# Negative Eigenvalue Estimates for the 1D Schrödinger Operator with Measure-Potential

## TL;DR

This paper studies negative eigenvalues of a 1D Schrödinger operator with a specific type of potential and provides new estimates for these eigenvalues.

## Contribution

The paper introduces new eigenvalue estimates using Otelbaev’s function for a Schrödinger operator with a measure-potential.

## Key findings

- Estimates for the eigenvalue counting function are derived.
- Individual eigenvalue bounds and Lieb–Thirring-type estimates are established.
- Otelbaev’s function is shown to be crucial for both proofs and result formulations.

## Abstract

We investigate the negative part of the spectrum of the operator \documentclass[12pt]{minimal}
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\begin{document}$$\mu \ge 0$$\end{document}μ≥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 \documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}μ, which is used both in the proofs and the formulation of most of the results.