Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature

TL;DR

This paper derives explicit fundamental solutions for Helmholtz operators on hyperbolic and spherical spaces of constant curvature, analyzes their properties, and develops Gegenbauer expansions using addition theorems.

Contribution

It provides closed-form fundamental solutions on curved spaces and introduces new addition theorems for Ferrers and Legendre functions for Gegenbauer expansions.

Findings

Explicit fundamental solutions for Helmholtz operators on hyperbolic and spherical spaces.

New addition theorems for Ferrers and Legendre functions.

Gegenbauer polynomial expansions in geodesic polar coordinates.

Abstract

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators $\big({-}\Delta\pm\beta^2\big)$ in $d$-dimensional, $R$-radius hyperbolic ${\mathbf H}_R^d$ and hyperspherical ${\mathbf S}_R^d$ geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of $\big({-}\Delta \pm \beta^2\big)$ on ${\mathbf H}_R^d$, $\big({-}\Delta+\beta^2\big)$ on ${\mathbf S}_R^d$, and present two candidate fundamental solutions for $\big({-}\Delta-\beta^2\big)$ on ${\mathbf S}_R^d$. Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.