Quantum Hamiltonians with weak random abstract perturbation. II. Localization in the expanded spectrum
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QUANTUM HAMILTONIANS WITH WEAK RANDOM ABSTRACTPERTURBATION. II. LOCALIZATION IN THE EXPANDED SPECTRUM
DENIS BORISOV , , , MATTHIAS T ¨AUFER , AND IVAN VESELI´C Abstract.
We consider multi-dimensional Schr¨odinger operators with a weak random per-turbation distributed in the cells of some periodic lattice. In every cell the perturbation isdescribed by the translate of a fixed abstract operator depending on a random variable. Therandom variables, indexed by the lattice, are assumed to be independent and identically dis-tributed according to an absolutely continuous probability density. A small global couplingconstant tunes the strength of the perturbation. We treat analogous random Hamiltoniansdefined on multi-dimensional layers, as well. For such models we determine the location ofthe almost sure spectrum and its dependence on the global coupling constant. In this paperwe concentrate on the case that the spectrum expands when the perturbation is switched on.Furthermore, we derive a Wegner estimate and an initial length scale estimate, which togetherwith Combes–Thomas estimate allows to invoke the multi-scale analysis proof of localization.We specify an energy region, including the bottom of the almost sure spectrum, which exhibitsspectral and dynamical localization. Due to our treatment of general, abstract perturbationsour results apply at once to many interesting examples both known and new. Department of Differential Equations, Institute of Mathematics with Computer Cen-ter, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky. st. 112,Ufa, 450008, Russia, Email: [email protected] Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Octoberrev. st. 3a, Ufa, 450000, Russia Faculty of Science, University of Hradec Kr´alov´e, Rokitansk´eho 62, 500 03, HradecKr´alov´e, Czech Republic School of Mathematical Sciences, Queen Mary University of London, United King-dom Fakult¨at f¨ur Mathematik, Technische Universit¨at Dortmund, 44227 Dortmund, Ger-many Introduction
Anderson localization refers to the physical phenomenon that waves do not propagate in certaintypes of disordered media. The occurrence of this phenomenon depends on the energy associatedwith the wave and the specifics of the randomness present in the medium. It was first studiedin the context of propagation or diffusion of electron wavepackets in quantum mechanical mod-els for condensed matter, but can also occur in models for the propagation of classical waves.More specifically, in the mathematical literature, spectral localization in a certain energy region I ⊂ R means that the Hamiltonian governing the equation of motion of the waves has neithersingular continuous nor absolutely continuous spectrum in I . Thus, any spectrum in I is purepoint spectrum. The term Anderson localization describes the situation of spectral localizationin I where all eigenfunctions with eigenvalues in I are exponentially decaying, even though thisadditional requirement is usually no restriction, in the sense that it occurs in most cases whenspectral localization does. Furthermore, there is the concept of dynamical localization referring tothe phenomenon that space-localized initial states do not spread, but remain localized uniformlyin time when evolving as determined by the Hamiltonian. In most situations, whenever one canprove spectral localization one can prove dynamical localization as well. Concerning the otherdirection, spectral localization is a direct consequence of dynamical localization. For a class of Mathematics Subject Classification.
Key words and phrases. random Hamiltonian, weak disorder, random geometry, quantum waveguide, low-lyingspectrum, asymptotic analysis, Anderson localization.GenLocLay-2020-3-1.tex. random Schr¨odinger operators the equivalence of these two notions has been proved in [GK04]. Inorder to understand the challenges posed by the models considered in this work let us emphasizethat the random perturbation operators considered in [GK04] are linear and monotone in therandom variables and consist of multiplication operators whereas we shall consider a more generalclass of operators.As a motivation for our results, let us start by noting that strictly speaking, the above mentionednotions of localization, which are ubiquitous in the literature, do not exclude the trivial situationwhere I is a subset of the resolvent set. At first glance, this seems to be a technical subtlety.On the one hand, methods to exclude any spectrum in an interval I are typically much simplerthan methods to prove that the entire spectrum in I is pure point. On the other hand, proofs oflocalization often only apply in certain specific energy intervals, which are usually near the edges ofthe spectrum of a non-random operator and which are sometimes subject to technical conditions.Showing that this region of localization indeed contains spectrum is a non-trivial task. We haveto perform a detailed analysis of the spectrum as a set, as well as of the region of localization, inorder to show that their intersection is non-empty – at least for certain parameter choices.The operators which we consider in this work are random perturbations of a Schr¨odinger op-erator. The perturbation however is general. It may be a differential, integral, multiplication orother operator, as long as it satisfies certain reasonable regularity conditions. Furthermore thefunctional dependence on the random parameters may be non-linear and non-monotone.Such perturbations are more difficult to analyse than the original Anderson or alloy type mod-els with fixed-sign single site potentials, for which the theory of localization has been developedfirst. Consequently, in subsequent developments, extensions of the methods have been invented inorder to be able to deal with non-monotone, non-linear and non-potential perturbations. The firststep in this direction was the study of alloy type potentials with sign-changing single site poten-tials, as carried out in for instance in [Klo95], [Sto00], [Ves01], [Klo02], [Ves02], [HK02], [KV06],[KN09], [Ves10b]. In fact, the Wegner estimate which we prove in this paper relies on the vectorfield method of [Klo95, HK02]. More recently also the discrete analogue of this model was stud-ied in [ETV10], [Ves10a], [ETV11], [TV10], [Kr¨u12], [CE12], [ESS14], [TV15]. Another relevantmodel without direct monotonicity is a random potential given by a Gaussian stochastic field withsign-changing covariance function, c.f. [HLMW01], [Uek04], [Ves11], [Tau19]. ElectromagneticSchr¨odinger operators with random magnetic field [Uek94], [Uek00], [HK02], [KNNN03], [Uek08],[Bou09], [EH12c], [EH12a], [EH12b], as well as Laplace-Beltrami operators with random metrics[LPV04], [LPPV08], [LPPV09] exhibit a non-monotonous parameter dependence which affects thehigher order terms of the differential operator. Another model where geometric randomness entersnaturally is the random displacement model, cf. e.g. [Klo93, BLS08, GK10, KLNS12]. This isalso the case for random waveguide Laplacians which we discuss in the next paragraph.The paper builds upon, improves and complements several earlier papers of two of us (partiallywith other coauthors). In [BHEV16] and [BHEV18] we have analysed what type of expansion ratefor the spectrum is possible for weak disorder random Hamiltonians. For a large class of randomoperators we were able to show that the expansion is either linear or quadratic, i. e. that otherrates do not occur. In the present paper we cover both of these two scenarios. Initial lengthscale estimates for randomly bent and randomly curved waveguides and more general geometricperturbations have been proved in [BV11], [BV13], and [BGV16] as well as [Bor17], respectively.However none of these paper proved Wegner estimates, which we provide here.Let us list the main achievements of the paper together with the theorems where they arespelled out and the sections where they are proven.(i) A characterisation of the almost sure spectrum as a set and its minimum, for a class ofgeneral weak disorder Hamiltonians. (Theorem 2.3, proof in Section 5.)(ii) Characterisation of linear and quadratic movement of the spectral minimum as a function ofthe weak disorder parameter. (Corollary 2.4)(iii) An improved variational lower bound for the principal eigenvalue of Hamiltonians on large,finite segments or cubes: The deviation of the random parameter configuration from the optimal one determines explicitly how much the eigenvalue is lifted. (Theorems 2.5 and 6.1,proof in Section 6.)(iv) A Wegner estimate for general weak disorder random Hamiltonians specifying an energy × disorder regime where it holds. (Theorem 2.6 and 7.1, proof in Section 7.)(v) Two results with explicit estimates on the location and size of a localization regime in thespectrum. One of them corresponds to models with linear shift of the spectral minimum,the other to models with a quadratic one, with respect to the small disorder parameter.(Theorem 2.7 and 2.8.)(vi) An broad list of examples which are covered by the general class or random Hamiltonianswhich we consider, presented in Section 3 and 4.One of the complementary motivations of this paper is to understand [HK02, Theorem 6.1],where a Wegner estimate in the weak disorder regime is presented. While we were able to generalizethe proof to our situation, we learned that it has to be complemented by a detailed analysis ofthe region in the energy × disorder plane where the Wegner bound holds and an analysis about theexpansion of the spectrum. For certain models, like weak disorder magnetic fields with no magneticfield in the unperturbed operator, the considered method of proof yields a Wegner estimate onlyin the resolvent set! We shall discuss this in more detail for the specific example of a compactlysupported magnetic field single site potential in Section 4.4 and Appendix B.An important technical assumption in our models is that each single site perturbation acts onfunctions supported in one individual periodicity cell. Overlapping single site perturbations wouldlead to additional higher order terms, which would change the perturbation analysis which wecarry out.The proof of our theorems on localization is split into two parts. First, one identifies anenergy interval I , where it is possible to prove an appropriate Wegner estimate and an initiallength scale estimate. Thus in this region one can perform the multi-scale analysis. Consequently,the spectrum in this energy region must be pure point spectrum with exponentially decayingeigenfunctions. However, this does note yet exclude the situation that I is part of the resolventset. For this reason we prove separately in Section 8 that a nonempty subset I ⊂ I belongsto the almost sure spectrum Σ ε . This is not trivial, because one has to analyse how fast thespectrum expands with emerging disorder ε . At the same time, the method of proof we use forthe Wegner estimate requires that one has a certain security distance away from the spectrumof the unperturbed operator H . Thus, I ⊂ Σ ε is a set which is close, but not too close to theunperturbed spectrum Σ .2. Definitions and results for operators on multidimensional layers
We shall first explain the abstract framework and our main results for operators on multidi-mensional layers before treating operators on R n in Section 3. We fix a dimension n ∈ N andwrite x ′ = ( x , . . . , x n ) and x = ( x ′ , x n +1 ) for Cartesian coordinates in R n and R n +1 , respec-tively. For d >
0, we define Π := { x : 0 < x n +1 < d } ⊂ R n +1 , the multidimensional layerof width d . Let Γ be a periodic lattice in R n with basis e , . . . , e n . Its elementary cell is (cid:3) ′ := { x ′ : x ′ = n P i =1 a i e i , a i ∈ (0 , } and we also define (cid:3) := (cid:3) ′ × (0 , d ) ⊂ Π.Let Π ∋ x V ( x ) = V ( x n +1 ) ∈ R be a measurable and bounded potential depending onlyon the transversal variable x n +1 . We consider the self-adjoint operator H := − ∆ + V in L (Π)with either Dirichlet or Neumann boundary conditions on ∂ Π, abbreviated by(2.1) B u = 0 where B u = u or B u = ∂u∂x n +1 . It is possible to choose different types of boundary conditions on the two (“upper” and “lower”)components of ∂ Π. The domain of this operator is(2.2) D ( H ) := { u ∈ W , (Π) : (2.1) is satisfied on ∂ Π } . BORISOV, T¨AUFER, AND VESELI´C
Let
T > L ( t ), t ∈ [ − T, T ]. These operators are assumed to act from W , ( (cid:3) ) into L ( (cid:3) ) and are defined as L ( t ) := t L + t L + t L ( t ) , where L i : W , ( (cid:3) ) → L ( (cid:3) ) are bounded symmetric linear operators. The operator L ( t ) andits derivative with respect to t are assumed to be bounded uniformly in t ∈ [ − T, T ]. Recall that W , ( (cid:3) ) and L ( (cid:3) ) are canonically embedded into W , (Π) and L (Π), respectively. For each u ∈ H (Π), the restriction of this function on (cid:3) is an element of u ∈ H ( (cid:3) ). Hence, the function L i u is a well-defined element of L ( (cid:3) ) and we may extend this function by zero in Π \ (cid:3) . Afterthe extension this function is an element of L (Π). In view of the described continuation, in whatfollows, we regard the operators L i as unbounded operators in L (Π) with domain W , (Π).Let { ξ k } k ∈ Z n be a sequence of numbers with values in [ − , S ( k ) the shiftoperator in L (Π), i.e. ( S ( k ) u )( x ) = u ( x ′ + k, x n +1 ). For ε ∈ (0 , T ], we define the unboundedoperator(2.3) H ε ( ξ ) := − ∆ + V + L ε ( ξ ) , L ε ( ξ ) := X k ∈ Γ S ( k ) L ( εξ k ) S ( − k ) , on the domain D ( H ε ( ξ )) = D ( H ) ⊂ L (Π).By k · k X → Y we denote the norm of a bounded operator acting from a Banach space X into aBanach space Y . Convention . To ensure a number of further properties of the operator H ε ( ξ ) it will be neces-sary to chose ε from a smaller interval than (0 , T ]. There is a (small) number 0 < t < n , Γ, B , V , kL k W , ( (cid:3) ) → L ( (cid:3) ) , kL k W , ( (cid:3) ) → L ( (cid:3) ) , sup − T t T kL ( t ) k W , ( (cid:3) ) → L ( (cid:3) ) ,sup − T t T k ∂∂t L ( t ) k W , ( (cid:3) ) → L ( (cid:3) ) and the measure µ (which will be defined shortly) such that allresults in the paper hold for ε ∈ (0 , t ]. In proofs, the specific values of t may change from lineto line, nevertheless we shall not write each time possibly decreasing t further . The main point isthat t does not depend on ε nor on the scale N appearing below.If t > ε ∈ (0 , t ] the operator L ε is relatively bounded withrespect to the Laplacian on D ( H ε ( ξ )) with relative bound smaller than one. As a consequence,the operator H ε ( ξ ) is lower semi-bounded and, by the Kato-Rellich theorem, self-adjoint.Let now ω := { ω k } k ∈ Γ be a sequence of independent and identically distributed random vari-ables on the probability space (Ω , P ), where Ω := × k ∈ Γ [ − , P = N k ∈ Γ µ . The probabilitymeasure µ corresponding to each ω j is assumed to have an absolutely continuous probabilitydensity h with respect to the Lebesgue measure and to satisfy − b = min supp µ < max supp µ = 1 . We shall write E for the expectation with respect to P . Throughout the paper, the symbol H ε ( ω )will denote a realization of our random operator corresponding to the configuration ω ∈ Ω. Thenotation H ε ( ξ ) will be used if we study properties which are valid for every single ξ ∈ Ω.We now define the auxiliary operator H δ (cid:3) := − ∆ + V | (cid:3) + L ( δ ) , δ ∈ [ − t , t ] , in L ( (cid:3) ) with domain consisting of all functions in the Sobolev space W , ( (cid:3) ) satisfying theboundary condition (2.1) on ∂ (cid:3) ∩ ∂ Π and periodic boundary conditions on γ := ∂ (cid:3) \ ∂ Π. Theoperator H δ (cid:3) is again self-adjoint. For δ ∈ [ − t , t ], we denote by Λ δ the lowest eigenvalue of theoperator H δ (cid:3) and by Ψ δ an associated, appropriately normalized eigenfunction.In Lemma 5.1, we prove that the eigenvalue Λ δ and the eigenfunction Ψ δ are, in a specific sense,twice differentiable as functions of δ in a neighbourhood of δ = 0 with Taylor expansions(2.4) Λ δ = Λ + δ Λ + δ Λ + O ( δ ) , and Ψ δ = Ψ + δ Ψ + δ Ψ + O ( δ ) , where Λ i and Ψ i , i ∈ { , , } , are explicitly given, cf. Lemma 5.1 for details.In the present paper we cover the following two scenarios: ( Case I ) The constant Λ is strictly negative.( Case II ) The constant Λ is zero and(2.5) b > − , η := − ( b + 1)Λ + (1 − b ) (cid:0) Φ − Ψ , L Ψ (cid:1) L ( (cid:3) ) > . where Φ ∈ W , ( (cid:3) ) is the unique solution to the problem(2.6) ( − ∆ + V − Λ )Φ = −L Ψ in (cid:3) , B Φ = 0 on ∂ (cid:3) ∩ ∂ Π , ∂ Φ ∂ν = 0 on γ, orthogonal to Ψ in L ( (cid:3) ). Remark . (1) Note that ( Case II ) implies that the vector L Ψ is orthogonal to Ψ itself,cf. Lemma 5.1, the unique ground state of H (cid:3) = − ∆ + V . Hence, L Ψ is orthogonal tothe entire spectral subspace of Λ . The operator H (cid:3) commutes with this subspace and isinvertible on the orthogonal complement. Thus, there is indeed a unique solution Φ to(2.6).(2) It was shown in [Bor17] that (cid:0) Re(Φ − Ψ ) , Re L Ψ (cid:1) L ( (cid:3) ) = (cid:0) Φ − Ψ , L Ψ (cid:1) L ( (cid:3) ) , thus the second inequality in condition (2.5) can be rewritten as η := − ( b + 1)Λ + (1 − b ) (cid:0) Re(Φ − Ψ ) , Re L Ψ (cid:1) L ( (cid:3) ) > , (3) Condition (2.5) is in some sense equivalent to Λ <
0. Namely, it was shown in [Bor17]that (cid:0)
Re(Φ − Ψ ) , Re L Ψ (cid:1) L ( (cid:3) ) . Hence, the second term in the definition of η is negative and we can satisfy (2.5) only ifΛ <
0. On the other hand, if Λ <
0, then condition (2.5) is satisfied at least for b closeenough to 1.(4) In a companion paper we plan to discuss the case Λ >
0. Currently we do not know howto treat the cases Λ = 0, Λ < b = −
1, Λ > , Λ > Case I ) and (
Case II ) and provided that we establish in Lemma 5.2 thefollowing minimization property ∀ ε ∈ (0 , t ] : min δ ∈ [ εb,ε ] Λ δ = Λ ε . We use the corresponding ground state Ψ ε to define a function ρ ε on the lateral boundary γ , givenby ρ ε := 1Ψ ε ∂ Ψ ε ∂ν , where ν is the outward normal vector. Note that a priori, one might have to verify whether thisdefinition of ρ ε is admissible. For instance, one needs to exclude that Ψ ε vanishes on a nonemptyopen subset of γ . The issue of well-definedness of ρ ε will be addressed by Assumption M below.We shall also assume that for all ε ∈ (0 , t ], the boundary term ρ ε can be uniformly approximatedby polynomial expressions in boundary terms derived from Ψ , Ψ , and Ψ , that appear in theTaylor expansion of Lemma 5.1. This is made precise in the following assumption, that is satisfiedfor many examples as discussed in Section 4. Assumption M.
