Level spacing and Poisson statistics for continuum random Schrödinger operators

Abstract

For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.