Asymptotics of the cut discontinuity and large order behaviour from the instanton singularity: The case of lattice Schr�dinger operators with exponential disorder

Abstract

We discuss the relation between the singularity structure of the Borel transform, the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory. In an explicit example — a tight binding model with exponential disorder — we show how to obtain the first instanton singularity from a cluster expansion in the Borel variable, and as an application we determine the exact decay of the density of states asE → ∞. The method opens some perspectives for similar problems arising from different models of Mathematical Physics.