For all ε ∈ [0 , t ] the function ρ ε is piecewise continuous on γ . Furthermore, • If ( Case I ) holds, then the asymptotic identity (2.7) sup γ (cid:12)(cid:12) ρ ε − ερ (cid:12)(cid:12) = O ( ε ) holds true, where (2.8) ρ := 1Ψ ∂ Ψ ∂ν is a piecewise continuous function on γ . BORISOV, T¨AUFER, AND VESELI´C • If ( Case II ) holds, then the asymptotic identity sup γ (cid:12)(cid:12) ρ ε − ερ − ε ρ (cid:12)(cid:12) = O ( ε ) holds true, where ρ is given by (2.8), ρ := 1Ψ ∂ Ψ ∂ν − Ψ Ψ ∂ Ψ ∂ν , and the functions ρ , ρ are piecewise continuous on γ . Since Ψ ε satisfies periodic boundary conditions, the function ρ ε inherits this periodicity. Moreprecisely, after periodically extending ρ ε from γ to L k ∈ Γ S ( k ) γ the values of ρ ε on two touchingboundaries of elementary cells of Γ will sum up to zero. The function ρ ε will be used to definerestrictions of the operator H ε ( ω ) onto finite parts of the layer Π with so-called Mezincescu bound-ary conditions . These are a special choice of Robin boundary conditions, which ensures that theground state of finite volume restrictions coincides with the infimum of the spectrum of the infinitevolume operator.Our first result describes the location of the spectrum of σ ( H ε ( ω )). Theorem 2.3.
Let t > be sufficiently small. For all ε ∈ [0 , t ] there exists a closed set Σ ε ⊂ R such that σ ( H ε ( ω )) = Σ ε P − a.s. This set Σ ε is equal to the closure of the union of spectra for all periodic realizations of H ε ( · ) : (2.9) Σ ε = [ N ∈ N [ ξ is N Γ -periodic σ (cid:0) H ε ( ξ ) (cid:1) , where the second union is taken over all sequences ξ : Γ → supp µ , which are periodic with respectto the sublattice N Γ := { N q : q ∈ Γ } .In particular, if we are in ( Case I ) or in ( Case II ) and Assumption M holds, then, assuming t > sufficiently small, we have (2.10) min Σ ε = Λ ε . A consequence of this theorem and Lemmata 5.1 and 5.2 is
Corollary 2.4.
Let t > be sufficiently small. If ( Case I ) holds, then for all ε ∈ (0 , t ]min Σ ε = Λ ε Λ − cε, where c = − Λ / > . If ( Case II ) holds, then for all ε ∈ (0 , t ] we have min Σ ε = Λ ε Λ − cε , where c = − Λ / > . This ensures that the random perturbation will indeed create new spectrum below the one ofthe unperturbed operator H . The newly created spectrum expands at least by order ε or ε awayfrom the original spectrum, respectively. This is crucial to make sure that our results below onlocalization are not trivial.Having proved a lower bound on the expansion of the spectrum below Λ , we then identify aregion where we can prove localization. The next theorem is the first ingredient for this purpose.It provides a lower bound on the distance of the ground state energy of finite volume restrictions H ε ( ξ ) (with Mezincescu boundary conditions) to the minimum over all configurations Λ ε in termsof an average over expressions containing the ξ k . Theorem 2.5 (See Theorem 6.1) . For N ∈ N , we denote by Π N the open interior of the set ˜Π N := { x ∈ R n +1 : x ′ = n X j =1 a j e j , a j ∈ [0 , N ) , < x n +1 < d } , Γ N := ˜Π N ∩ Γ , and by Λ εN ( ξ ) the lowest eigenvalue of the operator H εN ( ξ ) := − ∆ + V + L εN ( ξ ) , L εN ( ξ ) := X k ∈ Γ N S ( k ) L ( εξ k ) S ( − k ) on Π N subject to boundary condition (2.1) on ∂ Π ∩ ∂ Π N and to the Mezincescu boundary condition ∂u∂ν = ρ ε u on γ N := ∂ Π N \ ∂ Π .Let t be sufficiently small. Then there exist constants N ∈ N and c > , depending exclusivelyon the operators L ( t ) , t ∈ [ − T, T ] , and on V , such that for all N ∈ N with N > N , ξ ∈ Ω , and ε ∈ (0 , t ] , we have Λ εN ( ξ ) − Λ ε > ε | Λ | N n X k ∈ Γ N (1 − ξ k ) for ε < c N − in ( Case I ) , Λ εN ( ξ ) − Λ ε > ηε N n X k ∈ Γ N (1 − ξ k ) for ε < c N − in ( Case II ) , where Λ is given in Lemma 5.1 and η is given in (2.5) . It is a canonical step to combine this theorem with large deviation estimates, and so-calledCombes-Thomas estimates to deduce a so-called initial scale estimate. Initial scale estimates areprobabilistic bounds on the decay of the resolvent of finite volume restrictions of H ε ( ω ). Theyserve as an induction anchor in the multi-scale analysis. The initial scale estimates can be foundin Theorem 6.3 in Section 6.The second ingredient in a multi-scale analysis proof of localization are Wegner estimates. Theybound the probability of finding an eigenvalue of a finite volume restriction of H ε ( ω ) in an intervaland serve as a non-resonance condition in the induction step in the multi-scale analysis. The nexttheorem is a Wegner estimate in an ε -dependent regime below Λ . Theorem 2.6 (See Theorem 8.7) . Assume that t > is sufficiently small. There are D dependingexclusively on the operators L ( t ) , t ∈ [ − T, T ] , C n,h depending exclusively on n and h , as well as C n,V depending merely on n , V , and the lattice Γ , such that for all ε ∈ (0 , t ] , all α ∈ Γ , and all N ∈ N the following hold:(i) For all E Λ − Dε and all κ Dε / we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n,V | (cid:3) | N n ] · κ N n . (ii) Assume that L . Then for all E Λ − Dε and all κ Dε / , we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n,V | (cid:3) | N n ] · κ N n . It is paramount to notice that in Theorem 2.6 the intervals [Λ ε , Λ − Dε ] or [Λ ε , Λ − Dε ],respectively, from which E is chosen, must have a sufficiently large distance to Λ . These intervalscan only be non-empty if ε Λ ε expands (anti-tonically) at a sufficient fast rate below Λ . It isCorollary 2.4 that ensures this.The initial scale estimate and the Wegner estimate are the crucial ingredients to start the multi-scale analysis as in [GK03] and conclude spectral and dynamical localization. To implement thisone needs actually a number of several additional, rather general properties. We show in Section 8that they hold, provided the following assumption is satisfied. Assumption R. ( R.1 ) There exists a constant c > such that for any function ϕ : (cid:3) ′ → [0 , , ϕ ∈ C ∞ ( (cid:3) ′ ) , there exists a constant c = c ( k ϕ k C ( (cid:3) ) ) satisfying the estimatefor all u ∈ W , ( (cid:3) ) , i = 1 , , t ∈ [ − T,T ] (cid:12)(cid:12) ( L i ( t ) u, ϕu ) L ( (cid:3) ) (cid:12)(cid:12) c ( ϕ ∇ u, ∇ u ) L ( (cid:3) ) + c k u k L ( (cid:3) ) . For i = 1 , , the above supremum can be omitted. The constant c is allowed to depend on ϕ only through its norm k ϕ k C ( (cid:3) ) in such a way that k ϕ k C ( (cid:3) ) c is non-decreasing.( R.2 ) For any function ϕ : (cid:3) ′ → [0 , , ϕ ∈ C ∞ ( (cid:3) ′ ) , there exists a constant c = c ( k ϕ k C ( (cid:3) ) ) such that the estimatefor all u ∈ W , ( (cid:3) ) , i = 1 , , t ∈ [ − T,T ] kL i ( t ) ϕu − ϕ L i ( t ) u k L ( (cid:3) ) c k u k W , ( (cid:3) ) BORISOV, T¨AUFER, AND VESELI´C holds true, where for i = 1 , , the above supremum can be omitted. The constant c isallowed to depend on ϕ only through its norm k ϕ k C ( (cid:3) ) in such a way that k ϕ k C ( (cid:3) ) c is non-decreasing. Finally, this allows us formulate our results on Anderson localization in a small neighbourhoodof Σ ε . Theorem 2.7.
Assume that we are in ( Case I ) , Assumptions M , R hold, and t > is suffi-ciently small. Then there exists C > such that for all ε ∈ (0 , t ] the set Σ ε ∩ [min Σ ε , min Σ ε + Cε ] is almost surely non empty and exhibits dynamical localization. Theorem 2.8.
Assume that we are in ( Case II ) , Assumptions M , R hold, and t > is suffi-ciently small. Assume also that the operator L is non-positive. Then there exists C > suchthat for all ε ∈ (0 , t ] the set Σ ε ∩ [min Σ ε , min Σ ε + Cε ] is almost surely non empty and exhibitsdynamical localization. The proof of Theorems 2.7 and 2.8 follows the classical strategy of multi-scale analysis proofsof localization, but some extra attention is required due to the interplay between the scale N andthe disorder ε . These details are explained in Section 8.3. Reformulation for operators acting on the entire space
We now explain the necessary modifications in order to also treat operators living on the wholespace R n . We use the same approach as in [BGV16, Sect. 3.7]. Let us define the operators L ′ ( t ) := t L ′ + t L ′ + t L ′ ( t ) , where L ′ i : W , ( (cid:3) ′ ) → L ( (cid:3) ′ ) are bounded symmetric linear operators and the operator L ′ ( t ) aswell as its derivative with respect to t are bounded uniformly in t ∈ [ − t , t ]. We then introducein L ( R n ) the operator H ′ ε ( ξ ) := − ∆ x ′ + X k ∈ Γ S ′ ( k ) L ′ ( εξ k ) S ′ ( − k )where ∆ x ′ denotes the Laplacian in R n and S ′ ( k ) is the shift operator: ( S ′ ( k ) u )( x ′ ) = u ( x ′ + k ).The operator H ′ ε ( ω ) is a random self-adjoint operator in L ( R n ).In order to use results of the previous section, we extend this construction from R n to the layerΠ ⊂ R n +1 where we set its height d = π and impose Neumann boundary conditions on ∂ Π, i.e. B u = ∂u∂x n +1 . More precisely, we extend the operators L ′ i constantly to the ( n + 1)th dimension, i.e. we extendthem to operators L i : W , ( (cid:3) ) → L ( (cid:3) ) via( L i u )( x ′ , x n +1 ) = (cid:0) L ′ i u ( · , x n +1 ) (cid:1) ( x ′ ) . This allows to introduce H ε ( ξ ) in L (Π) as before by (2.3) with V = 0. Now, we have in particularthat the lowest eigenvalue Λ of H ⊥ := d d x n +1 on (0 , π )with Neumann boundary condition is Λ = 0, see Section 5. The associated eigenfunction Ψ ,extended to (cid:3) , and with the normalization as in Section 5 is simply a constant functionΨ = p π | (cid:3) ′ | , ( x ′ ) ≡ . The spectral properties of H ε ( ξ ) can be analysed by separation of variables ( x ′ and x n +1 ). Moreprecisely we have(3.1) H ε ( ξ ) = H ′ ε ⊗ H ⊥ , This implies(3.2) σ ( H ε ( ξ )) = ∞ [ m =0 σ ( H ′ ε ( ξ )) + m . and that H ε ( ξ ) χ I ( H ε ( ξ )) and H ′ ε ( ξ ) χ I ( H ′ ε ( ξ )) are unitarily equivalent for I := [min σ ( H ′ ε ( ξ )) , min σ ( H ′ ε ( ξ )) + 1] . Since Theorem 2.3, Lemma 5.1, and Lemma 5.2 concern only properties in a small neighbourhoodof the spectral bottom, they immediately imply analogous statements for the operator H ′ ε ( ξ ).More precisely, the analogue of the auxiliary operator H δ (cid:3) is introduced as H ′ δ (cid:3) ′ := − ∆ + L ′ ( δ ) , δ ∈ [ − ε, ε ] , in L ( (cid:3) ′ ) subject to periodic boundary conditions on ∂ (cid:3) ′ . By Λ ′ δ we denote again the lowesteigenvalue of the operator H ′ δ (cid:3) and by Ψ ′ δ an appropriately normalized associated eigenfunction.Similarly as in the previous section, we can expand them in a Taylor seriesΛ ′ δ = δ Λ ′ + δ Λ ′ + O ( δ ) and Ψ δ ( x ′ ) = 1 p | (cid:3) ′ | (cid:0) + δ Ψ ′ ( x ′ ) + δ Ψ ′ ( x ′ ) (cid:1) + O ( δ )around δ = 0. The expressions Λ ′ i and Ψ ′ i , i ∈ { , } are given by analogous formulas as thecorresponding Λ i and Ψ i in Lemma 5.1. For convenience, we explicitly formulate these expressionsin Lemma 5.3.Analogously to ( Case I ) and (
Case II ) we treat the following cases:(
Case I’ ) The constant Λ ′ is strictly negative.( Case II’ ) The constant Λ ′ is zero and b > − , η := − ( b + 1)Λ ′ + (1 − b ) (cid:0) Re(Φ ′ − Ψ ′ ) , Re L ′ (cid:1) L ( (cid:3) ′ ) > , where Φ ′ ∈ W , ( (cid:3) ′ ) is the unique solution to the problem − ∆Φ ′ = −L in (cid:3) ′ , ∂ Φ ′ ∂ν = 0 on ∂ (cid:3) ′ , orthogonal to in L ( (cid:3) ′ ).Lemma 5.2 carries over verbatim which means that for sufficiently small t we have ∀ ε ∈ (0 , t ] : min δ ∈ [ εb,ε ] Λ ′ δ = Λ ′ ε . We also introduce on the lateral boundary ∂ (cid:3) ′ the analogue of the function ρ ε : ρ ′ ε := 1Ψ ′ ε ∂ Ψ ′ ε ∂ν , where ν is the outward normal.The place of Assumption M will be taken by Assumption M’.
For all ε ∈ [0 , t ] the function ρ ′ ε is piecewise continuous on ∂ (cid:3) ′ . Furthermore, • If ( Case I’ ) holds, then the asymptotic identity sup ∂ (cid:3) ′ (cid:12)(cid:12) ρ ′ ε − ερ ′ (cid:12)(cid:12) = O ( ε ) holds true, where (3.3) ρ ′ := ∂ Ψ ′ ∂ν , is piecewise continuous on ∂ (cid:3) ′ . • If ( Case II’ ) holds, then the asymptotic identity sup ∂ (cid:3) ′ (cid:12)(cid:12) ρ ′ ε − ερ ′ − ε ρ ′ (cid:12)(cid:12) = O ( ε ) holds true, where ρ ′ is given by (3.3), ρ := ∂ Ψ ′ ∂ν − Ψ ′ ∂ Ψ ′ ∂ν , and ρ ′ , ρ ′ are piecewise continuous on ∂ (cid:3) ′ . Assumption R’.
Assumptions
R.1 and
R.2 remain almost unchanged: we just replace (cid:3) by (cid:3) ′ in their formulation. Combining identity (3.2) with the results of the previous section, we arrive at the followinganalogous results on the almost sure spectrum and on Anderson localization near the bottom ofthe spectrum of H ′ ε ( ω ). Theorem 3.1.
