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

We prove a probabilistic level-spacing estimate at the bottom of the spectrumfor continuum alloy-type random Schr\"odinger operators, assumingsign-definiteness of a single-site bump function and absolutely continuousrandomness. More precisely, given a finite-volume restriction of the randomoperator onto a box of linear size $L$, we prove that with high probability theeigenvalues below some threshold energy $E_{\rm sp}$ keep a distance of atleast $e^{-(\log L)^\beta}$ for sufficiently large $\beta>1$. This impliessimplicity of the spectrum of the infinite-volume operator below $E_{\rm sp}$.Under the additional assumption of Lipschitz-continuity of the single-siteprobability density we also prove a Minami-type estimate and Poisson statisticsfor the point process given by the unfolded eigenvalues around a referenceenergy $E$.