## 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^{−}(log ^{L)β} for sufficiently large β > 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.

Original language | English |
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Pages (from-to) | 1257-1293 |

Number of pages | 37 |

Journal | Journal of the European Mathematical Society |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics