On the construction of composite Wannier functions
aa r X i v : . [ m a t h - ph ] M a r On the construction of composite Wannier functions
March 9, 2016
Horia D. Cornean , Ira Herbst , Gheorghe Nenciu Abstract
We give a constructive proof for the existence of an N -dimensional Bloch basis which isboth smooth (real analytic) and periodic with respect to its d -dimensional quasi-momenta,when 1 ≤ d ≤ N ≥
1. The constructed Bloch basis is conjugation symmetric whenthe underlying projection has this symmetry, hence the corresponding exponentially localizedcomposite Wannier functions are real. In the second part of the paper we show that byadding a weak, globally bounded but not necessarily constant magnetic field, the existence ofa localized basis is preserved.
Contents P ( k ) satisfying conjugationsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Construction of composite Wannier functions . . . . . . . . . . . . . . . . . . . . . 61.4 The non-zero magnetic field case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 CS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Some facts about matrices having CS or CS ′ symmetry . . . . . . . . . . . . . . . 122.6 The case d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 The case d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7.1 When N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7.2 When the eigenvalues of β never cross . . . . . . . . . . . . . . . . . . . . . 172.7.3 When a global Cayley transform exists . . . . . . . . . . . . . . . . . . . . . 182.7.4 When β ( · ) is real analytic and β (0) and β (1 /
2) have nondegenerate spectrum 192.7.5 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7.6 Proof of Proposition 2.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania Introduction and main results
The Wannier functions (as bases of localized functions spanning subspaces corresponding to energybands in periodic solids) have been playing a central role since their introduction in 1937 in bothqualitative and quantitative aspects of the one-electron theory of solid state physics. In particular,they are the key ingredient in obtaining effective tight-binding Hamiltonians, a topic related to thePeierls-Onsager substitution. After the seminal paper of Marzari and Vanderbilt [MV], they havebecome a powerful tool in ab initio computational studies of electronic properties of materials (see[N2], [MMYSV], [FMP] and references given there).The existence and construction of exponentially localized
Wannier functions (ELWF) has beenone of the fundamental problems in solid state physics. Let us remind the reader that prior tothe crucial observation made by Thouless [Th] that the very existence of ELWF implies that thequantum Hall current of the corresponding band vanishes, the fact that ELWF might not existwas largely overlooked by physicists.It turns out that due to several mathematical subtleties, the proof of exponential localizationis difficult and depends on d (the dimension of the configuration space) among other things.Accordingly, the progress in proving the existence of ELWF in all cases of physical interest wasslow. For example, the existence of ELWF for composite energy bands of time-reversal invariantHamiltonians in two and three dimensions has been only recently achieved [P], [BPCMM].The first rigorous result about the existence of ELWF in one dimension was obtained by Kohnin his 1959 classic paper [K]. Using a technique based on ordinary differential equations, Kohnconsidered simple bands in crystals with a center of inversion and obtained a complete solution:he constructed Wannier functions which were real, symmetric or antisymmetric with respect to anappropriate reflection and with an optimal exponential decay. Kohn’s method does not generalizeto higher dimensions and the next major step was made by des Cloizeaux. In a couple of basicpapers [dCl1], [dCl2] in 1964 which are still worth reading, he showed that the existence of ELWFfor composite bands in crystals of arbitrary dimension can be achieved in two steps:I . To each isolated energy band one associates a family P ( k ) , k ∈ R d , of rank N orthogonalprojections ( N = 1 for simple bands and 1 < N < ∞ for composite bands) which are Z d -periodic and jointly analytic in a complex tubular neighborhood I of R d . These projectionslive in a separable Hilbert space H .II . Show that the vector bundle B := { ( k , v ) : k ∈ T d , v ∈ Ran P ( k ) } is trivial (= thesubspace admits a continuous orthonormal basis). More precisely, one has to show there exist N vectors { Ξ j ( k ) } Nj =1 ⊂ H which are continuous, Z d -periodic and form an orthonormal basisof Ran( P ( k )) for all k ∈ R d . Step I. is quite general and it is based on the Bloch-Floquet-Gelfand transform and analyticperturbation theory. It works for all standard Schr¨odinger operators with periodic potentials,including their discrete and pseudo-differential variants (see e.g.[dCl1],[N2], [FMP], [dN-G] as wellas section 1.2 below).Step II. is a subtle and more demanding mathematical problem, due to the fact that vectorbundles can be non-trivial. A well known example is the tangent space to any even-dimensionalunit sphere: the existence of a smooth orthonormal basis would contradict the Hairy-Ball Theorem.des Cloizeaux realized that in proving step II the obstructions have topological origin and showedthat they are not present for simple bands ( N = 1) in the absence of a magnetic field or when theperiodic electric potential has a center of inversion. The restriction to crystals with a center ofinversion was removed in 1983 by Nenciu [N0] who proved by operator analytic methods that atthe abstract level, for arbitrary d and N = 1 the obstructions to the triviality of the vector bundle B are absent if P ( k ) satisfies θP ( k ) θ = P ( − k ) (1.1)2or some antiunitary involution θ . The proof is constructive and leads to optimally localized, realWannier functions [N2]. A simpler proof was later given by Helffer and Sj¨ostrand [HS]; however,it seems that it does not provide the optimal exponential decay for the Wannier functions.For a Schr¨odinger operator with a real symbol (i.e. without a magnetic potential), (1.1)is insured by the fact that it commutes with complex conjugation i.e. it obeys time reversalsymmetry. Concerning the case N >
1, it has been sugested [N2], [NN2] that exploiting (1.1) inthe context of characteristic classes theory in combination with some deep results in the theoryof analytic functions of several complex variables might lead to the existence of ELWF. Indeed, if d = 2 , P ( k ) satisfies (1.1), the triviality of the bundle was proved by Panati [P], providinga non-constructive proof of the existence of ELWF for all cases of physical interest [BPCMM]. Inhigher dimensions, examples of non-trivial bundles have been given in [dN-G].While concerning the non-constructive existence of ELWF the situation is satisfactory, thereare still many interesting questions left open in the area. Below we list a few of them:1. As it stands in [P], the triviality of the bundle generated by P ( k ) for d = 2 , N > d = 2 and N >
1. Ourmethod is quite different from the one in [FMP]; in particular, it does not require a preliminaryreduction to N = 2, and builds on the operator analytic approach in [N0], [N2]. Note that theresults of our paper are not applicable to the so-called ‘fermionic’ case related to Z topologicalinsulators [SV, FMP2]. However, our methods can be adapted in order to deal with the twodimensional fermionic case ([CMT], in preparation).The discussion above is about the case when no magnetic fields are present. While theexistence of Wannier bases for real and periodic Schr¨odinger operators can be reduced to the studyof the existence of smooth and periodic Bloch bases, for non-periodic systems the situation is morecomplicated. Nevertheless, for a large class of non-periodic perturbations of periodic systems itcan be proved that localized bases still exist, see [NN2]; the main idea behind it is based on a‘continuity’ argument which interpolates between the periodic and non-periodic operators.Our second main result gives the construction of bases of exponentially localized functions forthe case when a weak, globally bounded magnetic field is added, taking for granted the existenceof such bases for the zero magnetic field case.The content of the paper is as follows. In paragraph 1.2 we outline how one arrives at the family P ( k ) starting from a periodic tight-binding Hamiltonian/ Schr¨odinger operator. In paragraph 1.3we formulate the main results. Sections 2 and 3 are devoted to proofs.3 .2 From time reversal symmetric Hamiltonians to periodic P ( k ) satis-fying conjugation symmetry Let
B ⊂ R d be a finite set containing D points and letΛ := B + Z d be the discrete configuration space. We assume that every point y ∈ Λ can be uniquely writtenas: y = y + γ, y ∈ B , γ ∈ Z d . Consider the Hilbert space l (Λ) and let H be a bounded self-adjoint operator acting on thisspace, which commutes with the translations acting on Z d . It is uniquely determined by the’matrix elements’ H ( y, γ ; y ′ , γ ′ ) = H ( y, γ − γ ′ ; y ′ , , ∀ y, y ′ ∈ B , ∀ γ, γ ′ ∈ Z d . Consider the unitary Bloch-Floquet-Gelfand transform l (Λ) ∋ Ψ ( G Ψ)( y ; k ) := X γ ∈ Z d e πi k · γ Ψ( y + γ ) ∈ L ([ − / , / d ; C D ) . We have GH G ∗ = Z ⊕ [ − / , / d h ( k ) d k , h ( k ) ∈ L ( C D ) , h ( y, y ′ ; k ) := X γ ∈ Z d e πi k · γ H ( y, γ ; y ′ , . The D × D matrix h ( k ) will have D eigenvalues (Bloch energies), whose range determine thespectrum of H . We see that h ( k ) can be periodically extended to the whole of R d and thatthe decay of H ( y, γ ; y ′ ,
0) as function of γ tells us how regular h ( k ) is. For example, exponentiallocalization of H gives real analyticity for h . Moreover, if H is a real kernel, then we immediatelyobtain: h ( y ′ , y ; k ) = h ( y, y ′ ; k ) = h ( y, y ′ ; − k ) , h ( k ) f = h ( − k ) f , a property which from now on will be called conjugation symmetry. We will see in Lemma 2.5that for a self-adjoint matrix, conjugation symmetry is equivalent to: t h ( k ) = h ( − k ) , k ∈ R d , where t h ( k ) means transposition.Assume that the union of the ranges of N < D eigenvalues λ j ( k ) of h ( · ) is separated from therange of the other eigenvalues, i.e. there exists a gap in the spectrum of H . We can define P ( k )to be the Riesz projection associated to these N eigenvalues. P ( · ) is periodic and has the sametype of smoothness as h . Moreover, because t { ( h ( k ) − z ) − } = ( h ( − k ) − z ) − , it follows that P ( · ) has the same conjugation property.Consider an orthonormal system of N vectors Ξ j ( k ) ∈ C D which form a basis for Ran( P ( k )),continuous in k ∈ [ − / , / d . Then we can define: w j ( y ) = w j ( y + γ ) := Z [ − / , / d e − πi k · γ Ξ j ( y ; k ) d k , ≤ j ≤ N, y ∈ B , γ ∈ Z d . (1.2)It is known that the set obtained by translating them as follows: { w j ( · − γ ) : γ ∈ Z d , ≤ j ≤ N } ⊂ l (Λ)4orms an orthonormal basis for the range of the spectral projection of H corresponding to thepart of the spectrum given by σ := N [ j =1 Ran λ j ( · ) . A similar problem can be formulated for continuous operators. Let V be a Z d -periodic, realand bounded potential. The operator H = − ∆ + V has purely absolutely continuous spectrum.Let Ω := [ − / , / d . Consider the Bloch transform L ( R d ) ∋ Ψ ( F Ψ)( y ; k ) := X γ ∈ Z d e πi k · γ Ψ( y + γ ) ∈ L (Ω; L (Ω)) . We have (in the formula below, k -BC is a short-cut for k -dependent boundary conditions f ( y + γ ) = e − πi k · γ f ( y ), when both y and y + γ belong to the boundary of Ω): F HF ∗ = Z ⊕ Ω h k d k , h k = − ∆ + V with k − BC in L (Ω) . Each fiber Hamiltonian h k has purely discrete spectrum. Assume that the range of N eigenvaluesform an isolated spectral island σ . Using Combes-Thomas exponential estimates and ellipticregularity (see Proposition 3.1 in [CN2]), one can show that the Riesz projection Π correspondingto σ has a real integral kernel Π( x , x ′ ) which is jointly continuous and exponentially localizedaround the diagonal, i.e. there exists α > | Π( x , x ′ ) | ≤ e − α | x − x ′ | . Then F Π F ∗ = Z ⊕ Ω P k d k , P k ( y, y ′ ) = X γ ∈ Z d e πi k · γ Π( y + γ ; y ′ ) in L (Ω) . Again, we see that P k has the conjugation symmetry, is real analytic and Z d -periodic in k . Notethat (1.2) also makes sense in the continuous case, with the only difference that y ∈ Ω.In the continuous case one can also consider the Bloch-Zak transform L ( R d ) ∋ Ψ ( F Z Ψ)( y ; k ) := X γ ∈ Z d e πi k · ( y + γ ) Ψ( y + γ ) ∈ L (Ω; L (Ω)) . At fixed k and for Ψ ∈ C ∞ ( R d ), the function ( F Z Ψ)( · ; k ) is C ∞ (Ω) and periodic. We have: F Z HF ∗ Z = Z ⊕ Ω ˜ h k d k , ˜ h k = ( − i ∇ + 2 π k ) + V with periodic − BC in L (Ω) . The Riesz projection Π is decomposed as: F Z Π F ∗ Z = Z ⊕ Ω Π k d k , Π k ( y, y ′ ) = X γ ∈ Z d e πi k · ( y + γ − y ′ ) Π( y + γ ; y ′ ) in L (Ω) . We see that the range of Π k consists of Ω-periodic functions in y , but Π k is no longer periodic in k . Instead, denoting by τ λ the unitary operator acting on L (Ω) given by ( τ λ f )( y ) = e πiλ · y f ( y )we have the identity: Π k + λ = τ λ Π k τ − λ , ∀ λ ∈ Z d . If the Ξ j ’s are eigenvectors of h ( k ), then the corresponding w j ’s (see (1.2)) are called Wannierfunctions. In general, the Wannier functions are just square integrable, but in many physicalapplications it is important to construct better spatially localized Wannier functions. For example,if we can simultaneously choose the Ξ j ’s to be periodic and real-analytic, then a standard Paley-Wiener argument shows that the Wannier functions are exponentially localized. Moreover, if wealso have Ξ j ( k ) = Ξ j ( − k ), then the Wannier functions are real.If the Ξ j ’s are not eigenvectors for the fibre Hamiltonian but they form an orthonormal basisfor Ran( P ( k )), we can still define the corresponding w j ’s. They are called composite Wannierfunctions. 5 .3 Construction of composite Wannier functions If R d ∋ k P ( k ) ∈ L ( H ) is a projection valued continuous map in some separable complexHilbert space H , then the dimension of Ran( P ( k )) is constant. We list four basic problems. Problem 1.1.
