Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators
aa r X i v : . [ c ond - m a t . o t h e r] F e b Optimally localized Wannier functions for quasione-dimensional nonperiodic insulators
H. D. Cornean , A. Nenciu , G. Nenciu Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej7G, DK-9220 Aalborg, Denmark Faculty of Applied Sciences University “Politehnica” of Bucharest,Splaiul Independentei 313, RO-060042 Bucharest, Romania Faculty of Physics, University of Bucharest,P.O. Box MG 11, RO-077125 Bucharest, RomaniaandInstitute of Mathematics of the Romanian Academy,P.O. Box 1-764, RO-014700 Bucharest, Romania
Abstract
It is proved that for general, not necessarily periodic quasi one dimensional systems, theband position operator corresponding to an isolated part of the energy spectrum has discretespectrum and its eigenfunctions have the same spatial localization as the corresponding spec-tral projection. As a consequence, an eigenbasis of the band position operator provides a basisof optimally localized (generalized) Wannier functions for quasi one dimensional systems, andthis proves the strong Marzari-Vanderbilt conjecture. If the system has some translation sym-metries (e.g. usual translations, screw transformations), they are ”inherited” by the Wannierbasis.
Wannier functions (WF) were introduced by Wannier in 1937 [1] as bases in subspaces of statescorresponding to energy bands in solids, bases consisting of exponentially localized functions (local-ized orbitals). For periodic crystals they are defined as Fourier transform of Bloch functions of thecorresponding bands. Since then WF proved to be a key tool in quantum theory of solids as theyprovide a tight binding description of the electronic band structure of solids. At the conceptuallevel they lay at the foundation of all effective mass type theories e.g the famous Peierls-Onsagersubstitution describing the dynamics of Bloch electrons in the presence of an external magneticfield (see e.g.[2]and references therein). At the quantitative level, especially after the seminal pa-per by Marzari and Vanderbilt [3], WF become an effective tool in ab initio computational studiesof electronic properties of materials. Moreover during the last decades WF proved to be an es-sential ingredient in the study of low dimensional nanostructures such as linear chains of atoms,nanowires, nanotubes etc (see e.g. [4],[5]). In particular WF are essential for most formulationsof transport phenomena using real space Green’s function method based on Landauer-B¨uttikerformalism both at rigorous [6] and computational levels [7],[4].A few remarks are in order here. The first one is that realistic low dimensional systems arenot strictly one (two) dimensional but rather quasi one (two) dimensional and one has to takeinto account the (restricted) motion along perpendicular directions. This adds specific features asfor example the screw symmetry in nanotubes and nanowires absent in strictly one dimensionalsystems. The second one is that realistic systems, due to the presence of defects, boundaries,randomness etc, do not have usually full translation symmetry and this ask for a theory of WF1ot based on Bloch formalism. Finally let us remind that contrary to a widespread opinion (seee.g. the discussion in [2]) that WF always exist for isolated band in solids this is not true. Moreprecisely, in more than one dimension there are subtle topological obstructions and these are relatedto the QHE [8], [9], [10]: a band for which WF are known to exist gives no contribution to thequantum Hall current. It is then crucial to have rigorous proofs of the existence of exponentiallylocalized WF.For one dimensional periodic systems the existence of exponentially localized WF has beenproved by Kohn in his classic paper [11] about analytic structure of Bloch functions. An extensionof Kohn analysis to quasi one dimensional systems has been done recently by Prodan [12]. As forhigher dimensions it was known since the work by des Cloizeaux [13] [14] that there are obstructionsto the existence of exponentially localized WF and that these obstructions are of topological origin(more precisely as explicitly stated in [15] these obstructions are connected to the topology of avector bundle of orthogonal projections). The fact that for simple bands of time reversal invariantsystems the obstructions are absent was proved by des Cloizeaux [13] [14] under the additionalcondition of the existence of centre of inversion and by Nenciu [15] in the general case. Whilethe proofs in [13] [14], [15] did not use the vector bundle theory it was suggested in [2],[16] thatthe characteristic classes theory in combination with some deep results in the theory of analyticfunctions of several complex variables (Oka principle) can be used to give alternative proof ofthe above results and to extend them to composite bands of time reversal symmetric systems.This has been substantiated recently in [17], [10] where the existence of exponentially localizedWannier functions has been proved for composite bands of time reversal symmetric systems intwo and three dimensions settling in the affirmative a long standing conjecture. In conclusion thesituation is satisfactory as far as periodic time reversal symmetric Hamiltonians are considered (asalready mentioned for Hamiltonians which are not time reversal symmetric exponentially localizedWannier functions might not exists).As already said above both the theory and applications of Wannier functions boosted sinceMarzari and Vanderbilt [3], introduced studied and proposed methods to compute the so calledmaximally localized Wannier functions (MLWFs) defined by the fact that they minimize theposition mean square deviation. It was conjectured in [3] that they can be chosen to be realfunctions and that they have ”optimal” exponential localization in the sense that they have thesame exponential localization as the integral kernel of the projection operator of the correspondingband. MLWFs proved to be an invaluable tool in the theory of electronic properties of periodicmedia especially in the modern theory of electronic polarizability (see e.g. [18] and referencestherein).In the one dimensional case the theory of MLWFs is much more developed. It is known [3]that MLWFs are identical to the eigenfunctions of the ”band position” operator and then they areunique (up to uninteresting phases) and can be chosen to be real