Mean Green's function of the Anderson model at weak disorder with an infra-red cut-off

Abstract

We develop a polymer expansion with large/small field conditions for the mean resolvent of a weakly disordered system. Then we show that we can apply our result to a two-dimensional model, for energies outside the unperturbed spectrum or in the free spectrum provided the potential has an infra-red cut-off. This leads to an asymptotic expansion for the density of states. We believe this is an important first step towards a rigourous analysis of the density of states in the free spectrum of a random Schr\"odinger operator at weak disorder.