Poisson Statistics for Eigenvalues of Continuum Random Schr\"odinger Operators

TL;DR

This paper demonstrates that in the localization region of continuum Anderson Hamiltonians, eigenvalues behave like a Poisson process with no repulsion, confirming their statistical independence and simplicity.

Contribution

It establishes Poisson statistics for eigenvalues in the continuum setting and provides a new proof of Minami's estimate for both continuum and discrete models.

Findings

Eigenvalues follow a Poisson point process in the localization region

Eigenvalues are simple and exhibit no level repulsion

Derived a Minami estimate for continuum Anderson Hamiltonians

Abstract

We show absence of energy levels repulsion for the eigenvalues of random Schr\"odinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum Anderson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model.