functions. Moreover the phasesof the corresponding Bloch functions are related to the parallel transport procedure [3], [19].Recently a detailed study of Wannier functions, including their exponential decay, emphasizingthe difference between the cases with and without inversion symmetry appeared in [20]. In thesame paper there are pointed out situations in which the Wannier functions could decay slowerthan the kernel of the projector, which shows that choosing the optimal phase is not a trivial task.Our results show that that by choosing the right phase one must always obtain an optimal decay.Motivated by the great interest in nonperiodic structures much effort has been devoted toextend the results about existence of exponentially localized bases for isolated bands in nonperiodicsystems. The basic difficulty stems from the fact that for nonperiodic systems one cannot defineWannier functions as Fourier transforms of the Bloch functions. One way out of the difficultyis to start from the periodic case or tight-binding limit where the Wannier functions are knownto exist and and use perturbation or “continuity” arguments. The basic idea is that since theobstructions are of topological origin the existence of exponentially localized WF is stable againstperturbations. Indeed along these lines it has been possible to prove the existence of (generalized)WF for a variety of nonperiodic systems [21], [22], [23], [2], [16]. Since in the periodic casethe obstructions to the existence of exponentially localized WF are absent [13],[14],[15] in onedimension it was naturally to conjecture [16],[24] that in one dimension WF exist for all isolated2ands irrespective of periodicity properties.The first problem to be solved was to find an alternative definition of WF. The basic idea goesback to Kivelson [25], who proposed to define the generalized WF as eigenfunctions of the “bandposition” operator. To substantiate the idea one has to prove that the band position operatoris self-adjoint, has discrete spectrum and its eigenfunctions are exponentially localized. For theparticular case of a periodic one dimensional crystal with one defect Kivelson proved that theeigenfunctions of the band position operator are indeed exponentially localized and asked for ageneral proof. In the general case, by a bootstrap argument, Niu [24] argued that the eigenfunctionsof the band position operator (if they exist) are at least polynomially localized. In full generalitythe fact that for all isolated parts of the spectrum the band position operator is self-adjoint, hasdiscrete spectrum and its eigenfunctions are exponentially localized has been proved in [26].In this paper we extend the results in [26] to quasi one-dimensional systems i.e. three dimen-sional systems for which the motion extends to infinity only in one direction. In addition we addthe result (which is new even in the strictly one dimensional case) that (see Theorem 2 below fordetails) the “density” of WF is uniformly bounded. While the main ideas of the proof are the sameas in [26] there are major differences both at the technical and physical level. In particular forquasi one dimensional systems with screw symmetry the constructed WF inherits this symmetrya property which is very useful in computational applications. Finally let us point out that as inthe periodic case, generalized WF defined as eigenfunctions of the band position operator havevery nice properties e.g. they are (up to uninteresting phases) uniquely defined and for real (i.e.time reversal invariant) Hamiltonians they can be chosen to be real functions and this solves forthe general quasi one dimensional case the “strong conjecture” in Section V. of [3]. As for theirexponential localization we have the following ”optimality” result (see Proposition 3 for a precisestatement) which seems to be new even in the one dimensional periodic case: the eigenfunctionsof the band position operator have the same exponential localization as the integral kernel of theprojection operator of the corresponding band.
Consider in L ( R ) the following Hamiltonian describing a particle subjected to a scalar potential V : H = P + V, P = − i ∇ , sup x ∈ R Z | x − y |≤ | V ( y ) | d y < ∞ (2.1)which, as is well known (see [27]), is essentially self-adjoint on C ∞ ( R ). We have already said inthe introduction that we are interested in potentials V which tend to zero as the distance fromthe Ox axis tends to infinity. Let us now be more precise. The notation x = ( x , x ⊥ ) will beused throughout the paper. For any R >
0, define: I V ( R ) := sup x ∈ R , | x ⊥ |≥ R Z | x − y |≤ | V ( y ) | d y . (2.2)The decay assumption for V will be: lim R →∞ I V ( R ) = 0 . (2.3)It is easy to see that [0 , ∞ ) ⊂ σ ( H ) (using a Weyl sequence argument), thus the only region where H might have an isolated spectral island is below zero. Now suppose that σ is such an isolatedpart of the spectrum and define: − E + := sup { E : E ∈ σ } < . (2.4)If Γ is a positively oriented contour of finite length enclosing σ , then the spectral subspacecorresponding to σ is: K := Ran( P ) , P = i π Z Γ ( H − z ) − dz. (2.5)3t a heuristic level, due to the fact that the wave packets from K cannot propagate in theclassically forbidden region (see (2.4) and (2.3)), at negative energies the motion is confined nearthe Ox axis, i.e. the system has a quasi one dimensional behavior. The following proposition states the ”localization” properties of P . On one hand, this give aprecise meaning to the previously discussed quasi one dimensional character, and on the otherhand it provides some key ingredients to the proof of exponential localization of eigenfunctions ofthe band position operator.Let a ∈ R , and let h X k ,a i be the multiplication operator corresponding to: g a ( x ) := p ( x − a ) + 1 , (2.6)and h X ⊥ i be multiplication operator given by: g ⊥ ( x ) := p | x ⊥ | + 1 . (2.7) Proposition 1.
There exist α k > , α ⊥ > , M < ∞ such that: sup a ∈ R k e α k h X k ,a i P e − α k h X k ,a i k≤ M, and (2.8) k e α ⊥ h X ⊥ i P e α ⊥ h X ⊥ i k≤ M. (2.9)The proof of Proposition 1 will also give values for α k and α ⊥ . In particular α ⊥ can be any numberstrictly smaller than p E + .We now can formulate the main technical result of this paper. To emphasize its generality westress that its proof only uses the decay condition (2.3) and the existence of an isolated part ofthe spectrum satisfying (2.4). Theorem 2.
