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

TL;DR

This paper provides estimates for the negative eigenvalues of a 1D Schrödinger operator with a measure-based potential, extending classical spectral bounds to more general measure potentials.

Contribution

It introduces new eigenvalue estimates for the Schrödinger operator with measure potentials, utilizing Otelbaev's function for the first time in this context.

Findings

Eigenvalue counting function estimates derived

Individual eigenvalue bounds established

Lieb-Thirring type inequalities obtained

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 $\mu \geq 0$ 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.