Level spacing for continuum random Schrödinger operators with applications

Adrian Dietlein, Alexander Elgart

Published 2017-12-11Version 1

For continuum alloy-type random Schr\"odinger operators with sign-definite 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 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 the threshold energy. 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 $E$.