Let d ≥ and let R d ∋ k P ( k ) ∈ L ( H ) be an orthogonal projection valued mapwhich is Z d -periodic and belongs to C n ( R d ) , n ≥ . Let dimRan( P ( k )) = N < ∞ . Can one finda system of N vectors { Ξ j ( k ) } Nj =1 which form an orthonormal basis in Ran( P ( k )) and which areboth in C n ( R d ) and Z d -periodic? Problem 1.2. If d ≥ , let R d ∋ k P ( k ) ∈ L ( H ) be an orthogonal projection valued map, Z d -periodic and jointly analytic in a tubular complex neighborhood I of R d . Let dimRan( P ( k )) = N < ∞ . Can one find a system of N vectors { Ξ j ( k ) } Nj =1 which are jointly analytic in a (possiblysmaller than I ) tubular complex neighborhood I ′ of R d , are Z d -periodic and form an orthonormalbasis of Ran( P ( k )) when k ∈ R d ? Most of the results of this paper will only be valid if some conjugation symmetries hold. Hereis a list of definitions:
Definition 1.3.
We say that a family H ( · ) of linear operators acting on a complex Hilbert space H has the CS property if there exists an anti-unitary operator K such that K = Id , KH ( k ) K = H ( − k ) , k ∈ R d . (1.3)The most common example of such a K in l ( Z ) is the complex conjugation Kf = f : H ( k ) f = H ( − k ) f , k ∈ R d . (1.4)In fact, up to a change of basis, every such K is given by a complex conjugation. Definition 1.4.
We say that a family of invertible operators U ( · ) acting on a complex Hilbertspace H has the CS ′ property if there exists an anti-unitary operator K such that K = Id , KU ( k ) K = [ U ( − k )] − , k ∈ R d . (1.5)The third problem is the following: Problem 1.5.
Assume that P ( · ) has the CS property. Can we construct a basis as in the previoustwo problems, which also obeys the property K Ξ j ( k ) = Ξ j ( − k ) ? Finally, let us formulate a more general problem as in [P, FMP], motivated by the Bloch-Zaktransform from the continuous case. Assume that we have some separable complex Hilbert space H with a conjugation K as in (1.3), and a family of unitary operators τ λ with λ ∈ Z d such that τ = Id , τ λ τ µ = τ λ + µ (= τ µ τ λ ) , τ ∗ λ = τ − λ (= τ − λ ) , Kτ λ = τ − λ K. (1.6) Problem 1.6. If d ≥ , let R d ∋ k Π( k ) be an orthogonal projection valued map in H suchthat: Π( k + λ ) = τ λ Π( k ) τ ∗ λ , K Π( k ) K = Π( − k ) , ∀ k ∈ R d , ∀ λ ∈ Z d . Assume that the map is jointly analytic in a tubular complex neighborhood I of R d . Assume thatthe dimension of Ran(Π( k )) equals N < ∞ . Can one find a system of N vectors { Ψ j ( k ) } Nj =1 which are jointly analytic in a (possibly smaller than I ) tubular complex neighborhood I ′ of R d ,which form an orthonormal basis of Ran(Π( k )) when k ∈ R d , obey τ λ Ψ j ( k ) = Ψ j ( k + λ ) and K Ψ j ( k ) = Ψ j ( − k ) for all k ∈ R d and λ ∈ Z d ? emark 1.7. We will show in Subsection (2.1) that this more general problem can be reducedto the periodic case. However, other more subtle properties like the existence of a canonical Berryconnection only appear in the Zak picture, see [FCG] for details.Here is our first main result.
Theorem 1.8.
The following statements hold true: (i)
Problem 1.6 can be reduced to the first three; see Subsection 2.1; (ii)
All problems can be reduced to finite dimensional Hilbert spaces; see Subsection 2.2; (iii)
Assume that the family P ( k ) is real analytic. If we can construct a continuous solution toour problems, then it can be made real analytic , preserving all its other properties; see Lemmas2.3 and 2.4; (iv) If d = 1 , a solution exists even in the absence of the CS property; see (2.14) and (2.17) ; (v) If d = 2 , a solution can be constructed if the CS property holds true. The main technical resultis Proposition 2.16. Remark 1.9.
Let us try to explain very roughly the main ideas behind the proof of (v). Theconstruction is inductive in d ; using (iv) we can construct some basis vectors Ψ j ( k , k ) whichare continuous in k but periodic only in k . Due to the periodicity of the projection P ( k ), thebasis at k = 1 / k = − / N × N unitary matrix β ( k ) which is continuous and periodic in k , and whose matrix elements have the symmetry[ β ( k )] mn = [ β ( − k )] nm . The main question is how to rotate each Ψ j ( k , k ) using β ( k ) suchthat the new vectors to be equal at k = ± /
2. This is done in Proposition 2.16, a result interestingin itself. Also, a related question is finding sufficient conditions for a matrix like β ( k ) to admit aperiodic and continuous logarithm in k . The main obstacle to the existence of such a logarithmis the possible crossing of eigenvalues of β ( k ) when k varies, see (2.25) for a generic example.One of the crucial ingredients in order to circumvent the crossing problem is the Analytic RellichTheorem [R-S4] which allows us to continuously follow the eigenvalues through crossings, alsopreserving periodicity in k . Remark 1.10.
The lack of a Rellich Theorem in more than one dimension is also the mainreason for which our proof cannot be generalized to d = 3. We conjecture though that a variant ofProposition 2.16 remains true if d = 3, but the proof becomes much more technical and seems tobe necessary to appeal to ‘avoided crossing’ techniques involving Sard’s lemma. This constructionwill be done elsewhere. Our second main result of this paper concerns the question as to whether the existence of Wannier-type localized bases is stable under the application of a (weak) non-decaying magnetic field. Forsimplicity, we only work in R . The setting is as follows. We assume that there exists a backgroundmagnetic field, orthogonal to the plane, whose only non-zero component is denoted with B ( x )and obeys: || B || C ( R ) < ∞ . (1.7)We can always construct its associate transverse gauge: A ( x ) := Z sB ( s x ) ds x ⊥ , x ⊥ := ( x , − x ) . (1.8)Let V be a bounded, real scalar potential and consider the Hamiltonian H = ( − i ∇ − A ) + V. (1.9)7ssume that the Hamiltonian H has an isolated spectral island σ and let P be its correspondingspectral projection. We also assume that the subspace Ran( P ) has an (exponentially localized)generalized Wannier basis; let us explain in detail the meaning of this. Let Γ ⊂ R be an infiniteset of points such that inf {|| γ − γ ′ || : γ, γ ′ ∈ Γ , γ = γ ′ } > . The above condition implies that Γ is countable; in particular, Γ can be a periodic lattice. Let
N < ∞ and assume that the set of functions w j,γ ∈ Ran( P ) , ≤ j ≤ N, γ ∈ Γform an orthonormal basis of Ran( P ). We also assume that there exist α > M < ∞ suchthat sup j,γ Z R | w j,γ ( x ) | e α || x − γ || d x < M. (1.10)Now let us consider a magnetic field perturbation given by B ( x ) such that || B || C ( R ) ≤ , A ( x ) := Z sB ( s x ) ds ( x , − x ) , (1.11)and introduce the perturbed Hamiltonian: H b := ( − i ∇ − A − b A ) + V, b ∈ R . (1.12)Below we list a few known facts about the resolvent of H b , see for details [N3, C, CN]: • the resolvent ( H b − z ) − has a Schwartz kernel denoted by ( H b − z ) − ( x , x ′ ) which is jointlycontinuous outside the diagonal, has a singularity of the type − ln( || x − x ′ || ) near the di-agonal, and there exist some C ( z ) < ∞ and α ( z ) > | ( H b − z ) − ( x , x ′ ) | ≤ C ( z ) e − α ( z ) || x − x ′ || if || x − x ′ || ≥ • Fix a compact K ⊂ ρ ( H ). Then there exist b > α > C < ∞ such that for every0 ≤ | b | ≤ b we have K ⊂ ρ ( H b ) and uniformly in x = x ′ :sup z ∈ K (cid:12)(cid:12)(cid:12) ( H b − z ) − ( x , x ′ ) − e ibφ ( x , x ′ ) ( H − z ) − ( x , x ′ ) (cid:12)(cid:12)(cid:12) ≤ C | b | e − α || x − x ′ || , (1.13)where φ ( x , x ′ ) := Z A ( x ′ + s ( x − x ′ )) · ( x − x ′ ) ds = − φ ( x ′ , x ) . (1.14)In particular, if | b | is small enough then H b has an isolated spectral island σ b close to σ , see also[N1] for a different proof of this fact. Denote by P b the corresponding projection. Here is thesecond main result of our paper: Theorem 1.11. (i) . If | b | is sufficiently small, then there exist a constant C < ∞ , a family of continuous functions Ξ j,γ,b and α > satisfying | Ξ j,γ,b ( x ) | ≤ Ce − α || x − γ || (1.15) such that { Ξ j,γ,b } j ∈{ ,...,N } ,γ ∈ Γ is an orthonormal basis of Ran( P b ) and sup γ,j || Ξ j,γ,b − e ibφ ( · ,γ ) w j,γ || ≤ C | b | . (1.16)(ii) . Moreover, if Γ is a periodic lattice, B = 0 , V is Γ -periodic, B is constant and w j,γ ( x ) = w j, ( x − γ ) = w j,γ ( x ) , (1.17)8 hen Ξ j,γ,b ( x ) = e ibφ ( x ,γ ) Ξ j, ,b ( x − γ ) (1.18) and Ξ j, ,b ( x ) = Ξ j, , − b ( x ) . (1.19) Remark 1.12.
