Bounds on the density of states for Schr\" odinger operators

TL;DR

This paper derives deterministic bounds on the density of states measure for Schr"odinger operators, introducing a new outer-measure that bounds the measure and proving its regularity in multiple dimensions.

Contribution

It introduces a density of states outer-measure that always exists and provides bounds without requiring the measure's existence, along with continuity results.

Findings

Established bounds on the density of states measure

Proved log-H"older continuity in multiple dimensions

Introduced a new outer-measure framework

Abstract

We establish bounds on the density of states measure for Schr\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\"odinger operators, and in any dimension for discrete Schr\"odinger operators.