Localization of the 1D Non-Stationary Anderson Model

TL;DR

This paper proves spectral and dynamical localization for a class of 1D non-stationary Anderson models with independent, possibly unbounded potentials, using a Furstenberg-type theorem for non-stationary matrix products.

Contribution

It establishes localization results for non-stationary Anderson models with broad potential distributions, extending previous stationary results.

Findings

Spectral localization with exponentially decaying eigenfunctions.

Dynamical localization for the model.

Applicability to potentials from any compact set away from deterministic distributions.

Abstract

This paper considers the family of Schr\"odinger operators on $\ell^2(\mathbb{Z})$ given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the choice of distributions to come from any compact set away from deterministic distributions. With these assumptions we prove spectral localization with exponentially decaying eigenfunctions as well as dynamical localization. One of the main tools is a Furstenberg-type theorem for non-stationary matrix products.