Extremal eigenvalue statistics and spectrum of d-dimensional random Schrödinger operator (Spectral and Scattering Theory and Related Topics)

Abstract

This is a review of joint works [5, 6] with K. Kawaai and Y. Maruyama (Tohoku University). We consider Schrödinger operator with random decaying potential on 𝘭²(Zᵈ) and (i) we showed that IDS coincides with that of free Laplacian in general cases, (ii) we show some examples, with heavy-tailed single-site distribution, such that the set of rescaled extremal eigenvalues converges to a inhomogeneous Poisson process, and positive real axis belongs to the essential spectrum, (iii) we show the other examples with light-tailed single-site distribution such that the Hamiltonian is bounded almost surely, and the essential spectrum coincides with that of the free Laplacian.