Existence of the density of states for one-dimensional alloy-type potentials with small support

TL;DR

This paper proves the existence and boundedness of the density of states for one-dimensional alloy-type Schrödinger operators with small support potentials, using a Wegner estimate applicable across all bounded energy intervals.

Contribution

It introduces a Wegner estimate for alloy-type potentials with small support, establishing the density of states' existence and boundedness in one-dimensional Schrödinger operators.

Findings

Wegner estimate proven for non-negative potentials with small support

Density of states exists and is locally uniformly bounded

Estimate valid for all bounded energy intervals

Abstract

We study spectral properties of Schr\"odinger operators with random potentials of alloy type on $L^2(\RR)$ and their restrictions to finite intervals. A Wegner estimate for non-negative single site potentials with small support is proven. It implies the existence and local uniform boundedness of the density of states. Our estimate is valid for all bounded energy intervals. Wegner estimates play a key role in an existence proof of pure point spectrum.