On the connection coefficients for linear differential systems with applications to the spheroidal and ellipsoidal wave equation

TL;DR

This paper derives an asymptotic formula for connection coefficients in 2x2 linear differential systems with applications to computing eigenvalues of spheroidal and ellipsoidal wave equations, aiding numerical analysis in physics.

Contribution

It introduces a new asymptotic formula for connection coefficients and develops algorithms for eigenvalue computation in spheroidal and ellipsoidal wave equations.

Findings

Asymptotic formula for connection coefficients derived.

Algorithms for eigenvalue computation presented.

Applicable to spectral problems in mathematical physics.

Abstract

This paper is concerned with the connection coefficients between the local fundamental solutions of a $2\times 2$ linear ordinary differential system with two neighboring regular singular points at $z=0$ and $z=1$. We derive an asymptotic formula for the connection coefficients which can be used for numerical calculations and, in particular, for determining the eigenvalues of some spectral problems arising in mathematical physics. As an application, new algorithms for computing the eigenvalues of the ellipsoidal wave equation and the spheroidal wave equation are presented.