# Eigenvalue statistics for Schr\"odinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

## TL;DR

This paper demonstrates that in the localization regime, the eigenvalue distribution of multi-dimensional Schr"odinger operators with random point interactions follows a Poisson process, highlighting a significant example in continuum models.

## Contribution

It establishes Poisson eigenvalue statistics for multi-dimensional continuum Schr"odinger operators with point interactions, a novel result in the localization regime.

## Key findings

- Eigenvalue statistics follow a Poisson process at energy E.
- The density of states determines the intensity measure.
- The proof leverages the structure of the resolvent and Minami estimate.

## Abstract

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schr\"odinger operators with random point interactions on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schr\"odinger operators in the continuum. The special structure of resolvent of Schr\"odinger operators with point interactions facilitates the proof of the Minami estimate for these models.