The Density of States and the Spectral Shift Density of Random Schroedinger Operators

TL;DR

This paper extends the analysis of random Schrödinger operators by developing a general theory of the spectral shift density across arbitrary dimensions, linking it to the density of states and scattering theory.

Contribution

It introduces an alternative construction of the density of states using Krein's spectral shift function applicable in any dimension, extending previous one-dimensional results.

Findings

Existence of spectral shift density in arbitrary dimensions

Equivalence of spectral shift density and difference of densities of states

Application to interactions concentrated near hyperplanes

Abstract

In this article we continue our analysis of Schroedinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.