Let X k be the operator of multiplication with x in L ( R ) and consider in K theoperator ˆ X k := P X k P (2.10) defined on D ( ˆ X k ) = D ( X k ) ∩ K . Then i . ˆ X k is self-adjoint on D ( ˆ X ) ; ii . ˆ X k has purely discrete spectrum; iii . Let g ∈ G := σ ( ˆ X k ) be an eigenvalue, m g its multiplicity, and { W g,j } ≤ j ≤ m g an orthonormalbasis in the eigenspace of ˆ X corresponding to g . Then for all β ∈ [0 , , there exists M < ∞ independent of g , j and β such that: Z R e − β ) α k | x − g | e βα ⊥ | x ⊥ | | W g,j ( x ) | d x ≤ M , (2.11) where α k and α ⊥ are the same exponents as those provided by the proof of Proposition 1; iv . Let a ∈ R and L ≥ . Denote by N ( a, L ) the total multiplicity of the spectrum of ˆ X k containedin [ a − L, a + L ] . Then there exists M < ∞ such that N ( a, L ) ≤ M · L. (2.12)Finally, we turn to the question of optimal localization properties of our Wannier functions.Theorem 2 provides an optimal exponential decay on the transverse direction, but in the paralleldirection it only implies a decay which is bound by the maximal decay of the resolvent in thegap. The conjecture on optimal exponential decay, as stated in Section V of [3], is whether the4 g,j ’s have the same exponential decay as the integral kernel P ( x , y ) of P (which can be largerthan the maximal decay of the resolvent in the gap; we are indebted to one of the referees forpointing this to us). Concerning this issue, we have the following result showing the optimality ofthe ”parallel” decay of W g,j at the exponential level. Proposition 3.
Assume that for all α < α we are given an a priori bound sup a ∈ R k e α h X k ,a i P e − α h X k ,a i k < ∞ . (2.13) Then for all α < α there exists M ( α ) , independent of g and j , such that Z R e α | x − g | | W g,j ( x ) | d x ≤ M ( α ) . (2.14) Remark . Here α is the ”exact” exponential decay of P ( x , y ). In certain particular periodiccases one might obtain a power-like asymptotic behavior of e α | x − y | P ( x , y ) in the variables x , y . We cannot say anything about an eventual asymptotic behavior of e α | x − g | W g,j ( x ). Butdue to the generality of the setting, we consider our result to be optimal. We come now to the case when V (hence H ) has additional symmetries. The point here is thatalthough the Wannier functions are not eigenfunctions of H , one would like them to inherit insome sense the symmetries of H . The reason is that usually the Wannier basis is used in order towrite down an effective Hamiltonian in K , and one would like this effective Hamiltonian to inheritas much as possible the symmetries of H .First we comment on time reversal invariance. Since V ( x ) is real, H commutes with the anti-unitary operator induced by complex conjugation. It follows (see (2.5)) that P and ˆ X k are alsoreal, thus the eigenfunctions of ˆ X k can be chosen to be real. Hence Theorem 2 provides us witha Wannier basis which is time reversal invariant.Second we consider the so called ”screw-symmetry” along the Ox -axis, of much interest inthe physics of carbon nanotubes. Namely, writing x ⊥ = ( r, θ ) , r ≥ , θ ∈ [0 , π ) , (2.15)one assumes that for some θ ∈ [0 , π ) we have: V ( x , r, θ ) = V ( x + 1 , r, θ + θ ) . (2.16)Here θ + θ has to be understood modulo 2 π . Defining the screw-symmetry operators T θ n by:( T θ n f )( x , r, θ ) := f ( x − n, r, θ − nθ ) , (2.17)one has a (unitary!) representation of Z in L ( R ). Taking into account (2.16) and the fact that[ − ∆ , T θ n ] = 0 (use cylindrical coordinates to prove this), one obtains:[ H, T θ n ] = 0 , (2.18)and then from functional calculus and (2.5):[ P , T θ n ] = 0 . (2.19)In particular, this implies that the family { T θ n } n ∈ Z induces a unitary representation of Z in K .Moreover, from (2.10) and 2.19) one obtains:[ T θ n , ˆ X k ] = nT θ n . (2.20)5et p < ∞ be the number of eigenvalues of ˆ X k in the interval [0 , { g j } pj =1 be the distincteigenvalues (each with multiplicity m j < ∞ ). We have:ˆ X k W g j ,α j = g j W g j ,α j , α j = 1 , , ..., m g j . (2.21)From (2.20 ) and (2.21) one obtains that for all g j , α j , n ∈ Z :ˆ X k T θ n W g j ,α j = ( g j + n ) T θ n W g j ,α j . (2.22)Conversely, for every other g ∈ σ ( ˆ X k ), choose an eigenvector W g . We can find n ∈ Z such that g + n ∈ [0 , X k T θ n W g = ( g + n ) T θ n W g , it means that g + n must be one of the g j ’sconsidered above. Therefore we proved the following corollary: Corollary 4.
