Sur le spectre des opérateurs aux différences finies aléatoires

Abstract

We study a class of random finite difference operators, a typical example of which is the finite difference Schrödinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrödinger operator with a random potential has pure point spectrum and developps no static conductivity.