Level spacing and Poisson statistics for continuum random Schrödinger operators

2017/12/11 by Dietlein, Adrian, Elgart, Alexander · 1 citation
#47B80 #60H25 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1712.03925

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_{\rm 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_{\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.

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