The spectrum of ˆ X k consists of a union of p ladders: G = ∪ pj =1 G j , G j = { g : g = g j + n, n ∈ Z } , j ∈ { , , ..., p } , (2.23) and an orthonormal basis in K can be chosen as: W n,g j ,α j := W g j + n,α j := T θ n W g j ,α j , (2.24) n ∈ Z , j ∈ { , , ..., p } , α j ∈ { , , ..., m g j } . It is interesting to express the effective Hamiltonian P HP as an infinite matrix with the helpof the Wannier basis. For notational simplicity we relabel the pair ( g j , α j ) as l ∈ { , , ..., N c = P pj =1 m g j } and write the Wannier basis as { W n,l } n ∈ Z , l ∈{ , ,...,N c } . Note that N c is nothing thatthe number of Wannier functions per unit cell [0 , h θ l,k ( m, n ) := h W m,l , HW n,k i . (2.25)The important fact is that in spite of a rotation with an angle θ for which it might happen that θ π to be irrational, from (2.18) and (2.24) one obtains (with the usual abuse of notation): h θ l,k ( m, n ) = h θ l,k ( m − n ) . (2.26)Then a standard computation gives the effective Hamiltonian as an operator in ( l ) N c which is of standard translation invariant tight binding type:( h θ eff φ ) l ( m ) := X k,n h θ l,k ( m − n ) φ k ( n ) . (2.27)This is another consequence of the quasi one-dimensional character of the motion for negativeenergies. More precisely, it reflects the fact that for arbitrary values of θ , since T θ n is a unitaryrepresentation of Z , one can still develop a Bloch type analysis but with a more complicated formof ”Bloch” functions: Ψ k ( x ) = e ikx u k ( x ) , u k ( x ) = T θ n u k ( x ) . (2.28)However, due to the complicated symmetry of the resulting Bloch functions (which does not allowto represent the fiber Hamiltonian as a differential operator on the unit cell with ”simple” boundaryconditions), the analysis gets much harder. The Bloch analysis reduces to the standard one (witha larger unit cell) for rational values of θ π . This section is devoted to the proof of Proposition 1, Theorem 2 and Proposition 3. A certainnumber of unimportant finite positive constants appearing during the proof will be denoted by M . One of the key ingredients in the proofs is the exponential decay of the integral kernel of theresolvent of Schr¨odinger operators. This is an elementary result in the Combes-Thomas-Agmontheory of weighted estimates. We summarize the needed result in:6 emma 5. Let W be a potential such that sup x ∈ R R | x − y |≤ | W ( y ) | d y < ∞ . Define K := P + W ( x ) as an operator sum, and let h be a real function satisfying: h ∈ C ∞ ( R ) , sup x ∈ R {|∇ h ( x ) | + | ∆ h ( x ) |} = m < ∞ . (3.1) Fix z ∈ ρ ( H ) . Then there exists α z > such that k e α z h ( K − z ) − e − α z h k ≤ M, (3.2) k e α z h P j ( K − z ) − e − α z h k ≤ M, (3.3) where P j = − i ∂∂x j , j ∈ { , , } . Without giving the details of the proof of Lemma 5, for later use we write down a key identityin (3.5): under the condition1 + α z ( ± i P · ∇ h ± i ∇ h · P − α z |∇ h | )( K − z ) − invertible (3.4)one has e ± α z h ( K − z ) − e ∓ α z h (3.5)= ( K − z ) − [1 + α z ( ± i P · ∇ h ± i ∇ h · P − α z |∇ h | )( K − z ) − ] − . Then (3.4) holds true if for example α z > Take Γ in (2.5) a contour of finite length enclosing σ and satisfyingdist(Γ , σ ( H )) = 12 dist( σ , σ ( H ) \ σ ) . (3.6)Then since |∇ g a | ≤ | ∆ g a | ≤
2, the estimate (2.8) follows directly from Lemma 5 by taking α k sufficiently small such that for all z ∈ Γ: k α k ( i P · ∇ g a + i ∇ g a · P − α k |∇ g a | )( K − z ) − k ≤ b < . We now prove (2.9). If
R >
0, define: H R = − ∆ + (1 − χ R ) V, (3.7)where χ R ( x ) = (cid:26) | x ⊥ | ≤ R | x ⊥ | > R . (3.8)From (2.3) it follows that lim R →∞ inf σ ( H R ) = 0 . In particular, for sufficiently large R , ( H R − z ) − is analytic inside Γ. Since H − H R = χ R V , thenusing resolvent identities we obtain:( H − z ) − = ( H R − z ) − (3.9) − ( H R − z ) − χ R V ( H R − z ) − + ( H R − z ) − χ R V ( H − z ) − χ R V ( H R − z ) − . From (2.5), (3.9) and the fact that ( H R − z ) − is analytic inside Γ one has P = i π Z Γ ( H R − z ) − χ R V ( H − z ) − χ R V ( H R − z ) − . (3.10)7otice that for all α >