When N = 1 (the case of a simple band), Theorem 1.11(ii) was proved in [N2].In the current manuscript we use a strategy which is related to the one of [N2]; what we add newis a significant generalization of the results and a substantial simplification of the proof due to theregularized magnetic perturbation theory as developed in [C, N3]. In fact, the estimate (1.13) isthe main new ingredient of the proof of Theorem 1.11.Theorem 1.11 is interesting in itself but it also plays an important role in the so-called Peierls-Onsager substitution at small magnetic fields [Pe, Lu, PST, FT]. These aspects will be treatedelsewhere. We first show that there exists a family of unitary operators u k with k ∈ C d such that u λ = τ λ if λ ∈ Z d , the map C d ∋ k u k ∈ L ( H ) is entire, and: u = Id , u k u k ′ = u k + k ′ (= u k ′ u k ) , u ∗ k = u − k (= u − k ) , Ku k = u − k K = u − k K, ∀ k , k ′ ∈ R d . (2.1)Denote by f j the vectors of the standard basis in R d . The operator τ f j is unitary hence thespectral theorem implies that it can be written as τ f j = Z ( − π,π ] e iφ dE j ( φ ) . (2.2)The operator M j = R ( − π,π ] φ dE j ( φ ) is bounded, self-adjoint, and τ f j = e iM j .If F is any 2 π -periodic smooth function we define e F ( e it ) = F ( t ) and we have e F ( τ f j ) = X m ∈ Z ˆ F ( m )( τ f j ) m , ˆ F ( m ) := 12 π Z π − π F ( t ) e − itm dt. (2.3)We have Kτ mf j K = ( τ f j ) − m . If F is real, then ˆ F ( m ) = ˆ F ( − m ). This leads to K e F ( τ f j ) K = e F ( τ f j )for all 2 π -periodic smooth and real functions F . By a limiting argument we conclude that thesame remains true for the spectral measure of τ f j , hence KM j K = M j . Moreover, because theset { τ f j } dj =1 consists of commuting operators, the same is true for their spectral measures and for { M j } dj =1 . Now if k = [ k , ..., k d ] ∈ R d we define u k := e i ( k M + ... + k d M d ) . One can verify that (2.1) is obeyed and because the generators M j are bounded, the map is alsoentire. Now if we define P ( k ) := u − k Π( k ) u k we see that P ( · ) is Z d -periodic, has the same smooth-ness properties as Π( · ) and KP ( k ) K = P ( − k ). Assuming that we have a solution { Ξ j ( k ) } Nj =1 forProblem 1.5, then we can easily check that Ψ j ( k ) := u k Ξ j ( k ) is a solution for Problem 1.6. Lemma 2.1.
Fix d and assume that Problem 1.1 can be solved if the complex Hilbert space has afinite dimension. Then it can also be solved in any infinitely dimensional separable complex Hilbertspace. roof. Let L ≥ L + 1) d points N L := { k n ∈ [ − / , / d : k n := [ n / (2 L ) , . . . n d / (2 L )] , − L ≤ n j ≤ L, n j ∈ Z , ≤ j ≤ d } . Since P ( · ) is continuous and Z d periodic, then for every ǫ > L ǫ large enoughsuch that for every k ∈ R d there exists some k n ∈ N L ǫ such that || P ( k ) − P ( k n ) || ≤ ǫ . Now definethe finite dimensional subspace (to be understood as a sum of linear subspaces) H ǫ := X k n ∈N Lǫ Ran( P ( k n )) , and denote by Π ǫ its corresponding orthogonal projection. Clearly, Π ǫ P ( k n ) = P ( k n )Π ǫ = P ( k n )for all k n . Define P ǫ ( k ) := Π ǫ P ( k )Π ǫ . For every k ∈ R d we have (the choice of k n below is suchthat || P ( k ) − P ( k n ) || ≤ ǫ ): P ǫ ( k ) − P ( k ) = Π ǫ { P ( k ) − P ( k n ) } Π ǫ + P ( k n ) − P ( k )hence || P ǫ ( k ) − P ( k ) || ≤ ǫ . It means that P ǫ ( k ) is close to a projection. If ǫ is small enough,then the operator: ˜ P ǫ ( k ) := − πi Z | z − | =1 / ( P ǫ ( k ) − z ) − dz is an orthogonal projection whose range is contained in H ǫ (to see this use ( P ǫ ( k ) − z ) − + z − = z − P ǫ ( k )( P ǫ ( k ) − z ) − in the above formula). Using the resolvent formula˜ P ǫ ( k ) − P ( k ) = 12 πi Z | z − | =1 / ( P ǫ ( k ) − z ) − ( P ǫ ( k ) − P ( k ))( P ( k ) − z ) − dz we obtain || ˜ P ǫ ( k ) − P ( k ) || ≤ const ǫ . The dimension of P ( k ) is constant and equal to N , thus thedimension of ˜ P ǫ ( k ) also equals N for all k , provided ǫ is small enough. This is true because wecan construct the Sz.-Nagy unitary which intertwines between ˜ P ǫ ( k ) and P ( k ): U ǫ ( k ) := { P ( k ) ˜ P ǫ ( k ) + (Id − P ( k ))(Id − ˜ P ǫ ( k )) }{ Id − ( ˜ P ǫ ( k ) − P ( k )) } − / , (2.4)with P ( k ) U ǫ ( k ) = U ǫ ( k ) ˜ P ǫ ( k ).To summarize: we constructed a projection valued map ˜ P ǫ ( · ) with dimension N which enjoysthe same smoothness and periodicity properties as P ( · ), and lives in the finite dimensional space H ǫ . Our assumption was that we can solve Problem 1.1 in finite dimensional Hilbert spaces.Hence let { ˜Ψ j,ǫ ( k ) } Nj =1 be an orthonormal basis of Ran( ˜ P ǫ ( k )) consisting of smooth and periodicvectors. Then a solution to our problem in the infinite dimensional Hilbert space will be given bythe vectors: Ψ j ( k ) := U ǫ ( k ) ˜Ψ j,ǫ ( k ) . Lemma 2.1 showed how to reduce the problem to a finite dimensional Hilbert space. We willnow show that the conjugation symmetry is also preserved by the previous construction.
Lemma 2.2.
We refer to the quantities defined in Lemma 2.1. Assume that P ( · ) has the CS property. Then both ˜ P ǫ ( · ) and U ǫ ( · ) have the same property. Moreover, if K ˜Ψ j,ǫ ( k ) = ˜Ψ j,ǫ ( − k ) ,the same property holds for Ψ j ( k ) .Proof. First we prove that the subspace H ǫ is invariant with respect to the action of K , i.e. Π ǫ commutes with K . We constructed the grid N L ǫ so that both k n and − k n belong to it for all n . Due to the CS property we have P ( − k n ) K = KP ( k n ), which shows that K Ran( P ( k n )) ⊂ Ran( P ( − k n )), hence K ( H ǫ ) ⊂ H ǫ . Using K = Id we also obtain H ǫ ⊂ K ( H ǫ ), thus K ( H ǫ ) = H ǫ .10ow let f ∈ H ǫ and g ∈ H ⊥ ǫ be arbitrary. Then using the anti-unitarity of K together with Kf ∈ H ǫ , we obtain: h Kg | f i = h Kf | g i = 0 . This shows that K ( H ⊥ ǫ ) ⊂ H ⊥ ǫ and in fact K ( H ⊥ ǫ ) = H ⊥ ǫ . Then:Π ǫ K (Id − Π ǫ ) = (Id − Π ǫ ) K Π ǫ = 0 . This implies that Π ǫ K = K Π ǫ . It follows that P ǫ ( k ) = Π ǫ P ( k )Π ǫ has the CS property. We seethat ( P ǫ ( k ) − z ) − K Ψ = K ( P ǫ ( − k ) − z ) − Ψwhich after integration leads to the conclusion that ˜ P ǫ ( k ) has the CS symmetry.The last thing we need to prove is that the Sz.-Nagy unitary U ǫ ( k ) also has CS symmetry. Thisis immediately implied by KP ( k ) K = P ( − k ) and K ˜ P ǫ ( k ) K = ˜ P ǫ ( − k ), which ends the proof.It is possible to construct an orthonormal basis in H ǫ such that Kf s = f s . If Ψ = P s c s f s then K Ψ = P s c s f s , i.e. K becomes just a complex conjugation. Thus we can assume withoutloss of generality that the Hilbert space is C D with 2 ≤ D < ∞ and K is complex conjugation.In this case, all operators are characterized by finite dimensional matrices. The next lemma states that if the map P ( · ) is real analytic, then a solution for Problem 1.1 canalways be turned into a solution for Problem 1.2. Lemma 2.3.
Let R d ∋ k P ( k ) ∈ L ( C D ) be a Z d -periodic, orthogonal projection valued mapwith dim(Ran P ( k )) = N < D for all k ∈ R d . Assume that all the components { P mn ( · ) } ≤ m,n ≤ D in the standard basis of C D have a Z d -periodic, bounded jointly analytic extension to a tubularcomplex neighborhood I of R d . Assume that { Ψ j ( k ) } Nj =1 is an orthonormal basis of Ran P ( k ) for all k ∈ R d , Z d -periodic, and continuous . Then there exists { Ξ j ( · ) } Nj =1 holomorphic on atubular complex neighborhood R d ⊂ I ′ ⊂ I , Z d -periodic, which is an orthonormal basis of Ran P ( k ) if k ∈ R d .Proof. The function g ( k ) = π − d Q dj =1 (1 + k j ) − obeys g ( k ) = g ( − k ), it is analytic on the strip { z ∈ C d : | Im( z j ) | < , j ∈ { , ..., d }} , and R R d g ( k ) d k = 1. If δ > g δ ( k ) = δ − d g ( δ − k ). Let us considerΨ j,δ ( k ) = Z R d g δ ( k − k ′ )Ψ j ( k ′ ) d k ′ , which defines a D dimensional vector which is Z d -periodic.Let us show that Ψ j,δ admits an analytic extension to a strip { z = ( z , ...z d ) ∈ C d : | Im( z j ) | < δ } . Due to periodicity, it is enough to find an extension to a bounded open neighbourhood O of [0 , d of the form O = O × ... × O d , O j = { z j = x j + iy j ∈ C : − < x j < , | y j | < δ ′ < δ } . For every z = ( x + iy , ..., x d + iy d ) ∈ O and every k ′ ∈ R d we have the estimate: | g δ ( z − k ′ ) | ≤ π − d δ d d Y j =1 δ − δ ′ + ( x j − k ′ j ) . g δ ( k − k ′ ) = 1(2 πi ) d Z ∂O ... Z ∂O d g δ ( ζ − k ′ ) Q dj =1 ( ζ j − k j ) d ζ , k ∈ [0 , d , k ′ ∈ R d . Because the Ψ j ’s are bounded on R d , the Fubini Theorem and the Cauchy Integral Formulaallow one to analytically extend each Ψ j,δ ( · ) to O , and by periodicity, to a strip around R d .Let us now continue the construction. A standard approximation argument implies:lim δ → (cid:26) max ≤ j ≤ N sup k ∈ R d || Ψ j,δ ( k ) − Ψ j ( k ) || (cid:27) = 0 . Denote by Φ j,δ ( k ) := P ( k )Ψ j,δ ( k ). Then for a given 0 < ǫ ≪ δ small enough suchthat: max ≤ j ≤ N sup k ∈ R d || Φ j,δ ( k ) − Ψ j ( k ) || < ǫ. Define the selfadjoint N × N Gram-Schmidt matrix h kj ( k ) = h Φ j,δ ( k ) , Φ k,δ ( k ) i = Z R d g δ ( k − y ) g δ ( k − y ′ ) h P ( k )Ψ j ( y ) , Ψ k ( y ′ ) i d y d y ′ . (2.5)Since { Ψ j ( k ) } Nj =1 is an orthonormal system, by choosing δ small enough we can insure that || h ( k ) − Id || ≤ / k ∈ R d . Moreover, h ( · ) is bounded, Z d -periodic and real analytic,and the same holds true for h ( · ) − / . Then the vectorsΞ j ( k ) := N X k =1 h h ( k ) − / i kj Φ k,δ ( k )form a basis with all the required properties. CS Lemma 2.4.
