# Radial positive definite functions and spectral theory of the Schr\"odinger operators with point interactions

## TL;DR

This paper extends the classical Schoenberg theorem for radial positive definite functions and applies it to analyze the spectral properties of Schrödinger operators with point interactions, showing they have purely absolutely continuous spectrum.

## Contribution

It completes the Schoenberg representation theorem for radial positive definite functions and applies it to spectral analysis of Schrödinger operators with point interactions.

## Key findings

- All realizations have purely absolutely continuous non-negative spectrum.
- Extended Schoenberg theorem for radial positive definite functions.
- Applied spectral analysis to Schrödinger operators with point interactions.

## Abstract

We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schr\"odinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.