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

TL;DR

This work develops an asymptotic framework to compute connection coefficients $\Theta$ between local fundamental solutions of a $2\times2$ linear ODE system with neighboring regular singular points at $z=0$ and $z=1$, enabling efficient eigenvalue computations in spectral problems. By transforming to holomorphic series and constructing a sequence $\Theta_k$ with provable convergence and an a posteriori error bound, the authors provide a practical algorithm whose convergence improves with an adjustable parameter $n$ and, in cases with integer $\delta$, through tailored handling. The method is then applied to ellipsoidal and spheroidal wave equations, turning their eigenvalue problems into zeros of $\Theta$ (and its transformed counterpart) and yielding new, accelerated algorithms with numerical demonstrations including Lamé and prolate spheroidal cases. The results offer robust, verifiable procedures for computing eigenvalues and eigenfunctions in these classical wave problems, with potential extensions to higher-dimensional systems and generalized Heun-type equations.

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.

Figures (6)

• Figure 1: The connection coefficient $\Theta$ for fixed $\gamma=4$, $c=1.6$, $\rho=1$, $\sigma=0$ as a function of the eigenvalue parameters $(\lambda,\mu)\in[-30,90]\times[-30,90]$. The contour lines indicate the zeros of the function $\Theta(\lambda,\mu)$.
• Figure 2: The zeros of the connection coefficients $\Theta$ (black) and $\hat{\Theta}$ (gray) for the case $\gamma=4$, $c=1.6$, $\rho=1$, $\sigma=0$, $\tau=1$ in the domain $(\lambda,\mu)\in[-30,90]\times[-90,30]$. The intersection points of these curves, which are marked by small circles, correspond to the eigenvalues of the ellipsoidal wave equation for the parameters given above.
• Figure 3: The ellipsoidal wave functions in the case $\gamma=4$, $c=1.6$ and $\rho=1$, $\sigma=0$, $\tau=1$ on the interval $(0,c)$ with none, one and two zeros, each normalized to a maximum magnitude $1$. In the legend, the corresponding pairs of eigenvalues $(\lambda,\mu)$ are listed with an accuracy of eight decimal places.
• Figure 4: The eigenfunction of the Lamé wave equation \ref{['Lame']} for the parameters $(\rho,\sigma,\tau)=(1,1,0)$, $\omega^2=100$, $k^2=0.5$ and the eigenvalues $H=599.43708$, $L=629.53546$ satisfying the normalization condition \ref{['Norm']}.
• Figure 5: The normalized eigenfunction of \ref{['Lame']} satisfying \ref{['Norm']} for the parameters $(\rho,\sigma,\tau)=(1,0,1)$, $\omega^2=1$, $k^2=0.9$ and the eigenvalues $(H,L)=(465.05152, 456.80932)$.

Theorems & Definitions (12)

• Remark 1
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• Remark 7