Let P ( k ) , { Ψ j ( k ) } Nj =1 and { Ξ j ( · ) } Nj =1 as constructed in Lemma 2.3. Assume that P ( k ) has the CS property, and that Ψ j ( k ) = Ψ j ( − k ) for all j . Then Ξ j ( k ) = Ξ j ( − k ) on R d forall j .Proof. We will refer to the already defined objects in the proof of Lemma 2.3. First, due to thesymmetry of g it is easy to check that Ψ j,δ ( k ) = Ψ j,δ ( − k ). This implies that Φ j,δ ( k ) = Φ j,δ ( − k ).This leads to the identity h jk ( k ) = h jk ( − k ). Since h ( k ) − / can be written as a power series in h ( k ) − Id, we have that [ h ( k ) − / ] kj = [ h ( − k ) − / ] kj . CS or CS ′ symmetry Lemma 2.5.
Let θ denote complex conjugation and consider a self-adjoint matrix family α ( k ) ∈L ( C N ) having CS symmetry, i.e. θα ( k ) θ = α ( − k ) . Then: [ α ( k )] mn = [ α ( − k )] nm , σ ( α ( k )) = σ ( α ( − k )) . (2.6) Proof.
We know that [ α ( k )] mn = [ α ( k )] nm from self-adjointness and [ α ( k )] mn = [ α ( − k )] mn from CS symmetry. The symmetry of the spectrum follows from the identitydet( A − z Id) = det( t A − z Id) . emma 2.6. Let θ denote the complex conjugation and consider a unitary matrix family β ( k ) ∈L ( C N ) with the CS ′ symmetry, i.e. θβ ( k ) θ = β ( − k ) − for all k . Then: [ β ( k )] mn = [ β ( − k )] nm , σ ( β ( k )) = σ ( β ( − k )) . (2.7) Proof.
We know that [ β ( − k ) − ] mn = [ β ( − k )] nm from unitarity and [ β ( − k ) − ] mn = [ β ( k )] mn fromthe CS ′ symmetry. The proof of the symmetry of the spectrum is the same as in the previouslemma. Remark 2.7.
We note the fact that both the CS property for self-adjoint matrices and the CS ′ property for unitary matrices boil down to the same identity, only involving transposition: t α ( k ) = α ( − k ) . Lemma 2.8.
Let k = [ k , k ] ∈ R and let α ( k ) be self-adjoint with CS symmetry. Then the uni-tary family β ( k ) := e ik α ( k ) has CS symmetry, while the family γ ( k ) := e iα ( k ) has CS ′ symmetry.Proof. We have θβ ( k ) θ = e − ik θα ( k ) θ = e − ik α ( − k ) = β ( − k ) . In a similar way: θγ ( k ) θ = e − iθα ( k ) θ = [ γ ( − k )] − . d = 1 Let us start by stating without proof a well-known fundamental lemma essentially due to Kato,formulated in a form which will be convenient for us in what follows.
Lemma 2.9.
Let Q ( x ) = Q ∗ ( x ) = Q ( x ) be a C -norm differentiable family of orthogonal projec-tions. Define K ( x ) := i (Id − Q ( x )) Q ′ ( x ) = i [ Q ′ ( x ) , Q ( x )] , K ( x ) = K ∗ ( x ) . (2.8) Given y ∈ R , one can define a family of unitary operators A ( x, y ) such that i∂ x A ( x, y ) = K ( x ) A ( x, y ) , A ( y, y ) = Id , (2.9) given by: A ( x, y ) = Id + X n ≥ ( − i ) n Z xy ds Z s y ds ... Z s n − y ds n K ( s ) ... K ( s n ) . (2.10) Furthermore, we have: i∂ y A ( x, y ) = − A ( x, y ) H ( y ) , [ A ( x, y )] − = A ( y, x ) , A ( x , x ) A ( x , x ) = A ( x , x ) . (2.11) The key intertwining identity is: Q ( x ) = A ( x, x ) Q ( x ) A ( x , x ) . (2.12) For fixed x ∈ R , if either Q ( · ) ∈ C n ( R ) with n ≥ or Q ( · ) admits an analytic extension to thestrip I a := { z ∈ C : | Im( z ) | < a } , then the same holds true for A ( · , x ) and A ( x , · ) .
13n the case in which the above family is Z -periodic, i.e. Q ( x + 1) = Q ( x ) for all x ∈ R ,then K ( x ) is Z -periodic and we observe that A ( x + 1 , x + 1) must equal A ( x, x ) (see (2.9)).Consider the unitary operator A (1 , M with spectrum in ( − π, π ] such that A (1 ,
0) = e iM . Also, due tothe intertwining relation Q (1) A (1 ,
0) = A (1 , Q (0) and because Q (1) = Q (0) due to periodicity,we conclude that A (1 ,
0) commutes with Q (0) and so does M .Define the unitary operator: U ( k ) := A ( k, e − ikM , k ∈ R . (2.13)One can verify that U ( · ) is Z -periodic: U ( k + 1) = A ( k + 1 , e − i ( k +1) M = A ( k + 1 , A (1 , e − i ( k +1) M = A ( k, e − ikM = U ( k )where we used A ( k + 1 ,
1) = A ( k, d = 1. Consider any orthonormal basis { Ψ j (0) } Nj =1 inthe range of P (0). Construct U ( k ) as above starting from P ( k ). We will now prove that thevectors: Ξ j ( k ) := U ( k )Ψ j (0) , k ∈ R , ≤ j ≤ N (2.14)give a solution for Problem 1.1. These vectors are clearly periodic and orthogonal, thus we onlyneed to prove that they live in the range of P ( k ). But since M commutes with P (0) and due tothe intertwining relation P ( k ) A ( k,
0) = A ( k, P (0), it follows that P ( k ) U ( k ) = U ( k ) P (0) and weare done.We note that the above construction has not used a possible CS property of P ( · ). If such aproperty does hold, i.e. besides smoothness and periodicity we also have θP ( k ) θ = P ( − k ) for all k , then let us show that we can construct a basis which solves Problem 1.5.Since θP (0) = P (0) θ , we have that θ Ψ j (0) belongs to the range of P (0), hence the realvectors (Ψ j (0) + θ Ψ j (0)) and i (Ψ j (0) − θ Ψ j (0)) have the same property. After a Gram-Schmidtprocedure we obtain a real orthonormal basis of Ran( P (0)), denoted by { Ξ j (0) } Nj =1 . Thus theonly thing we need to check is that U ( k ) has the CS property, i.e. θU ( k ) θ = U ( − k ).First, due to the identity P ′ ( − k ) = − θP ′ ( k ) θ we have θ K ( k ) θ = K ( − k ), thus θA ( k, θ = A ( − k,
0) because they solve the same initial value problem.Second, taking k = 1 in the last identity we get θA (1 , θ = A ( − ,
0) = A (0 ,
1) = [ A (1 , − i.e. θe iM θ = e − iM , (2.15)where M = M ∗ commutes with P (0) and has its spectrum in ( − π, π ]. Since the underlying Hilbertspace is C D , all operators are represented by D × D matrices, hence M has discrete eigenvalues andcan be written as M = P − π<φ ≤ π φ Π φ where Π φ are spectral orthogonal projections correspondingto different eigenvalues. From (2.15) we immediately obtain that θ Π φ θ = Π φ for all eigenvalues,hence θM θ = M . Thus: θe − ikM θ = e ikθMθ = e ikM , k ∈ R , hence θU ( k ) θ = U ( − k ) and the basis vectors we are looking for are given by Ξ j ( k ) := U ( k )Ξ j (0). Remark 2.10.
Define Ψ m ( k ) := A ( k, m (0) and consider M jk := h M Ξ k (0) | Ξ j (0) i , i.e. thematrix elements of P (0) M P (0) in the real basis Ξ(0). Using the fact that M commutes with P (0) = P (1) leads toΨ m ( k + 1) = A ( k, P (0) e iM P (0)Ξ m (0) = N X n =1 [ e i M ] nm Ψ n ( k ) (2.16)and Ξ m ( k ) = U ( k )Ξ m (0) = A ( k, P (0) e − ikM Ξ m (0) = N X n =1 [ e − ik M ] nm Ψ n ( k ) . (2.17)In other words, we have found a unitary, k -dependent change of basis in the range of P ( k ), whichtransforms the Ψ( k )’s into a periodic basis also preserving the CS symmetry and the regularityof the original projection. 14 .7 The case d = 2 From now on (if not otherwise stated) we assume that the underlying Hilbert space is C D andthe family of projections P ( · ) has the CS property, i.e. θP ( k ) θ = P ( − k ) for all k ∈ R . Theidea is to proceed recursively. Consider P (0 , k ). Using the result from d = 1 we can construct anorthonormal basis { Ξ j (0 , k ) } Nj =1 of Ran P (0 , k ) such thatΞ j (0 , k + 1) = Ξ j (0 , k ) . θ Ξ j (0 , k ) θ = Ξ j (0 , − k ) , ∀ k ∈ R . (2.18)Now keeping k fixed, consider the orthogonal projection family Π( · ) := P ( · , k ). We can applyLemma 2.9 with Q replaced by Π and obtain a family of unitary operators denoted by A k ( x, y )generated by the self-adjoint operators K k ( k ) := i [ ∂ P ( k , k ) , P ( k , k )] . Reasoning as in the case d = 1 we obtain the intertwining relation: P ( k , k ) A k ( k ,
0) = A k ( k , P (0 , k ) , (2.19)and θ K k ( k ) θ = K − k ( − k ) , θA k ( k , θ = A − k ( − k , . (2.20)We also note the periodicity in k : K k +1 ( k ) = K k ( k ) , A k +1 ( k ,
0) = A k ( k , . (2.21)Due to the intertwining relation (2.19) we have that the set of vectorsΨ m ( k ) := A k ( k , m (0 , k )form an orthonormal basis in Ran P ( k ), having all the right properties but one: they are notperiodic in k . Using A k ( k + 1 ,
1) = A k ( k ,
0) we observe that:Ψ m ( k + 1 , k ) = A k ( k , A k (1 , m (0 , k ) . From the intertwining relation (2.19) and due to P (1 , k ) = P (0 , k ) we have that A k (1 , P (0 , k ) hence:Ψ m ( k + 1 , k ) = N X n =1 h A k (1 , m (0 , k ) | Ξ n (0 , k ) i Ψ n ( k , k ) . (2.22)The N × N matrix family defined by β nm ( k ) := h A k (1 , m (0 , k ) | Ξ n (0 , k ) i (2.23)is unitary in C N and we have:Ψ m ( k + 1 , k ) = N X n =1 β nm ( k ) Ψ n ( k , k ) . (2.24)This identity closely resembles (2.16) but the important difference is that the matrix β dependson k . The important question we have to answer is whether we can find a k -dependent unitaryrotation, like we did in (2.17), which transforms the Ψ’s into a Z -periodic basis, preserving atleast its continuity and CS symmetry. We note that β ( k ) is similar to the ‘obstruction unitary’ U obs ( k ) introduced in [FMP]. Lemma 2.11.
The N × N matrix family defined by (2.23) is unitary, Z -periodic, inherits theregularity properties of P (0 , k ) and has the CS ′ symmetry. roof. We only prove the statement related to the CS ′ symmetry. Since A − k (1 ,
0) = θA k ( − , θ = θA k (0 , θ = θ [ A k (1 , ∗ θ and because θ Ξ m (0 , − k ) = Ξ m (0 , k ) we have: β nm ( − k ) = h θ [ A k (1 , ∗ Ξ m (0 , k ) | θ Ξ n (0 , k ) i = h Ξ n (0 , k ) | [ A k (1 , ∗ Ξ m (0 , k ) i = β mn ( k ) , which according to Lemma 2.6 is equivalent with the CS ′ symmetry. Remark 2.12.
