Relative Oscillation Theory for Sturm-Liouville Operators Extended

TL;DR

This paper extends relative oscillation theory to Sturm-Liouville operators with varying coefficients, linking the zeros of Wronskians to spectral shift functions within spectral gaps.

Contribution

It introduces an extension of relative oscillation theory to operators with different p functions, connecting zeros of solutions to spectral shift functions.

Findings

Weighted zeros of Wronskians equal spectral shift function values

Extension applies to Sturm-Liouville operators with variable coefficients

Establishes a link between oscillation counts and spectral properties

Abstract

We extend relative oscillation theory to the case of Sturm--Liouville operators $H u = r^{-1}(-(pu')'+q u)$ with different $p$'s. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.