Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

TL;DR

This paper establishes Wegner estimates and proves localization for alloy-type Schrödinger operators with sign-changing, exponentially decaying single-site potentials, extending previous results to more general potential shapes.

Contribution

It introduces a Wegner estimate for alloy-type models with sign-changing potentials and applies it to prove localization via multiscale analysis.

Findings

Wegner estimate polynomial in volume and linear in energy interval

Localization proven for models with sign-changing potentials

Applicable to continuum and lattice Schrödinger operators

Abstract

We study Schr\"odinger operators on $L^2 (\RR^d)$ and $\ell^2(\ZZ^d)$ with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation a Wegner estimate which is polynomial in the volume of the box and linear in the size of the energy interval holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.