Limiting distribution of extremal eigenvalues of d-dimensional random Schrödinger operator

Abstract

We consider the Schrödinger operator with random decaying potential on [Formula: see text] and showed that, (i) Integrated Density of States (IDS) coincide with that of free Laplacian in general cases, (ii) the set of extremal eigenvalues, after rescaling, converges to an inhomogeneous Poisson process, under certain condition on the single-site distribution, and (iii) there are “border-line” cases, such that we have Poisson statistics in the sense of (ii) above if the potential does not decay, while we do not if the potential does decay.