Level spacing and Poisson statistics for continuum random Schrödinger operators

Adrian Dietlein, Alexander Elgart

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 Esp 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 Esp. 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.

Original languageEnglish
Pages (from-to)1257-1293
Number of pages37
JournalJournal of the European Mathematical Society
Issue number4
StatePublished - 1 Jan 2021
Externally publishedYes


  • Anderson localization
  • Level statistics
  • Minami estimate
  • Poisson statistics of eigenvalues

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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