Regular Trace Formula of Eigenvalues of a Discontinuous One-point Boundary Value Problem with Retarded Argument

TL;DR

This paper derives a regular trace formula for eigenvalues of a second-order differential boundary value problem with discontinuity, eigenparameter-dependent boundary and interface conditions, advancing spectral analysis in such complex systems.

Contribution

It introduces a novel regular trace formula for eigenvalues of a discontinuous boundary value problem with eigenparameter-dependent conditions and an interior discontinuity.

Findings

Derived a regular trace formula for the eigenvalues.

Handled a second-order differential equation with interior discontinuity.

Incorporated eigenparameter-dependent boundary and interface conditions.

Abstract

In this study, we found a regular trace formula for the eigenvalues of the boundary value problem, which we created with the second-order differential equation with eigen parameter and discontinuity at x ={\pi}/2, which is an interior point of the finite range [0, {\pi}], and boundary conditions that also contain eigen parameter, and interface conditions.