Weak disorder expansions for the Anderson model on a one dimensional strip at the center of the band
type: Item
instance of: scholarly article
title: Weak disorder expansions for the Anderson model on a one dimensional strip at the center of the band
zbMATH Open document ID: 0772.60102
DOI: 10.1007/BF02096942
author: Athanasios Speis
published in: Communications in Mathematical Physics
publication date: 1993-01-17T00:00:00Z
review text: Let \(H_ \lambda=-\Delta/2+\lambda V\) be a family of random Schrödinger operators on a discrete strip of width \(l\), indexed by the real parameter \(\lambda\), such that the random potentials \(V\) are i.i.d. at each site \(x=(n,i)\); \(n\in\mathbb{Z}\); \(i=1,\dots,l\). For a real energy \(E\) one considers the boundary value of the averaged Green matrix defined by \[ J(\lambda,E)=\lim_{\varepsilon\to 0}\mathbb{E}\{G_ \lambda(E+i\varepsilon,n,n)\}. \] It can be proved that under some mild conditions on the probability distribution of the potential this limit exists for all \((E,\lambda)\) and is a smooth function of \(E\) for \(\lambda\neq 0\). The problem is to find an asymptotic expansion of \(J(\lambda,E)\) when \(\lambda\to 0\). In one dimension (that is for \(l=1)\) it has already been proved that such an expansion is in general valid but is inadequate for some special values of the energy \(E\). The same result is proved here, by using the supersymmetric method in order to rewrite the Green matrix in a suitable way, but the definition of the exceptional energies have to be adapted to this situation. This result can be used to obtain similar expansions for the density of states and the sum of the nonnegative Lyapunov exponents.
reviewed by: Jean Lacroix
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 46N55
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 97872
zbMATH Keywords: random Schrödinger operators
zbMATH Keywords: random potentials
zbMATH Keywords: probability distribution of the potential
zbMATH Keywords: Green matrix
zbMATH Keywords: Lyapunov exponents
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02096942
OpenAlex ID: W2060451532
description: scientific article; zbMATH DE number 97872