Approximating the density of states for Poisson distributed random Schroedinger operators

TL;DR

This paper develops a method to approximate the density of states for Poisson-distributed random Schrödinger operators, valid for any disorder strength, by expanding around the free Laplacian and controlling the coefficients in the infinite volume limit.

Contribution

It introduces a power series expansion for the density of states of Poisson random Schrödinger operators, applicable at all disorder levels, including small disorder regimes.

Findings

Expectations of resolvent matrix elements can be approximated arbitrarily well.

Expansion coefficients are explicitly given via Neumann series.

Boundary values of the spectral parameter exist in the infinite volume limit.

Abstract

We consider a Schroedinger operator with random potential distributed according to a Poisson process. We show that expectations of matrix elements of the resolvent as well as the density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. One can control these expansion coefficients in the infinite volume limit. We show that for this limit the boundary value as the spectral parameter approaches the real axis exists as well. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.