − ∗ Proof. (a) First observe that, by (5.11), Q n ˇ S ( ∗ ) n +1 ˇ Q ∗ n +1 ˇ Q n +1 = Q n L ∗ D ( ∗ ) n +1 − Q ∗ n +1 ∆ ( n +1)( ∗ ) L − ∗ = ( Q n D − n ( ∗ ) Q ∗ n ) Q ∗ ˇ∆ ( n +1)( ∗ ) (6.5)Using (6.5), the operator A ψ ( ∗ ) θ ( ∗ ) = ( aL − Q ∗ Q + Q n ) − (cid:8) aL − Q ∗ + Q n Q n D − n ( ∗ ) ˇ Q ∗ n +1 ˇ∆ ( n +1)( ∗ ) (cid:9) = Q − n ( Q ∗ Q Q − n + aL − − Q ∗ + A ψ,φ D − n ( ∗ ) ˇ Q ∗ n +1 ˇ∆ ( n +1)( ∗ ) = Q − n Q ∗ ( Q Q − n Q ∗ + aL − − + A ψ,φ D − n ( ∗ ) ˇ Q ∗ n +1 ˇ∆ ( n +1)( ∗ ) = Q − n Q ∗ ˇ Q n +1 + A ψ,φ D − n ( ∗ ) ˇ Q ∗ n +1 ˇ∆ ( n +1)( ∗ ) ∂ ν A ψ,φ = ∂ ν (1l + aL − Q − n Q ∗ Q ) − Q n = ∂ ν Q n − ∂ ν aL − Q − n Q ∗ (1l + aL − Q Q − n Q ∗ ) − QQ n = Q − n,ν ∂ ν − aL − Q − n Q (+)+ ,ν (1l + aL − Q Q − n Q ∗ ) − Q ( − )+ ,ν Q ( − ) n,ν ∂ ν = (cid:2) − Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν (cid:3) Q − n,ν ∂ ν Therefore by part (a), (3.1), (6.2) and Remark 2.5, ∂ ν A ψ ( ∗ ) θ ( ∗ ) = ∂ ν (cid:2) Q − n Q ∗ ˇ Q n +1 + A ψ,φ L ∗ D − ∗ ) n +1 Q ∗ n +1 ∆ ( n +1)( ∗ ) L − ∗ (cid:3) = A ψ ( ∗ ) θ ( ∗ ) ν ∂ ν since ∂ ν L ∗ = L ν L ∗ ∂ ν by [7, Remark 2.2.a,b]. To get ∂ ν A ψ ( ∗ ) θ ( ∗ ) ( µ ) = A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) ∂ ν when µ = 0, write, using (6.2), A ψ ( ∗ ) θ ( ∗ ) ( µ ) = A ψ ( ∗ ) θ ( ∗ ) + µA ψ,φ ˇ S ( ∗ ) n +1 ˇ S n +1 ( µ ) ( ∗ ) ˇ Q ∗ n +1 ˇ Q n +1 = A ψ ( ∗ ) θ ( ∗ ) + L µA ψ,φ L ∗ S ( ∗ ) n +1 S n +1 ( L µ ) ( ∗ ) Q ∗ n +1 Q n +1 L − ∗ and use (5.1), Remark 2.5 and the fact that Q n +1 is fully translation invariant.The operators of principal interest, A ψ ∗ ,θ ∗ and A ψ,θ , act from L ( X crs ) = H ( n +1) − to L ( X fin ) = H ( n )0 with X fin = X ( n )0 X crs = X ( n +1) − B = B + We now give a bunch of Fourier transforms (in the sense of [5, (7) and (8)] – but weshall suppress the ˆ from the notation). All of the operators above are periodized inthe sense of [5, Definition 2]. As before we denote • momenta dual to the L –lattice L Z × L Z by k ∈ (cid:0) R / πL Z (cid:1) × (cid:0) R / πL Z (cid:1) , • momenta dual to the unit lattice Z × Z by k ∈ (cid:0) R / π Z (cid:1) × (cid:0) R / π Z (cid:1) anddecompose k = k + ℓ or k = k + ℓ ′ with k in a fundamental cell for (cid:0) R / πL Z (cid:1) × (cid:0) R / πL Z (cid:1) and ℓ, ℓ ′ ∈ (cid:0) πL Z / π Z (cid:1) × (cid:0) πL Z / π Z (cid:1) = ˆ B + and • momenta dual to the ε j –lattice ε j Z × ε j Z by p j ∈ (cid:0) R / πε j Z (cid:1) × (cid:0) R / πε j Z (cid:1) anddecompose p j = k + ℓ j with k in a fundamental cell for (cid:0) R / π Z (cid:1) × (cid:0) R / π Z (cid:1) and ℓ j ∈ (cid:0) π Z / πε j Z (cid:1) × (cid:0) π Z / πε j Z (cid:1) = ˆ B j . Here 1 ≤ j ≤ n .51he Fourier transform of A ψ ( ∗ ) θ ( ∗ ) is (cid:0) A ψ ( ∗ ) θ ( ∗ ) (cid:1) k ( ℓ ) = X ℓ ′ ∈ ˆ B + (cid:0) aL Q ∗ Q + Q n (cid:1) − k ( ℓ, ℓ ′ ) n aL Q ∗ k ( ℓ ′ ) + Q n ( k + ℓ ′ ) (cid:0) Q n D − n ( ∗ ) Q ∗ n (cid:1) ( k + ℓ ′ ) Q ∗ k ( ℓ ′ ) ˇ∆ ( n +1)( ∗ ) ( k ) o (6.6)where, by Remark 2.1, Remark 6.2.a and (6.2) (cid:0) aL − Q ∗ Q + Q n (cid:1) k ( ℓ, ℓ ′ ) = Q n ( k + ℓ ) δ ℓ,ℓ ′ + aL − u + ( k + ℓ ) q u + ( k + ℓ ′ ) q and Q ∗ k ( ℓ ) = u + ( k + ℓ ) q Q n ( k ) = a h n − X j =1 P ℓ j ∈ ˆ B j L j u j ( k + ℓ j ) q i − ( Q n D − ∗ ) n Q ∗ n )( k ) = X ℓ n ∈ ˆ B n u n ( k + ℓ n ) q D − ∗ ) n ( k + ℓ n )ˇ∆ ( n +1)( ∗ ) ( k ) = ˇ Q n +1 ( k )1 + ˇ Q n +1 ( k ) P ℓ n ∈ ˆ B n ℓ ∈ ˆ B + u n ( k + ℓ + ℓ n ) q u + ( k + ℓ ) q D − ∗ ) n ( k + ℓ + ℓ n )ˇ Q n +1 ( k ) = aL h P ℓ ∈ ˆ B + aL u + ( k + ℓ ) q Q n ( k + ℓ ) − i − Lemma 6.3.
Let | Im k ν | ≤ L ν for each ≤ ν ≤ . There is a constant Γ , dependingonly on q , such that the following hold for all L > Γ .(a) We have aL ≤ | ˇ Q n +1 ( k ) | ≤ aL and Re ˇ Q n +1 ( k ) ≥ a L .(b) We have (cid:12)(cid:12)(cid:12)(cid:0) aL − Q ∗ Q + Q n (cid:1) − k ( ℓ, ℓ ′ ) − Q n ( k + ℓ ) − δ ℓ,ℓ ′ (cid:12)(cid:12)(cid:12) ≤ aL Q ν =0 (cid:0) L ν | ℓ ν | + π (cid:1) q Q ν =0 (cid:0) L ν | ℓ ′ ν | + π (cid:1) q for all ℓ Q ≤ ν ≤ ℓν =0 L ν | k ν | if ℓ = 052 c) Let ≤ ν ≤ . Then (cid:12)(cid:12)(cid:2) Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν (cid:3) k ( ℓ, ℓ ′ ) (cid:12)(cid:12) ≤ e L (cid:0) π (cid:1) Q ν =0 (cid:0) L ν | ℓ ν | + π (cid:1) q − Q ν =0 (cid:0) L ν | ℓ ′ ν | + π (cid:1) q Proof. (a) is proven much as Proposition 2.4.a was.(b) The straight forward Neumann expansion gives (cid:12)(cid:12)(cid:12)(cid:0) aL Q ∗ Q + Q n (cid:1) − k ( ℓ, ℓ ′ ) − Q n ( k + ℓ ) − δ ℓ,ℓ ′ (cid:12)(cid:12)(cid:12) ≤ aL (cid:12)(cid:12)(cid:12) Q n ( k + ℓ ) − u + ( k + ℓ ) q u + ( k + ℓ ′ ) q Q n ( k + ℓ ′ ) − (cid:12)(cid:12)(cid:12) ∞ X j =0 (cid:20) X l ∈ ˆ B + aL (cid:12)(cid:12)(cid:12) u + ( k + l ) q Q n ( k + l ) (cid:12)(cid:12)(cid:12)(cid:21) j ≤ aL (cid:0) a (cid:1) (cid:12)(cid:12) u + ( k + ℓ ) q (cid:12)(cid:12) Q ν =0 (cid:0) L ν | ℓ ′ ν | + π (cid:1) q ∞ X j =0 (cid:2) ′ L (cid:3) j ≤ aL Q ν =0 (cid:0) L ν | ℓ ν | + π (cid:1) q Q ν =0 (cid:0) L ν | ℓ ′ ν | + π (cid:1) q ℓ Q ≤ ν ≤ ℓν =0 L ν | k ν | if ℓ = 0by part (a), Lemma 2.3.a,b and Proposition 2.4.a, with L satisfying the conditionsof part (a).(c) The specified bound follows from (2.12), part (a), Proposition 2.4.a and Lemmas2.6.b, 2.3.a. Lemma 6.4.
