Relative oscillation theory and essential spectra of Sturm--Liouville operators

TL;DR

This paper extends relative oscillation theory to general Sturm-Liouville operators, analyzing how perturbations in coefficients, including the weight function, affect the essential spectrum and operator invariance.

Contribution

It introduces a novel approach to perturbation analysis for Sturm-Liouville operators, accommodating changes in the weight function and different Hilbert spaces.

Findings

Established invariance of essential spectra under perturbations.

Proved perturbation results for Sturm-Liouville operators.

Extended oscillation theory to include weight function perturbations.

Abstract

We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form \[ \frac{1}{r}\left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) \] and prove perturbation results and invariance of essential spectra in terms of the real coefficients $p$, $q$, $r$. The novelty here is that we also allow perturbations of the weight function $r$ in which case the unperturbed and the perturbed operator act in different Hilbert spaces.