Negative eigenvalue estimates for the 1D Schr{ö}dinger operator with measure-potential

Abstract

We investigate the negative part of the spectrum of the operator $-{\partial }^2-\mu$ on $L^2(\mathbb{R})$, where a locally finite Radon measure $\mu \ge 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb-Thirring type. A crucial tool for our estimates is Otelbaev's function, a certain average of the measure potential $\mu$, which is used both in the proofs and the formulation of most of the results.