For all k = L − ( k ) ∈ (cid:0) R / πL Z (cid:1) × (cid:0) R / πL Z (cid:1) k ∈ (cid:0) R / π Z (cid:1) × (cid:0) R / π Z (cid:1) p = L − ( p ) ∈ (cid:0) R / πε n Z (cid:1) × (cid:0) R / πε n Z (cid:1) p ∈ (cid:0) R / πε n +1 Z (cid:1) × (cid:0) R / πε n +1 Z (cid:1) ℓ n +1 ∈ (cid:0) π Z / πε n +1 Z (cid:1) × (cid:0) π Z / πε n +1 Z (cid:1) we have(a) u + (cid:0) L − ( k ) (cid:1) = u ( k ) and u n (cid:0) L − ( k ) (cid:1) u + (cid:0) L − ( k ) (cid:1) = u n +1 ( k ) for all n ∈ N .(b) ˇ Q n +1 (cid:0) L − ( k ) (cid:1) = L Q n +1 ( k ) (c) D − n (cid:0) L − ( p ) (cid:1) = L D − n +1 ( p ) 53 d) ˇ∆ ( n +1)( ∗ ) (cid:0) L − ( k ) (cid:1) = L ∆ ( n +1)( ∗ ) ( k ) (e) ˇ S n +1 , L − ( k ) (cid:0) L − ( ℓ n +1 ) , L − ( ℓ ′ n +1 ) (cid:1) = L ˆ S n +1 ,k ( ℓ n +1 , ℓ ′ n +1 ) Proof.
These all follow from (6.2), Remark 6.2.a and • Q = L − ∗ Q L ∗ by (2.2) • D n +1 = L L − ∗ D n L ∗ by (3.1)and [5, Lemmas 15.b and 16.b]. Corollary 6.5.
There are constants m > and Γ such that, for all L > Γ , k L − ∗ ˇ S n +1 L ∗ k m ≤ L Γ k L − ∗ ˇ S n +1 ˇ Q ∗ n +1 L ∗ k m ≤ L Γ Proof.
Set m = min { m , m } . Just combine [5, Lemmas 15.b and 16.b] and parts(a) and (e) of Lemma 6.4 to yield k L − ∗ ˇ S n +1 L ∗ k m = L k S n +1 k m k L − ∗ ˇ S n +1 ˇ Q ∗ n +1 L ∗ k m = L k S n +1 Q n +1 k m and then apply Proposition 5.1 and Lemma 5.6. Lemma 6.6.
Assume that
L > Γ , the constant of Lemma 6.3. There are constants m > and Γ such that the following hold for all k ∈ C with | Im k | ≤ m .(a) (cid:12)(cid:12)(cid:12)(cid:0) L − ∗ A ψ ( ∗ ) θ ( ∗ ) L ∗ (cid:1) k ( ℓ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) A ψ ( ∗ ) θ ( ∗ ) (cid:1) L − ( k ) ( L − ( ℓ )) (cid:12)(cid:12)(cid:12) ≤ Γ
19 3 Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q (b) If ℓ = 0 , then (cid:12)(cid:12)(cid:12)(cid:0) L − ∗ A ψ ( ∗ ) θ ( ∗ ) L ∗ (cid:1) k ( ℓ ) (cid:12)(cid:12)(cid:12) ≤ Γ | k | Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q Proof.
