Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function

TL;DR

This paper explores the spectral characteristics of Sturm-Liouville operators with measure potentials, providing estimates and criteria for spectrum discreteness and resolvent properties using Otelbaev's function.

Contribution

It introduces new spectral estimates and criteria for measure potential Sturm-Liouville operators leveraging Otelbaev's function, advancing understanding of their spectral behavior.

Findings

Two-sided estimates for spectral distribution function

Criteria for spectrum discreteness and Schatten class membership

Bounds for the essential spectrum

Abstract

We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness of the spectrum and a criterion for the membership of the resolvents to Schatten classes. We give two side estimates for the lower bound of the essential spectrum. Our main tool in achieving this is Otelbaev's function.