Localization via fractional moments for models on $\mathbb{Z}$ with single-site potentials of finite support

TL;DR

This paper introduces a new fractional moment method variant to prove exponential decay of Green's function and localization for one-dimensional alloy-type models with finite support and sign-changing potentials.

Contribution

It develops a novel fractional moment approach tailored for 1D alloy-type models with finite support and sign-changing potentials, extending localization results.

Findings

Proves exponential decay of Green's function

Establishes localization for models with sign-changing potentials

Extends fractional moment method to new class of models

Abstract

One of the fundamental results in the theory of localization for discrete Schr\"odinger operators with random potentials is the exponential decay of Green's function and the absence of continuous spectrum. In this paper we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.