0: sup x ∈ R Z | x − y |≤ | ( e αg ⊥ χ R V )( y ) | d y < ∞ . (3.11)Take now α ⊥ > z ∈ Γ, K = H R , h = g ⊥ and α z = α ⊥ . That islet us suppose that1 + α ⊥ ( ± i P · ∇ g ⊥ ± i ∇ g ⊥ · P − α ⊥ |∇ g ⊥ | )( H R − z ) − is invertible (3.12)uniformly on Γ. Then we can rewrite P as: P = e − α ⊥ h X ⊥ i (cid:26) i π Z Γ h e α ⊥ h X ⊥ i ( H R − z ) − e − α ⊥ h X ⊥ i i(cid:2) e α ⊥ g ⊥ χ R V ( H − z ) − (cid:3) (cid:2) e α ⊥ g ⊥ χ R V ( H R − z ) − (cid:3) (3.13) (cid:2) α ⊥ ( − i P · ∇ g ⊥ − i ∇ g ⊥ · P − α ⊥ |∇ g ⊥ | )( H R − z ) − (cid:3) − dz o e − α ⊥ h X ⊥ i . Due to (3.11) the operator under the integral sign is uniformly bounded in z and the proof ofProposition 1 is finished provided we can show why we can choose α ⊥ as close to p E + as wewant. The argument is as follows. Choose 0 ≤ α ⊥ < p E + . Choose a contour Γ which is veryclose to σ , at a distance δ >
0, infinitesimally small. Using the spectral theorem (or in this casethe Plancherel theorem), there exists δ small enough such that the following estimates hold true:sup z ∈ Γ (cid:13)(cid:13) ( P − z ) − (cid:13)(cid:13) ≤ const , sup z ∈ Γ max j ∈{ , , } (cid:13)(cid:13) P j ( P − z ) − (cid:13)(cid:13) ≤ const . (3.14)Hence we can find δ small enough and R large enough such that the operator in (3.12) is invertibleif 1 + α ⊥ ( ± i P · ∇ g ⊥ ± i ∇ g ⊥ · P − α ⊥ |∇ g ⊥ | )( P − ℜ ( z )) − is invertible (3.15)uniformly on Γ. Now the operator in (3.15) is invertible if1 ± iα ⊥ ( P − ℜ ( z )) − ( P · ∇ h + ∇ h · P )( P − ℜ ( z )) − − α ⊥ ( P − ℜ ( z )) − |∇ h | ( P − ℜ ( z )) − (3.16)is invertible (by a resummation of the Neumann series and analytic continuation). Now assumethat uniformly on Γ we have:0 < α ⊥ ( P − ℜ ( z )) − |∇ h | ( P − ℜ ( z )) − ≤ α ⊥ −ℜ ( z ) < , which can be achieved if α ⊥ < E + and δ is chosen to be small enough. Define S := (cid:16) − α ⊥ ( P − ℜ ( z )) − |∇ h | ( P − ℜ ( z )) − (cid:17) − , and T = T ∗ := S ( P − ℜ ( z )) − ( P · ∇ h + ∇ h · P )( P − ℜ ( z )) − S. Then the operator in (3.16) is invertible if 1 ± iα ⊥ T is invertible, which is always the case:(1 ± iα ⊥ T ) − = (1 ∓ iα ⊥ T )(1 + α ⊥ T ) − . Therefore Proposition 1 is proved. 8 .2 Proof of Theorem 2
Proof of (i). First we recall an older result (see e.g. [28, 2, 29]), according to which the commutator[ X k , P ] defined on D ( X k ) has a bounded closure on L ( R ). We seek an approximate resolventof ˆ X k by defining for µ > R ± µ = P ( X k ± iµ ) − P . (3.17)Since one can rewrite ˆ R ± µ asˆ R ± µ = ( X k ± iµ ) − P + ( X k ± iµ ) − [ X k , P ]( X k ± iµ ) − P it follows that ˆ R ± µ K ⊂ D ( ˆX k ) and by a straightforward computation (as operators in K )( ˆ X k ± iµ ) ˆ R ± µ = P ( X k ± iµ ) P ( X k ± iµ ) − P = 1 K + ˆ A ± µ (3.18)with ˆ A ± µ = P [ X k , P ]( X k ± iµ ) − P . (3.19)Since [ X k , P ] is bounded and k ( X k ± iµ ) − k ≤ µ , it follows that for sufficiently large µ : k ˆ A ± µ k ≤ . (3.20)Then again as operators in K : ( ˆ X ± iµ ) ˆ R ± µ (1 K + ˆ A ± µ ) − = 1 K (3.21)This implies that ˆ X ± iµ is surjective on ˆ R ± µ (1 K + ˆ A ± µ ) − K ⊂ D ( ˆ X ) . By the fundamentalcriterion of self-adjointness [27] ˆ X is self-adjoint in K on D ( ˆ X ). In addition, from (3.21) oneobtains the following formula for the resolvent of ˆ X k :( ˆ X k ± iµ ) − = ˆ R ± µ (1 K + ˆ A ± µ ) − . (3.22) Proof of (ii). We will show that ˆ R ± µ is compact in K which implies (see (3.22)) that ˆ X k hascompact resolvent, thus purely discrete spectrum. Consider a cut-off function φ N which equals 1if | x | ≤ N and is zero if | x | ≥ N . For N ≥ R ± µ = P ( X k ± iµ ) − φ N P + P ( X k ± iµ ) − (1 − φ N ) P . (3.23)Writing φ N P = { φ N ( P + 1) − }{ ( P + 1) P } we see that φ N P is compact (even Hilbert-Schmidt) in L ( R ) (the first factor is Hilbert-Schmidtwhile the second one is bounded). Now if 0 < α is small enough, we know that e αg ⊥ P is bounded(see (2.9)). Since lim N →∞ (cid:13)(cid:13) ( X k ± iµ ) − (1 − φ N ) e − αg ⊥ (cid:13)(cid:13) = 0 , we have shown: lim N →∞ (cid:13)(cid:13)(cid:13) ˆ R ± µ − P ( X k ± iµ ) − φ N P (cid:13)(cid:13)(cid:13) = 0 , thus ˆ R ± µ equals the norm limit of a sequence of compact operators, therefore it is compact. Ac-cordingly, since the self-adjoint operator ˆ X k has compact resolvent it has purely discrete spectrum[27]: σ ( ˆ X k ) = σ disc ( ˆ X k ) =: G, (3.24)and the proof of the second part of Theorem 2 is finished.9 roof of (iii). Now we will consider the exponential localization of eigenfunctions of ˆ X k . Let g ∈ G be an eigenvalue, m g its multiplicity and W g,j , ≤ j ≤ m g be an orthonormal basis in theeigenspace of ˆ X k corresponding to g . We shall prove that uniformly in g and j k e α k h X k ,g i W g,j k ≤ M and (3.25) k e α ⊥ h X ⊥ i W g,j k ≤ M. (3.26)Taking (3.25) and (3.26) as given, one can easily obtain (2.11) by a simple convexity argument:the function f ( x ) = a − x b x ; a, b > R , and for 0 ≤ β ≤ βe α k g a ( x ) + (1 − β ) e α ⊥ g ⊥ ≥ e − β ) α k g a ( x ) e βα ⊥ g ⊥ , (3.27)which together with (3.25) and (3.26) it proves (2.11) with M = M . Since (3.26) follows directlyfrom (2.9) and W g,j = P W g,j we are left with the proof of (3.25).Although the proof of (3.25) mimics closely the proof in the one dimensional case [26], we giveit here for completeness. In order to emphasize the main idea of the proof let us remind one of thesimplest proofs of the exponential decay of eigenfunctions of Schr¨odinger operators correspondingto discrete eigenvalues (assuming that the potential V is bounded and has compact support).Namely assume that for some E > − ∆ + V + E )Ψ = 0, which can be rewritten asΨ = − ( − ∆ + E ) − V Ψ . (3.28)Since for | α | < √ E , e α |·| ( − ∆ + E ) − e − α |·| and e α |·| V are bounded:Ψ = − e − α |·| n e α |·| ( − ∆ + E ) − e − α |·| o ( e α |·| V )Ψwhich proves the exponential localization of Ψ. The main idea in proving (3.25) is to rewrite theeigenvalue equation for ˆ X k in a form similar to (3.28) and and then to use (2.8).Let us start with some notation. If b > a ∈ R , define: f a,b ( x ) := b f (cid:18) x − ab (cid:19) (3.29)where f is a real C ∞ ( R ) cut-off function satisfying 0 ≤ f ( y ) ≤ f ( y ) = (cid:26) | y | ≤ | y | ≥ . Define the function h a,b by: h a,b ( x ) := x − a + if a,b ( x ) . (3.30)Note that by construction, h a,b only depends on x , and obeys: | h a,b ( x ) | ≥ b . (3.31)Moreover, its first two derivatives are uniformly bounded:sup x ∈ R sup a ∈ R sup b ≥ {|∇ h a,b ( x ) | + | ∆ h a,b ( x ) |} = K < ∞ . (3.32)The eigenvalue equation for W g,j reads as P ( ˆ X k − g ) P W g,j = 0. Using (3.30) it can be rewrittenas: P h g,b P W g,j = iP f g,b P W g,j . (3.33)We now prove that P h g,b P is invertible. Like in the proof self-adjointness of ˆ X k we compute P h − g,b P P h g,b P = 1 K + P h − g,b [ P , h g,b ] P . (3.34)10he key remark is that [ P , h g,b ] is bounded. Indeed we have the identity:[ P , h g,b ] = − π Z Γ ( H − z ) − { P · ∇ h g,b + ∇ h g,b · P } ( H − z ) − dz = − π Z Γ ( H − z ) − {− i ∆ h g,b + 2 ∇ h g,b · P } ( H − z ) − dz. (3.35)It follows that [ P , h g,b ] is uniformly bounded in g ∈ R and b ≥ B g,b = P h − g,b [ P , h g,b ] P : K → K (3.36)satisfies k ˆ B g,b k ≤
12 (3.37)if b ≥ b for some large enough b < ∞ . It follows that 1+ ˆ B g,b is invertible and then the eigenvalueequation (see (3.33), (3.34) and (3.36)) takes the form W g,j = i (cid:16) B g,b (cid:17) − P h − g,b P f g,b P W g,j (3.38)which is the analog of (3.28). By construction (see the definition of f g,b in (3.29)): k e α k h X k ,g i f g,b k ≤ be α k ( b +1) . Moreover, e α k h X k ,g i P h − g,b P e − α k h X k ,g i = n e α k h X k ,g i P e − α k h X k ,g i o h − g,b n e α k h X k ,g i P e − α k h X k ,g i o is bounded due to (2.8). Thus the only thing it remains to be proved is the existence of a b largeenough such that the following bound holds:sup g ∈ R (cid:13)(cid:13)(cid:13)(cid:13) e α k h X k ,g i (cid:16) B g,b (cid:17) − e − α k h X k ,g i (cid:13)(cid:13)(cid:13)(cid:13) < ∞ . (3.39)Using the Neumann series for (cid:16) B g,b (cid:17) − , it follows that it suffices to prove thatlim b →∞ sup g ∈ R (cid:13)(cid:13)(cid:13) e α k h X k ,g i ˆ B g,b e − α k h X k ,g i (cid:13)(cid:13)(cid:13) = 0 . (3.40)Since (see (3.31)) lim b →∞ k h − g,b k = 0 (uniformly in g ∈ R ), for (3.40) to holds true it is sufficientto show: sup g ∈ R (cid:13)(cid:13)(cid:13) e α k h X k ,g i [ P , h g,b ] e − α k h X k ,g i (cid:13)(cid:13)(cid:13) ≤ const . (3.41)But this easily follows from (3.35), (3.32), (3.2) and (3.3) where we take K = H , α z = α k and h = g g . The proof of (iii) is concluded. Proof of (iv). We start with a technical result:
Lemma 6.
