Schr\"odinger Operators with $\delta$-interactions in a Space of Vector-Valued Functions

TL;DR

This paper investigates the spectral characteristics of Schr"odinger operators with delta interactions on a semi-axis, linking their properties to boundary triplet theory and block Jacobi matrices, advancing understanding of their mathematical structure.

Contribution

It introduces a novel connection between spectral properties of Schr"odinger operators with point interactions and block Jacobi matrices using boundary triplet theory.

Findings

Established criteria for self-adjointness and semiboundedness.

Characterized the spectrum's discreteness and resolvent comparability.

Linked spectral properties to block Jacobi matrix features.

Abstract

We study spectral properties of Schr\"odinger operators with $\delta$-interactions on a semi-axis by using the theory of boundary triplets and the corresponding Weyl functions. We establish a connection between spectral properties (deficiency indices, self-adjointness, semiboundedness, discreteness of spectra, resolvent comparability etc.) of Schr\"odinger operators with point interactions and a special class of block Jacobi matrices.