Spectrum, trace and oscillation of a Sturm-Liouville type retarded differential operator with interface conditions

TL;DR

This paper derives a formula for regularized sums of eigenvalues and investigates oscillation properties of a Sturm-Liouville problem with retarded argument and interface conditions, advancing understanding of such differential operators.

Contribution

It introduces a new formula for eigenvalue sums and analyzes oscillation behavior in Sturm-Liouville problems with retarded arguments and discontinuities.

Findings

Derived a regularized sum formula for eigenvalues

Analyzed oscillation properties of the problem

Provided insights into differential operators with interface conditions

Abstract

In this study, a formula for regularized sums of eigenvalues of a Sturm-Liouville problem with retarded argument at the point of discontinuity is obtained. Moreover, oscillation properties of the related problem is investigated.