On the Probability of the Existenceof a Localized Basic State for a Discrete Schrödinger Equation with Random Potential, Perturbed by a Compact Operator

Abstract

Let $H_d$ be the difference Laplace operator in $l_2(\bZ^d)$ and $\bW$ be a discrete potential (a bounded diagonal operator). We search for the conditions on the spectrum of the operator $H_d+\bW$ under which the complete Hamiltonian $H_d+\bW+\bV(\og)$ with random potential ${\bV}(\omega)$ has a localized basic state (a) with positive probability and (b) with probability 1. We prove that the condition that the maximal point of the spectrum of $H_d+\bW$ is isolated from the remaining spectral points of the operator is sufficient for (a) to be true (if $\bW$ is a compact operator this condition is necessary). Respectively, the condition that the length of the random potential does not exceed the distance between the maximal point of the spectrum of $H_d+\bW$ and the rightmost point of its essential spectrum is a sufficient one for (b) to be true. It is shown that if $\bW$ is an operator of rank 1, then this condition is necessary.