# A self-adjointness criterion for the Schr\"odinger operator with infinitely many point interactions and its application to random operators

## TL;DR

This paper establishes a self-adjointness criterion for Schr"odinger operators with infinitely many point interactions, including random configurations, and analyzes their spectral properties.

## Contribution

It introduces a new self-adjointness criterion for Schr"odinger operators with infinitely many point interactions and applies it to random and Poisson configurations.

## Key findings

- Proves self-adjointness for operators with interactions in uniformly discrete clusters.
- Establishes self-adjointness for operators on random perturbations of lattices and Poisson configurations.
- Determines the spectrum of Schr"odinger operators with Poisson--Anderson type random point interactions.

## Abstract

We prove the Schr\"odinger operator with infinitely many point interactions in $\mathbb{R}^d$ $(d=1,2,3)$ is self-adjoint if the support of the interactions is decomposed into uniformly discrete clusters. Using this fact, we prove the self-adjointness of the Schr\"odinger operator with point interactions on a random perturbation of a lattice or on the Poisson configuration. We also determine the spectrum of the Schr\"odinger operators with random point interactions of Poisson--Anderson type.