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

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.