On the Multi-Dimensional Schr\"odinger Operators with Point Interactions

TL;DR

This paper analyzes multi-dimensional matrix Schr"odinger operators with point interactions, providing a comprehensive parametrization of their self-adjoint extensions, spectral characterization, and conditions for negative eigenvalues using boundary triplet techniques.

Contribution

It generalizes existing results by parametrizing all self-adjoint extensions and characterizing their spectra for multi-dimensional Schr"odinger operators with point interactions.

Findings

Parametrization of all self-adjoint extensions via boundary conditions

Spectral characterization of these extensions

Sufficient conditions for negative eigenvalues based on distances and intensities

Abstract

We study two- and three-dimensional matrix Schr\"odinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by the other authors in this field. For instance, we parametrize all self-adjoint extensions of the initial minimal symmetric Schr\"odinger operator by abstract boundary conditions and characterize their spectra. Particularly, we find a sufficient condition in terms of distances and intensities for the self-adjoint extension $H_{\alpha,X}^{(3)}$ to have $m'$ negative eigenvalues, i.e., $\kappa_-(H_{\alpha,X}^{(3)})=m'\leq m$. We also give an explicit description of self-adjoint nonnegative extensions.