Regularity of the Density of Surface States

TL;DR

This paper proves that the integrated density of surface states for Anderson-type operators is a measurable function, extending previous results and utilizing new spectral shift function bounds.

Contribution

It generalizes prior work by showing the density of surface states is a measurable function rather than a distribution, using novel $L^p$ bounds.

Findings

Integrated density of surface states is a measurable function.

Provides a simple proof of Hoelder continuity of bulk states.

Extends previous results to both continuous and discrete cases.

Abstract

We prove that the integrated density of surface states of continuous or discrete Anderson-type random Schroedinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalize our recent results on the existence of the integrated density of surface states in the continuous case and those of A. Chahrour in the discrete case. The proof uses the new $L^p$-bound on the spectral shift function recently obtained by Combes, Hislop, and Nakamura. Also we provide a simple proof of their result on the Hoelder continuity of the integrated density of bulk states.