One-dimensional Schr\"odinger operators with $\delta'$-interactions on a set of Lebesgue measure zero

TL;DR

This paper defines one-dimensional Schrödinger operators with delta-prime interactions on measure-zero sets, analyzing their spectral properties and providing examples with infinitely many negative eigenvalues.

Contribution

It introduces an abstract framework for Schrödinger operators with delta-prime interactions on arbitrary measure-zero sets and explores their spectral characteristics.

Findings

Number of negative eigenvalues is at least the count of negative interaction points.

Constructive examples show operators with infinitely many negative eigenvalues.

Spectral properties depend on the set and interaction intensities.

Abstract

We give an abstract definition of a one-dimensional Schr\"odinger operator with $\delta'$-interaction on an arbitrary set~$\Gamma$ of Lebesgue measure zero. The number of negative eigenvalues of such an operator is at least as large as the number of those isolated points of the set~$\Gamma$ that have negative values of the intensity constants of the $\delta'$-interaction. In the case where the set~$\Gamma$ is endowed with a Radon measure, we give constructive examples of such operators having an infinite number of negative eigenvalues.