One-dimensional Schr\"odinger operators with ${\delta }^{\mathrm{\prime }}$-interactions on a set of Lebesgue measure zero

We give an abstract definition of a one-dimensional Schr"odinger operator with ${\delta }^{\mathrm{\prime }}$-interaction on an arbitrary set~$\mathrm{\Gamma }$ 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~$\mathrm{\Gamma }$ that have negative values of the intensity constants of the ${\delta }^{\mathrm{\prime }}$-interaction. In the case where the set~$\mathrm{\Gamma }$ is endowed with a Radon measure, we give constructive examples of such operators having an infinite number of negative eigenvalues.