Some new estimates on the spectral shift function associated with random Schr\"{o}dinger operators

TL;DR

This paper establishes new pointwise bounds on the spectral shift function for random Schrödinger operators, enhancing understanding of spectral properties in both finite and infinite-volume settings.

Contribution

It introduces novel bounds and a Wegner estimate for random Schrödinger operators, along with a representation of the density of states in lattice models.

Findings

New pointwise-in-energy bounds on spectral shift functions

A new Wegner estimate for finite-volume operators

Representation of the density of states in lattice models

Abstract

We prove some new pointwise-in-energy bounds on the expectations of various spectral shift functions associated with random Schr\"{o}dinger operators in the continuum having Anderson-type random potentials in both finite-volume and infinite-volume. These estimates are a consequence of our new Wegner estimate for finite-volume random Schr\"{o}dinger operators. For lattice models, we also obtain a representation of the infinite-volume density of states in terms of a spectral shift function. For continuum models, the corresponding measure is absolutely continuous with respect to the density of states and agrees with it in certain cases. We present a variant of a new spectral averaging result and use it to prove a pointwise upper bound on the SSF for finite-rank perturbations.