On eigenvalue bounds for a general class of Sturm-Liouville operators

TL;DR

This paper establishes eigenvalue bounds for a broad class of Sturm-Liouville operators with measure-valued coefficients, using a local periodicity condition, and demonstrates the bounds' sharpness with quasiperiodic examples.

Contribution

It introduces a new eigenvalue bound for Sturm-Liouville operators with measure-valued coefficients under a local periodicity condition, extending previous results.

Findings

Eigenvalue bounds are proven for operators with measure-valued weights and potentials.

The bounds are shown to be sharp through quasiperiodic operator examples.

A local periodicity (Gordon-type) condition is used to derive the bounds.

Abstract

We consider Sturm-Liouville operators with measure-valued weight and potential, and positive, bounded diffusion coefficient which is bounded away from zero. By means of a local periodicity condition, which can be seen as a quantitative Gordon condition, we prove a bound on eigenvalues for the corresponding operator in $L_p$, for $1\leq p<\infty$. We also explain the sharpness of our quantitative bound, and provide an example for quasiperiodic operators.