Let t > be sufficiently small. Then, for all ε ∈ [0 , t ] there exists a closed set Σ ′ ε such that σ ( H ′ ε ( ω )) = Σ ′ ε P − a.s. The set Σ ′ ε is equal to the closure of the union of spectra for all periodic realizations of H ′ ε ( ω ) : Σ ′ ε = [ N ∈ N [ ξ is N Γ -periodic σ (cid:0) H ′ ε ( ξ ) (cid:1) , where the second union is taken over all sequences ξ : Γ → supp µ , which are periodic with respectto the sublattice N Γ := { N q : q ∈ Γ } .Assume that we are in ( Case I’ ) or in ( Case II’ ) and Assumption M’ holds, then, assuming t > sufficiently small, we have min Σ ′ ε = Λ ′ ε . Corollary 3.2.
Let t > be sufficiently small. If ( Case I’ ) holds, then for all ε ∈ (0 , t ]inf Σ ′ ε inf Σ ′ − cε, where c = − Λ ′ / > . If ( Case II’ ) holds, then for all ε ∈ (0 , t ] we have inf Σ ′ ε inf Σ ′ − cε where c = − Λ ′ / > . We also obtain the following analogue of Theorem 2.5:
Theorem 3.3.
For N ∈ N , we denote by (cid:3) ′ N the open interior of the set (cid:3) ′ N := { x ′ ∈ R n : x ′ = n X j =1 a j e j , a j ∈ [0 , N ) } , Γ N := (cid:3) ′ N ∩ Γ , and by Λ ′ εN ( ξ ) the lowest eigenvalue of the operator H ′ εN ( ξ ) := − ∆ x ′ L ′ εN ( ξ ) , L ′ εN ( ξ ) := X k ∈ Γ N S ( k ) L ′ ( εξ k ) S ( − k ) on (cid:3) ′ N subject to the Mezincescu boundary condition ∂u∂ν = ρ ′ ε u on ∂ (cid:3) ′ N .Let t > be sufficiently small. Then there exist constants N ∈ N and c > , dependingexclusively on the operators L ′ ( t ) , t ∈ [ − T, T ] , such that for all N ∈ N with N > N , ξ ∈ Ω , and ε ∈ (0 , t ] , we have Λ ′ εN ( ξ ) − Λ ′ ε > ε | Λ | N n X k ∈ Γ N (1 − ξ k ) for ε < c N − in ( Case I ) , (3.4) Λ ′ εN ( ξ ) − Λ ′ ε > ηε N n X k ∈ Γ N (1 − ξ k ) for ε < c N − in ( Case II )(3.5) Now, we can again combine Theorem 3.3 with Combes-Thomas estimates and deduce a corre-sponding initial scale estimate for the operator H ′ ε ( ω ). Finally, we note that the Wegner estimateof Theorem 2.6 can also be proved for the operator H ′ ε ( ω ). This is clear from the proof in Section 7since the fact that Theorem 2.6 is formulated on the layer Π only enters in the volume bound. Weconclude: Theorem 3.4.
Assume that t > is sufficiently small. There are D depending exclusively on theoperators L ′ ( t ) , t ∈ [ − T, T ] , C n,h depending exclusively on n , and h , as well as C n dependingmerely on n , and the lattice Γ , such that for all ε ∈ (0 , t ] , all α ∈ Γ , and all N ∈ N the followinghold:(i) For all E Λ − Dε and all κ Dε / we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n | (cid:3) ′ | N n ] · κ N n . (ii) Assume that L . Then for all E Λ − Dε and all κ Dε / , we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n | (cid:3) ′ | N n ] · κ N n . In conclusion, we have again all ingredients needed to start the multi-scale analysis and proveAnderson localization near the bottom of Σ ′ ε for sufficiently small ε > Theorem 3.5.
Assume that we are in ( Case I’ ) , Assumptions M’ and R’ hold, and t > issufficiently small. Then there exists C > such that for all ε ∈ (0 , t ] the set Σ ′ ε ∩ [Λ ′ ε , Λ ′ ε + Cε ] is almost surely non empty and only contains pure point spectrum with exponentially decayingeigenfunctions. Theorem 3.6.
Assume that we are in ( Case II’ ) , Assumptions M’ and R’ hold, and t > issufficiently small. Assume also that the operator L ′ is non-positive. Then there exists C > suchthat for all ε ∈ (0 , t ] the set Σ ′ ε ∩ [Λ ′ ε , Λ ′ ε + Cε ] is almost surely non empty and only containspure point spectrum with exponentially decaying eigenfunctions. Examples
All examples we present, concern operators defined on the layer as in Section 2, but immediatelyextended by formula (3.1) to the full space setting on R n as in Section 3.4.1. Random potential.
Our first example is the classical perturbation by a random potential.Here the operator L ( t ) is the multiplication by a potential W ( t ), which is defined as W ( t ) = tW + t W where W i = W i ( x ) are bounded measurable real functions. ¿From Lemma 5.1, we infer that theconstant Λ is given by Λ = Z (cid:3) W ( x ) | Ψ ( x n +1 ) | dx. If Λ <
0, we are in (
Case I ).If Λ = 0, then we infer from Lemma 5.1 that W Ψ = − ( H (cid:3) − Λ )Ψ whence Λ = Z (cid:3) W ( x )Ψ ( x n +1 ) dx + Z (cid:3) W ( x )Ψ Ψ dx = Z (cid:3) W ( x )Ψ ( x n +1 ) dx − Z (cid:3) Ψ ( H (cid:3) − Λ )Ψ dx < Z (cid:3) W ( x )Ψ ( x n +1 ) dx since H (cid:3) − Λ is a nonnegative operator and Ψ is not its (unique) ground state. We concludethat if we assume W
0, as needed in Theorem 2.8, then we automatically have Λ < Case II ) provided η = − ( b + 1)Λ + (1 − b ) Z (cid:3) W ( x )(Φ ( x ) − Ψ ( x ))Ψ ( x n +1 ) dx > . Assumption M is satisfied once the potentials V and W i are smooth enough, for instance V , W i ∈ C q ( (cid:3) ) for some q ∈ (0 , δ ,Ψ , Ψ belong to the H¨older space C q ( (cid:3) ) and the asymptotics (5.2) holds in the sense of thenorm in this space. Assumptions R.1 , R.2 are obviously true for the considered example.4.2.
Integral operator.
Our next example is a random integral operator corresponding to( L i u )( x ) = Z (cid:3) K i ( x, y ) u ( y ) dy, i = 1 , , where K i ∈ L ( (cid:3) × (cid:3) ) are symmetric kernels: K i ( y, x ) = K i ( x, y ) . Here, the constant Λ is given by the integralΛ = Z (cid:3) × (cid:3) K ( x, y )Ψ ( x n +1 )Ψ ( y n +1 ) dx dy, and if Λ <
0, we are in (
Case I ).If Λ = 0, conditions (5.3), (2.6) take the form( H (cid:3) − Λ )Ψ = − Z (cid:3) K ( · , y )Ψ ( y n +1 ) dy, and ( − ∆ + V − Λ )Φ = − Z (cid:3) K ( · , y )Ψ ( y n +1 ) dy in (cid:3) , B Φ = 0 on ∂ (cid:3) ∩ ∂ Π , ∂ Φ ∂ν = 0 on γ. The constants Λ and η read in this caseΛ = Z (cid:3) × (cid:3) K ( x, y )Ψ ( x n +1 )Ψ ( y n +1 ) dx dy,η = − ( b + 1)Λ + (1 − b ) Z (cid:3) × (cid:3) K ( x, y )(Φ ( y ) − Ψ ( y ))Ψ ( x n +1 ) dx dy. For Theorem 2.8, the kernel K needs to satisfy furthermore the inequality(4.1) Z (cid:3) × (cid:3) K ( x, y ) u ( x ) u ( y ) dx dy u ∈ L ( (cid:3) ) . A classical case where the last condition is satisfied is the following: If K ( x, y ) = K ( x − y )is translation invariant, continuous and satisfies the normalization K (0) = 1. Bochner’s theoremtells us that (4.1) holds if and only if there is a probability measure ν on R n , such that K ( x ) = − Z exp( iξ · x )d ν ( ξ ) . Other sufficient conditions for (4.1) can be found in [BCR84].Assumption M can be again ensured by sufficient smoothness of the kernels K i , e. g. K i ∈ C q ( (cid:3) × (cid:3) ). In this case the term L ( δ )Ψ δ in the eigenvalue equation for Ψ δ is holomorphic in δ in the sense of the C q ( (cid:3) )-norm. By Schauder estimates this implies that Ψ δ ∈ C q ( (cid:3) ) and Ψ δ is holomorphic in the sense of the norm in this space. This ensures that Assumption M holds.Assumptions R hold, since K i are Hilbert-Schmidt operators.4.3. Random second order differential operator.
It is possible to treat the case where L i are general symmetric second order differential operators:(4.2) L i = n +1 X k,j =1 ∂∂x k Q ( i ) kj ∂∂x j + i n +1 X j =1 (cid:18) Q ( i ) j ∂∂x j + ∂∂x j Q ( i ) j (cid:19) + Q ( i )0 , where Q ( i ) kj , Q ( i ) j are real-valued piecewise continuously differentiable functions, Q ( i )0 is a real mea-surable bounded function. We assume that(4.3) Q ( i ) jk = Q ( i ) kj in (cid:3) and Q ( i ) kj = Q ( i ) k = 0 on ∂ (cid:3) \ ∂ Π . The latter condition is sufficient for L i to be symmetric.An integration by parts gives the formula(4.4) Λ = − Z (cid:3) Q (1) n +1 n +1 ( x ) (cid:0) Ψ ( x n +1 ) (cid:1) dx + Z (cid:3) Q (1)0 ( x )Ψ ( x n +1 ) dx. If Λ <
0, we are in (
Case I ).If Λ = 0, then(4.5) L Ψ = n +1 X k =1 ∂∂x k Q (1) k n +1 Ψ + i Q (1) n +1 Ψ + i n +1 X j =1 ∂∂x j ( Q (1) j Ψ ) + Q (1)0 Ψ and this formula is to be substituted into the right hand side of the equations in (5.3), (2.6). Theconstant Λ is given by the identity similar to (4.4):(4.6) Λ = − Z (cid:3) Q (2) n +1 n +1 ( x ) (cid:0) Ψ ( x n +1 ) (cid:1) dx + Z (cid:3) Q (2)0 ( x )Ψ ( x n +1 ) dx. The formula for η is rewritten as(4.7) η := − ( b + 1)Λ + (1 − b ) − n +1 X k =1 (cid:18) ∂∂x k (Φ − Ψ ) , Q (1) k n +1 Ψ (cid:19) L ( (cid:3) ) + i(Φ − Ψ , Q (1) n +1 Ψ ) L ( (cid:3) ) − i n +1 X j =1 (cid:18) ∂∂x j (Φ − Ψ ) , Q (1) j Ψ (cid:19) L ( (cid:3) ) + (Φ − Ψ , Q (1)0 Ψ ) L ( (cid:3) ) . Assumption M again follows from Schauder estimates as above.Assumption R.1 can be checked easily. Indeed, by integration by parts we get that( L i u, ϕu ) L ( (cid:3) ) = − n +1 X k,j =1 (cid:18) Q ( i ) kj ∂u∂x j , ∂ϕu∂x i (cid:19) L ( (cid:3) ) + i n +1 X j =1 (cid:18) Q ( i ) j ∂u∂x j , ϕu (cid:19) L ( (cid:3) ) + i n +1 X j =1 (cid:18) Q ( i ) j u, ∂ϕu∂x j (cid:19) L ( (cid:3) ) + ( Q ( i )0 u, ϕu ) L ( (cid:3) ) . Substituting here the identities ∂ϕu∂x i = ϕ ∂u∂x i + u ∂ϕ∂x i , after obvious estimates we are led to Assumption R.1 .The proof of Assumption
R.2 is even simpler. It is clear that in the expression ϕ L i u − L i ϕu all second order derivatives of u cancel out and hence Assumption R.2 holds true.
Random electro-magnetic field.
Our next example is a random magnetic field, which isin fact a particular case of the previous example. The random operator is introduced as H ε ( ξ ) := (cid:0) i ∇ + A ε ( ξ ) (cid:1) + W ε ( ξ ) ,A ε ( ξ ) := ε X k ∈ Γ ξ k S ( k ) A S ( − k ) , W ε ( ξ ) := ε X k ∈ Γ ξ k S ( k )( W + εξ k W ) S ( − k ) . Here A : (cid:3) → R n +1 , is a real-valued magnetic potential, W i : (cid:3) → R are bounded measurablereal-valued potentials. We suppose that A is piecewise continuously differentiable and it vanisheson the lateral boundary of (cid:3) . The corresponding operators L i are given by the identities L = 2i A · ∇ + i div A + W , L = | A | + W , L = 0 . The case W = W = 0 was already considered in [BGV16, Sect. 3.3] and it was shown thatin this case Λ = 0 and Λ >
0. While this was stated in [BGV16] for V ≡
0, this also holds fornon-trivial V . An explicit calculation is provided in Appendix A. In view of Remark 2.2 (3), thisimplies that we are neither in ( Case I ) nor (
Case II ). For this reason random magnetic fieldsare a very instructive example to illustrate why the case Λ = 0 and Λ > W and W is non-zero. It was shown in [BGV16, Sect. 3.3] that Z (cid:3) Ψ (cid:0) A · ∇ + div A (cid:1) Ψ dx = 0and therefore, Λ = Z (cid:3) W Ψ dx. If Λ <
0, we are in (
Case I ).If Λ = 0, then L Ψ = 2i A n +1 Ψ ′ + iΨ div A + W Ψ , and Λ = Z (cid:3) ( | A | + W )Ψ dx + Z (cid:3) Ψ (2i A n +1 Ψ ′ + iΨ div A + W Ψ ) dx. The constant η is given by the identity η = − ( b + 1)Λ + (1 − b ) Z (cid:3) Re(Φ − Ψ ) W Ψ dx. The condition L | A | + W < . Finally, Assumption M is again guaranteed via the Schauder estimates, assuming sufficient smooth-ness of A and W i , for instance A ∈ C q ( (cid:3) ), W i ∈ C q ( (cid:3) ), q ∈ (0 , R.1 , R.2 hold as well.4.5.
Random metric.
One more particular case of the above described random second orderdifferential operator is provided by a random metric. The corresponding random operator isintroduced as H ε ( ξ ) = − ∆ + V + X k ∈ Γ S ( k ) n +1 X j,k =1 ∂∂x j ( εξ k Q (1) jk + ε ξ k Q (2) jk ) ∂∂x k and the associated operators L i , i = 1 ,
2, are given by (4.2) with Q ( i ) j = Q ( i )0 = 0, i = 1 , j = 1 , . . . , n + 1. The functions Q ( i ) jk are assumed to be piecewise continuously differentiable in (cid:3) ,real-valued and obeying condition (4.3). The operator L is supposed to be zero.Formula (4.4) for Λ takes the formΛ = − Z (cid:3) Q (1) n +1 n +1 ( x ) (cid:0) Ψ ( x n +1 ) (cid:1) dx. If Λ <
0, we are in (
Case I ).If Λ = 0, we just need to appropriately adapt formulae (4.5), (4.6), (4.7). To identify whetherwe are in ( Case II ) let us restrict here our attention to the particular case Q (1) n +1 n +1 = Q (1) k n +1 = Q n +1 k = 0 . Then Λ = 0 and moreover, by (4.5), we also have L Ψ = 0. This implies immediately Ψ =Φ = 0 and the formulae for Λ and η becomeΛ = − Z (cid:3) Q (2) n +1 n +1 ( x ) (cid:0) Ψ ( x n +1 ) (cid:1) dx, η = − ( b + 1)Λ . In order to satisfy simultaneously the condition L
0, we impose the assumption that n +1 X j,k =1 Q (2) jk ( x ) z j z k > , z j ∈ R , and Q (2) n +1 n +1 > , x ∈ (cid:3) . Then Λ < η > Q ( i ) jk ∈ C q ( (cid:3) ), q ∈ (0 , Random delta-potential.