Fix ≤ α ⊥ < p E + . Then there exists a bounded operator D such that P = e − α ⊥ h X ⊥ i ( P + 1) − D (3.42)11 roof . We use the notation and ideas of Proposition 1, and we rewrite P in a convenient form.First, for R > H − z ) − = ( H R − z ) − − ( H R − z ) − χ R V ( H − z ) − . Second, choose Γ close enough to σ and R large enough, such that ( H R − z ) − becomes analyticinside Γ and (3.12) holds true for all z ∈ Γ. Then we can write: P = − e − α ⊥ h X ⊥ i i π Z Γ (3.43)( H R − z ) − [1 + α ⊥ ( i P · ∇ g ⊥ + i ∇ g ⊥ · P − α ⊥ |∇ g ⊥ | )( H R − z ) − ] − e α ⊥ g ⊥ χ R V ( H − z ) − dz. Now by the closed graph theorem we have that ( P + 1)( H R + 1) − is bounded (here R is largeenough such that ( −∞ , − / ⊂ ρ ( H R )), and together with the spectral theorem:sup z ∈ Γ k ( P + 1)( H R − z ) − k < ∞ . Use this in (3.43) and we are done.We now have all the necessary ingredients for proving the last statement of our theorem. Forevery
L > a ∈ R , denote by χ L,a the characteristic function of the slab { x : | x − a | ≤ L } .Then define the operator B := χ L,a P . Using (3.42) let us show that B is Hilbert-Schmidt, andmoreover, uniformly in a ∈ R we have: k B k ≤ M · L, (3.44)for some M < ∞ . Indeed, since B = χ L,a e − α ⊥ h X ⊥ i ( − ∆ + 1) − D , a direct computation using theexplicit formula for the integral kernel of the free Laplacian gives: k χ L,a e − α ⊥ h X ⊥ i ( P + 1) − k ≤ const · L. It follows that the operator χ L,a P χ L,a = BB ∗ is trace class and | Tr( χ L,a P χ L,a ) | ≤ k B k ≤ M · L (3.45)for some M < ∞ independent of L and a .Now let P L,a be the orthogonal projection onto the subspace spanned by those W g,j for which g ∈ [ a − L, a + L ]: P L,a := X | g − a |≤ L m g X j =1 h· , W g,j i W g,j . (3.46)We can choose A sufficiently large such that (3.25) implies: Z | x − a |≥ A | W g,j ( x ) | d x ≤ , (3.47)uniformly in a and g ∈ [ a − L, a + L ]. Since P ≥ P L,a , from (3.45) one obtains: M · ( L + A ) ≥ Tr( χ L + A,a P χ L + A,a ) ≥ Tr( χ L + A,a P L,a χ L + A,a )= X | g − a |≤ L m g X j =1 Z R χ L + A,a ( x ) | W g,j ( x ) | d x ≥ X | g − a |≤ L m g X j =1
12 = 12 N ( a, L ) , (3.48)where in the last inequality we used (3.47). In particular, if L ≥
1, then uniformly in a ∈ R wehave N ( a, L ) ≤ M · (1 + A ) L and the proof is finished. 12 .3 Proof of Proposition 3 The only thing we have to prove is that (3.41) holds true for α k replaced by any α < α , where α is the a-priori given, ”exact” exponential localization.We introduce the multiplication operator given by { e α |·− t | f } ( x ) := e α | x − t | f ( x ). We start bynoticing that due to the bound e ± α ( √ s +1 −| s | ) ≤ e α we can replace (2.13) with:sup t ∈ R k e α |·− t | P e − α |·− t | k < ∞ . (3.49)The same replacement can be done in (3.41). Now the integral kernel A ( x , y ) of the operator A := e α |·− g | [ P , h g,b ] e − α |·− g | equals A ( x , y ) = P ( x , y ) e α ( | x − g |−| y − g | ) ( h g,b ( y ) − h g,b ( x )) . (3.50)We consider A as an operator on L ( R ) = L p ∈ Z L ([ p, p + 1] × R ). Let χ p be the characteristicfunction of the slab [ p, p + 1] × R . We have that A pp ′ := χ p Aχ p ′ is a bounded operator between L ([ p ′ , p ′ + 1] × R ) and L ([ p, p + 1] × R ), and we can write A = { A pp ′ } p,p ′ ∈ Z . We will boundthe norm of A with a Schur-Holmgren type estimate (see below Lemma 7): || A || ≤ sup p ′ ∈ Z X p ∈ Z || A pp ′ || sup p ∈ Z X p ′ ∈ Z || A pp ′ || . (3.51)For 0 ≤ x , y ≤
