Lifshitz tails for Schr\"odinger operators with random breather potential

TL;DR

This paper establishes Lifshitz tail bounds for the integrated density of states in a class of random Schrödinger operators with breather potentials, addressing non-monotonic and non-linear potential dependencies.

Contribution

It proves Lifshitz tail bounds for Schrödinger operators with non-monotone, non-linear breather potentials, extending previous results to more complex potential models.

Findings

Lifshitz tail bounds are established for the model.

The model handles non-star-shaped sets in the potential.

The results apply to non-monotone, non-linear potential dependencies.

Abstract

We prove a Lifshitz tail bound on the integrated density of states of random breather Schr\"odinger operators. The potential is composed of translated single site potentials. The single site potential is an indicator function of set $tA$ where $t$ is from the unit interval and $A$ is a measurable set contained in the unit cell. The challenges of this model are: Since $A$ is not assumed to be star-shaped the dependence of the potential on the parameter $t$ is not monotone. It is also non-linear and not differentiable.