Lifshitz tails for a class of Schr\"odinger operators with random breather-type potential

TL;DR

This paper establishes Lifshitz tail behavior for a class of Schrödinger operators with non-linearly dependent random potentials, including breather models, providing insights into spectral properties and localization phenomena.

Contribution

It introduces bounds on the integrated density of states for non-linear random potentials, extending Lifshitz tail analysis to breather-type models and related operators.

Findings

Lifshitz asymptotics near the spectrum bottom

Bounds on the integrated density of states for non-linear potentials

Implications for Anderson localization in specific regimes

Abstract

We derive bounds on the integrated density of states for a class of Schr\"odinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schr\"odinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.