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

SIGMA 14 (2018), 136, 45 pages      arXiv:1803.07149      https://doi.org/10.3842/SIGMA.2018.136
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

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

Howard S. Cohl a, Thinh H. Dang b and T.M. Dunster c
a) Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA
b) Department of Computer Science, George Washington University, Washington D.C. 20052, USA
c) Department of Mathematics & Statistics, San Diego State University, San Diego, CA 92182, USA

Received March 01, 2018, in final form December 14, 2018; Published online December 31, 2018

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.

Key words: hyperbolic geometry; hyperspherical geometry; fundamental solution; Helmholtz equation; Gegenbauer series; separation of variables; addition theorems; associated Legendre functions; Ferrers functions.

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