Our next example is a random delta interaction. The results ofthis paragraph have been presented in the announcement [BTV18].We introduce the random delta interaction as follows: Let M be a closed bounded C manifoldin (cid:3) ⊂ R n of codimension one. The outward normal vector to M is denoted by ν . The manifold M is assumed to be separated from the boundary ∂ (cid:3) by a positive distance. By M k , k ∈ Γ, wedenote the translate of M along Γ: M k := { x ∈ Π : x − k ∈ M } . We also let M := S k ∈ Γ M k .By y = ( y , . . . , y n − ) we denote some local coordinates on M , while ̺ is the distance in (cid:3) from a point to M measured along ν . Since M k are translations of M , the coordinates ( y, ̺ )are well-defined in every (cid:3) k , k ∈ Γ, and hence, in the whole of Π.By b = b ( y ) we denote a real function on M assuming that b ∈ C ( M ). We extent b periodically to the entire set M .Our random operator is introduced as the negative Laplacian˜ H ε ( ξ ) := − ∆ in L (Π) , whose domain consists of the functions u ∈ W , (Π \ M ) satisfying the boundary conditions[ u ] M k = 0 , (cid:20) ∂u∂̺ (cid:21) M k = − εb ξ k u (cid:12)(cid:12) M k , [ u ] M k := u (cid:12)(cid:12)(cid:12) ̺ =+0 y ∈ Mk − u (cid:12)(cid:12) ̺ = − y ∈ Mk . The introduced random operator ˜ H ε ( ξ ) cannot be represented as (2.3) since the domain of ˜ H ε ( ξ )is not a subspace in W , (Π). However, here we apply the approach proposed in [Bor07a, Sect.8.5], see also [Bor07b, Sect. 8.5]. This approach will allow us to transform our operator to anotherone obeying the assumptions of the present work. Let us describe this approach.First we denote by P the mapping describing the change of the variables x ( y, ̺ ): ( y, ̺ ) = P ( x ). This mapping is well-defined in a small neighbourhood of each M k , k ∈ Γ, and the shape of this neighbourhood is independent of k . Then in these neighbourhoods we introduce one moremapping: P ( x, t ) := P − (cid:0) y, ̺ + ̺ | ̺ | tb ( ξ ) (cid:1) , where t ∈ [ − t , t ], and t is the constant used in the definition of the operator L ( t ).Let χ = χ ( z ) be an infinitely differentiable cut-off function vanishing for | z | > | z | <
1. We define P ( x, t ) = (cid:18) − χ (cid:18) ̺δ (cid:19)(cid:19) x + χ (cid:18) ̺δ (cid:19) P ( x, t ) , where δ > P were proved: For sufficiently small δ , the mapping P is a C -diffeomorphism, maps (cid:3) ontoitself and it acts as the identity mapping outside some small neighbourhood of M . We define asimilar mapping on R n as P ( x, εξ ) := P ( x, εξ k ) on (cid:3) k . In view of the aforementioned properties of P , the mapping P is a C -diffeomorphism, mapsΠ onto itself, and it acts as the identity mapping outside a small neighbourhood of M . It alsofollows from [Bor07a, Sect. 8.5] that the operator( U u )( x ) := p − ( x ) u (cid:0) P − ( x, εξ ) (cid:1) is unitary in L ( R n ) and(4.8) H ε ( ξ ) := U ˜ H ε ( ξ ) U − = − p div x p − P ⊺ P ∇ x p , where P is the Jacobian matrix formed by the derivatives ∂ P ∂x i , and p := det P is the associatedJacobian.The domain of the operator H ε ( ξ ) coincides with space (2.2). The matrix P and the function pdoes not have continuous derivatives on M , however these derivatives are well-defined on Π \ M andthus have limits on both the inner and outer side of M . This is why the action of the differentialexpression in the right hand side in (4.8) should be treated as follows: this expression is appliedto a function in the domain of H ε ( ξ ) and the result is calculated in the sense of usual derivativesin Π \ M ; the values of a zero measure M are neglected.The operator H ε ( ξ ) satisfies (2.3); the corresponding operator L ( t ) acts as(4.9) L ( t ) := − p div x p − P ⊺ P ∇ x p + ∆ x , where in the right hand side we let ε := 1, ξ k := t . The operators L i can be obtained by expandingthe right hand side of (4.9) into the Taylor series as t →
0. The spectra of the operators H ε ( ξ )and ˜ H ε ( ξ ) coincide and therefore, the stated localization of the spectrum for H ε ( ξ ) implies thesame for that of ˜ H ε ( ξ ) provided we can satisfy the required assumptions.Reproducing literally the calculations of Section 8.5 in [Bor07b], one can check easily thatΛ = Z M b ( ξ )Ψ ( x ) dx. If Λ <
0, we are in (
Case I ). Assumption M then holds due to standard smoothness improvingtheorems. Indeed, since the perturbation is localized on the manifold M , which is separated from γ , asymptotics (5.2) holds true in the vicinity of γ in H p -norm for each p >
1. This impliesimmediately the asymptotics (2.7).Condition R here can be checked in the same way as this was done for a second order differentialoperator in Section 4.3.5. Perturbation theory and minimum of the spectrum
In this section we provide details on the Taylor approximations (2.4) for Λ δ and Ψ δ , prove somepreliminary perturbation estimates and prove Theorem 2.3. Taylor expansion of the ground state.
Let Λ be defined as the smallest eigenvalue ofthe operator − d dx n +1 + V on (0 , d )subject to boundary condition (2.1).The function Ψ is defined as follows: Let Ψ = Ψ ( x n +1 ) be the unique positive eigenfunctioncorresponding to Λ with normalization chosen such that k Ψ k L (0 ,d ) = 1 p | (cid:3) ′ | . We extend Ψ to (cid:3) by Ψ ( x ′ , x n +1 ) = Ψ ( x n +1 ) and use the same symbol for this extension. Theresulting function then belongs to W , ( (cid:3) ) and is the unique, non-negative, normalized groundstate of H (cid:3) . Furthermore, we note that the function Ψ satisfies Neumann as well as periodicboundary conditions on γ = ∂ (cid:3) \ ∂ Π.The first lemma describes some properties of Λ δ and Ψ δ (lowest eigenvalue and eigenfunctionof the operator H δ (cid:3) ). Lemma 5.1.
The eigenvalue Λ δ is simple and twice continuously differentiable with respect tosufficiently small δ . The associated eigenfunction Ψ δ can be normalized to obey (Ψ δ , Ψ ) L ( (cid:3) ) = 1 and under such a normalization it is twice differentiable with respect to δ in the norm of W , ( (cid:3) ) .The first terms of the Taylor expansions for Λ δ and Ψ δ are Λ δ = Λ + δ Λ + δ Λ + O ( δ ) , (5.1) Ψ δ ( x ) = Ψ ( x ) + δ Ψ ( x ) + δ Ψ ( x ) + O ( δ ) , (5.2) where Λ i , Ψ i , i = 1 , , are uniquely determined by the conditions Λ := ( L Ψ , Ψ ) L ( (cid:3) ) , Λ := ( L Ψ , Ψ ) L ( (cid:3) ) + (Ψ , L Ψ ) L ( (cid:3) ) , ( H (cid:3) − Λ )Ψ = −L Ψ + Λ Ψ , (Ψ , Ψ ) L ( (cid:3) ) = 0 , (5.3) ( H (cid:3) − Λ )Ψ = −L Ψ − L Ψ + Λ Ψ + Λ Ψ , (Ψ , Ψ ) L ( (cid:3) ) = 0 . Proof.
The proof of this lemma is based on regular perturbation theory, see also the proof ofLemma 2.1 in [Bor17]. Namely, as δ →
0, the lowest eigenvalue of H δ (cid:3) converges to Λ and issimple. The eigenfunction associated with Λ δ can be chosen so that it converges to Ψ in W , ( (cid:3) )and we normalize Ψ δ as it is stated in the formulation of the lemma. Expansions (5.1), (5.2) andthe stated formulae for their coefficients are implied directly by standard perturbation theory. (cid:3) Lemma 5.2.
Assume that we are either in ( Case I ) or ( Case II ) . Let t > be sufficientlysmall. Then, for all ε ∈ (0 , t ] , the value δ ∈ [ bε, ε ] which minimizes [ bε, ε ] ∋ δ Λ δ is given by δ = ε .Proof. We differentiate formula (5.1): d Λ δ dδ = Λ + O ( δ )and we see that in ( Case I ) the sign of d Λ δ dδ coincides with that of Λ . Hence, for Λ >
0, theminimum of Λ δ is attained at δ = εb , while for Λ <
0, it is attained at δ = ε .In ( Case II ) we have d Λ δ dδ = 2Λ δ + O ( δ )Recall from Remark 1 and [Bor17] that in ( Case II ) the constant Λ is strictly negative andtherefore, sign d Λ δ dδ = − sign δ. Hence, the minimum is attained either at δ = ε or at δ = εb . Since b > −
1, due to the asymptotics(5.1), we conclude that the minimum is attained at δ = ε (decreasing possibly t further). (cid:3) In the situation of operators on the whole space as discussed in Section 3, we have the followinganalogue of Lemma 5.1. The proof carries over verbatim.
Lemma 5.3.
The eigenvalue Λ ′ δ is simple and twice continuously differentiable with respect tosufficiently small δ . The associated eigenfunction can Ψ ′ δ can be normalized as p | (cid:3) ′ | Z (cid:3) ′ Ψ ′ δ dx ′ = 1 and under such a normalization it is twice continuously differentiable with respect to δ in the normof W , ( (cid:3) ′ ) . The first terms of the Taylor expansions for Λ ′ δ and Ψ ′ δ are Λ ′ δ = δ Λ ′ + δ Λ ′ + O ( δ ) , Ψ δ ( x ′ ) = 1 p | (cid:3) ′ | (cid:0) + δ Ψ ′ ( x ′ ) + δ Ψ ′ ( x ′ ) (cid:1) + O ( δ ) , where Λ i , Ψ ′ i , i = 1 , , are uniquely determined by the conditions Λ := 1 | (cid:3) ′ | Z (cid:3) ′ L ′ dx ′ , Λ := 1 | (cid:3) ′ | Z (cid:3) ′ L dx ′ + 1 | (cid:3) ′ | (Ψ ′ , L ) L ( (cid:3) ′ ) , H ′ (cid:3) ′ Ψ ′ = −L ′ + Λ ′ , Z (cid:3) ′ Ψ ′ dx ′ = 0 , H ′ (cid:3) ′ Ψ ′ = −L ′ Ψ ′ − L ′ + Λ ′ Ψ ′ + Λ ′ , Z (cid:3) ′ Ψ ′ dx ′ = 0 . Mezincescu boundary condition.
In this subsection we discuss the restrictions of theoperator H ε ( ξ ) on large bounded subdomains of the layer Π with special boundary conditions.Given α ∈ Γ, N ∈ N , we denoteΠ α,N := { x : x ′ = α + n X j =1 a j e j , a j ∈ (0 , N ) , < x n +1 < d } , Γ α,N := (cid:26) x ′ ∈ Γ : x ′ = α + n X j =1 a j e j , a j = 0 , , . . . , N − (cid:27) and we obtain the obvious identity Π α,N = [ k ∈ Γ α,N (cid:3) k up to a zero measure set. By H εα,N ( ξ ) we denote the operator H εα,N ( ξ ) := − ∆ + V + L εα,N ( ξ ) , L εα,N ( ξ ) := X k ∈ Γ α,N S ( k ) L ( εξ k ) S ( − k )on Π α,N subject to boundary condition (2.1) on ∂ Π ∩ ∂ Π α,N and to boundary condition(5.4) ∂u∂ν = ρ ε u on γ α,N := ∂ Π α,N \ ∂ Π . In the context of random operators, this boundary condition was first used by Mezincescu [Mez87].This is why in what follows we refer to it as Mezincescu boundary condition. Proof of Theorem 2.3.
The larger part of Theorem 2.3 is a particular case of Theorem 2.1in [BHEV18] and it only remains to prove identity (2.10).Identity (2.9) and formula (5.17) in [BHEV18] with θ = 0 yield(5.5) min Σ ε ( H ε (cid:3) Ψ ε , Ψ ε ) L ( (cid:3) ) k Ψ ε k L ( (cid:3) ) = Λ ε . To obtain the converse estimate, let us introduce the quadratic forms D (cid:0) h ε,ξ (cid:1) ∋ v h ε,ξ ( v ) := k∇ v k L (Π) + ( V v, v ) L (Π) + ( L ε ( ξ ) v, v ) L (Π) D (cid:0) h ε,ξ,α,N (cid:1) ∋ v h ε,ξ,α,N ( v ) := k∇ v k L (Π α,N ) + ( V v, v ) L (Π α,N ) + ( L ε ( ξ ) v, v ) L (Π α,N ) − ( ρ ε v, v ) L ( γ α,N ) corresponding to H ε ( ξ ) and H εα,N ( ξ ), respectively. To specify the corresponding domains let usdenote by ∂ D Π the part of ∂ Π where (2.1) corresponds to Dirichlet boundary conditions. Set D (cid:0) h ε,ξ (cid:1) := { f ∈ C ∞ (Π) ∩ W , (Π) | dist(supp f, ∂ D Π) > } where the closure is taken w. r. t. the norm of W , (Π). Furthermore for α ∈ Γ and n ∈ N let D (cid:0) h ε,ξ,α,N (cid:1) := { f ∈ W , (Π α,N ) | there exists g ∈ D (cid:0) h ε,ξ (cid:1) such that f = g Π α,N } By the variational characterisation of the infimum of the spectruminf σ ( H ε ( ξ )) = inf = u ∈ D ( h ε,ξ )) h ε,ξ ( u ) k u k L (Π) = inf = u ∈ D ( h ε,ξ ) P α ∈ N Γ h ε,ξ,α,N ( u ) + ( ρ ε u, u ) L ( γ α,N ) P α ∈ N Γ k u k L (Π α,N ) Observe that by cancellation at the interfaces between elementary cells of N Γ we have X α ∈ N Γ ( ρ ε u, u ) L ( γ α,N ) = 0 , for u ∈ D (cid:0) h ε,ξ (cid:1) = M α ∈ N Γ D (cid:0) h ε,ξ,α,N (cid:1) , and for each u α ∈ D (cid:0) h ε,ξ,α,N (cid:1) h ε,ξ,α,N ( u α ) > k u α k L (Π α,N ) inf = v ∈ D (cid:0) h ε,ξ,α,N (cid:1) h ε,ξ,α,N ( v ) k v k L (Π α,N ) = k u α k L (Π α,N ) Λ εα,N ( ξ ) , where Λ εα,N ( ξ ) the lowest eigenvalues of the operator H εα,N ( ξ ). Hence, inf σ ( H ε ( ξ ) is lower boundedby inf = u ∈ D ( h ε,ξ ) P α ∈ N Γ h ε,ξ,α,N ( u ) P α ∈ N Γ k u k L (Π α,N ) > inf = u =( u α ) α ∈ N Γ ∈ L α ∈ N Γ D (cid:0) h ε,ξ,α,N (cid:1) P α ∈ N Γ k u α k L (Π α,N ) Λ εα,N ( ξ ) P α ∈ N Γ k u α k L (Π α,N ) . We now specialize to N = 1, observe that Λ ε , ( ξ ) = Λ εξ and Λ ε min ξ ∈ [ b, Λ ε , ( ξ ) byLemma 5.2, hence conclude(5.6) inf σ ( H ε ( ξ )) > Λ ε Minimizing over the configuration, we obtainmin Σ ε > Λ ε . This inequality and (5.5) prove (2.10). Initial length scale estimate
In this section we formulate and prove a slight generalization of Theorem 2.5. We follow thescheme proposed in [BV11], [BV13], [BGV16], [Bor17] for proving similar eigenvalue bounds frombelow. Thereafter we deduce an initial length scale estimate, which is the induction anchor forthe multi-scale induction proof of Anderson localization.
Theorem 6.1.
Let Assumption M hold. There exist constants N ∈ N and c > , dependingexclusively on the operators L ( t ) , t ∈ [ − T, T ] , and on V , such that for all α ∈ Γ , N ∈ N with N > N and ξ ∈ Ω the inequalities hold Λ εα,N ( ξ ) − Λ ε > ε | Λ | N n X k ∈ Γ α,N (1 − ξ k ) for ε < c N − in ( Case I ) , (6.1) Λ εα,N ( ξ ) − Λ ε > ηε N n X k ∈ Γ α,N (1 − ξ k ) for ε < c N − in ( Case II ) . (6.2)To prove this theorem, we shall need some preliminary notations and lemmata. By we denotethe constant sequence = { } k ∈ Γ . Let ˆΛ ε be the second eigenvalue of the operator H εα,N ( ). Sincethe operator L ( t ) has uniformly bounded derivative with respect to t , it satisfies the estimate(6.3) (cid:13)(cid:13)(cid:0) L ( t ) − L ( t ) (cid:1) u (cid:13)(cid:13) L ( (cid:3) ) C | t − t |k u k W , ( (cid:3) ) for all t , t ∈ [ − t , t ] and u ∈ W , ( (cid:3) ), where C is a constant independent of t , t , u . Repro-ducing literally the proof of Lemma 5.1 in [Bor17], one proves the following lemma. Lemma 6.2.
For sufficiently large N and small t > there exist constants C , C , C , dependingexclusively on the operators L , L , L ( t ) , t ∈ [ − T, T ] and on V such that for all α ∈ Γ and all ε ∈ (0 , t ] we have | ˆΛ ε − Λ − C N − | C ε, Λ εα,N ( ξ ) Λ + C ε in ( Case I ) , Λ εα,N ( ξ ) Λ + C ε in ( Case II ) . (6.4) Proof of Theorem 6.1.
