JournalsjemsVol. 23, No. 4pp. 1257–1293

Level spacing and Poisson statistics for continuum random Schrödinger operators

  • Adrian Dietlein

    Ludwig-Maximilians-Universität München, Germany
  • Alexander Elgart

    Virginia Tech, Blacksburg, USA
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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 LL, we prove that with high probability the eigenvalues below some threshold energy EspE_{sp} keep a distance of at least elogL)βe^{-\mathrm{log} L)^\beta} for sufficiently large β>1\beta > 1. This implies simplicity of the spectrum of the infinite-volume operator below EspE_{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 EE.

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