By (6.6), Lemma 6.4 and Remark 2.1.e, (cid:0) A ψ ( ∗ ) θ ( ∗ ) (cid:1) L − ( k ) ( L − ( ℓ ))= X ℓ ′ ∈ ˆ B (cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , L − ( ℓ ′ )) n aL Q ∗ L − ( k ) ( L − ( ℓ ′ ))+ Q n ( L − ( k + ℓ ′ )) (cid:0) Q n D − ∗ ) n Q ∗ n (cid:1) ( L − ( k + ℓ ′ )) Q ∗ L − ( k ) ( L − ( ℓ ′ )) ˇ∆ ( n +1)( ∗ ) ( L − ( k )) o = X ℓ ′ ∈ ˆ B (cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , L − ( ℓ ′ )) B ( k, ℓ ′ )= Q n (cid:0) L − ( k + ℓ ) (cid:1) − B ( k, ℓ ) + X ℓ ′ ∈ ˆ B C ( k, ℓ , ℓ ′ ) B ( k, ℓ ′ )54here B ( k, ℓ ′ ) = u ( k + ℓ ′ ) q n aL + Q n (cid:0) L − ( k + ℓ ′ ) (cid:1) X ℓ n ∈ ˆ B n B ( k, ℓ ′ , ℓ n ) o B ( k, ℓ ′ , ℓ n ) = u n ( L − ( k + ℓ ′ ) + ℓ n ) q D − ∗ ) n +1 (cid:0) k + ℓ ′ + L ( ℓ n ) (cid:1) ∆ ( n +1)( ∗ ) ( k ) C ( k, ℓ , ℓ ′ ) = (cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , L − ( ℓ ′ )) − Q n (cid:0) L − ( k + ℓ ) (cid:1) − δ ℓ ,ℓ ′ (a) Choose m = min { , m , ¯ m ( π ) } . Then (cid:12)(cid:12)(cid:12)(cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , L − ( ℓ ′ )) (cid:12)(cid:12)(cid:12) ≤ a δ ℓ ,ℓ ′ + aL Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q (by Lemma 6.3.b, Proposition 2.4.a) (cid:12)(cid:12) u ( k + ℓ ′ ) q (cid:12)(cid:12) ≤ Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q (by Lemma 2.2.a) (cid:12)(cid:12) Q n ( L − ( k + ℓ ′ )) (cid:12)(cid:12) ≤ a (by Proposition 2.4.a) (cid:12)(cid:12) u n ( L − ( k + ℓ ′ ) + ℓ n ) q (cid:12)(cid:12) ≤ Q ν =0 (cid:0) | ℓ n,ν | + π (cid:1) q (by Lemma 2.2.a) (cid:12)(cid:12) D − ∗ ) n +1 (cid:0) k + ℓ ′ + L ( ℓ n ) (cid:1)(cid:12)(cid:12) ≤ γ π if ( ℓ ′ , ℓ n ) = (0 ,
0) (by Lemma 3.2.d) (cid:12)(cid:12) ∆ ( n +1)( ∗ ) ( k ) (cid:12)(cid:12) ≤ a (by Lemma 4.2.c) (cid:12)(cid:12) D − ∗ ) n +1 ( k ) ∆ ( n +1)( ∗ ) ( k ) (cid:12)(cid:12) ≤ Γ (by Lemma 4.2.f)So (cid:12)(cid:12)(cid:12)(cid:0) A ψ ( ∗ ) θ ( ∗ ) (cid:1) L − ( k ) ( L − ( ℓ )) (cid:12)(cid:12)(cid:12) ≤ X ℓ ′ ∈ ˆ B n a δ ℓ ,ℓ ′ + aL Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q o Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q n aL + a X ℓ n ∈ ˆ B n Q ν =0 (cid:0) | ℓ n | + π (cid:1) q max (cid:8) Γ , aγ π (cid:9)o ≤ const X ℓ ′ ∈ ˆ B n a δ ℓ ,ℓ ′ + aL Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q o Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q ≤ Γ
19 3 Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q (cid:12)(cid:12) u ( k + ℓ ′ ) q (cid:12)(cid:12) ≤ | k | Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q if ℓ ′ = 0 (by Lemma 2.2.b) (cid:12)(cid:12) u n ( L − ( k + ℓ ′ ) + ℓ n ) q (cid:12)(cid:12) ≤ | L − ( k + ℓ ′ ) | Q ν =0 (cid:0) | ℓ n | + π (cid:1) q if ℓ n = 0 (by Lemma 2.2.b)we have, if ℓ ′ = 0, (cid:12)(cid:12) B ( k, ℓ ′ ) (cid:12)(cid:12) ≤ | k | Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q n aL + a X ℓ n ∈ ˆ B n Q ν =0 (cid:0) | ℓ n | + π (cid:1) q max (cid:8) Γ , aγ π (cid:9)o ≤ const | k | Q ν =0 (cid:0) | ℓ ′ ,ν | + π (cid:1) q and (cid:12)(cid:12) B ( k, (cid:12)(cid:12) ≤ Q ν =0 (cid:0) π (cid:1) q n aL + a X ℓ n ∈ ˆ B n Q ν =0 (cid:0) | ℓ n | + π (cid:1) q max (cid:8) Γ , aγ π (cid:9)o ≤ constUsing these bounds, the first bound of part (a) and Lemma 6.3.b, and assuming that ℓ = 0, (cid:0) A ψ ( ∗ ) θ ( ∗ ) (cid:1) L − ( k ) ( L − ( ℓ )) = X ℓ ′ ∈ ˆ B (cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , L − ( ℓ ′ )) B ( k, ℓ ′ )= (cid:0) aL Q ∗ Q + Q n (cid:1) − L − ( k ) ( L − ( ℓ ) , B ( k,
0) + O (cid:16) | k | Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q (cid:17) = aL Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q Q ν =0 (cid:0) π (cid:1) q Q ≤ ν ≤ ℓ ,ν =0 | k ν | B ( k,
0) + O (cid:16) | k | Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q (cid:17) = O (cid:16) | k | Q ν =0 (cid:0) | ℓ ,ν | + π (cid:1) q (cid:17) Proof of Proposition 6.1.Bound on (cid:13)(cid:13) L − ∗ A ψ,φ L ∗ (cid:13)(cid:13) m =1 : By [5, Lemma 15.b] and Lemma 6.3.b, if | Im k ν ′ | ≤ ≤ ν ′ ≤ (cid:12)(cid:12)(cid:2) L − ∗ (cid:8)(cid:0) aL − Q ∗ Q + Q n (cid:1) − − Q − n (cid:9) L ∗ (cid:3) k ( ℓ, ℓ ′ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:2)(cid:0) aL − Q ∗ Q + Q n (cid:1) − − Q − n (cid:3) L − k ( L − ℓ, L − ℓ ′ ) (cid:12)(cid:12) ≤ aL Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q Q ν =0 (cid:0) | ℓ ′ ν | + π (cid:1) q So, by [5, Lemma 12.b], (cid:13)(cid:13) L − ∗ (cid:8)(cid:0) aL − Q ∗ Q + Q n (cid:1) − − Q − n (cid:9) L ∗ k m =1 ≤ const q By Proposition 2.4.a, Lemma 2.2.a and [5, Lemmas 12.b,c], k Q n k m =1 , k Q − n k m =1 , k Q n k m =1 ≤ const q too. Now just apply [5, Lemmas 15.c and 16.c]. Bound on (cid:13)(cid:13) L − ∗ A ψ,φ,ν L ∗ (cid:13)(cid:13) m =1 : By [5, Lemma 15.b] and Lemma 6.3.c, if | Im k ν ′ | ≤ ≤ ν ′ ≤ (cid:12)(cid:12)(cid:2) L − ∗ (cid:0) Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν (cid:1) L ∗ (cid:3) k ( ℓ, ℓ ′ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:2) Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν (cid:3) L − k ( L − ℓ, L − ℓ ′ ) (cid:12)(cid:12) ≤ e L (cid:0) π (cid:1) Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q − Q ν =0 (cid:0) | ℓ ′ ν | + π (cid:1) q As q >
2, [5, Lemma 12.b] yields (cid:13)(cid:13) L − ∗ Q − n Q (+)+ ,ν ˇ Q n +1 Q ( − )+ ,ν L ∗ k m =1 ≤ const q By Lemma 2.6.b and [5, Lemma 12.c], k Q ( − ) n,ν k m =1 ≤ const q too, since q >
2. Nowjust apply [5, Lemma 16.c].
Bound on (cid:13)(cid:13) L − ∗ A ψ ( ∗ ) θ ( ∗ ) L ∗ (cid:13)(cid:13) m : This follows from Lemma 6.6.a by [5, Lemma 12.c].
Bound on (cid:13)(cid:13) L − ∗ aL C ( n ) ( µ ) ( ∗ ) Q ∗ L ∗ (cid:13)(cid:13) m : By (6.3) L − ∗ aL C ( n ) ( µ ) ( ∗ ) Q ∗ L ∗ = L − ∗ A ψ ( ∗ ) θ ( ∗ ) L ∗ + L µ L − ∗ A ψ,φ L ∗ S ( ∗ ) n +1 S n +1 ( L µ ) ( ∗ ) Q ∗ n +1 Q n +1 Now just apply Proposition 5.1, Lemma 2.2 and Proposition 2.4.c.
