Fundamental solutions and Gegenbauer expansions of Helmholtz operators on Riemannian spaces of constant curvature

Published

December 31, 2018

Author(s)

Howard S. Cohl, Thinh Dang, Dunster Mark

Abstract

We compute closed-form expressions for oscillatory and damped spherically symmetric fundamental solutions of the Helmholtz equation in $d$-dimensional hyperbolic and hyperspherical geometry. We are using the $R$-radius hypersphere and $R$-radius hyperboloid model of hyperbolic geometry. These models represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. 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. In geodesic polar coordinates, we derive Gegenbauer expansions for these fundamental solutions for $d\ge 3$, and azimuthal Fourier expansions in two-dimensions.

Citation

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Volume

14

Pub Type

Journals

Download Paper

https://doi.org/10.3842/SIGMA.2018.136

Local Download

Keywords

Hyperbolic geometry, Hyperspherical geometry, Fundamental solution, Helmholtz equation, Gegenbauer series, Separation of variables

Citation

Cohl, H. , Dang, T. and Mark, D. (2018), Fundamental solutions and Gegenbauer expansions of Helmholtz operators on Riemannian spaces of constant curvature, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), [online], https://doi.org/10.3842/SIGMA.2018.136 (Accessed October 14, 2025)