Limit theorems for basic states of the Anderson model
type: Item
instance of: scholarly article
title: Limit theorems for basic states of the Anderson model
zbMATH Open document ID: 0791.60101
DOI: 10.1007/BF01075060
publication date: 1994-05-18T00:00:00Z
review text: Let \(V=\{x=(x^ 1,\dots,x^ \nu) \in \mathbb{Z}^ \nu:| x^ i | \leq n\), \(i=1, \dots, \nu\}\), and let \(N=(2n+1)^ \nu\) be the number of points in \(V\), \(\nu \geq 1\). Let us consider the following Hamiltonian in \(L^ 2(V)\): \[ H_ V=\kappa \Delta_ V+\xi (x),\quad x \in V, \tag{1} \] where \(\Delta_ V\) is the discrete Laplacian on \(V\) with zero Dirichlet boundary conditions (the restriction of the operator \(\Delta \varphi (x)=\sum_{| y-x |=1} \varphi (y)\), \(x \in \mathbb{Z}^ \nu\), to \(V)\), \(| x |=| x^ 1 |+\cdots+| x^ \nu |\), \(\xi (x)\), \(x \in \mathbb{Z}^ \nu\), is a random field of independent identically distributed variables, \(\kappa\) is a positive constant. Let \(\lambda_{1,N}>\cdots>\lambda_{N,N}\) be (random) eigenvalues of the operator (1). This paper presents limit theorems for the first (normalized) eigenvalues \(\lambda_{1,N},\dots,\lambda_{i,N}\) under the following assumption on the distribution density \(p(t):=(d/dt) P(\xi (0)<t)\): \[ p(t)=\bigl( A \alpha+o(1) \bigr) t^{\alpha-\beta-1} \exp (- At^ \alpha) \quad (t \to \infty), \tag{2} \] where \(\alpha>0\), \(A>0\), \(- \infty<\beta<\infty\). In the theory of Lifshits tails, this class of distributions is of special importance due to existence of bifurcations with respect to \(\alpha\) in the Anderson model (1).
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 561007
zbMATH Keywords: discrete Laplacian
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: limit theorems
zbMATH Keywords: Lifshits tails
zbMATH Keywords: Anderson model
author: Arvydas Astrauskas
author: S. Molchanov
MaRDI profile type: Publication
cites work: Absence of diffusion in the Anderson tight binding model for large disorder or low energy
full work available at URL: https://doi.org/10.1007/bf01075060
OpenAlex ID: W2008725707
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 561007