Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions

A number of results on radial positive definite functions on ${\mathbb R^n}$ related to Schoenberg's integral representation theorem are obtained. They are applied to the study of spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with countably many point interactions. In particular, we find conditions on the configuration of point interactions such that any self-adjoint realization has purely absolutely continuous non-negative spectrum. We also apply some results on Schrödinger operators to obtain new results on completely monotone functions.

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