Level spacing and Poisson statistics for continuum random Schrödinger operators

We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. 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 E_sp keep a distance of at least e⁻(log ^L)β for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below E_sp. 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.