We choose sufficiently large N and by T N we denote the circle in thecomplex plane of radius C N centered at the origin, where the constant C comes from Lemma 6.2.Due to our assumption on ε , asymptotics (5.1) and Lemma 6.2, this circle contains no spectralpoints of the operator H εα,N ( ) except for Λ ε anddist (cid:0) E, σ ( H εα,N ( )) \ { Λ ε } (cid:1) > C N . for all E ∈ T N . We rewrite the operator H εα,N ( ξ ) as(6.5) H εα,N ( ξ ) = H εα,N ( ) + ˆ L εα,N ( ξ ) , ˆ L εα,N ( ξ ) := L ε ( ξ ) − L ε ( ) = X k ∈ Γ α,N S ( k ) ˆ L ( εξ k ) S ( − k ) , ˆ L ( εζ ) := L ( εζ ) − L ( ε ) = ε ( ζ − L + ε ( ζ + 1) L ) + ε (cid:0) ζ L ( εζ ) − L ( ε ) (cid:1) . We observe that due to (6.3), for arbitrary u ∈ W , ( (cid:3) ) we have(6.6) k ε ( ζ L ( εζ ) − L ( ε )) u k L ( (cid:3) ) ε k ( ζ − L ( εζ ) u k + ε k ( L ( εζ ) − L ( ε )) u k L ( (cid:3) ) Cε | ζ − |k u k W , ( (cid:3) ) with a constant C depending only on the norms of L ( t ) and its t -derivative, for t ∈ [ − T, T ] (inparticular, independent of u , ε and ζ ).Proceeding as in [Bor17, Eqs. (5.19)–(5.26)], one can check easily that Λ εα,N ( ξ ) solves theequation Λ εα,N ( ξ ) − Λ ε = (cid:0) A εα,N ( ξ ) ˆ L εα,N ( ξ )Ψ ε , Ψ ε (cid:1) L (Π α,N ) N n k Ψ ε k L ( (cid:3) ) , (6.7) A εα,N ( ξ ) := (cid:0) I + ˆ L εα,N ( ξ ) ˆ R εα,N (Λ εα,N ( ξ )) (cid:1) − , where Ψ ε stands for the periodic extension of the function Ψ δ in Lemma 5.1 with δ = ε andˆ R εα,N ( E ) is the reduced resolvent of the operator H εα,N ( ) for E in the vicinity of Λ ε , specifically,for E ∈ T N . The operator ˆ R εα,N ( E ) is bounded as a map from L (Π α,N ) into W , (Π α,N ) isholomorphic in E ∈ T N and it satisfies the estimate(6.8) k ˆ R εα,N ( E ) f k L (Π α,N ) CN k f k L (Π α,N ) for all f ∈ L (Π α,N ), where C is a constant independent of f , ε , α , N .In ( Case II ), equation (6.7) was analysed in [Bor17, Sect. 5] and inequality (6.2) was provedthere. Hence, in the following we consider only (
Case I ). The operator A εα,N ( ξ ) can be representedas A εα,N ( ξ ) = I − ˆ L εα,N ( ξ ) ˆ R εα,N (Λ εα,N ( ξ )) A εα,N ( ξ ) . We substitute this representation into the right hand side of (6.7) and obtainΛ εα,N ( ξ ) − Λ ε = S + S , S := ( ˆ L εα,N ( ξ )Ψ ε , Ψ ε (cid:1) L (Π α,N ) N n k Ψ ε k L ( (cid:3) ) ,S := − (cid:0) ˆ L εα,N ( ξ ) ˆ R εα,N (Λ εα,N ( ξ )) A εα,N ( ξ ) ˆ L εα,N ( ξ )Ψ ε , Ψ ε (cid:1) L (Π α,N ) N n k Ψ ε k L ( (cid:3) ) . Our next step is to estimate S and S .First we observe that according Lemma 5.1, the function Ψ δ is twice differentiable in δ in the W , ( (cid:3) )-norm and hence, the norm k Ψ ε k W , ( (cid:3) ) is bounded uniformly in ε .We begin by estimating the term S . We substitute the asymptotics (5.2) for Ψ ε and formula(6.5) into the definition of S and employ estimate (6.6). This leads us to the identity S = 1 N n X k ∈ Γ α,N ε ( ξ k − (cid:0) Λ + S ( ε, N, ξ k , α ) (cid:1) , where S is some function satisfying the estimate | S ( ε, N, ξ k , α ) | Cε.
Here C is a constant depending only on the norms of L , L , L ( t ), ∂ t L ( t ), t ∈ [ − T, T ], inparticular independent of ε , N , ξ , α , k . Hence, since Λ <
0, for sufficiently small ε and all N , ξ , α , ξ k we have Λ + S ( ε, N, ξ k , α ) Λ . Thus, we obtain(6.9) S > | Λ | N n X k ∈ Γ α,N ε | ξ K − | . We estimate S in a rather rough way. Namely, in view of the definition of A εα,N and ˆ L εα,N , therelative boundedness of the operators L i ( t ), and estimate (6.8), we get that for each u ∈ L (Π α,N ) kA εα,N ( ξ ) u k L (Π α,N ) C k u k L (Π α,N ) with a constant C independent of ε , α , N , ξ , u . Thus, again by the definition of ˆ L εα,N ( ξ ), (6.5),(6.6), (6.4), (6.8), we get (cid:12)(cid:12)(cid:0) ˆ L εα,N ( ξ ) ˆ R εα,N (Λ εα,N ( ξ )) A εα,N ( ξ ) ˆ L εα,N ( ξ )Ψ ε , Ψ ε (cid:1) L (Π α,N ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) ˆ R εα,N (Λ εα,N ( ξ )) A εα,N ( ξ ) ˆ L εα,N ( ξ )Ψ ε , ˆ L εα,N ( ξ )Ψ ε (cid:1) L (Π α,N ) (cid:12)(cid:12) CN k ˆ L εα,N ( ξ )Ψ ε k L (Π α,N ) = CN X k ∈ Γ α,N k ˆ L εα,N ( ξ )Ψ ε k L ( (cid:3) k ) CN ε X k ∈ Γ α,N | ξ k − | CN ε X k ∈ Γ α,N | ξ k − | . Hence, | S | Cε N n − X k ∈ Γ α,N | ξ k − | . Combining this estimate with (6.9), we arrive at (6.1). This completes the proof. (cid:3)
Now we are in the position to derive the desired initial length scale estimate combining theCombes–Thomas bound with an elementary probabilistic estimate. For this purpose one employsTheorem 6.1, estimate (5.6) and proceeds as in [BV11] [BGV16], [Bor17], to obtain the nexttheorem.
Theorem 6.3.
Let β , β ∈ Γ α,N , m , m > be such that B := Π β ,m ⊂ Π α,N , B :=Π β ,m ⊂ Π α,N . Let τ ∈ N satisfy τ > in ( Case I ) or τ > in ( Case II ) . Let N and c beas defined in Theorem 6.1. Define for N > N the intervals J N := " p | Λ | E ( | ω | ) 1 N , c N τ in ( Case I ) J N := " √ p η E ( | ω | ) 1 N , c N τ in ( Case II ) . Then there is a constant c , depending on the measure µ only, c > independent of ε , α , N , β , β , m , m such that for all N > max { N τ , K τ } and for all ε ∈ J N , where K := c p | Λ | E ( | ω | ) ! τ − in ( Case I ) K := c s η E ( | ω | ) ! τ − in ( Case II ) we have the estimate (6.10) P (cid:18) ∀ E Λ ε + 12 √ N : k χ B ( H εα,N ( ξ ) − E ) − χ B k √ N exp (cid:18) − c dist( B , B ) √ N (cid:19)(cid:19) > − N n ( − τ ) exp (cid:0) − c N nτ (cid:1) . This estimate is also valid if H εα,N ( ω ) is equipped with Dirichlet boundary conditions on γ α,N .Remark . Theorem 6.3 will be used to start the multi-scale analysis which proves spectral anddynamical localization for all disorders ε ∈ J N and all energies E in some energy interval. A carefulanalysis is will be required to ensure that the perturbed operator actually has any spectrum inthe energy regions appearing in the probability in (6.10). Such a discussion will be performed inSection 8.Furthermore, we emphasize that for fixed N , Theorem 6.3 only allows for choices of ε in theinterval J N , i.e. there is a lower bound on the disorder ε . In order to have a meaningful statementfor all sufficiently small ε , we shall have to choose initial scales N := N ( ε ), depending on ε .Both issues will be discussed in Section 8.7. Wegner estimate
In this section we prove a Wegner estimate, Theorem 7.1, for the operator H ε ( ω ) defined in(2.3). It holds for restrictions H εα,N ( ω ) of H ε ( ω ) to Π α,N with Mezincescu boundary conditionson γ α,n = ∂ Π α,N \ ∂ Π. The results of this section hold equally if Mezincescu boundary conditionsare replaced by Dirichlet boundary conditions on γ α,n . For convenience we assume in this sectionthat ε > H ( ω ), L ( ω ), H α,N ( ω ) and L α,N ( ω ) instead of H ε ( ω ), L ε ( ω ), H εα,N ( ω ) and L εα,N ( ω ), respectively.Note that in the case of Mezincescu boundary conditions on γ α,n , we haveΛ = inf σ ( H ) = inf σ ( H α,N ) for all α ∈ Γ and N ∈ N , cf. for instance [KV10, Section 2], and for Dirichlet boundary conditionsΛ inf σ ( H α,N ) . Motivated by [Klo95], we define a vector field A acting on the probability space Ω = × k ∈ Γ [ − , A := X k ∈ Γ ω k ∂∂ω k . Our Wegner estimate reads as follows, where we refer to Section 5.3 for the notation on quadraticforms:
Theorem 7.1.
Let E < Λ and assume that there is C b > such that for all ω ∈ Ω , α ∈ Γ , N ∈ N , and all φ ∈ D (cid:0) h ε,ξ,α,N (cid:1) we have h φ, ( A L α,N ( ω ) − L α,N ( ω )) φ i h φ, ( H α,N − E ) φ i , and (7.2) |h φ, L α,N ( ω ) φ i| h φ, H α,N φ i + C b k φ k L (Π α,N ) . (7.3) Then there is C Weyl ∈ (0 , ∞ ) , depending only the dimension, such that for all E ∈ ( −∞ , E ] , all κ (Λ − E ) / , all α ∈ Γ , and all N ∈ N , we have (7.4) P (dist( σ ( H α,N ( ω )) , E ) κ ) C Weyl ( n ) k h k W , ( R ) Λ − E h C b + k V k ∞ ) n +12 | (cid:3) | N n i · κN n . This holds whether H α,N ( ω ) is equipped with Mezincescu or Dirichlet boundary conditions on γ α,N . We shall cast this theorem in a form suitable for the multi-scale analysis in Theorem 8.7 belowafter introducing the parameter ε . We stress again that it is a priori not clear whether H ( ω ) hasany spectrum below these E . Some analysis of the infimum of the spectrum of H ( ω ) will berequired. This discussion will be done in Section 8. Proof.
We adapt the strategy of [HK02] to our situation. Possibly by adding a constant to thepotential V , we may assume that Λ = 0 in which case we have | Λ − E | | E | and the assumptionon κ becomes κ | E | /
4. We shall frequently use that H α,N − E is a positive operator and that E E <
0. For
E < ω ∈ Ω, we define X := ( H α,N − E ) − ( L α,N ( ω )) ( H α,N − E ) − where we suppressed the dependence of X on α, N, E and ω . Note that due to (7.3) X is a boundedoperator, as can be seen explicitly by adapting the estimates in (7.9). If E σ ( H α,N ( ω )) we have( H α,N ( ω ) − E ) − = ( H α,N − E ) − (1 + X ) − ( H α,N − E ) − . Using k ( H α,N − E ) − k = dist { E, σ ( H α,N ) } − | E | − , we find k ( H α,N ( ω ) − E ) − k L (Π α,N ) → L (Π α,N ) k (1 + X ) − k L (Π α,N ) → L (Π α,N ) | E | . This translates into the probability estimate P (dist { σ ( H α,N ( ω )) , E } κ ) = P (cid:18) k ( H α,N ( ω ) − E ) − k L (Π α,N ) → L (Π α,N ) > κ (cid:19) P (cid:18) k (1 + X ) − k L (Π α,N ) → L (Π α,N ) > | E | κ (cid:19) P (cid:18) dist( σ ( X ) , − κ | E | (cid:19) E (Tr( χ I ϑ ( X )))where I ϑ = [ − − ϑ, − ϑ ] with ϑ := κ/ | E | . By assumption on κ , we have − ϑ − / − − ϑ, − ϑ ] ⊂ ( −∞ , − / For a smooth, antitone function ϕ such that ϕ ≡ −∞ , − ϑ ] and ϕ ≡ ϑ , ∞ ) we have Z ϑ − ϑ dd t ϕ ( x + 1 − t )d t = ϕ ( x + 1 − ϑ/ − ϕ ( x + 1 + 3 ϑ/ > χ I ϑ ( x )implying E (Tr( χ I ϑ ( X ))) E Tr Z ϑ − ϑ dd t ϕ ( X + 1 − t )d t = E Tr Z ϑ − ϑ − ϕ ′ ( X + 1 − t )d t . Bearing in mind that − ϕ ′ > [ t ∈ [ − ϑ , ϑ supp ϕ ′ ( · + 1 − t ) ⊂ [ − − ϑ, − ϑ ] , and X ( − ϑ ) > Id on Ran χ [ − − ϑ, − ϑ ] ( X ) , we find − ϕ ′ ( X + 1 − t ) = − ϕ ′ ( X + 1 − t ) χ [ − − ϑ, − ϑ ] ( X ) − ϕ ′ (( X + 1 − t ) χ [ − − ϑ, − ϑ ] ( X ) X − ϑ − ϕ ′ ( X + 1 − t ) X − ϑ for all t ∈ [ − ϑ , ϑ ]. This yields(7.5) E Tr Z ϑ − ϑ − ϕ ′ ( X + 1 − t )d t − ϑ E Tr Z ϑ − ϑ ϕ ′ ( X + 1 − t ) X d t . With the vector field A , defined in (7.1), we calculate A ϕ ( X + 1 − t ) = ϕ ′ ( X + 1 − t ) A X = ϕ ′ ( X + 1 − t ) X + ϕ ′ ( X + 1 − t )( A X − X )whence ϕ ′ ( X + 1 − t ) X = A ϕ ( X + 1 − t ) − ϕ ′ ( X + 1 − t )( A X − X ) . Plugging this into (7.5) and using 1 − ϑ > / E Tr Z ϑ − ϑ − ϕ ′ ( X + 1 − t )d t E Tr Z ϑ − ϑ A ϕ ( X + 1 − t ) − ϕ ′ ( X + 1 − t )( A X − X )d t . Combining inequality (7.2), positivity of − ϕ ′ ( X + 1 − t ), and the fact that E E we obtain − Tr ϕ ′ ( X + 1 − t )( A X − X ) −
14 Tr ϕ ′ ( X + 1 − t )( H α,N − E ) − ( H α,N − E )( H α,N − E ) − −
