# Level spacing and Poisson statistics for continuum random Schrödinger operators

### Adrian Dietlein

Ludwig-Maximilians-Universität München, Germany### Alexander Elgart

Virginia Tech, Blacksburg, USA

## Abstract

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 $e^{-\mathrm{log} L)^\beta}$ for sufficiently large $\beta > 1$. 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 $E$.

## Cite this article

Adrian Dietlein, Alexander Elgart, Level spacing and Poisson statistics for continuum random Schrödinger operators. J. Eur. Math. Soc. 23 (2021), no. 4, pp. 1257–1293

DOI 10.4171/JEMS/1033