A Self-adjointness Criterion for the Schrödinger Operator with Infinitely Many Point Interactions and Its Application to Random Operators

We prove the Schrödinger operator with infinitely many point interactions in R d ( d = 1 , 2 , 3 ) is self-adjoint if the support Γ of the interactions is decomposed into infinitely many bounded subsets { Γ j } j such that inf j ≠ k dist ( Γ j , Γ k ) > 0 . 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.