14 Tr ϕ ′ ( X + 1 − t ) . This allows to absorb the second summand on the right hand side of inequality (7.6) on the lefthand side. We have established so far P (dist { σ ( H α,N ( ω )) , E } < κ ) X j ∈ Γ α,N E Tr Z ϑ − ϑ ω j ∂∂ω j ϕ ( X + 1 − t )d t = 4 X j ∈ Γ α,N Z ϑ − ϑ E Tr (cid:20) ω j ∂∂ω j ϕ ( X + 1 − t )d t (cid:21) Interchanging integrals and the trace is justified by the fact that the trace is actually a finitesum, as will become explicit in (7.10). The sum over j ∈ Γ α,N will lead to a N n factor inIneq. (7.4). Therefore, it suffices to estimate every summand separately with a j -independentbound, proportional to κ · ( dN n + 1).We use the product structure of the probability space to perform the integration with respectto the particular random variable ω j first. Since the density λ h ( λ ) of the random variablesis absolutely continuous, so is λ ˜ h ( λ ) := λh ( λ ). Using integration by parts for absolutelycontinuous functions as well as the triangle inequality, we have (cid:12)(cid:12) Z b ˜ h ( ω j ) ∂∂ω j Tr { ϕ ( X + 1 − t ) } d ω j (cid:12)(cid:12) = (cid:12)(cid:12) ˜ h (1) Tr ϕ ( X ,j + 1 − t ) − ˜ h ( b ) Tr ϕ ( X b,j + 1 − t ) − Z b ˜ h ′ ( ω j ) Tr ϕ ( X ω j ,j + 1 − t )d ω j (cid:12)(cid:12) (cid:16) | ˜ h (1) | + | ˜ h ( b ) | + k ˜ h ′ k L ( b, (cid:17) max λ ∈ [ b, Tr ϕ ( X λ,j + 1 − t ) k h k W , ( R ) max λ ∈ [ b, Tr ϕ ( X λ,j + 1 − t )where X λ,j denotes the operator X with ω j replaced by λ . It remains to bound Z ϑ − ϑ Z [ b, | Γ α,N |− max λ ∈ [ b, Tr ϕ ( X λ,j + 1 − t ) Y i ∈ Γ α,N ,i = j h ( ω i )d ω i d t. (7.7)For this purpose, it suffices to establish(7.8) Tr ϕ ( X + 1 − t ) C Weyl ( n ) h C b + k V k ∞ ) n +12 | (cid:3) | N n i because then, the integration in (7.7) with respect to t will yield the κ/ | E | factor and the integra-tion with respect to the remaining random variables { ω i } i = j will amount to one. This will boundin the end (7.7) by C Weyl ( n ) h C b + k V k ∞ ) n +12 | (cid:3) | N n i κ | E | To prove (7.8), observe that ϕ
1, supp ϕ ⊂ ( −∞ , ϑ ], and ( −∞ , ϑ − t ] ⊂ ( −∞ , − ],consequently we have for every ω ∈ Ω and every t ∈ [ − ϑ , ϑ ]Tr ϕ ( X + 1 − t ) { Eigenvalues of X in ( −∞ , − / } =: k. We claim that this number k is bounded from above by the number of eigenvalues of ( H α,N − E ) − in [(4 C b ) − , ∞ ), where C b is the constant from Ineq. (7.3). In fact, if the k -th eigenvalue (countedfrom above) λ ← k (( H α,N − E ) − ) of ( H α,N − E ) − was smaller than (4 C b ) − , then for the k -th eigenvalue (counted from below) λ → k ( X ) of X α,N we would have(7.9) λ → k ( X ) = inf V ⊂ L (Π α,N )dim V = k sup φ ∈ V k φ k =1 D ( H α,N − E ) − φ, L α,N ( ω )( H α,N − E ) − φ E k φ k = inf W ⊂ D (cid:0) h ε,ξ,α,N (cid:1) dim W = k sup ψ ∈ W k ψ k =1 h ψ, L α,N ( ω ) ψ ik ( H α,N − E ) ψ k > inf W ⊂ D (cid:0) h ε,ξ,α,N (cid:1) dim W = k sup ψ ∈ W k ψ k =1 − h ψ, H α,N ψ i k ( H α,N − E ) ψ k − C b k ψ k k ( H α,N − E ) ψ k ! > − − C b · sup V ⊂ L (Π α,N )dim V = k inf φ ∈ V k φ k =1 (cid:10) φ, ( H α,N − E ) − φ (cid:11) k φ k = − − C b · λ ← k (( H α,N − E ) − ) > − . Since k ( H α,N − E ) − k = | E | − | Tr ϕ ( X + 1 − t ) | ♯ (cid:8) Eigenvalues of ( H α,N − E ) − in [(4 C b ) − , ∞ ) (cid:9) = ♯ (cid:8) Eigenvalues of ( H α,N − E ) − in [(4 C b ) − , | E | − ] (cid:9) By the spectral mapping theorem this is equal to ♯ (cid:8) Eigenvalues of H α,N in [0 , C b + E ] (cid:9) ♯ (cid:8) Eigenvalues of H α,N in [0 , C b ] (cid:9) (7.10)which is in turn bounded by the number of eigenvalues of the negative Neumann Laplacian onΠ α,N up to energy 4 C b + k V k ∞ . This can be bounded by˜ C Weyl ( n )(4 C b + k V k ∞ ) n +12 | (cid:3) | N n + 2 n +1 C Weyl ( n ) h C b + k V k ∞ ) n +12 | (cid:3) | N n i . (cid:3) Localization
In this section, we prove Theorems 2.7 and 2.8 on localization for operators on the multidimen-sional layer. Theorem 2.7 treats
Case I , where the spectrum expands linearly with ε . Theorem 2.8treats Case II , corresponding to a quadratic expansion of Σ ε .Theorems 3.5 and 3.6 for operators acting on the entire space then follow simply by formula(3.1). Throughout this section we assume that Assumptions M and R hold.To prove localization, we shall invoke the multi-scale analysis from [GK03]. For this purpose it isnecessary to check a number of basic properties of the model referred usually to as “Independenceat distance” (IAD), “Weyl asymptotics” (NE), “Simon-Lieb inequality” (SLI), “Eigenfunctiondecay inequality” (EDI), “Strong generalized resolvent expansion” (SGEE), “Wegner estimate”(W) as well as an “Initial scale estimate” (ISE). We shall address their validity in the subsequentsubparagraphs.Given α ∈ Γ and a natural N >
7, by Υ α,N we denote the setΥ α,N := Π α + e ,N − \ Π α +3 e ,N − , e := n X j =1 e j . The shape of this set is a ‘belt’ formed by cells (cid:3) k located along the lateral boundary of Π α,N .The width of this ‘belt’ is two cells in each direction e j and this ‘belt’ is separated from the lateralboundary of Π α,N by one cell. By χ α,N we denote the characteristic function of Υ α,N . Theorem 8.1 ([GK03, Theorem 2.4]) . Assume that the Assumptions (IAD), (NE), (SLI), (EDI),(SGEE), and (W) hold in an open interval I . Fix a length scale N ∈ N , N > max (cid:26) , ρ, η − [( + n )] − I (cid:27) , where ρ := ρ IAD is the parameter from (IAD), and η I is a parameter from (W). Then, all E ∈ Σ ∩ I such that the initial scale estimate (ISE) (8.1) P (cid:26) D I N ( n ) k χ Υ α,N ( H εα,N ( ω ) − E ) − χ α + N / e ,N / k < (cid:27) > − n holds with D I = 39 n max { · n Q I , } θ I where Q I is the constant in the upper bound of the Wegner estimate and θ I is the constant inthe Simon-Lieb inequality, belong to the region in which spectral and dynamical localization holds.Remark . η I is the maximal value of κ in the Wegner estimate and is equal to Dε / Case I )or Dε / Case II ).Let us check that all assumptions from Theorem 8.1 are satisfied, assuming now both Assump-tions M and R :8.1. Independence at a distance (IAD).
By construction, the restrictions H B ( ω ), H B ( ω )of H ( ω ) to disjoint open B , B ⊂ Π are independent random operators as soon as the distancebetween the sets B and B is larger than ρ IAD := diam( (cid:3) ).8.2.
Weyl asymptotics (NE).
From (7.9) and (7.10) we infer that the following Weyl asymp-totics holds: There is a constant C Weyl = C Weyl ( n ), depending only on the dimension n , and a con-stant C b , depending on the operator norms of L , L , L ( · ), L ′ ( · ) as operators W , ( (cid:3) ) → L ( (cid:3) )such that for all ω ∈ Ω, N ∈ N , α ∈ Γ, ε ∈ (0 , t ] ⊂ [0 ,
1] we haveTr χ ( −∞ , ( H εα,N ( ω )) C Weyl h C b + k V k ∞ ) n +12 k (cid:3) | N n i . In particular, for every interval
I ⊂ ( −∞ ,
0] condition (NE) holds with C I C Weyl h C b + k V k ∞ ) n +12 i . Simon-Lieb inequality (SLI).
We recall that H εα,N ( ξ ) and L εα,N ( ξ ) are the restrictions of H ε ( ξ ) and L ε ( ξ ) on L (Π α,N ), where H εα,N ( ξ ) is equipped with Dirichlet or Mezincescu conditions(5.4) on the lateral boundary γ α,N and with condition (2.1) on ∂ Π ∩ ∂ Π α,N .We choose L, ℓ ∈ N , ℓ > L >
7, and α, β ∈ Γ such that Π β,ℓ ⊂ Π α +3 e ,L − . For brevity wedenote U := Π β +3 ε ,ℓ − and χ U denotes the characteristic function of this set. We denote c := 3 min i =1 ,...,n − | e i | , c := 2 max | V | + 2 . We are going to prove the (SLI) for both types of the boundary conditions on γ α,N . Lemma 8.3.
Suppose that the operators H εα,L ( ξ ) and H εβ,ℓ ( ξ ) are equipped with Dirichlet or Mez-incescu conditions (5.4) on the lateral boundaries γ α,L and γ β,ℓ . if E σ ( H εα,L ( ξ )) ∪ σ ( H εβ,ℓ ( ξ )) ,the estimate holds: (cid:13)(cid:13) χ α,L (cid:0) H εα,L ( ξ ) − E (cid:1) − χ U (cid:13)(cid:13) L (Π β,ℓ ) → L (Υ α,L ) θ I (cid:13)(cid:13) χ β,ℓ (cid:0) H εβ,ℓ ( ξ ) − E (cid:1) − χ U (cid:13)(cid:13) L (Π β,ℓ ) → L (Υ β,ℓ ) (cid:13)(cid:13) χ α,L (cid:0) H εα,L ( ξ ) − E (cid:1) − χ β,ℓ (cid:13)(cid:13) L (Υ β,ℓ ) → L (Υ α,L ) , where θ I := ( c + 1) p | E | + c + 4 nc + 1 . In the notation of [GK03, Theorem 2.4] the constant θ I is called κ I . Proof.
We choose f ∈ L (Π β,ℓ ) arbitrary and denote u α,L := (cid:0) H εα,L ( ξ ) − E (cid:1) − χ U f, u β,ℓ := (cid:0) H εβ,ℓ ( ξ ) − E (cid:1) − χ U f. By χ in = χ in ( x ′ ), χ out = χ out ( x ′ ), we denote an infinitely differentiable cut-off function such that(8.2) χ in ≡ β +3 e ,ℓ − , χ in ≡ β,ℓ \ Π β + e ,ℓ − ,χ out ≡ β,ℓ \ Υ β,ℓ , χ out ( x ) = 1 where χ in ( x ) = 0 and χ in ( x ) = 1 , χ ♭ , (cid:12)(cid:12)(cid:12)(cid:12) ∂χ ♭ ∂x i (cid:12)(cid:12)(cid:12)(cid:12) c , (cid:12)(cid:12)(cid:12)(cid:12) ∂ χ ♭ ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) c on Υ β,ℓ , ♭ ∈ { in , out } . Such functions obviously exist.We extend the function u β,ℓ by zero outside Π β,ℓ and denote v := u α,L − χ in u β,ℓ . In view ofΠ β,ℓ ⊂ Π α +3 e ,L − , the function v satisfies the identity(8.3) v = u α,L on Υ α,L , belongs to the domain of H εα,L ( ξ ) and solves the equation(8.4) (cid:0) H εα,L ( ξ ) − E (cid:1) v = g = χ β,ℓ g,g := 2 ∇ χ in · ∇ u β,ℓ + u β,ℓ ∆ χ in + (cid:0) χ in L εβ,ℓ ( ξ ) − L εβ,ℓ ( ξ ) χ in (cid:1) u β,ℓ . The function g is supported in the set Υ β,ℓ . Indeed, this is obvious for the first two termsin its definition thanks to properties of χ . And due to the definition of L εβ,ℓ ( ξ ), the difference (cid:0) χ in L εβ,ℓ ( ξ ) − L εβ,ℓ ( ξ ) χ in (cid:1) u β,ℓ vanishes on each cell (cid:3) k , on which χ in is identically constant.It follows from (8.3), (8.4) that k χ α,L u α,L k L (Υ α,L ) = k χ α,L v k L (Υ α,L ) = (cid:13)(cid:13) χ α,L (cid:0) H εα,L ( ξ ) − E (cid:1) − χ β,ℓ g (cid:13)(cid:13) L (Υ α,L ) (cid:13)(cid:13) χ α,L (cid:0) H εα,L ( ξ ) − E (cid:1) − χ β,ℓ (cid:13)(cid:13) L (Υ β,ℓ ) → L (Υ α,L ) k g k L (Υ β,ℓ ) . (8.5)It remains to estimate k g k L (Υ β,ℓ ) . Applying Assumption R.2 and the estimates for the deriva-tives in (8.2), we see immediately that(8.6) k g k L (Υ β,ℓ ) c k∇ u β,ℓ k L (supp χ in ) + c k u β,ℓ k L (supp χ in ) + εc k u β,ℓ k W , (supp χ in ) , where c is some absolute constant independent of ε and u β,ℓ .To estimate k u β,ℓ k W , (supp χ in ) , we write the integral identity associated with the equation for u β,ℓ choosing χ u β,ℓ as a test function:(8.7) ( ∇ u β,ℓ , ∇ χ u β,ℓ ) L (Υ β,ℓ ) + ( V u β,ℓ , u β,ℓ ) L (Υ β,ℓ ) − E k χ out u β,ℓ k L (Υ β,ℓ ) + ( L εβ,ℓ ( ξ ) u β,ℓ , χ u β,ℓ ) L (Υ β,ℓ ) = ( f χ U , χ u β,ℓ ) L (Υ β,ℓ ) = 0 , where we have also employed that the supports of χ U and χ out are disjoint. We transform thefirst term in the above identity as follows:(8.8)( ∇ u β,ℓ , ∇ χ u β,ℓ ) L (Υ β,ℓ ) = k χ out ∇ u β,ℓ k L (Υ β,ℓ ) + 2( ∇ u β,ℓ , u β,ℓ χ out ∇ χ out ) L (Υ β,ℓ ) = k χ out ∇ u β,ℓ k L (Υ β,ℓ ) − (cid:0) ( |∇ χ out | + χ out ∆ χ out ) u β,ℓ , u β,ℓ ) L (Υ β,ℓ ) . The third term in the left hand side in (8.7) can be estimated by means of Assumption
R.1 :(8.9) (cid:12)(cid:12) ( L εβ,ℓ ( ξ ) u β,ℓ , χ u β,ℓ ) L (Υ β,ℓ ) (cid:12)(cid:12) ε (cid:16) c k χ out ∇ u β,ℓ k L (Υ β,ℓ ) + c k u β,ℓ k L (Υ β,ℓ ) (cid:17) , where c is some absolute constant independent of ε and u β,ℓ . We substitute the obtained relationsinto (8.7) and arrive at the inequality(8.10) k χ out ∇ u β,ℓ k L (Υ β,ℓ ) + (cid:0) ( E + V − |∇ χ out | − χ out ∆ χ out ) u β,ℓ , u β,ℓ ) L (Υ β,ℓ ) − ε (cid:16) c k χ out ∇ u β,ℓ k L (Υ β,ℓ ) + c k u β,ℓ k L (Υ β,ℓ ) (cid:17) , and for sufficiently small ε we get:(8.11) k χ out ∇ u β,ℓ k L (Υ β,ℓ ) | E | + 1 + max | V | + 2 nc ) k u β,ℓ k L (Υ β,ℓ ) . Since χ out ≡ |∇ χ in | ∪ supp ∆ χ out , by the latter identity and (8.6) we arrive at the finalestimate for g :(8.12) k g k L (Π β,ℓ ) θ I k u β,ℓ k L (Υ β,ℓ ) θ I (cid:13)(cid:13) χ β,ℓ (cid:0) H εβ,ℓ ( ξ ) − E (cid:1) − χ U (cid:13)(cid:13) L (Π β,ℓ ) → L (Υ β,ℓ ) k f k L (Π β,ℓ ) . The obtained estimate and (8.5) imply the statement of the lemma. (cid:3)
Strong generalized eigenfunction expansion (SGEE).
Here we show that condition(SGEE) is satisfied under our assumptions. We denote D ε + ( ξ ) := (cid:8) u : u ∈ D ( H ε ( ξ )) ∩ L (Π) , H ε ( ξ ) u ∈ L (Π) (cid:9) ,L (Π) := u : Z Π | u | (1 + | x ′ | ) m dx , where m > n is some fixed integer number to be chosen later. Let T be an operator in L (Π)of multiplication by (1 + | x ′ | ) − m and P ε ⊥ ( ξ ) be the orthogonal projector in L (Π) onto theorthogonal complement to the kernel of H ε ( ξ ), that is, to the closure of the range of H ε ( ξ ).According to the formulation of (SGEE) in [GK01, Sect. 2.3] and equation (2.36) in [GK01], it issufficient to prove the following lemma. Lemma 8.4.