Bound on (cid:13)(cid:13) L − ∗ A ψ ( ∗ ) θ ( ∗ ) ν L ∗ (cid:13)(cid:13) m : It suffices to bound • L − ∗ A ψ,φ,ν L ∗ as above, 57 bound L − ∗ Q − n Q (+)+ ,ν ˇ Q n +1 L ∗ using (cid:12)(cid:12)(cid:2) L − ∗ (cid:0) Q − n Q (+)+ ,ν ˇ Q n +1 (cid:1) L ∗ (cid:3) k ( ℓ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:2) Q − n Q (+)+ ,ν ˇ Q n +1 (cid:3) L − k ( L − ℓ ) (cid:12)(cid:12) ≤ e L (cid:0) π (cid:1) Q ν =0 (cid:0) | ℓ ν | + π (cid:1) q − (by [5, Lemma 16.b], (2.12), Proposition 2.4.a and Lemmas 2.6.b, 2.3.a, 6.3.a) and[5, Lemma 12.c], and • bound D − ∗ ) n +1 Q (+) n +1 ,ν ∆ ( n +1)( ∗ ) using (cid:12)(cid:12)(cid:0) ˆ D − ∗ ) n +1 Q (+) n +1 ,ν ∆ ( n +1)( ∗ ) (cid:1) k ( ℓ n +1 ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ˆ D − ∗ ) n +1 (cid:0) k + ℓ n +1 ) (cid:1) ζ (+) n +1 ,ν ( k, ℓ n +1 ) u (+) n +1 ,ν ( k + ℓ n +1 ) u n +1 ( k + ℓ n +1 ) q − ˆ∆ ( n +1)( ∗ ) ( k ) (cid:12)(cid:12) (by (2.11)) and (cid:12)(cid:12) ζ (+) n +1 ,ν ( k, ℓ n +1 ) u (+) n +1 ,ν ( k + ℓ n +1 ) (cid:12)(cid:12) ≤ e (cid:0) π (cid:1) (by Lemma 2.6.b) (cid:12)(cid:12) u n +1 ( k + ℓ n +1 ) q − (cid:12)(cid:12) ≤ Q ν =0 (cid:0) | ℓ n +1 | + π (cid:1) q − (by Lemma 2.2.a) (cid:12)(cid:12) ˆ D − ∗ ) n +1 (cid:0) k + ℓ n +1 (cid:1)(cid:12)(cid:12) ≤ γ π if ℓ n +1 = 0 (by Lemma 3.2.d) (cid:12)(cid:12) ˆ∆ ( n +1)( ∗ ) ( k ) (cid:12)(cid:12) ≤ a (by Lemma 4.2.c) (cid:12)(cid:12) ˆ D − ∗ ) n +1 ( k ) ˆ∆ ( n +1)( ∗ ) ( k ) (cid:12)(cid:12) ≤ Γ (by Lemma 4.2.f)and [5, Lemma 12.c]. Bound on (cid:13)(cid:13) L − ∗ A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) L ∗ (cid:13)(cid:13) m : This follows from the previous bounds of thisProposition, Remark 6.2.b, Proposition 5.1, Lemma 2.6.c and Proposition 2.4.c.58
Trigonometric Inequalities
Lemma A.1. (a) For x, y real with | x | ≤ π , (cid:12)(cid:12) sin( x + iy ) (cid:12)(cid:12) ≥ √ π | x + iy | (b) For x, y real with | y | ≤ , | sin( x + iy ) || x + iy | ≤ (cid:8) , | x + iy | (cid:9) (cid:12)(cid:12)(cid:12) Im sin( x + iy ) x + iy (cid:12)(cid:12)(cid:12) ≤ | y | min (cid:8) | x | , | x + iy | (cid:9) (c) For < ε ≤ and x, y real with | εx | ≤ π , | y | ≤ (cid:12)(cid:12)(cid:12)(cid:12) sin ( x + iy ) ε sin ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:26) , | x | (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) Im sin ( x + iy ) ε sin ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | y | min (cid:26) | x | , | x | (cid:27) (d) For x real with | x | ≤ π , π ≤ sin xx ≤ (e) For any complex number z obeying | z | ≤ , (cid:12)(cid:12) sin zz − (cid:12)(cid:12) ≤ | z | Proof.
By the standard trig identitysin( x + iy ) = sin( x ) cos( iy ) + cos( x ) sin( iy ) = sin( x ) cosh( y ) + i cos( x ) sinh( y )(a) For x, y real with | x | ≤ π (cid:12)(cid:12) Re sin( x + iy ) (cid:12)(cid:12) = (cid:12)(cid:12) sin( x ) cosh( y ) (cid:12)(cid:12) ≥ | sin( x ) | ≥ π | x | (cid:12)(cid:12) Re sin( x + iy ) (cid:12)(cid:12) = (cid:12)(cid:12) sin( x ) cosh( y ) (cid:12)(cid:12) ≥ (cid:12)(cid:12) sin( x ) sinh( y ) (cid:12)(cid:12) ≥ | sin( x ) | | y | (cid:12)(cid:12) Im sin( x + iy ) (cid:12)(cid:12) = (cid:12)(cid:12) cos( x ) sinh( y ) (cid:12)(cid:12) ≥ | cos( x ) | | y | so that (cid:12)(cid:12) sin( x + iy ) (cid:12)(cid:12) ≥ max (cid:8) π | x | , | y | (cid:9) ≥ √ π | x + iy | (b) For x, y