A self-adjointness criterion for the Schrödinger operator with infinitely many point interactions and its application to random operators

Abstract

We prove the Schrödinger 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 Schrödinger operator with point interactions on a random perturbation of a lattice or on the Poisson configuration. We also determine the spectrum of the Schrödinger operators with random point interactions of Poisson--Anderson type.

32 pages, 11 figures