For m > n , the set D ε + ( ξ ) is dense in L (Π) and an operator core for H ε ( ξ ) , forany ξ ∈ Ω . Furthermore, there exists a bounded continuous function f strictly positive on thespectrum of H ε ( ξ ) such that Tr (cid:0) T f ( H ε ( ξ )) P ε ⊥ ( ξ ) T (cid:1) c , where c is a some constant independent of ε and ξ .Proof. Let D be the set of functions u ∈ C ∞ (Π) obeying boundary conditions (2.1) and vanishingfor | x ′ | large enough. It is clear that D is a subset of D ε + ( ξ ), is dense in L (Π) and is an operatorcore of the operator H . By the assumed properties, the operator L ε ( ξ ) is H -bounded with arelative bound less than one as soon as ε is small enough. Then by Kato-Rellich theorem, the set D is also a core for H ε ( ξ ) and this proves the first statement of the lemma.We proceed to prove the bound for the trace. For each u ∈ D ε + ( ξ ) and each bounded, continuous,positive function f we have: (cid:0) T f ( H ε )( ξ ) P ε ⊥ ( ξ ) T u, u (cid:1) L (Π) = (cid:0)p f ( H ε )( ξ ) T u, P ε ⊥ ( ξ ) p f ( H ε )( ξ ) T u (cid:1) L (Π) (cid:0) T u, f ( H ε )( ξ ) T u (cid:1) L (Π) = (cid:0) T f ( H ε )( ξ ) T u, u (cid:1) L (Π) . Thus by the definition of the trace,Tr (cid:0) T f ( H ε ( ξ )) P ε ⊥ ( ξ ) T (cid:1) Tr (cid:0) T f ( H ε ( ξ )) T (cid:1) . Thanks to Assumption
R.1 with ϕ ≡
1, we obtain for all u ∈ D ( H ε ( ξ )) (cid:0) ( H ε ( ξ ) + c ) u, u (cid:1) L (Π) > k∇ u k L (Π) + ( V u, u ) L (Π) + c k u k L (Π) − ε (cid:16) c k∇ u k L (Π) + c k u k L (Π) (cid:17) . If we set c := 1 + k V k ∞ + 2 T c and choose ε t (2 c ) − then we have for all ξ ∈ Ω H ε ( ξ ) + c >
12 ( H B + 1) , where H B is the operator − ∆ in L (Π) subject to boundary conditions (2.1); its domain is thesame as of H .Now set f ( x ) := ( x + c ) − m , with m ∈ N a constant, to be chosen later. Since f is monotonedecreasing on [0 , + ∞ ) by the trace properties we conclude that(8.13) Tr (cid:0) T f ( H ε ( ξ )) T (cid:1) m Tr( T ( H B + 1) − m T ) = 2 m Tr (cid:0) (( H B + 1) − m T ) ∗ ( H B + 1) − m T (cid:1) . Let us show that ( H B + 1) − m T is as Hilbert-Schmidt operator. We consider the operator − d dx n +1 in (0 , d ) subject to boundary conditions (2.1). By λ q , q ∈ N , we denote its eigenvaluestaken in increasing order and by ϕ q = ϕ q ( x n +1 ) the associated eigenfunctions orthonormalized in L (0 , d ). For all possible choices of the operator B in (2.1), the eigenvalues λ q and eigenfunctions ϕ q can be found explicitly. These formulae imply the lower bounds(8.14) λ q > π ( q − d , q ∈ N . For each u ∈ L (Π), we have the representation u ( x ) = ∞ X q =1 u q ( x ′ ) ϕ q ( x n +1 ) , k u k L (Π) = ∞ X q =1 k u q k L ( R n ) and hence,(8.15) ( H B + 1) − m T u = ∞ X q =1 (cid:0) ( − ∆ x ′ + λ q + 1) − m T ′ u q (cid:1) ϕ q . Here − ∆ x ′ is the Laplacian and T ′ is the multiplication operator by (1 + | x ′ | ) − m , both acting in L ( R n ). As in the proof of Theorem 4.1 in [Sim05, Ch. 4], the operator ( − ∆ x ′ + λ q + 1) − m T ′ isintegral and its kernel is (2 π ) − n (1 + | y ′ | ) − m g q ( x ′ − y ′ ), where g q is the inverse Fourier transformof the function z ( | z | + λ q + 1) − m , z ∈ R n . Hence, by (8.15), the operator ( H B + 1) − m T isalso integral with the kernel K ( x, y ) := (2 π ) − n ∞ X q =1 ϕ q ( x n +1 )(1 + | y ′ | ) − m g q ( x ′ − y ′ ) ϕ q ( y n +1 ) . Let us find the L (Π × Π)-norm of this kernel. By | S n − | we denote the area of the unit sphere in R n . Thanks to identity (4.7) in the proof of Theorem 4.1 in [Sim05, Ch. 4], we have: k K k L (Π × Π) =(2 π ) − n ∞ X q =1 Z R n × R n (1 + | y ′ | ) − m | g q ( x ′ − y ′ ) | dx ′ dy ′ =(2 π ) − n Z R n | y ′ | ) m ∞ X q =1 Z R n | g q ( x ′ − y ′ ) | dx ′ dy ′ =(2 π ) − n Z R n dy ′ (1 + | y ′ | ) m ∞ X q =1 Z R n dx ′ ( | x ′ | + λ q + 1) m =(2 π ) − n | S n − | ∞ Z r n − dr (1 + r ) m ∞ X q =1 + ∞ Z s n − ds ( s + λ q + 1) m =(2 π ) − n | S n − | ∞ Z r n − dr (1 + r ) m + ∞ Z t n − dt ( t + 1) m ∞ X q =1 ( λ q + 1) − m + n , where we have passed to the spherical coordinates and made the rescaling s = t p λ q + 1 to obtainthe last identity. Now we employ estimate (8.14) and Theorem 2.11 in [Sim05, Ch. 2] and weconclude thatTr (cid:0) ( H B + 1) − m T ) ∗ ( H B + 1) − m T (cid:1) = k K k L (Π × Π) = (2 π ) − n | S n − | ∞ Z r n − dr (1 + r ) m + ∞ Z r n − dr ( r + 1) m ∞ X q =1 ( λ q + 1) − m + n < ∞ for m > n . The obtained estimate and (8.13) complete the proof. (cid:3) Eigenfunction decay inequality (EDI).
Next we prove (EDI). Let ψ be a generalizedeigenfunction of H ε ( ξ ) associated with some E ∈ σ ( H ε ( ξ )). Then for each α ∈ Γ, L >
7, therestriction of ψ on Π α,L (still denoted by ψ ) belongs to W , (Π α,L ) and solves the equation − ∆ ψ + V ψ + L εα,L ( ξ ) ψ = Eψ, which is treated as the identity for two functions in L (Π α,L ). In this subsection we continue usingthe notation from Section 8.3. The (EDI) condition is established by the next Lemma 8.5.
Suppose that the operator H εα,L ( ξ ) is equipped with Dirichlet or Mezincescu condi-tions (5.4) on the lateral boundary γ α,L . Then, if E ∈ σ ( H εα,L ( ξ )) , the estimate holds: k χ U ψ k L (Π α,L ) θ I (cid:13)(cid:13) χ α,L (cid:0) H εα,L ( ξ ) − E (cid:1) − χ U (cid:13)(cid:13) L (Π β,ℓ ) → L (Υ α,L ) (cid:13)(cid:13) χ α,L ψ k L (Υ α,L ) . Proof.
We introduce cut-off functions χ in , χ out as in the proof of Lemma 8.3 with properties (8.2),where β , ℓ are replaced by α , L . Since E / ∈ σ (cid:0) H εα,L ( ξ ) (cid:1) and ψχ in ∈ D (cid:0) H εα,L ( ξ ) (cid:1) , we have thefollowing chain of identities: χ U ψ = χ U χ in ψ = χ U (cid:0) H εα,L ( ξ ) − E (cid:1) − (cid:0) H εα,L ( ξ ) − E (cid:1) χ in ψ = χ U (cid:0) H εα,L ( ξ ) − E (cid:1) − g = χ U (cid:0) H εα,L ( ξ ) − E (cid:1) − χ α,L g, where g is defined by the formula in (8.4) with u β,ℓ replaced by ψ . Then we immediately get:(8.16) k χ U ψ k L (Π α,L ) (cid:13)(cid:13) χ U (cid:0) H εα,L ( ξ ) − E (cid:1) − χ α,L (cid:13)(cid:13) L (Υ α,L ) → L (Π α,L ) k g k L (Υ α,L ) . The norm k g k L (Υ α,L ) satisfies estimate (8.6) with u β,ℓ and Υ β,ℓ replaced by ψ and Υ α,L . Estimates(8.7), (8.8), (8.9), (8.10), (8.11) remain true with u β,ℓ and Υ β,ℓ replaced by ψ and Υ α,L . As in(8.12), we then obtain k g k L (Υ α,L ) θ I k χ α,L ψ k L (Υ α,L ) . Substituting this inequality into (8.16), we complete the proof. (cid:3)
Wegner estimate (W).
We formulate now a version of Theorem 7.1 where the disorderparameter ε > ω j by εω j . The following lemma verifiesthe hypotheses of Theorem 7.1 for sufficiently small t . Lemma 8.6.
There is ε max > , depending only on the norms from W , ( (cid:3) ) into L ( (cid:3) ) of theoperators L , L , L ( t ) , and ∂ t L ( t ) , t ∈ [ − T, T ] , Let t be sufficiently small, depending only onthe norms from W , ( (cid:3) ) into L ( (cid:3) ) of the operators L , L , L ( t ) , and Then there exists D > depending only on the norms from W , ( (cid:3) ) into L ( (cid:3) ) of the operators L , L ( t ) , and ∂ t L ( t ) , t ∈ [ − T, T ] , such that for all ε ∈ (0 , t ] (i) we have |h φ, L α,N ( ω ) ε φ i| h φ, H α,N φ i + k φ k L (Π α,N ) ; (ii) for all E Λ − Dε , we have (cid:10) φ, (cid:0) A L εα,N ( ω ) − L εα,N ( ω ) (cid:1) φ (cid:11) h φ, ( H α,N − E ) φ i ; (iii) if L , then for all E Λ − Dε , we have (cid:10) φ, (cid:0) A L εα,N ( ω ) − L εα,N ( ω ) (cid:1) φ (cid:11) h φ, ( H α,N − E ) φ i . This holds whether H α,N is equipped with Mezincescu or Dirichlet boundary conditions on γ α,N .Proof. Statement (i) follows from the fact that L ε ( t ) = ε ( L + ε L + ε L ( εt )) is relatively bounded,uniformly with a bound of order ε .To see (ii), we note that by the definition of L ( t ) and an elementary calculation we have t ∂∂t L ε ( t ) − L ε ( t ) = ε t L + 2 ε t L ( εt ) + ε t L ′ ( εt ) = ε ( t L + 2 εt L ( εt ) + ε t L ′ ( εt )) . The operators t L , εt L ( εt ) and ε t L ′ ( εt ) are bounded from W , ( (cid:3) ) into L ( (cid:3) ) uniformly for t ∈ [ b,
1] and ε ∈ (0 , H (cid:3) − E . Therefore, A L εα,N ( ω ) − L εα,N ( ω ) is also relatively bounded with respect to the positivedefinite operator H α,N − E uniformly for ω ∈ Ω and ε ∈ (0 ,
1] with relative bound proportional to ε . Relative boundedness implies relative form boundedness, cf. [RS75, Thm. X.18], thus, thereare D a , D b > ε ∈ (0 ,
1] such that for all φ ∈ D (cid:0) h ε,ξ,α,N (cid:1) we have (cid:10) φ, ( A L εα,N ( ω ) − L εα,N ( ω )) φ (cid:11) ε (cid:0) D a h φ, ( H α,N − E ) φ i + D b k φ k (cid:1) Note that for both Mezincescu and Dirichlet boundary conditions H α,N > Σ . Choosing D := 8 D b ,we find for E Λ − Dε ε D b k φ k ε D b Λ − E h φ, ( H α,N − E ) φ i ε D b Dε h φ, ( H α,N − E ) φ i = 18 h φ, ( H α,N − E ) φ i and obtain (cid:10) φ, ( A L εα,N ( ω ) − L εα,N ( ω )) φ (cid:11) (cid:18) D a ε + 18 (cid:19) h φ, ( H α,N − E ) φ i . For sufficiently small ε > ε D a / L as in claim (iii) we have t ∂∂t L ε ( t ) − L ε ( t ) ε (2 t L ( εt ) + εt L ′ ( εt ))We find for E Λ − D b ε and sufficiently small ε > (cid:10) φ, ( A L εα,N ( ω ) − L εα,N ( ω )) φ (cid:11) (cid:18) D a ε + 18 (cid:19) h φ, ( H α,N − E ) φ i h φ, ( H α,N − E ) φ i . (cid:3) Combining Lemma 8.6 and Theorem 7.1, we find the following Wegner estimate:
Theorem 8.7.
