Level spacing and Poisson statistics for continuum random Schr\"odinger operators

We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schr\"odinger 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 $E_{\rm sp}$ keep a distance of at least $e^{-(\log L)^\beta}$ for sufficiently large $\beta>1$. This implies simplicity of the spectrum of the infinite-volume operator below $E_{\rm sp}$. 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 $E$.