  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 Schr\"odinger 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 Schr\"odinger
operators to obtain new results on completely monotone functions.
