Wegner estimates for sign-changing single site potentials

TL;DR

This paper establishes Wegner estimates for certain random Schrödinger operators with sign-changing potentials, showing linear bounds and Lipschitz continuity of the integrated density of states.

Contribution

It proves Wegner estimates for models with sign-changing single site potentials of a generalized step function form, extending previous results to more general potentials.

Findings

Wegner estimate is linear in volume and energy interval length.

The integrated density of states is Lipschitz continuous.

Applicable to models with sign-changing potentials of a generalized step function form.

Abstract

We study Anderson and alloy type random Schr\"odinger operators on $\ell^2(\ZZ^d)$ and $L^2(\RR^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states.