Assume that t > is sufficiently small. Then, there exists D > (carrying thesame dependencies as in Lemma 8.6), C n,h depending exclusively on n and h , as well as C n,V depending merely on n and V , such that for all ε ∈ (0 , t ] , all α ∈ Γ , and all N ∈ N the followinghold:(i) For all E Λ − Dε and all κ Dε / we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n,V | (cid:3) | N n ] · κ N n . (ii) Assume that L . Then for all E Λ − Dε and all κ Dε / , we have P (dist( σ ( H εα,N ( ω )) , E ) κ ) C n,h Dε [1 + C n,V | (cid:3) | N n ] · κ N n . We note that the constant in the Wegner estimate is Q I = C n,h D [1 + C n,V | (cid:3) | N n ] · ( ε − in ( Case I ) ,ε − in ( Case II ) . In particular, we conclude that the constant D I in Theorem 8.1 is of order ε − or ε − , respectively.8.7. Initial scale estimate (8.1) . The initial scale estimates in Theorem 6.3 hold for ranges of ε which depend on the scale N . We aim for a localization statement for all sufficiently small ε whence we shall choose a sufficiently small t and for every ε ∈ (0 , t ] a corresponding ε -dependentinitial scale N ( ε ).We shall first discuss ( Case I ). Choosing τ = 5 in Thm. 6.3 the initial scale estimate thereofholds for all ε ∈ J N = " p | Λ | E ( | ω | ) 1 N , c N =: (cid:20) ˜ c N , c N (cid:21) . Let us now additionally require that t min n c / (2˜ c ) , ˜ c / √ o For ε ∈ (0 , t ], define then N := N ( ε ) := &(cid:18) ˜ c ε (cid:19) ' where ⌈ x ⌉ denotes the least multiple of six larger or equal than x . With this choice, one cancheck that every ε ∈ (0 , t ] satisfies ε ∈ J N .Now, if t is furthermore chosen so small that for all ε ∈ (0 , t ] we have N > N + (cid:18) ˜ c c (cid:19) = N τ + K τ > max { N τ , K τ } where N and K are the parameters from Theorem 6.3, then this theorem and (2.10) imply P (cid:18) ∀ E Λ ε + 14 ε ˜ c : k χ B ( H εα,N ( ξ ) − E ) − χ B k p N e − c B ,B √ N (cid:19) > − N n e − c N n . (Recall that min Σ ε = Λ ε .) We now choose B = Π α + 12 N e , N and B = Π β, where β is a lattice point in Υ α,N . Assuming N ∈ N , N >
12, which can be ensured by thechoice of t , we have in particular dist( B , B ) > C Γ N for a constant C Γ only depending on thelattice Γ. Thus dist( B ,B ) √ N > C Γ √ N . Since (cid:13)(cid:13)(cid:13) χ Υ α,N ( H εα,N ( ξ ) − E ) − χ Π α + N e , N (cid:13)(cid:13)(cid:13) X β ∈ Γ ∩ Υ α,N (cid:13)(cid:13)(cid:13) χ Π β, ( H εα,N ( ξ ) − E ) − χ Π α + N e , N (cid:13)(cid:13)(cid:13) and the number of lattice points in Υ α,N can be bounded by 4 nN n − , we have by a union bound P (cid:18) ∀ E Λ ε + 14 ε ˜ c : k χ B ( H εα,N ( ξ ) − E ) − χ Υ α,N k p N e − c B ,B √ N (cid:19) > − nN n − N n e − c N n = 1 − nN n − e − c N n In order to obtain inequality (8.1) in Theorem 8.1 it suffices to ensure that e c C Γ √ N > D I N n and 4 nN n − < n e c N n . Since N ∼ ε − and D I ∼ ε − , this is true for all ε ∈ (0 , t ] as soon as t is chosen sufficientlysmall, depending on n , h , V , | (cid:3) ′ | , L ( · ), Λ , E | ω | , θ I . Thus, in ( Case I ), we obtain Andersonlocalization for a sufficiently small t > ε ∈ (0 , t ] in the energy region I ε := (cid:0) −∞ , Λ − Dε (cid:3) ∩ (cid:18) −∞ , Λ ε + ε c (cid:21) ∩ Σ ε . By Corollary 2.4 Λ ε Λ + Λ ε and for sufficiently small t we have the equality of sets I ε = h Λ ε , min { Λ − Dε , Λ ε + ε c } i ∩ Σ ε . We see that I ε ⊃ (cid:20) Λ ε , min { Λ ε + | Λ | ε − Dε , Λ ε + ε c } (cid:21) ∩ Σ ε ∋ Λ ε . Thus, there exists
C > ε ∈ (0 , t ] the set Σ ε ∩ [Λ ε , Λ ε + Cε ] is almost surely nonempty and exhibits dynamical localization, which proves Theorem 2.7.In ( Case II ), we proceed analogously. We set τ = 17 in Theorem 6.3, require t min n c / (2˜ c ) , (5 / / ˜ c o , where ˜ c = √ p η E ( | ω | )and define N := N ( ε ) := &(cid:18) ˜ c ε (cid:19) '
64 BORISOV, T¨AUFER, AND VESELI´C
By analogous calculations as in (
Case I ), we find Anderson localization for every ε ∈ (0 , t ] in theenergy interval I e := (cid:0) −∞ , Λ − Dε (cid:3) ∩ (cid:18) −∞ , Λ ε + ε c (cid:21) ∩ Σ ε Note that Λ − Dε > Λ ε + | Λ | ε − Dε . Thus, there exists a constant C > ε ∈ (0 , t ] I ε ⊃ (cid:2) Λ ε , Λ ε + Cε (cid:3) ∩ Σ ε and the latter set is non-empty and exhibits dynamical localization, almost surely. This completesthe proof of Theorem 2.8. Appendix A. Random magnetic field with non-zero electric potential
Remark
A.1 . We show here that for a random magnetic field as in Section 4.4, an arbitrarymeasurable and bounded V and no random electric potential (i.e. W = W = 0 in the notationof Section 4.4) we have Λ = 0 and Λ >
0, i.e. we are neither in (
Case I ) nor (
Case II ). Forthat purpose, we recall parts of the calculation in [BGV16, Sec. 3.3] and study the effect of addinga non-zero background potential V . Here we also assume that the considered magnetic field isnon-trivial in the sense that it can not be removed by an appropriate gauge transformation. As itis known, this is equivalent to assuming that the magnetic potential A is not a gradient of somescalar function.Recall that Λ is the smallest eigenvalue of the operator − d dx n +1 + V on (0 , d )subject to Dirichlet or Neumann boundary condition and Ψ = Ψ ( x n +1 ) is the associated positiveeigenfunction, extended to (cid:3) by Ψ ( x ′ , x n +1 ) = Ψ ( x n +1 ), and normalized appropriately. We thenhave Λ = ( L Ψ , Ψ ) L ( (cid:3) ) , Λ = ( L Ψ , Ψ ) L ( (cid:3) ) + (Ψ , L Ψ ) L ( (cid:3) ) , where Ψ is the unique solution to( H (cid:3) − Λ )Ψ = −L Ψ + Λ Ψ , (Ψ , Ψ ) L ( (cid:3) ) = 0 . As in Section 4.4, the operators L and L can be written as(A.1) L = i[ ∇ · A + A · ∇ ] = 2i A · ∇ + i div A, L = | A | . We observe that the formula for L implies in particular that the function Ψ is pure imaginary.Since Λ is a ground state of a 1-dimensional Schr¨odinger equation, it is non-degenerate andthe corresponding eigenfunction Ψ is up to a phase real-valued (real part and imaginary part arelinearly dependent because else they would yield two linearly independent ground states). Thus,without loss of generality, we assume that Ψ is real-valued and since A is also real-valued, we cancalculate by employing integration by parts:(A.2) Λ = Z (cid:3) iΨ [ ∇ · A + A · ∇ ] Ψ d x = i Z (cid:3) (cid:0) ( A Ψ ) · ∇ Ψ − ∇ Ψ · ( A Ψ ) (cid:1) d x = 0 . Hence, also for non-zero V , random perturbations consisting purely of magnetic fields alwaysimply Λ = 0. Thus, we are not in ( Case I )Worse, we also have Λ >
0, such that we are not even in (
Case II ). To see this, let us firstnote thatΛ = ( L Ψ , Ψ ) L ( (cid:3) ) + (Ψ , L Ψ ) L ( (cid:3) ) = Z (cid:3) | A | · | Ψ | − A · ∇ Ψ − iΨ Ψ div A d x. Now, to see that Λ > = Z (cid:3) | Ψ | (cid:12)(cid:12)(cid:12)(cid:12) A + i ∇ Ψ Ψ (cid:12)(cid:12)(cid:12)(cid:12) d x. Indeed, the right hand side is obviously non-negative and the sum A + i ∇ Ψ Ψ cannot vanish sinceotherwise A would be the gradient of a real-valued scalar function because Ψ is purely imaginary.Identity (A.3) is proved by calculations which are explicitly performed in [BGV16, Section 3.3]using in particular that Ψ ∇ Ψ Ψ = ∇ Ψ − Ψ Ψ ∇ Ψ Note that since Ψ is a ground state, it is bounded away from zero by Harnack’s inequality andwe can divide by Ψ without any trouble. Appendix B. Chasing the Wegner estimate: An example
Remark
B.1 . In [HK02], operators with random magnetic fields are studied. In particular, Sec-tion 6 of [HK02] treats random operators of the form(B.1) H ε ( ω ) = (cid:0) i ∇ + A + A ε ( ω ) (cid:1) + V , A ε ( ω ) := ε X k ∈ Γ ω k A ( x ′ − k, x n ) , with a random magnetic field A ε ( ω ), a deterministic magnetic field A and a deterministic electricpotential V . We assume that A : Π → R n +1 , A : Π → R n +1 and V : Π → R are (cid:3) -periodic,the potential A vanishes on the boundary of (cid:3) , and all these functions are twice continuouslydifferentiable.Note that in this case, the zero disorder limit of the random operator is the magnetic Schr¨odingeroperator (i ∇ + A ) + V . This is a more general situation than considered in the main body ofthis paper, where the corresponding limit operator is − ∆ + V . In [HK02, Theorem 6.1.(a)], aWegner estimate is proved, however only in an energy region strictly below the infimum of thespectrum of the unperturbed operator and at small disorder.In such a situation it is crucial to investigate whether at small disorder there is any spectrum inthe region where the Wegner estimate can be proven. Else it would concern the resolvent set andwould be a trivial statement. Unfortunately from the discussion in Appendix A it follows that atleast in the special case where A = 0, we have Λ = 0 and Λ >
0. Thus, in this special case, atsmall disorder there is no spectrum at all below the infimum of the unperturbed operator.Furthermore, even if the spectrum expanded below the infimum of the deterministic operator,another issue would arise:More precisely, Theorem 6.1.(a) of [HK02] states the following: Fix parameters E < min Σ :=min σ ( H ) and η ∈ (0 , η sup ), where η sup = dist( E , Σ ) / H = ( i ∇ + A ) + V denotes theunperturbed operator. Then there exists ε > E and η ) such that for all disorderstrengths ε ε , we have a Wegner estimate in [ E − η, E + η ]. Clearly, since the spectrumexpands continuously with the disorder strength and since η < dist( E , Σ ) /
2, the region of theWegner estimate [ E − η, E + η ] will contain no spectrum for small ε . Whether there is a parameter ε ∈ (0 , ε ] such that region where the Wegner estimate holds contains any spectrum at all dependson the rate of expansion of the spectrum with respect to ε and on the interplay of η , E , and ε .In fact, in the proof of [HK02, Theorem 6.1 (a)] (text between Formulas (6.13) and (6.14)) once E < min Σ is chosen, the disorder strength ε must satisfy(B.2) ε ε := 1 − η/ dist( E , Σ )2 k R ( E ) L R ( E ) k = η sup − η η sup k R ( E ) L R ( E ) k where R ( E ) = ( H − E ) − . If we choose η close to η sup , we have ε ∼
0, such that σ ( H ε ( ω ))does not intersect [ E − η, E + η ], cf. Figure 1This shows that it is more natural to choose the interval [ E − η, E + η ] depending on thedisorder ε , in particular E = E ( ε ) and η = η ( ε ). To make sure that this interval intersectsΣ ε we need to have min Σ < E ( ε ) + η ( ε ). Possibly adding a constant to H , we can assumew.l.o.g min Σ = 0. Since η ( ε ) < η sup = | E ( ε ) | / E ( ε ) + η ( ε ) E ( ε ) / In [HK02, Theorem 6.1 (a)], the disorder is denoted by λ and the maximal disorder strength by λ . We callthem ε and ε here in order to be consistent with the notation in the rest of the paper ε min Σ E ( ε )( E − η )( ε ) ( E + η )( ε ) Wegner holds here Σ ε Localization ismeaningful here
Figure 1.
The maximal possible expansion of the spectrum as a function of η and the area where the Wegner estimate holdsLet us first consider the case that min Σ ε decreases linearly for small ε >
0, i.e. there is an c ℓ > ε − c ℓ ε (in analogy to Case (I) in the main body of the paper). In orderto ensure E ( ε ) + η ( ε ) ∈ Σ ε , we choose E ( ε ) := − c ℓ ε and η ( ε ) = c ℓ ε . Then η sup = dist( E ( ε ) , / c ℓ ε > η ( ε )and, indeed, E ( ε ) + η ( ε ) = − c ℓ ε > − c ℓ ε > min Σ ε . Hence for this choice of η ( ε ), in the lightof (B.2), [HK02, Theorem 6.1 (a)] allows disorder strengths ε k R ( E ( ε )) L R ( E ( ε )) k . Let us bound the denominator, using representation (A.1) k R ( E ( ε )) L R ( E ( ε )) k k R ( E ( ε )) k kL k = | E ( ε ) | − k| A | k ∞ = sup | A | c ℓ ε Thus Ineq. (B.2) is satisfied if ε c ℓ ε/ (4 sup | A | ). This is true for sufficiently small ε .Let us turn to the case that min Σ ε decreases quadratically for small ε >
0, i.e. there is an c q > ε − c q ε (in analogy to Case (II) in the main body of the paper).In order to ensure E ( ε ) + η ( ε ) ∈ Σ ε , we choose E ( ε ) := − c q ε and η ( ε ) = c q ε . Then η sup = dist( E ( ε ) , / c q ε > η ( ε )and, indeed, E ( ε ) + η ( ε ) = − c q ε > − c q ε > min Σ ε . Hence for this choice of η ( ε ), in the lightof (B.2), [HK02, Theorem 6.1 (a)] allows disorder strengths ε k R ( E ( ε )) L R ( E ( ε )) k . Let us bound the denominator, using representation (A.1) k R ( E ( ε )) L R ( E ( ε )) k k R ( E ( ε )) k kL k = | E ( ε ) | − k| A | k ∞ = sup | A | c q ε Thus Ineq. (B.2) is satisfied if ε c q ε / (4 sup | A | ), i. e. 4 sup | A | c q To decide whether thiscondition holds, one needs to provide a lower bound on the expansion coefficient c q .In any case, merely assuming small disorder is not sufficient to ensure that [HK02, Theorem 6.1(a)] is a non-trivial statement. Additional arguments or assumptions are required. To elucidatethis further we want to exhibit situations where indeed we have linear expansion, i.e. min Σ ε − c ℓ ε with c ℓ > Remark
B.2 . The operator (B.1) does not fit the assumptions of the present paper. Indeed, for ε = 0 the unperturbed operator becomes H = (i ∇ + A ) , possibly with A = 0. Nevertheless,the issue on how the spectrum expands can be analyzed in this situation, as well, by applying thegeneral results of [BHEV18]. In this context Theorem 2.3 is replaced by Theorem 2.1 in [BHEV18]still ensuring that there exist an almost sure spectral set Σ ε . To fit the setting of [BHEV18] weintroduce some notation and assumptions. We begin with the Floquet-Bloch expansion for theunperturbed operator, namely, we consider the operator H per := (i ∇ + A ) on the periodicitycell (cid:3) subject to θ -quasiperiodic boundary conditions on the (lateral) boundaries γ and denoteits lowest eigenvalue by E ( θ ). We assume further that there exists a unique quasimomentum θ such that E ( θ ) = min θ E ( θ ) = Λ := min Σ ε , that Λ is a simple eigenvalue of H on (cid:3) with θ -periodic boundary conditions on γ and Ψ is the associated eigenfunction normalized in L ( (cid:3) ).Continuous dependence of the Floquet eigenvalues on the quasimomentum implies that there isclosed ball U around θ such that min θ ∈ U dist( E ( θ ) , σ ( H ) \{ E ( θ ) } ) >
0. Such a scenario indeedoccurs for electromagnetic Schr¨odinger operators, see e.g. [Sht05, FK18]. Since here we focus onthe effects of magnetic vector potentials we will assume in the following V ≡
0. Assume also thatthe function ̟ := Ψ | Ψ | belongs to C ( (cid:3) ).Then one can prove along the lines of [BHEV18] that the bottom of the spectrum of the operator H ε ( ω ) satisfies the asymptotic formula(B.3) inf σ (cid:0) H ε ( ω ) (cid:1) Λ + ε min { b Λ , Λ } + O ( ε ) , where the constant Λ is determined by a formula similar to (A.2):(B.4) Λ = Z (cid:3) Ψ [(i ∇ + A ) · A + A · (i ∇ + A )] Ψ d x. We also note that if the function ∂ Ψ ∂ν belongs to C ( ∂ (cid:3) ) then according Theorem 2.3 in[BHEV18], the inequality in (B.3) can be replaced by the identity.Integrating by parts in (B.4) and taking into consideration that A vanishes on the boundary of (cid:3) , we obtain:(B.5) Λ = Z (cid:3) (cid:16) Ψ A · (i ∇ + A ) + Ψ A · (i ∇ + A )Ψ (cid:17) d x = 2 Re (cid:0) Ψ A, (i ∇ + A )Ψ (cid:1) L ( (cid:3) ) . Thanks to the assumed smoothness of A , A and V and by the Schauder estimates, we haveΨ ∈ C ( (cid:3) ). Then we can rewrite formula (B.5) as(B.6) Λ = 2 Re (cid:0) A, Ψ (i ∇ + A )Ψ (cid:1) L ( (cid:3) ) . This representation for Λ shows that once A is fixed, the vector function Ψ (i ∇ + A )Ψ is fixedas well, and we can vary the potential A to achieve Λ = 0. Let us prove this fact rigorously.We are going to show that there exists at least one potential A such the constant Λ defined by(B.6) is non-zero. We argue by contradiction assuming that Λ vanishes for all A . Then formula(B.6) implies that(B.7) 0 ≡ Re Ψ (i ∇ + A )Ψ = − Im Ψ ∇ Ψ + A | Ψ | = 0 in (cid:3) . Employing the definition of the function ̟ , it is straightforward to check that the above identityis rewritten equivalently as(B.8) A = − i ̟ ∇ ̟. The right hand side in the above formula is a real-valued vector function. Indeed, since ̟̟ ≡ ̟ ∇ ̟ + ̟ ∇ ̟ = ̟ ∇ ̟ + ̟ ∇ ̟ = ∇ ̟̟ ≡ ̟ ∇ ̟ is a pure imaginary vector function.On L ( (cid:3) ), we introduce a unitary transformation by the formula U u := ̟u , and U − u = ̟u .It is straightforward to check that the operator H per is unitary equivalent to U − H per U = (cid:0) i ∇ + i ̟ ∇ ̟ + A (cid:1) = − ∆ , where the differential expressions in the right hand side are treated as operators in L ( (cid:3) ) subjectto the same boundary conditions as H per . This means that Λ can vanish simultaneously for all A only in the above discussed case A = 0. Once A describes a non-trivial magnetic field, thereexists at least one potential A , for which Λ is non-zero. Once such potential A is found and fixed,we can consider small variation of A of the form A + δA and in view of the continuity of Λ in A ,we conclude immediately that Λ is also non-zero for A + δA provide δA is small enough. Hence,there exist infinitely many examples of A , for which Λ is non-zero.Moreover, given some A , A , for which Λ is non-zero, we easily see that replacing A by − A ,we change the sign of Λ . This means that in general, Λ is non-zero and can be both negativeand positive. Hence, returning back to formula (B.3), we conclude that in general, the term oforder O ( ε ) in this formula is non-zero and can be both positive and negative.Let us discuss the assumption Ψ | Ψ | = ̟ ∈ C ( (cid:3) ). This is surely true once we know that Ψ does not vanish in (cid:3) and behaves “well enough” at the boundaries of (cid:3) . Apriori, we failed tryingto prove this fact for general A , but the set of potentials A obeying this assumption is non-empty. Indeed, we can start from A ≡
0, then, as it was discussed in Appendix A, the groundstate Ψ is real and sign, hence in this case the function ̟ is identically constant: ̟ ≡
1. Now,consider a (sufficiently) small non-trivial magnetic potential A . Its ground state will be a smallperturbation of the one for A ≡
0, i. e. a (sufficiently) small variation of the function ̟ ≡ C ( (cid:3) ). Acknowledgments
The scientific contribution by D.I.B. in the results presented in Sections 2, 4, 5, 7, 8 is financiallysupported by Russian Science Foundation (project No. 17-11-01004).
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