Spectral theory of Schr\"odinger operators with infinitely many point interactions and radial positive definite functions

TL;DR

This paper investigates the spectral properties of Schr"odinger operators with infinitely many point interactions, utilizing radial positive definite functions and Schoenberg's theorem, revealing conditions for purely absolutely continuous spectra.

Contribution

It introduces new conditions on point interaction configurations ensuring purely absolutely continuous spectra for Schr"odinger operators, linking spectral theory with positive definite functions.

Findings

Conditions for purely absolutely continuous spectrum identified

New connections between positive definite functions and spectral properties established

Results extend understanding of Schr"odinger operators with infinitely many interactions

Abstract

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.