New basis functions for wave equation

TL;DR

This paper introduces new basis functions for the wave equation derived from an extended Differential Transfer Matrix Method, simplifying calculations near turning points and enabling precise eigenvalue evaluation.

Contribution

It extends the Differential Transfer Matrix Method to the complex plane, providing a new set of basis functions that are easier to use and handle singularities at turning points.

Findings

New basis functions are less accurate than WKB but easier to work with.

Basis functions exactly satisfy initial conditions and handle turning points without divergence.

High-precision eigenvalue evaluation demonstrated through examples.

Abstract

The Differential Transfer Matrix Method is extended to the complex plane, which allows dealing with singularities at turning points. The result for real-valued systems are simplified and a pair of basis functions is found. These bases are a bit less accurate than WKB solutions but much easier to work with because of their algebraic form. Furthermore, these bases exactly satisfy the initial conditions and may go over the turning points without the divergent behavior of WKB solutions. The findings of this paper allow explicit evaluation of eigenvalues of confined modes with high precision, as demonstrated by few examples.