Let us assume that there exists an N × N self-adjoint matrix family h ( k ) whichis continuous, Z -periodic, with the CS symmetry, such that β ( k ) = e ih ( k ) . Note that this wasthe case when d = 1 (there h ( k ) = M , see (2.16)). Then the matrix e − ik h ( k ) has the CS ′ symmetry (see Lemma 2.8) and the vectorsΞ m ( k ) := N X n =1 [ e − ik h ( k ) ] nm Ψ n ( k )give the periodic, continuous and CS symmetric basis we are looking for. Indeed, in this case wehave: Ξ m ( k + 1 , k ) = N X n =1 [ e − i ( k +1) h ( k ) ] nm N X j =1 [ e ih ( k ) ] jn Ψ j ( k ) = Ξ m ( k ) . Unfortunately, finding a ‘good’ logarithm for β ( k ) (i.e. a matrix h ( k ) which is self-adjoint,periodic, continuous, CS symmetric and β ( k ) = e ih ( k ) ) is not always possible, as it can be seenfrom the following example. Let β ( k ) = (cid:20) cos(2 πk ) sin(2 πk ) − sin(2 πk ) cos(2 πk ) (cid:21) = cos(2 πk )Id + i sin(2 πk ) σ , k ∈ R (2.25)be a family of unitary matrices, real analytic, Z -periodic, and which obeys the CS ′ symmetry.Note that β ( k ) has degenerate eigenvalues at k = 0 and k = ± /
2, and the range of its spectrumcovers the whole unit circle.Let us show that one cannot find a continuous self-adjoint family h ( k ) which is also Z -periodic,has the CS symmetry and β ( k ) = e ih ( k ) for all k . Assume the contrary. From det( β ( k )) = 1 = e i Tr( h ( k )) we have that k π Tr( h ( k )) is a continuous, integer valued function. Hence there existsan integer m such that Tr( h ( k )) = 2 πm for all k . Let F j ( k ) := Tr( σ j h ( k ))2 ; then we have h ( k ) = Tr( h ( k ))2 Id + X j =1 F j ( k ) σ j = mπ Id + F ( k ) · σ. Thus β ( k ) = ( − m e i F ( k ) · σ = ( − m (cid:18) cos( | F ( k ) | )Id + i sin( | F ( k ) | ) | F ( k ) | F ( k ) · σ (cid:19) . Comparing with the definition of β ( k ), both F and F must be identically zero. Hence F ( k ) =[0 , F ( k ) ,
0] and | F ( k ) | = | F ( k ) | . Therefore β ( k ) = ( − m (cos( F ( k ))Id + i sin( F ( k )) σ ) = cos( F ( k ) + mπ )Id + i sin( F ( k ) + mπ ) σ . We conclude that there exists an integer-valued function n ( k ) such that F ( k )+ mπ = 2 πk +2 πn ( k ).Since F must be continuous, it follows that n ( k ) is constant and equals some n . Hence F ( k ) =2 πk + π (2 n − m ) for all k . In order to make sure that β pq ( k ) = β qp ( − k ) we must have that16 ( k ) = − F ( − k ), i.e. m = 2 n , ( − m = 1 and F ( k ) = 2 πk . This shows that F cannot beperiodic.Fortunately, the lack of a ‘good’ logarithm can be circumvented by a two step rotation pro-cedure (see (2.34)), where for each step a ‘good’ logarithm can be constructed. In what follows,we will discuss four particular situations in which a good logarithm can be constructed and thenshow how they can be combined in order to deal with the general case. N = 1Here the unitary matrix β ( k ) is just a complex number on the unit circle, which due to the CS ′ property (see (2.7)) is also even: β ( k ) = β ( − k ). Lemma 2.13.
There exists a unique real and even function φ p ( k ) = φ p ( − k ) , with the samesmoothness properties as β , Z -periodic, φ p (0) = 0 , such that β ( k ) = β (0) e iφ p ( k ) .Proof. Without loss of generality we may assume that β (0) = 1. We start by first proving theexistence of a continuous phase φ ( t ) such that β ( t ) = e iφ ( t ) and φ (0) = 0.The map β is uniformly continuous on R , thus there exists δ > | β ( t ) − β ( s ) | = | β ( t ) /β ( s ) − | ≤ / | t − s | ≤ δ. If Ln denotes the principal branch of the natural logarithm, then in a neighbourhood [ − δ, δ ] wecan define φ ( t ) = − i Ln(1 + ( β ( t ) − φ ( − t ) where β ( t ) = e iφ ( t ) , φ (0) = 0, φ is real andcontinuous.We can repeat this construction in the interval ( δ, δ ] by defining φ ( t ) = φ ( δ ) − i Ln[1 + ( β ( t ) /β ( δ ) − . A similar formula holds on [ − δ, − δ ), the extension being continuous at ± δ and symmetric. Aftera finite number of steps we can cover any bounded interval of R . Thus we obtain a continuous,symmetric, real function φ with φ (0) = 0, not necessarily periodic, such that β ( t ) = e iφ ( t ) . Weonly need to prove its Z -periodicity.We know that β is Z -periodic, thus the function F ( t ) := φ ( t + 1) − φ ( t )2 π must be both continuous and integer valued. Thus F ( t ) equals some constant n ∈ Z . But wehave n = F ( − /
2) = 0 because φ (1 /
2) = φ ( − / φ ( t + 1) = φ ( t ) for all t . Theregularity properties of φ are the same as those of β , which can be seen from the formula φ ( t ) = φ ( t ) − i Ln[1 + ( β ( t ) /β ( t ) − t with | t − t | small enough. In particular, if β is analytic in a strip containing R , thesame holds true for φ .The uniqueness of such a phase is a consequence of the following fact: if φ is continuous, φ (0) = 0, and e iφ ( t ) = 1 for all t , then φ must be a constant equal to zero. β never cross Here we assume that
N > β ( k ) has nondegenerate spectrum for all k , besides beingcontinuous, Z -periodic and with CS ′ symmetry. Lemma 2.14.
There exists a self-adjoint N × N matrix family h ( k ) which is continuous and Z -periodic, has the CS property and β ( k ) = e ih ( k ) . roof. β is uniformly continuous on [ − / , / ǫ > δ > || β ( k ) − β ( k ′ ) || ≤ ǫ whenever | k − k ′ | ≤ δ . Because the eigenvalues never cross, there must existsome positive distance ǫ > k . Let δ > || β ( k ) − β ( k ′ ) || ≤ ǫ /
10 whenever | k − k ′ | ≤ δ .Because β (0) has N nondegenerate eigenvalues, we can order them according to their argu-ments. Thus we have − π < φ < φ < ... < φ N ≤ π and λ j (0) := e iφ j . We know that || β ( k ) − β (0) || ≤ ǫ /
10 whenever | k | ≤ δ . Regular perturbation theory implies that each set σ ( β ( k )) ∩ B ǫ / ( λ j (0)) consists of exactly one point when | k | ≤ δ . The operators P j ( k ) := i π Z | z − λ j (0) | = ǫ / ( β ( k ) − z ) − dz, | k | ≤ δ , are the orthogonal spectral projections of β on | k | ≤ δ . They are continuous. Moreover, λ j ( k ) := Tr( β ( k ) P j ( k )) , | k | ≤ δ are N continuous eigenvalues. Also, because the spectrum of β is symmetric (see (2.7)), we musthave λ j ( k ) = λ j ( − k ) on | k | ≤ δ . Now we can repeat the procedure on [ − δ , − δ ] ∪ [ δ , δ ]starting from k = ± δ .After a finite number of steps we obtain some continuous eigenvalues λ j ( k ) = λ j ( − k ) on − / ≤ k ≤ /
2, together with their one dimensional Riesz projections. The disk | z − λ j ( k ) | ≤ ǫ / β ( k ) besides λ j ( k ), uniformly in | k | ≤ / j , and P j ( k ) = i π Z | z − λ j ( k ) | = ǫ / ( β ( k ) − z ) − dz, | k | ≤ / . Using the fact that t [( β ( k ) − z ) − ] = ( β ( − k ) − z ) − (from the CS ′ property) together with λ j ( k ) = λ j ( − k ), we have t P j ( k ) = t P j ( − k ). Moreover, P j ( − /
2) = P j (1 /
2) due to the periodicityof β .Now any k ∈ R can be uniquely written as k = n + k with n ∈ Z and − / < k ≤ /
2. DefineΛ j ( k ) := λ j ( k ) , P j ( k ) := P j ( k ) . Since the spectrum of β is periodic (as a set), the Λ j ’s are the periodic and continuous eigenvaluesof β on R , which for n > j ( n + k ) = λ j ( k ) = λ j ( − k ) = Λ j ( − n − k ) , i.e. they are also even. From Lemma 2.13 it follows that we can find φ j periodic, continuousand even such that Λ j ( k ) = e iφ j ( k ) . The corresponding projections, P j ( k ), (as given by the Rieszformula) are continuous, periodic, and obey P j ( n + k ) = P j ( k ) = t P j ( − k ) = t P j ( − n − k ) , ∀ n > . Thus h ( k ) := P Nj =1 φ j ( k ) P j ( k ) obeys all the necessary conditions. Again we assume
N > β ( · ) continuous, Z -periodic, and with CS ′ symmetry. We also assumethat there exists some φ ∈ ( − π, π ] such that e iφ never belongs to the spectrum of β ( k ). Thenthe statement of Lemma 2.14 remains true, although the proof is rather different, as we will showin what follows.The matrix γ ( k ) := e i ( π − φ ) β ( k ) has the property that − s ( k ) := i { Id − γ ( k ) }{ Id + γ ( k ) } − (2.26)18s self-adjoint, continuous, periodic and θs ( k ) θ = s ( − k ), hence it has the CS symmetry. Byelementary algebra we get: γ ( k ) = { Id + is ( k ) }{ Id − is ( k ) } − . Consider f ( z ) = (1 + iz ) / (1 − iz ) defined on a (narrow) strip S containing the (real) spectrum of s ( k ) for all k . Then: h ( k ) := − π + φ − i Ln γ ( k ) = − π + φ + 12 π Z ∂S (Ln f ( z ))( s ( k ) − z ) − dz (2.27)obeys all the properties. β ( · ) is real analytic and β (0) and β (1 / have nondegenerate spectrum This is the last particular situation in which we prove the existence of a ‘nice’ logarithm withthe properties listed in Lemma 2.14. The real analyticity of β ( · ) allows us to relax the strictnon-crossing condition we had in Lemma 2.14, now we only demand nondegenerate spectrum at k = 0 and k = 1 /
2. Note that the periodicity of β ( · ) implies, in particular, that the spectrum at k = − / β ( · ) consists of real analytic eigenvalues. Letus construct them. Assume that e iφ is not in the spectrum of β (0) and define γ ( k ) := e i ( π − φ ) β ( k ).Then Id + γ ( k ) is invertible for k near 0 and we can proceed as in (2.26) and (2.27) and constructa self-adjoint matrix h ( · ) which is analytic in a small open disk containing k = 0 such that β ( k ) = e i ( − π + φ + h ( k )) near 0. We can label the eigenvalues of β ( k ) as { λ j ( k ) = e iφ j ( k ) } Nj =1 choosing for example that at k = 0 we have − π < φ (0) < ... < φ N (0) ≤ π .The second step is to extend these projections and eigenvalues to | k | ≤ / β never cross, then the projections and eigenvalueswe constructed in Lemma 2.14 are the objects we are looking for. But here the eigenvalues cancross, and we need to be able to analytically continue the one dimensional projections throughcrossing points. By applying a finite number of times the Analytic Rellich Theorem [R-S4] tothe ‘local self-adjoint logarithms’ (2.27) we can follow analytically each projection P j ( k ) and eachcorresponding eigenvalue λ j ( · ) to | k | ≤ /
2. Because β is periodic and β (1 /
2) = β ( − /
2) arenon-degenerate, this construction can be extended to the whole of R .Near k = 0 the spectrum of β is nondegenerate and we can write each P j ( k ) as a Riesz integralon a k -independent contour: P j ( k ) = i π Z | z − λ j (0) | = r ( β ( k ) − z ) − dz, hence t [ P j ( k )] = P j ( − k ) near zero, identity which remains true on R through analytic continuation.Also, because λ j ( k ) = λ j ( − k ) near k = 0 (due to the CS ′ symmetry and the non-degeneracyof the spectrum of β (0)), this equality remains true on R by analytic continuation.The only thing we still need to prove is the periodicity of eigenvalues and their eigenprojections.We know that λ j ( − /
2) = λ j (1 /
2) = a j from the symmetry, all of them being nondegenerate sinceby assumption β (1 / β ( − / r > | z − a j | ≤ r does not contain other eigenvalues. Then if k is small enough we have: i π Z | z − a j | = r ( β ( − / k ) − z ) − dz = i π Z | z − a j | = r ( β (1 / k ) − z ) − dz due to the periodicity of β , thus P j ( − / k ) = P j (1 / k ) near k = 0, hence they are equaleverywhere through real analyticity. This proves that each P j is periodic. Finally, for small enough k we have: λ j ( − / k ) = Tr { P j ( − / k ) β ( − / k ) } = Tr { P j (1 / k ) β (1 / k ) } = λ j (1 / k )which must hold everywhere so they are periodic. Now we can apply Lemma 2.13 for each λ j obtaining λ j ( k ) = e iφ j ( k ) where φ j is continuous, Z -periodic and even. Then the solution is h ( k ) := P Nj =1 φ j ( k ) P j ( k ). 19 .7.5 The general case Let us go back to (2.24), the starting point of our discussion. Remember that the matrix family β ( · ) has the CS ′ symmetry and we want to unitarily rotate the Ψ m ’s so that the new basis stays CS symmetric, remains at least continuous and is also periodic.Let us rewrite (2.24) by setting k equal to − / m (1 / , k ) = N X n =1 [ β ( k )] nm Ψ n ( − / , k ) . (2.28)We will see in the next lemma that the only relevant interval is k ∈ [ − / , / Lemma 2.15.
