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

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.