real with | y | ≤ | sin( x + iy ) | = (cid:12)(cid:12) sin( x ) cosh( y ) + i cos( x ) sinh( y ) (cid:12)(cid:12) ≤ cosh(1) (cid:12)(cid:12) sin( x ) + i cos( x ) (cid:12)(cid:12) = cosh(1) 59nd, since (cid:12)(cid:12) sinh( y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) P ∞ n =0 y n +1 (2 n +1)! (cid:12)(cid:12)(cid:12) ≤ | y | P ∞ n =0 | y | n (2 n )! = | y | cosh( y ) ≤ cosh(1) | y | , | sin( x + iy ) | = (cid:12)(cid:12) sin( x ) cosh( y ) + i cos( x ) sinh( y ) (cid:12)(cid:12) ≤ cosh(1) (cid:12)(cid:12) | x | + i | y | (cid:12)(cid:12) = cosh(1) | x + iy | Thus | sin( x + iy ) || x + iy | ≤ cosh(1) min (cid:8) , | x + iy | (cid:9) ≤ (cid:8) , | x + iy | (cid:9) giving the first bound.For the second bound (cid:12)(cid:12)(cid:12) Im sin( x + iy ) x + iy (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) x Im sin( x + iy ) − y Re sin( x + iy ) x + y (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) x cos( x ) sinh( y ) − y sin( x ) cosh( y ) x + y (cid:12)(cid:12)(cid:12) Using cos x − − Z x dt sin t = − Z x dt Z t ds cos s sin x − x = Z x dt [cos t −
1] = − Z x dt Z t ds Z s du cos u cosh y − Z y dt sinh t = Z y dt Z t ds cosh s sinh y − y = Z y dt [cosh t −
1] = Z y dt Z t ds Z s du cosh u and cosh(1) <
2, we havecos x = 1 + α ( x ) x sin x = x + β ( x ) x cosh y = 1 + γ ( y ) y sinh y = y + δ ( y ) y with, for | y | ≤ | α ( x ) | , | β ( x ) | , | γ ( y ) | , | δ ( y ) | ≤
1. Consequently, (cid:12)(cid:12)(cid:12) Im sin( x + iy ) x + iy (cid:12)(cid:12)(cid:12) ≤ | xy | Alternatively, using | sin( x ) | ≤ | x | , | sinh( y ) | ≤ | y | and | cosh( y ) | ≤ (cid:12)(cid:12)(cid:12) Im sin( x + iy ) x + iy (cid:12)(cid:12)(cid:12) ≤ | xy | x + y ≤ | y | √ x + y (c) For 0 < ε ≤ x, y real and | εx | ≤ π , | y | ≤ (cid:12)(cid:12)(cid:12)(cid:12) sin ( x + iy ) ε sin ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin ( x + iy ) ( x + iy )sin ε ( x + iy ) ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π √ cosh(1) min (cid:26) , | x + iy | (cid:27) ≤ (cid:26) , | x | (cid:27) (cid:12)(cid:12)(cid:12)(cid:12) Im sin ( x + iy ) ε sin ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) Im sin ( x + iy ) ( x + iy ) Re sin ε ( x + iy ) ε ( x + iy ) − Re sin ( x + iy ) ( x + iy ) Im sin ε ( x + iy ) ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) sin ε ( x + iy ) ε ( x + iy ) (cid:12)(cid:12) ≤ | y | min (cid:8) | x | , | x + iy | (cid:9) + 4 min (cid:8) , | x | (cid:9)(cid:12)(cid:12) Im sin ε ( x + iy ) ε ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) sin ε ( x + iy ) ε ( x + iy ) (cid:12)(cid:12) ≤ | y | min (cid:8) | x | , | x + iy | (cid:9) + 4 min (cid:8) , | x | (cid:9) ε | y | min (cid:8) ε | x | , ε | x + iy | (cid:9) √ π ≤ π √ | y | min (cid:8)(cid:0) + 2 ε (cid:1) | x | , | x + iy | (cid:9) ≤ | y | min (cid:8) | x | , | x + iy | (cid:9) (d) and (e) are standard. 61 Lattice and Operator Summary
The following table gives, for most of the operators considered in this paper, • the definition of the operator • a reference to where in [7, 8], the operator is introduced and • the translation invariance properties of the operator.A later table will specify where, in this paper, bounds on the operators are proven.Operator Definition Tiwrt D = − e − h ∂ + (cid:2) − e − h (cid:3) : H → H § X D n = L n L − n ∗ D L n ∗ : H n → H n Def 1.5.a X n Q n = a (cid:0)
