Spherical Schrödinger operators with δ-type interactions

Abstract

We investigate spectral properties of spherical Schrödinger operators (also known as Bessel operators) with δ-point interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be self-adjoint, lower-semibounded and also we investigate their spectra. We also extend the classical Bargmann estimate to such Hamiltonians. In certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities. We apply our results to Schrödinger operators in \documentclass[12pt]{minimal}\begin{document}$L^2(\mathbb {R}^n)$\end{document}L2(Rn) with a singular interaction supported by an infinite family of concentric spheres.