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 {Γj}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.