On asymptotic expansions of the density of states for Poisson distributed random Schr\"odinger operators

TL;DR

This paper derives asymptotic expansions for the density of states and resolvent matrix elements of a Poisson-distributed random Schrödinger operator in the small disorder limit, providing explicit coefficient estimates.

Contribution

It introduces explicit asymptotic expansions and estimates for the density of states of Poisson random Schrödinger operators, including their finite volume limits.

Findings

Asymptotic expansions are established for small disorder.

Explicit estimates for expansion coefficients are provided.

Finite volume limits of coefficients are shown to be finite.

Abstract

We study expectation values of matrix elements for boundary values of the resolvent as well as the density of states for a random Schr\"odinger operator with potential distributed according to a Poisson process. Asymptotic expansions for these quantities in the limit of small disorder are derived. Explicit estimates for the expansion coefficients are given and we show that their infinite volume limits are in fact finite as the spectral parameter approaches the spectrum of the free Laplacian.