On the Negative Spectrum of One-Dimensional Schr\"odinger Operators with Point Interactions

TL;DR

This paper introduces a new method using boundary triplets and Weyl functions to analyze the negative spectra of one-dimensional Schrödinger operators with point interactions, providing algorithms and generalizations of previous results.

Contribution

It develops a novel approach to study negative spectra of 1D Schrödinger operators with point interactions, extending existing results and offering an algorithm for counting negative eigenvalues.

Findings

Algorithm for counting negative eigenvalues with delta interactions

Number of negative squares equals the negative strengths for delta' interactions

Generalization of previous spectral results using boundary triplet technique

Abstract

We investigate negative spectra of 1--D Schr\"odinger operators with $\delta$- and $\delta'$-interactions on a discrete set in the framework of a new approach. Namely, using technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of S. Albeverio and L. Nizhnik. For instance, we propose the algorithm for determining the number of negative squares of the operator with $\delta$-interactions. We also show that the number of negative squares of the operator with $\delta'$-interactions equals the number of negative strengths.