The Spectrum of Schr\"odinger Operators with Randomly Perturbed Ergodic Potentials

TL;DR

This paper analyzes Schr"odinger operators with potentials combining ergodic and random components, providing an explicit description of the almost sure spectrum as a finite union of intervals, generalizing classical Anderson model results.

Contribution

It offers a novel explicit characterization of the spectrum for Schr"odinger operators with combined ergodic and random potentials, extending classical results.

Findings

Almost sure spectrum is a finite union of intervals.

Spectrum derived explicitly from unperturbed spectrum and support of distribution.

Generalizes classical Anderson model spectrum formula.

Abstract

We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a connected compact metric space and a continuous sampling function, we show that the almost sure spectrum arises in an explicitly described way from the unperturbed spectrum and the topological support of the single-site distribution. In particular, assuming that the latter is compact and contains at least two points, this explicit description of the almost sure spectrum shows that it will always be given by a finite union of non-degenerate compact intervals. The result can be viewed as a far reaching generalization of the well known formula for the spectrum of the classical Anderson model.