Localisation for non-monotone Schroedinger operators

TL;DR

This paper investigates how strong disorder causes localisation in non-monotone Schrödinger operators on lattices, providing bounds on transition amplitudes and a nearly optimal Wegner estimate.

Contribution

It introduces new methods to establish dynamical localisation and finite fractional moments for non-monotone potentials, including the discrete alloy-type Anderson model.

Findings

Proves dynamical localisation with exponential decay bounds.

Derives finite fractional moments of the resolvent.

Provides a nearly optimal Wegner estimate.

Abstract

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.