Assume that the vectors { e Ξ n ( x, k ) } Nn =1 form an orthonormal basis for P ( x, k ) and are continuous on [ − / , / × R , have the CS symmetry e Ξ n ( x, k ) = e Ξ n ( − x, − k ) andmoreover, e Ξ n ( − / , k ) = e Ξ n (1 / , k ) . Given k ∈ R , we can uniquely write it as k = n + x with n ∈ Z and x ∈ ( − / , / . Then the vectors defined by Ξ n ( k , k ) := e Ξ n ( x, k ) are continuous on R , CS symmetric and Z -periodic.Proof. The periodicity is a consequence of the identity Ξ n ( x + n, k ) = e Ξ n ( x, k ) which is validfor all n ∈ Z and x ∈ ( − / , / CS symmetry. Wehave: Ξ n ( x + n, k ) = e Ξ n ( x, k ) = e Ξ n ( − x, − k ) = Ξ n ( − x − n, − k ) . The main technical result needed for the proof of Theorem 1.8 is the following:
Proposition 2.16.
There exists a family of unitary matrices u ( x, k ) continuous on [ − / , / × R and Z -periodic in k , such that: [ u ( − / , k )] − β ( k ) u (1 / , k ) = Id , (2.29) which also has CS symmetry: θu ( x, k ) θ = u ( − x, − k ) . (2.30) Moreover, the vectors: e Ξ m ( x, k ) := N X m =1 [ u ( x, k )] nm Ψ n ( x, k ) (2.31) obey the conditions needed in Lemma 2.15. Before anything else, let us relate the proposition to the cases in which we can write β ( k ) = e ih ( k ) with some ‘nice’ h ( k ). In all those cases we can put u ( x, k ) = e − ixh ( k ) and we immediately seethat (2.29) means no more than: e − i h ( k ) β ( k ) e − i h ( k ) = Id . Since a ‘good’ logarithm might not always exist, the new important idea is to realize that onecan straighten up the initial basis by performing two successive unitary rotations. First, given β as in the theorem, we will show that we can always find some ‘good’ (i.e. Z -periodic, with CS symmetry and continuous) h ( k ) such thatsup k ∈ R || e − i h ( k ) β ( k ) e − i h ( k ) − Id || ≤ / . (2.32)20hen rotating the Ψ-basis with e − ixh ( k ) we get e Ψ m ( x, k ) := P Nn =1 [ e − ixh ( k ) ] nm Ψ n ( x, k ) whichobey: e Ψ m (1 / , k ) = N X n =1 [ e − i h ( k ) β ( k ) e − i h ( k ) ] nm e Ψ n ( − / , k ) . (2.33)But now the new matching unitary matrix˜ β ( k ) := e − i h ( k ) β ( k ) e − i h ( k ) = Id + e − i h ( k ) [ β ( k ) − e ih ( k ) ] e − i h ( k ) has the CS ′ property, is periodic, continuous and always has − h ( k ) such that ˜ β ( k ) = e i ˜ h ( k ) .By rotating e Ψ n ( x, k ) with e − ix ˜ h ( k ) we obtain our e Ξ n ( x, k ). This also gives an expression forthe unitary u : u ( x, k ) = e − ixh ( k ) e − ix ˜ h ( k ) , e − i ˜ h ( k ) e − i h ( k ) β ( k ) e − i h ( k ) e − i ˜ h ( k ) = Id . (2.34)Now let us give the precise statements. Lemma 2.17.
Assume that we can find M ≥ self-adjoint N × N matrix families H j ( k ) whichare Z -periodic, continuous and with CS symmetry so that: e − i H ( k ) · · · · · e − i H M ( k ) β ( k ) e − i H M ( k ) · · · · · e − i H ( k ) = Id . (2.35) Then u ( x, k ) := e − ixH M ( k ) ...e − ixH ( k ) is unitary, Z -periodic in the second variable and obeys (2.29) and (2.30) . Moreover, the vectors defined in (2.31) obey the conditions needed in Lemma2.15.Proof. Unitarity of u and its periodicity in k are obvious, while the CS symmetry is a consequenceof Lemma 2.8. Also, (2.29) is a direct consequence of (2.35).Now let us consider the vectors in (2.31). The only thing left to be proved is that they are thesame when x = ± /
2. We have:Ψ m ( x, k ) = N X s =1 [ u ( x, k ) − ] sm e Ξ s ( x, k ) . (2.36)Then: e Ξ j (1 / , k ) = N X r =1 [ u (1 / , k )] rj Ψ r (1 / , k ) = N X r =1 [ u (1 / , k )] rj N X m =1 β mr ( k )Ψ m ( − / , k )= N X r =1 [ u (1 / , k )] rj N X m =1 β mr ( k ) N X s =1 [ u ( − / , k ) − ] sm e Ξ s ( − / , k )= N X s =1 [ u ( − / , k ) − β ( k ) u (1 / , k )] sj e Ξ s ( − / , k )= e Ξ j ( − / , k ) , (2.37)where in the last line we used (2.29).The last difficult technical problem we still have to solve is the construction of the approx-imation given in (2.32). When this is done, (2.34) provides the unitary we are looking for inProposition 2.16, which finishes the construction.The following lemma solves the approximation problem and is of independent interest:21 emma 2.18. Let β ( k ) be a continuous family of unitary matrices, Z -periodic and satisfying the CS ′ property. Then there exists a sequence of real analytic families β n ( k ) of unitary matrices, Z -periodic, with nondegenerate spectrum at and / and satisfying the CS ′ symmetrysuch that: lim n →∞ sup k ∈ R || β n ( k ) − β ( k ) || = 0 . (2.38) Proof.
Before giving the proof, let us remark that each β n satisfies the conditions from Subsection2.7.4, hence they can be written as β n ( k ) = e ih n ( k ) where each h n is continuous, periodic and with CS symmetry. This implies (2.32) since || β n ( k ) − β ( k ) || = || Id − e − ih n ( k ) / β ( k ) e − ih n ( k ) / || .Now let us prove the lemma. We split the proof into two parts. First, we construct a continuousfamily β n , with a completely nondegenerate spectrum at 0 and 1 / Z -periodic and with CS ′ symmetry. Second, we show how it can be made real analytic by preserving all other properties. Step 1.
Assume that the spectrum of β (0) consists of 1 ≤ p ≤ N clusters of degenerateeigenvalues labeled as { λ (0) , ..., λ p (0) } in the increasing order of their principal arguments. If s > | k | < s , due to the continuity of β ( · ) we know that the spectrumof β ( k ) will also consist of well separated clusters of eigenvalues. Let C j be a simple, positivelyoriented contour, independent of | k | < s , enclosing the j ’th cluster and no other eigenvalues. LetΠ j ( k ) be the spectral projection of β ( x ) corresponding to the j ’th cluster. We have:Π j ( k ) = i π Z C j ( β ( k ) − z ) − dz, t Π j ( k ) = Π j ( − k ) , lim k → || Π j ( k ) − Π j (0) || = 0 , | k | < s. (2.39)The matrix β ( k ) is block diagonal with respect to the decomposition C N = L p j =1 Π j ( k ) C N , i.e. β ( k ) = P p j =1 Π j ( k ) β ( k )Π j ( k ) . Define e β ( k ) := p X j =1 λ j (0)Π j ( k ) = e i P p j =1 Arg( λ j (0))Π j ( k ) , | k | < s. Clearly, e β ( k ) is unitary, commutes with β ( k ) and t e β ( k ) = e β ( − k ) if | k | < s . Define γ ( k ) := e β − ( k ) β ( k ); we have that γ ( k ) is unitary, commutes with β ( k ), t γ ( k ) = γ ( − k ) if | k | < s andlim k → γ ( k ) = Id. Using the Cayley transform as in Subsection 2.7.3, see formulas (2.26)-(2.27)with φ = π , we can find a self-adjoint matrix e h ( k ) such that γ ( k ) = e i e h ( k ) , e h (0) = 0 , t e h ( k ) = e h ( − k ) , [Π j ( k ) , e h ( k )] = 0 , | k | < s. (2.40)Putting everything together and using that β ( k ) = β ( k + m ) for all m ∈ Z we obtain: β ( k + m ) = e i ( P p j =1 Arg( λ j (0))Π j ( k )+ e h ( k )) , | k | < s, ∀ m ∈ Z . (2.41)We proceed similarly for k ∈ {| k ± / | < s } . Notice that due to the periodicity of β , there willbe the same number of eigenvalue clusters at k = ± / λ j (1 /
2) = λ j ( − /
2) for all 1 ≤ j ≤ p / ≤ N ; note that there is no connection withthe numbering at k = 0. From the Riesz integral formula, choosing a contour ˜ C j around each λ j (1 /
2) = λ j ( − /
2) and using that β ( k ) = β ( k + 1) we obtain Π j ( − /
2) = Π j (1 /
2) and moreover: t Π j ( k ) = Π j ( − k ) , t Π j (1 /
2) = Π j ( − /
2) = Π j (1 / , , | k ± / | < s. (2.42)As in (2.41) we get: β ( k + m ) = e i ( P p / j =1 Arg( λ j (1 / j ( k )+ e h ( k )) , | k ± / | < s, ∀ m ∈ Z . (2.43)In particular, Π j ( ± /
2) are real and symmetric matrices.22he main idea is to locally perturb β around all integers and half-integers such that thespectrum is nondegenerate at these points and all symmetry properties are left unchanged. Letus now describe the perturbation. First, consider k = 0. Since Π j (0) ∗ = Π j (0) = t Π j (0),these matrices are also real and symmetric. Accordingly, Ran(Π j (0)) is invariant under complexconjugation and we can find a real orthonormal basis. Let 1 ≤ r j (0) ≤ N be the multiplicity of λ j (0), i.e. the dimension of Ran(Π j (0)). We have:Π j (0) = r j (0) X l j =1 P j,l j (0) , dim Ran( P j,l j (0)) = 1 , P j,l j = P ∗ j,l j = t P j,l j = P j,l j . (2.44)Define: A j (0) := r j (0) X l j =1 ( l j − P j,l j (0) , ≤ j ≤ p . (2.45)By construction, we have t A j (0) = A j (0). Seen as an operator acting on Ran(Π j (0)), the spectrumof A j (0) is nondegenerate and equals { , , ..., r j (0) − } .Now consider k = ± /
2. Since Π j (1 /
2) = Π j ( − /
2) are real for all 1 ≤ j ≤ p / (see (2.42)),we can mimic the construction in (2.45) and obtain: A j (1 /
2) := r j (1 / X m j =1 ( m j − P j,m j (1 / , ≤ j ≤ p / . (2.46)Now let g s : R [0 , g s ( k ) = g s ( − k )), g s (0) = 1, supp( g s ) ⊂ [ − s/ , s/ v s ( k ) := s X m ∈ Z g s ( k + m ) p X j =1 A j (0) + s X m ∈ Z g s ( k + m + 1 / p / X j =1 A j (1 / . (2.47)From (2.41) and (2.43) we see that β ( k ) = e ih ( k ) on the set:Ω s := [ m ∈ Z {| k − m | < s } ∪ {| k − m − / | < s } . We see that Ω s is symmetric with respect to the origin and consists of a union of disjoint openintervals centered around all integers and half-integers. Also, the support of v s is contained inΩ s , it is periodic and has the CS symmetry. Define β s ( k ) to be e i ( h ( k )+ v s ( k )) if k ∈ Ω s , and let β s ( k ) = β ( k ) outside Ω s . The family β s obeys all the required properties and converges uniformlyin norm to β when s → Step 2.