1, the kernel of A pp ′ can be written as: A pp ′ ( x , y ) = P ( x + p, x ⊥ ; y + p ′ , y ⊥ ) e α ( | x + p − g |−| y + p ′ − g | ) ( h g,b ( y + p ′ ) − h g,b ( x + p ))= P ( x + p, x ⊥ ; y + p ′ , y ⊥ ) e α ( | x + p − g |−| y + p ′ − g | ) ( h g,b ( p ′ ) − h g,b ( p ))+ P ( x + p, x ⊥ ; y + p ′ , y ⊥ ) e α ( | x + p − g |−| y + p ′ − g | ) ( h g,b ( y + p ′ ) − h g,b ( p ′ ))+ P ( x + p, x ⊥ ; y + p ′ , y ⊥ ) e α ( | x + p − g |−| y + p ′ − g | ) ( − h g,b ( x + p ) + h g,b ( p ))=: A (1) pp ′ ( x , y ) + A (2) pp ′ ( x , y ) + A (3) pp ′ ( x , y ) . (3.52)The last two kernels can be analyzed with the same methods as the first one, thus we only estimatethe norm of A (1) pp ′ . The crucial observation is that we can write this operator as a product of threeoperators having the corresponding kernels: A (1) pp ′ ( x , y ) = e α ( | x + p − g |−| p − g | ) · e α ( | p − g |−| p ′ − g | ) P ( x + p, x ⊥ ; y + p ′ , y ⊥ )( h g,b ( p ′ ) − h g,b ( p )) · e − α ( | y + p ′ − g |−| p ′ − g | ) . (3.53)The kernel in the middle corresponds to the operator χ p P χ ′ p times some coefficients dependingon p, p ′ .Using the triangle inequality to bound the exponentials, and (3.32) in order to write | h g,b ( y ) − h g,b ( x ) | ≤ K | x − y | , we have: || A (1) pp ′ || ≤ Ke α e α | p − p ′ | | p − p ′ | · || χ p P χ ′ p || . Using t = p ′ and ( α + α ) / || χ p P χ ′ p || ≤ Ce − ( α + α ) | p − p ′ | / , thus || A (1) pp ′ || ≤ C ′ | p − p ′ | e − ( α − α ) | p − p ′ | / which is summable in the sense of (3.51). The same strategy can be applied in the case of A (2) pp ′ and A (3) pp ′ . The last thing to be done is to prove the Schur-Holmgren estimate:13 emma 7. The estimate (3.51) holds true.Proof.
Let ψ ∈ L ( R ) with compact support and || ψ || = 1. We write: || Aψ || = X p ∈ Z || χ p Aψ || . (3.54)But || χ p Aψ || ≤ X p ′ ∈ Z q || A pp ′ || q || A pp ′ || || χ p ′ ψ || ≤ X p ′ ∈ Z || A pp ′ || X p ′ ∈ Z || A pp ′ || || χ p ′ ψ || ≤ ( sup s ∈ Z X t ∈ Z || A st || ) X p ′ ∈ Z || A pp ′ || || χ p ′ ψ || (3.55)where in the second inequality we used Cauchy-Schwarz with respect to p ′ . Introduce this in (3.54)and the bound follows after the use of P p ′ ∈ Z || χ p ′ ψ || = 1. Acknowledgements.
Part of this work was done during a visit of G. Nenciu at the Departmentof Mathematical Sciences, Aalborg University; both hospitality and financial support are gratefullyacknowledged. H. Cornean acknowledges support from Danish F.N.U. grant
Mathematical Physicsand Partial Differential Equations . A. Nenciu and G. Nenciu were partially supported by CEEXGrant 05-D11-45/2005. We also thank the first referee for his/hers most valuable commentsregarding Proposition 3.
References [1] Wannier G H 1937
Phys. Rev. , 191-197[2] Nenciu G 1991 Rev. Mod. Phys. , 91-127[3] Marzari N and Vanderbilt D 1997 Phys. Rev.
B56
Phys. Rev.
B69 , 03518[5] Chang E, Bussi G, Ruini A, Molinari E 2004
Phys. Rev. Lett. , 196401[6] Cornean H, Jensen A, Moldoveanu V 2005 J. Math. Phys. Phys. Rev.
B60
J. Phys. C: Solid State Phys. , L325-L327[9] Cornean H, Nenciu G, Pedersen T G 2006 J. Math. Phys. Phys. Rev. Lett. ,046402[11] Kohn W 1959 Phys. Rev.
Phys. Rev. B , 035128[13] des Cloizeaux J 1964 Phys. Rev.
A685-A697[14] des Cloizeaux J 1964
Phys. Rev.
A698-A7071415] Nenciu G 1983
Commun. Math. Phys. , 81-85[16] Nenciu A and Nenciu G 1993 Phys. Rev.
B47 , 10112-10115[17] Panati G 2007
Ann. Henri Poincar´e , 995-1011.[18] Wu X, Dieguez O, Rabe K.M., Vanderbilt D 2006 Phys. Rev. Lett. , 107602.[19] Bhattacharjee J., Waghmare 2005 Phys. Rev.
B 71 045106.[20] Bruno-Alfonso A, Nacbar D.R. 2007
Phys. Rev.
B 75 , 115428.[21] Kohn W and Onffroy J 1973
Phys. Rev. B8 Phys. Rev.
B10 , 448-455[23] Geller M R and Kohn W 1993
Phys. Rev.
B48
Modern Physics Letters
B14,15 , 923-931[25] Kivelson S 1982
Phys. Rev.
B26
Commun. Math. Phys. , 541-548[27] Reed M and Simon B 1975
Methods of Modern Mathematical Physics: II. Fourier Analysis.Self-adjointness. (New York, Academic Press)[28] Avron J E 1979
J. Phys. A: Math. Gen. , 2393-2398[29] Nenciu A and Nenciu G 1981 J. Phys. A: Math. Gen.14