Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder.
type: Item
instance of: scholarly article
title: Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder.
zbMATH Open document ID: 1035.82023
author: W.-M. Wang
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: The author considers the model described by a discrete random Schrödinger operator of the following type \(H = \Delta + \gamma V\). It is defined on \(\ell^2 (\mathbb{Z}^d)\). Here \(\gamma>0\) is a parameter and the matrix elements (in the canonical base) of \(\Delta\) and \(V\) are as \(\Delta_{ij} = \delta_{| i - j| ,1}\), \(V_{ij} = \delta_{ij} v_i\), where \(| i-j| \) is the \(\ell^1\)-distance in \(\mathbb{Z}^d\) and \(\{v_i \}_{i \in \mathbb{Z}^d}\) is a family of independent identically distributed random variables with distribution \(g\). It is well-known that the Laplacian \(\Delta\) has an absolutely continuous spectrum \(\sigma (\Delta)= [-2d , 2d]\). It is also known that \[ \sigma(H) = [-2d , 2d] + \gamma \cdot \text{supp\,}g, \quad \text{a.s.} \] For \(\gamma \gg 1\) and \(d\), \(\sigma (H)\) is an almost surely pure point (Anderson localization). It is believed that for \(d\geq 3\) in the region of small \(\gamma\) there is a transition from the point to continuous spectrum (Anderson transition). This is the reason why multidimensional Anderson models with weak disorder attract much attention in the recent years. In the present article it is proven that under certain conditions on \(g\) there exists \(\gamma_0>0\) such that for all \(\gamma \in (0, \gamma_0)\), with probabil\-ity~1 \(\sigma (H) \cap I(\gamma) = \sigma_{\text{pp}} (H)\). Here \(I(\gamma)\) is a nonempty sub-interval of \([-2d , 2d]\) depending on \(\gamma \) and \(g\). It is also proven that the eigenfunctions \(\psi_E\) corresponding to \(E \in I(\gamma)\) decay at least exponentially with the mean-field rate \(\sqrt{\gamma}\). The statistics of the eigenvalue spacing is proven to be asymptotically Poissonian. The proofs rely on supersymmetric arguments.
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1777219
zbMATH Keywords: Anderson localization
zbMATH Keywords: pure point spectrum
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: supersymmetry approach
reviewed by: Yuri Kozitsky
MaRDI profile type: Publication
full work available at URL: s002220100169
OpenAlex ID: W1979755504
DOI: 10.1007/S002220100169
description: scientific article; zbMATH DE number 1777219