Here we will show how we can make β s real analytic without affecting the other requiredproperties. In order to simplify notation, we drop the subscript s and we assume that β ( · ) iscontinuous, with CS ′ symmetry, Z -periodic and with completely nondegenerate spectrum at 0and 1 / f ( x ) = π − (1 + x ) − . If ν > f ν ( x ) := ν − f ( x/ν ). The function f is positive, symmetric f ( x ) = f ( − x ), f ν is an approximationof the Dirac distribution and is analytic on the strip {| Im( k ) | < ν } . Then the matrix family µ ν ( k ) := Z R f ν ( k − k ′ ) β ( k ′ ) dk ′ (2.48)is real analytic and Z -periodic. A-priori µ ν ( k ) is neither self-adjoint nor unitary, but it obeys t µ ν ( k ) = µ ν ( − k ). Also, µ ν ( · ) converges uniformly to β ( · ) when ν →
0, which implies:lim ν → sup k ∈ R || µ ν ( k ) µ ν ( k ) ∗ − Id || = 0 . (2.49)23f ν is small enough, the operator µ ν ( k ) µ ν ( k ) ∗ is self-adjoint and positive, hence we can define: β ν ( k ) := { µ ν ( k ) µ ν ( k ) ∗ } − / µ ν ( k ) , β ν ( k ) β ν ( k ) ∗ = Id . (2.50)Clearly, β ν ( · ) is unitary, Z -periodic and converges uniformly to β ( · ) when ν →
0. This guaranteesthat β ν has completely nondegenerate spectrum at 0 and ± / ν small enough. We still needto prove that β ν ( · ) has the CS ′ symmetry and it is real analytic.If ν is small enough we can write: β ν ( k ) = i π Z | z − | =1 / z − / ( µ ν ( k ) µ ν ( k ) ∗ − z ) − µ ν ( k ) dz (2.51)Now the matrix family γ ν ( k ) := µ ν ( k ) µ ν ( k ) ∗ = Z R f ν ( k − k ′ ) f ν ( k − k ′′ ) β ( k ′ ) β ( k ′′ ) ∗ dk ′ dk ′′ has a holomorphic extension to the strip {| Im( k ) | < ν } . If ν is small enough, due to (2.49)it follows that ( µ ν ( k ) µ ν ( k ) ∗ − z ) − exists for all | z − | = 1 / {| Im( k ) | < α } with 0 < α ≪ ν . Together with (2.48) this proves that β ν has a holomorphicextension to {| Im( k ) | < α } .The last thing we need to prove is t β ν ( k ) = β ν ( − k ), which would imply the CS ′ property.From (2.50) we see that if ν is small enough one can write: β ν ( k ) = X n ≥ a n [ µ ν ( k ) µ ν ( k ) ∗ − Id] n µ ν ( k ) , (1 + x ) − / = X n ≥ a n x n . We have already seen that t µ ν ( k ) = µ ν ( − k ) and this leads to t [ µ ν ( k ) ∗ ] = µ ν ( − k ) ∗ . Then: t { [ µ ν ( k ) µ ν ( k ) ∗ − Id] n µ ν ( k ) } = µ ν ( − k )[ µ ν ( − k ) ∗ µ ν ( − k ) − Id] n = [ µ ν ( − k ) µ ν ( − k ) ∗ − Id] n µ ν ( − k )where the second identity can be proved by induction with respect to n . The proof of the lemmais over. We will use the generic notation α and C for a finite number of positive constants which willappear during the proof.The difficulty stems from the fact that since P b is not norm continuous in b , one cannot use thecontinuity approach of [NN2]. The first step in overcoming this difficulty is to use an old ansatz byPeierls [Pe] and Luttinger [Lu] and to couple it with the regularized magnetic perturbation theoryin order to construct (out of the w j,γ ’s) an intermediate orthogonal projection Π b such thatlim b → || Π b − P b || = 0 . (3.1)The second step is to construct a localized orthonormal basis for Ran(Π b ), while the last step isto transfer it to Ran( P b ) by the ‘continuity’ method of [N2, NN2].If | b | is small enough, the orthogonal projection P b can be written as a Riesz integral (see(1.13)): P b = i π Z C ( H b − z ) − dz where C is a simple, positively oriented contour which surrounds σ (hence σ b ) and no other partof the spectrum of H . By writing P b = i π Z C [( H b − z ) − − ( H b − i ) − ] dz = i π Z C ( z − i )( H b − z ) − ( H b − i ) − dz P b has a jointlycontinuous integral kernel P b ( x , x ′ ) and there exist α > C < ∞ such that | P b ( x , x ′ ) | ≤ Ce − α || x − x ′ || , ∀ x , x ′ ∈ R , | b | ≤ b . (3.2)In particular, this holds true for b = 0. Since w j,γ ( x ) = R R P ( x , x ′ ) w j,γ ( x ′ ) d x ′ , we have thateach w j,γ is continuous and by applying the Cauchy-Schwarz inequality and (3.2) we obtain theexistence of some constants C ′ < ∞ and α ′ < α such thatsup x ∈ R ,γ ∈ Γ e α ′ || x − γ || | w j,γ ( x ) | ≤ C ′ . (3.3)This implies the identity: P ( x , x ′ ) = N X j =1 X γ ∈ Γ w j,γ ( x ) w j,γ ( x ′ ) (3.4)where the above series is absolutely convergent; let us show why. The estimate (3.3) can bereinterpreted as | w j,γ ( x ) | ≤ Ce − α || x − γ || for some C < ∞ and α >
0, uniformly in γ and j . Usingthe triangle inequality we obtain: N X j =1 X γ ∈ Γ | w j,γ ( x ) | | w j,γ ( x ′ ) | ≤ const e − α || x − x ′ || / . (3.5)Later we will need the following bound which holds true due to our sparsity assumption on Γ:sup c ∈ R X γ ∈ Γ || c − γ || k e − α || c − γ || < ∞ , ∀ k ≥ . (3.6) (i) . Applying the Riesz integral formula in (1.13) we obtain: (cid:12)(cid:12)(cid:12) P b ( x , x ′ ) − e ibφ ( x , x ′ ) P ( x , x ′ ) (cid:12)(cid:12)(cid:12) ≤ C | b | e − α || x − x ′ || . (3.7)Define: b P b ( x , x ′ ) := e ibφ ( x , x ′ ) P ( x , x ′ ) , (3.8)and denote by b P b the corresponding operator. Applying a Schur-Holmgren estimate using (3.7)leads us to: || b P b − P b || ≤ C | b | , where | P b ( x , x ′ ) − b P b ( x , x ′ ) | ≤ C | b | e − α || x − x ′ || . (3.9)Define the kernel: e P b ( x , x ′ ) = N X j =1 X γ ∈ Γ e ibφ ( x ,γ ) w j,γ ( x ) e ibφ ( x ′ ,γ ) w j,γ ( x ′ ) , (3.10)and by e P b the corresponding operator. The series is absolutely convergent (see the estimate in(3.5)) and defines a jointly continuous function. Moreover, we have: | e P b ( x , x ′ ) | ≤ C e − α || x − x ′ || . (3.11)From the definition of φ ( x , x ′ ) (see (1.14) and (1.11)) it follows that it equals the magnetic fluxthrough the triangle with corners situated at 0, x and x ′ .25et f l ( x , y , x ′ ) denote the magnetic flux through the triangle with corners situated at x , y and x ′ . Using Stokes’ theorem and the fact that | B ( x ) | ≤ f l ( x , y , x ′ ) = φ ( x , y ) + φ ( y , x ′ ) − φ ( x , x ′ ) , | f l ( x , y , x ′ ) | ≤ | x − y | | y − x ′ | . (3.12)Using the various definitions, the triangle and Cauchy-Schwarz inequality we obtain: | b P b ( x , x ′ ) − e P b ( x , x ′ ) | ≤ N X j =1 X γ ∈ Γ | sin( b f l ( x , γ, x ′ ) / | | w j,γ ( x ) | | w j,γ ( x ′ ) |≤ | b | N C e − α || x − x ′ || / sup c ∈ R X γ ∈ Γ || c − γ || e − α || c − γ || , which leads to the existence of a C < ∞ and α > | b P b ( x , x ′ ) − e P b ( x , x ′ ) | ≤ C | b | e − α || x − x ′ || , || b P b − e P b || ≤ C | b | . (3.13)Consider the closed subspace S b := closure { Span { e ibφ ( · ,γ ) w j,γ ( · ) : γ ∈ Γ , j ∈ { , , ..., N }}} and its associated Gram-Schmidt matrix M b ( γ, j ; γ ′ , j ′ ) := D e ibφ ( · ,γ ′ ) w j ′ ,γ ′ ( · ) | e ibφ ( · ,γ ) w j,γ ( · ) E L ( R ) . (3.14) Lemma 3.1.
Consider the self-adjoint operator M b acting on the space l (Γ) ⊗ C N given by thematrix elements M b ( γ, j ; γ ′ , j ′ ) . If b is small enough, then there exists some α > and C < ∞ such that if | b | ≤ b : | δ jj ′ δ γγ ′ − M b ( γ, j ; γ ′ , j ′ ) | ≤ C | b | e − α || γ − γ ′ || . (3.15) Moreover, M b ≥ / , and if β ∈ { / , } then | δ jj ′ δ γγ ′ − M − βb ( γ, j ; γ ′ , j ′ ) | ≤ C | b | e − α || γ − γ ′ || . (3.16) Finally, the vectors: Ψ j,γ,b ( x ) := X γ ′ ∈ Z N X j ′ =1 [ M − / b ]( γ ′ , j ′ ; γ, j ) e ibφ ( x ,γ ′ ) w j ′ ,γ ′ ( x ) (3.17) form an orthonormal basis of S b , being uniformly exponentially localized around γ , i.e. | Ψ j,γ,b ( x ) | ≤ C e − α || x − γ || , | Ψ j,γ,b ( x ) − e ibφ ( x ,γ ) w j,γ ( x ) | ≤ C | b | e − α || x − γ || . (3.18) Proof.
