Schr\"odinger operators with measure-valued potentials: semiboundedness and spectrum

We study the 1-D Schr"odinger operators in Hilbert space $L^2(\mathbb{R})$ with real-valued Radon measure $q^{\mathrm{\prime }}(x)$, $q\in {\mathrm{B}\mathrm{V}}_{loc}(\mathbb{R})$ as potentials. New sufficient conditions for minimal operators to be bounded below and selfadjoint are found. For such operators a criterion for the discreteness of the spectrum is proved, which generalizes Molchanov’s, Brinck’s, and the Albeverio-Kostenko-Malamud criteria. The quadratic forms corresponding to the investigated operators are described.