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

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.