Define D b := M b − Id. Since δ jj ′ δ γγ ′ = h w j ′ ,γ ′ ( · ) | w j,γ ( · ) i L ( R ) and φ ( γ, γ ) = 0 we have: D b ( γ, j ; γ ′ , j ′ ) = e ibφ ( γ ′ ,γ ) Z R d w j ′ ,γ ′ ( x ) w j,γ ( x ) (cid:16) e ib fl ( γ ′ , x ,γ ) − (cid:17) d x and | D b ( γ, j ; γ ′ , j ′ ) | ≤ | b | e − α ′ || γ − γ ′ || Z R d || γ ′ − x || e α ′ || γ ′ − x || | w j ′ ,γ ′ ( x ) | || γ − x || e α ′ || γ − x || | w j,γ ( x ) | d x for some small enough α ′ >
0. Using the Cauchy-Schwarz inequality we find: | D b ( γ, j ; γ ′ , j ′ ) | ≤ | b | Ce − α || γ − γ ′ || . (3.19)26f n ≥ D nb ( γ, j ; γ ′ , j ′ ) = X j ,γ ... X j n − ,γ n − D b ( γ, j ; γ , j ) ...D b ( γ n − , j n − ; γ ′ , j ′ )and (3.19) implies: e α || γ − γ ′ || | D nb ( γ, j ; γ ′ , j ′ ) | ≤ | b | CN n − sup k,j ′′ ,γ ′′ X c ∈ Γ | D b ( γ ′′ , j ′′ ; c , k ) | e α || γ ′′ − c || ! n − ≤ | b | n ˜ C n , ∀ n ≥ . (3.20)The estimate (3.19) also implies that D b goes to zero with b in the operator norm, hence M b isclose to the identity operator, thus M − βb = (Id + D b ) − β exists and can be expressed as a norm-convergent power series around zero. Then (3.20) implies (3.16). Finally, it is straightforwardto check that the vectors defined in (3.17) are orthogonal and span S b , while (3.18) is an easyconsequence of (3.16) and the triangle inequality.The orthogonal projection associated to the subspace S b isΠ b := X γ ∈ Γ N X j =1 | Ψ j,γ,b ih Ψ j,γ,b | . (3.21)Under the conditions of Lemma 3.1, using (3.17) and the self-adjointness of M b we obtain that Π b has a jointly continuous integral kernel given by:Π b ( x , x ′ ) = X γ ′ ,γ ′′ ∈ Γ N X j ′ ,j ′′ =1 M − b ( γ ′ , j ′ ; γ ′′ , j ′′ ) e ibφ ( x ,γ ′ ) w j ′ ,γ ′ ( x ) e ibφ ( γ ′′ , x ′ ) w j ′′ ,γ ′′ ( x ′ )where the series converges absolutely. Using (3.16) and (3.10) we find that there exists some α > | Π b ( x , x ′ ) − e P b ( x , x ′ ) | ≤ C | b | e − α || x − x ′ || . (3.22)Using (3.9) and (3.13) we obtain: | Π b ( x , x ′ ) − P b ( x , x ′ ) | ≤ C | b | e − α || x − x ′ || , || Π b − P b || ≤ C | b | , (3.23)which finishes the proof of (3.1).The next step is to consider the Sz.-Nagy unitary operator U b which intertwines P b with Π b ,i.e. P b U b = U b Π b , given (as in (2.4)) by the formula: U b := { P b Π b + (Id − P b )(Id − Π b ) }{ Id − (Π b − P b ) } − / . (3.24)A consequence of (3.23) is that for small enough b , the operator U b − Id has a jointly continuousintegral kernel which obeys the same estimate as in (3.23). Then the set of vectors:Ξ j,γ,b := U b Ψ j,γ,b = X γ ′ ∈ Γ N X j ′ =1 M − / b ( γ ′ , j ′ ; γ, j )[ U b e ibφ ( · ,γ ′ ) w j ′ ,γ ′ ( · )] (3.25)form an orthonormal basis of Ran( P b ) satisfying (1.15) and (1.16).27 .2 Proof of (ii) . Remember that here the magnetic field is constant with magnitude B ( x ) = 1 and Γ is a periodiclattice embedded in R . In this case, the phase is not just antisymmetric but it is also bilinear,since now φ ( x , x ′ ) = ( x ′ x − x x ′ ) or, with the usual abuse of language, φ ( x , x ′ ) = B · ( x ′ ∧ x ).The identity ( x − y ) ∧ ( x ′ − y ) = − x ′ ∧ x + y ∧ x + x ′ ∧ y implies: φ ( x , y ) + φ ( y , x ′ ) − φ ( x , x ′ ) = φ ( x ′ − y , x − y ) = φ ( x ′ − y , x − x ′ ) . (3.26)Denote by τ bγ the unitary magnetic translation given by:[ τ bγ f ]( x ) = e ibφ ( x ,γ ) f ( x − γ ) . Using (1.17), (3.26) and (3.14) we obtain: M b ( γ, j ; γ ′ , j ′ ) = h τ bγ ′ ( w j ′ , ) | τ bγ ( w j, ) i L ( R ) =: e ibφ ( γ,γ ′ ) M b ( γ − γ ′ ; j, j ′ ) , where M b ( γ ′′ ; j, j ′ ) = Z R e − ibφ ( y ,γ ′′ ) w j, ( y − γ ′′ ) w j ′ , ( y ) d y = h w j ′ , | τ bγ ′′ ( w j, ) i L ( R ) . (3.27) Lemma 3.2.
There exists some b small enough such that the operator M − / b has a matrix ofthe form M − / b ( γ, j ; γ ′ , j ′ ) =: e ibφ ( γ,γ ′ ) T b ( γ − γ ′ ; j, j ′ ) , where |T b ( γ ′′ ; j, j ′ ) | ≤ Ce − α | γ ′′ | for every ≤ b ≤ b .Proof. In the proof of Lemma (3.1) we introduced the operator D b = M b − Id in l (Γ) ⊗ C N andshowed that if | b | is small enough then M − / b = (Id + D d ) − / exists as a power series in D b . Thematrix elements of D b inherit the symmetry properties of M b : D b ( γ ′′ ; j, j ′ ) := M b ( γ ′′ ; j, j ′ ) − δ γ ′′ δ jj ′ , D b ( γ, j ; γ ′ , j ′ ) = e ibφ ( γ,γ ′ ) D b ( γ − γ ′ ; j, j ′ ) . Then using (3.26) and the fact that Γ is a lattice we obtain: D b ( γ, j ; γ ′ , j ′ ) = N X j ′′ =1 X γ ′′ ∈ Γ e ib ( φ ( γ,γ ′′ )+ φ ( γ ′′ ,γ ′ )) D b ( γ − γ ′′ ; j, j ′′ ) D b ( γ ′′ − γ ′ ; j ′′ , j ′ )= e ibφ ( γ,γ ′ ) N X j ′′ =1 X γ ′′ ∈ Γ e ibφ ( γ ′ − γ ′′ ,γ − γ ′ ) D b ( γ − γ ′′ ; j, j ′′ ) D b ( γ ′′ − γ ′ ; j ′′ , j ′ )=: e ibφ ( γ,γ ′ ) D (2) b ( γ − γ ′ ; j, j ′ ) , (3.28)where D (2) b (˜ γ ; j, j ′ ) = N X j ′′ =1 X γ ′′ ∈ Γ e ibφ (˜ γ,γ ′′ ) D b (˜ γ − γ ′′ ; j, j ′′ ) D b ( γ ′′ ; j ′′ , j ′ ) . If n ≥ D nb ( γ, j ; γ ′ , j ′ ) := e ibφ ( γ,γ ′ ) D ( n ) b ( γ − γ ′ ; j, j ′ ) where: D ( n ) b (˜ γ ; j, j ′ ) = N X j ′′ =1 X γ ′′ ∈ Γ e ibφ (˜ γ,γ ′′ ) D ( n − b (˜ γ − γ ′′ ; j, j ′′ ) D b ( γ ′′ ; j ′′ , j ′ ) . This structure will be preserved for any convergent series in D b , while the exponential localizationis a direct consequence of (3.16). 28sing the above expression for M − / b ( γ, j ; γ ′ , j ′ ) in (3.17) we obtain:Ψ j,γ,b ( x ) = X γ ′ ∈ Γ X j ′ =1 e ibφ ( γ ′ ,γ ) T b ( γ ′ − γ ; j ′ , j ) e ibφ ( x ,γ ′ ) w j ′ , ( x − γ ′ ) . (3.29)We note the identity φ ( γ − γ ′ , x − γ ′ ) = φ ( x − γ, γ ′ − γ )which together with (3.26) gives: φ ( x , γ ′ ) + φ ( γ ′ , γ ) − φ ( x , γ ) = φ ( γ − γ ′ , x − γ ′ ) = φ ( x − γ, γ ′ − γ ) . Replacing this in (3.29) we obtain:Ψ j,γ,b ( x ) = e ibφ ( x ,γ ) X γ ′ ∈ Γ N X j ′ =1 T b ( γ ′ − γ ; j ′ , j ) e ibφ ( x − γ,γ ′ − γ ) w j ′ , ( x − γ ′ )= e ibφ ( x ,γ ) X γ ′′ ∈ Γ N X j ′ =1 T b ( γ ′′ ; j ′ , j ) e ibφ ( x − γ,γ ′′ ) w j ′ , ( x − γ − γ ′′ )= τ bγ X γ ′′ ∈ Γ N X j ′ =1 T b ( γ ′′ ; j ′ , j ) e ibφ ( · ,γ ′′ ) w j ′ , ( · − γ ′′ ) ( x ) . (3.30)Denote by: u j,b := X γ ′′ ∈ Γ N X j ′ =1 T b ( γ ′′ ; j ′ , j ) τ bγ ′′ ( w j ′ , ) . (3.31)From (3.30) we have Ψ j,γ,b = τ bγ ( u j,b ). The last technical result we need is the following: Lemma 3.3.
Both P b and Π b commute with the magnetic translations τ bγ , and so does U b .Proof. Since H b commutes with τ bγ , so does P b . We only need to prove this for Π b . Using theformula Ψ j,γ,b = τ bγ ( u j,b ), we can express the kernel of Π b (see (3.21) as:Π b ( x , x ′ ) = X γ ′ ∈ Γ N X j =1 e ibφ ( x ,γ ′ ) u j,b ( x − γ ′ ) e − ibφ ( x ′ ,γ ′ ) u j,b ( x ′ − γ ′ ) . We have:[ τ bγ Π b ]( x , x ′ ) = X γ ′ ∈ Γ N X j =1 e ibφ ( x ,γ ) e ibφ ( x − γ,γ ′ ) u j,b ( x − γ ′ − γ ) e − ibφ ( x ′ ,γ ′ ) u j,b ( x ′ − γ ′ ) (3.32)and [Π b τ bγ ]( x , x ′ ) = X γ ′ ∈ Γ N X j =1 e ibφ ( x ,γ ′ ) u j,b ( x − γ ′ ) e − ibφ ( x ′ + γ,γ ′ ) e ibφ ( x ′ + γ,γ ) u j,b ( x ′ − γ ′ + γ ) . Changing γ ′ with γ ′ + γ in the second formula we obtain:[Π b τ bγ ]( x , x ′ ) = X γ ′ ∈ Γ N X j =1 e ibφ ( x ,γ ′ + γ ) u j,b ( x − γ ′ − γ ) e − ibφ ( x ′ + γ,γ ′ + γ ) e ibφ ( x ′ + γ,γ ) u j,b ( x ′ − γ ′ ) . (3.33)Using the bi-linearity and antisymmetry of φ we see that the magnetic phases in (3.32) and (3.33)are the same and the two kernels coincide. 29enote by e u j,b := U b u j,b . Since τ bγ commutes with U b , we obtain thatΞ j,γ,b = U b Ψ j,γ,b = U b τ bγ ( u j,b ) = τ bγ ( e u j,b )and: P b = N X j =1 X γ ∈ Γ | τ bγ ( e u j,b ) ih τ bγ ( e u j,b ) | , Ξ j, ,b = e u j,b . (3.34)In particular, this proves (1.18). The last thing we need to prove is (1.19), or equivalently: e u j,b = e u j, − b . (3.35)This last identity would be implied by two others: u j,b = u j, − b and U b f = U − b f , ∀ f ∈ L ( R ) . (3.36)In order to prove u j,b = u j, − b , we see from (3.31) and (1.17) that the only thing we need for thisis T b ( γ ′′ ; j ′ , j ) = T − b ( γ ′′ ; j ′ , j ) . (3.37)From (3.27) and (1.17) we obtain that M b ( γ ′′ ; j ′ , j ) = M − b ( γ ′′ ; j ′ , j ), hence M b ψ = M − b ψ for all ψ ∈ l (Γ) ⊗ C N . By functional calculus, the same identity is obeyed by M − / b which proves (3.37),hence u j,b = u j, − b . Since Ψ j,γ,b = τ bγ ( u j,b ) we immediately get Ψ j,γ,b = Ψ j,γ, − b . From (3.21) weobtain that Π b f = Π − b f for all f ∈ L ( R ). Moreover, since H b g = H − b g for all g ∈ C ∞ ( R ), astandard argument shows that P b f = P − b f for all f ∈ L ( R ). Now using the formula (3.24) weobtain the second identity in (3.36) and the proof of Theorem 1.11 is over. Acknowledgements
H.C. acknowledges financial support from Grant 4181-00042 of the Danish Council for IndependentResearch | Natural Sciences, and from a Bitdefender Invited Professor Scholarship with IMAR,Bucharest. Both I.H. and G.N. acknowledge financial support from VELUX Visiting ProfessorProgram and from Aalborg University.The authors thank D. Monaco and G. Panati for several very useful discussions.
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