Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian H _ω =Δ+V _ω , where for all x∈ $\mathbb{Z}$ , V _ω(2x)=V _ω(2x+1) and where the V _ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ ²( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P _I(H _ω) is the spectral projector of H _ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H _ω)].     

Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ? _N , the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as \({\frac{\pi^{2}}{(\ell_{N} +1)^{2}}} \) in the sense that the ratio of the quantities goes to one.     

Anderson Localization for Time Quasi-Periodic Random Schrödinger and Wave Equations

We prove that at large disorder, with large probability and for a corresponding set of Diophantine frequencies of large measure, Anderson localization in ^d is stable under localized time quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. This paper paves the way for the construction of time quasi-periodic KAM type of solutions of non linear random Schrödinger equations in [BW].Wei-Min Wang thanks A. Soffer and T. Spencer for many useful conversations and for initiations to the subject. She also thanks M. Combescure and J. Sjöstrand for helpful discussions on the quasi-energy operator formulation of time dependent Schrödinger equations. The support of NSF grant DMS 9729992 is gratefully acknowleged.     

Lifshitz Tail for Schrödinger Operator¶with Random Magnetic Field

We study the behavior of the density states at the lower edge of the spectrum for Schr?dinger operators with random magnetic fields. We use a new estimate on magnetic Schr?dinger operators, which is similar to the Avron–Herbst–Simon estimate but the bound is always nonnegative. Received: 3 January 2000 / Accepted: 18 April 2000     

Long Time Anderson Localization for the Nonlinear Random Schrödinger Equation

We prove long time Anderson localization for the nonlinear random Schrödinger equation for arbitrary ? ² initial data, hence giving an answer to a widely debated question in the physics community. The proof uses a Birkhoff normal form type transform to create a barrier where there is essentially no propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat “rough” data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.     

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

Spectrum of the Schrödinger Operator in a Perturbed Periodically Twisted Tube

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of “slowing down” the twisting in the mean gives rise to a non-empty discrete spectrum Mathematical Subject Classifications: 35P05, 81Q10.     

On the Spectrum of the Schrödinger Operator with Periodic Surface Potential

We consider a discrete Schrödinger operator H=–+V acting in l ²( ^d ), with periodic potential V supported by the subspace surface {0}× ^d ₂. We prove that the spectrum of H is purely absolutely continuous, and that surface waves oscillate in the longitudinal directions to the surface. We also find an explicit formula for the generalized spectral shift function introduced by the author in Helv. Phys. Acta. 72 (1999), 93–122.     

The Discrete Spectrum¶of the Perturbed Periodic Schrödinger Operator¶in the Large Coupling Constant Limit

Let A be a periodic Schr?dinger operator and let V ₀≥ 0 be a decaying potential. We study the number of the eigenvalues of the operator A(α) =A−αV ₀ inside a fixed interval (λ₁,λ₂). We obtain an asymptotic formula for as α→∞. Received: 12 September 2000 / Accepted: 22 November 2000     

On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

On the d-dimensional lattice \({\mathbb{Z}^d}\) and the r-regular tree \({T^r}\), an exact expression for the number of discrete eigenvalues of a discrete Laplacian with a finitely supported potential is described in terms of the support and the intensities of the potential on each case. In particular, the number of eigenvalues less than the infimum of the essential spectrum is bounded by the number of negative intensities.     

A Lower Bound for the Lyapounov Exponents of the Random Schrödinger Operator on a Strip

We consider the random Schrödinger operator on a strip of width W, assuming the site distribution of bounded density. It is shown that the positive Lyapounov exponents satisfy a lower bound roughly exponential in ?W for W→∞. The argument proceeds directly by establishing Green’s function decay, but does not appeal to Furstenberg’s random matrix theory on the strip. One ingredient involved is the construction of ‘barriers’ using the random Schrödinger operator theory on $\mathbb{Z}$ .     

Localization for some continuous random Schrödinger operators

We study the spectrum of random Schrödinger operators acting onL ²(R ^d ) of the following type . The are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.

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Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL ²(R ^d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.

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U.R.A. 760 C.N.R.S.     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials

We consider discrete one-dimensional random Schrödinger operators with decaying matrix-valued, independent potentials. We show that if the ?²-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schrödinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schrödinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon et al. (Ann. Inst. H. Poincaré Phys. Théor. 42(3):283–309, 1985).     

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.     

Symmetries of Schrödinger Operator with Point Interactions

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling W are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.     

Localization for a class of one dimensional quasi-periodic Schrödinger operators

We prove for small and satisfying a certain Diophantine condition the operator

L p -Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator

Given a one dimensional perturbed Schrödinger operator H =  ? d ²/dx ² + V(x), we consider the associated wave operators W  _± , defined as the strong L ² limits $\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}Given a one dimensional perturbed Schr?dinger operator H = − d ²/dx ² + V(x), we consider the associated wave operators W _± , defined as the strong L ² limits . We prove that W _± are bounded operators on L ^p for all 1 < p < ∞, provided , or else and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.