1l + P n − j =1 1 L j Q j Q ∗ j (cid:1) − : H ( n )0 → H ( n )0 Def 1.5.b X ( n )0 ∆ (0) = D : H → H (1.14) X ∆ ( n ) = (cid:0)
1l + Q n Q n D − n Q ∗ n (cid:1) − Q n , n ≥ H ( n )0 → H ( n )0 (1.14) X ( n )0 C ( n ) = (cid:0) aL Q ∗ Q + ∆ ( n ) (cid:1) − : H ( n )0 → H ( n )0 (1.15) X ( n +1) − S − n = D n + Q ∗ n Q n Q n : H n → H n Thm 1.13 X ( n )0 S n ( µ ) − = D n + Q ∗ n Q n Q n − µ : H n → H n Thm 1.13 X ( n )0 ˇ Q n +1 = (cid:0) L a
1l + Q Q − n Q ∗ (cid:1) − : H ( n +1) − → H ( n +1) − Lem 2.4.b X ( n +1) − ˇ S n +1 ( µ ) = (cid:8) D n + ˇ Q ∗ n +1 ˇ Q n +1 ˇ Q n +1 − µ (cid:9) − : H n → H n (5.1) X ( n +1) − A ψ,φ : H n → H ( n )0 Prop 5.1 X ( n +1) − The references in the above table are to [7, 8] and “Tiwrt” stands for “translationinvariant with respect to”.Operator Definition Tiwrt S n Q ∗ n = D − n Q ∗ n (cid:0)
1l + Q n Q n D − n Q ∗ n (cid:1) − : H ( n )0 → H n Lemma 5.6 X ( n )0 ˇ S n +1 = ˇ S n +1 (0) : H n → H n after (6.2) X ( n +1) − A ψ,φ : H n → H ( n )0 (6.4) X ( n +1) − A ψ,φ,ν : H n → H ( n )0 Remark 6.2 X ( n +1) − A ψ ( ∗ ) ,θ ( ∗ ) = aL C ( n )( ∗ ) Q ∗ : H ( n +1) − → H ( n )0 before Rmk 6.2 X ( n +1) − A ψ ∗ ,θ ∗ ,ν , A ψ,θ,ν , A ψ ∗ ,θ ∗ ,ν ( µ ) , A ψ,θ,ν ( µ ) : H ( n +1) − → H ( n )0 Remark 6.2 X ( n +1) − X n = (cid:0) ε n Z /ε n L tp Z (cid:1) × (cid:0) ε n Z /ε n L sp Z (cid:1) ˆ X n = (cid:0) πε n L tp Z / πε n Z (cid:1) × (cid:0) πε n L sp Z / πε n Z (cid:1) X ( n )0 = (cid:0) Z /ε n L tp Z (cid:1) × (cid:0) Z /ε n L sp Z (cid:1) ˆ X ( n )0 = (cid:0) πε n L tp Z / π Z (cid:1) × (cid:0) πε n L sp Z / π Z (cid:1) X ( n +1) − = (cid:0) L Z /ε n L tp Z (cid:1) × (cid:0) L Z /ε n L sp Z (cid:1) ˆ X ( n +1) − = (cid:0) πε n L tp Z / πL Z (cid:1) × (cid:0) πε n L sp Z / πL Z (cid:1) where ε n = L n . The “single period” lattices are B n = (cid:0) ε n Z / Z (cid:1) × (cid:0) ε n Z / Z (cid:1) ˆ B n = (cid:0) π Z / πε n Z (cid:1) × (cid:0) π Z / πε n Z (cid:1) B + = (cid:0) Z /L Z (cid:1) × (cid:0) Z /L Z (cid:1) ˆ B + = (cid:0) πL Z / π Z (cid:1) × (cid:0) πL Z / π Z (cid:1) The following table specifies where, in this paper, bounds on the various operatorsare proven. Operator Bound Q n Lemma 2.2.a Q n Proposition 2.4 Q ( ± ) n,ν Lemma 2.6 D n Lemma 3.2∆ ( n ) Lemma 4.2 C ( n ) Corollary 4.5 D ( n ) Corollary 4.5 S n ( µ ) , S n Proposition 5.1 S ( ± ) n,ν ( µ ) , S ( ± ) n,ν Proposition 5.1 A ψ,φ Proposition 6.1 A ψ,φ,ν Proposition 6.1 A ψ ( ∗ ) θ ( ∗ ) Proposition 6.1 A ψ ( ∗ ) θ ( ∗ ) ν ( µ ) Proposition 6.1ˇ S n +1 Corollary 6.563 eferences [1] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. A Functional IntegralRepresentation for Many Boson Systems. I: The Partition Function.
AnnalesHenri Poincar´e , 9:1229–1273, 2008.[2] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. A Functional IntegralRepresentation for Many Boson Systems. II: Correlation Functions.
AnnalesHenri Poincar´e , 9:1275–1307, 2008.[3] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. Power Series Represen-tations for Complex Bosonic Effective Actions. I. A Small Field RenormalizationGroup Step.
Journal of Mathematical Physics , 51:053305, 2010.[4] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. The Temporal UltravioletLimit for Complex Bosonic Many-body Models.