Negative Eigenvalue Estimates for the 1D Schrödinger 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 \(\mu \ge 0\) serves 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.