Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature (1803.07149v2)

Published 19 Mar 2018 in math.AP, math.CA, and math.DG

Abstract: We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators $\big({-}\Delta\pm\beta2\big)$ in $d$-dimensional, $R$-radius hyperbolic ${\mathbf H}_Rd$ and hyperspherical ${\mathbf S}_Rd$ 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 $\big({-}\Delta \pm \beta2\big)$ on ${\mathbf H}_Rd$, $\big({-}\Delta+\beta2\big)$ on ${\mathbf S}_Rd$, and present two candidate fundamental solutions for $\big({-}\Delta-\beta2\big)$ on ${\mathbf S}_Rd$. 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.

Summary

We haven't generated a summary for this paper yet.