Differential Transfer Matrix Solution of Generalized Eigenvalue Problems

TL;DR

This paper introduces an extended differential transfer matrix method to analytically solve a broad class of second-order differential equations with variable eigenvalues, applicable in quantum mechanics and optics.

Contribution

The paper presents a novel extension of the differential transfer matrix method for solving generalized eigenvalue problems with arbitrary eigenvalue functions.

Findings

Successfully applied to boundary and initial value problems

Demonstrated effectiveness through multiple examples

Provides analytical solutions for complex differential equations

Abstract

We report a new analytical method for solution of a wide class of second-order differential equations with eigenvalues replaced by arbitrary functions. Such classes of problems occur frequently in Quantum Mechanics and Optics. This approach is based on the extension of the previously reported differential transfer matrix method with modified basis functions. Applications of the method to boundary value and initial value problems, as well as several examples are illustrated.