Bounds on the spectral shift function and the density of states

TL;DR

This paper establishes bounds on the spectral shift function for Schrödinger operators, demonstrating decay properties and deriving a Wegner estimate that ensures the H"older continuity of the density of states in random alloy-type models.

Contribution

It provides new bounds on the spectral shift function and applies these to prove a Wegner estimate for alloy-type random Schrödinger operators.

Findings

Almost exponential decay of singular values of semigroup differences.

Bounds on the spectral shift function for operator pairs.

Hölder continuity of the integrated density of states.

Abstract

We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values $\mu_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schr\"odinger operators. The single site potential $u$ is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be H\"older continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies H\"older continuity of the integrated density of states.