2018/03/31 by Howard S. Cohl, Thinh H. Dang, T. M. Dunster · 1 citation
Mathematics · #math.AP #math.CA #math.DG #msc:31C12 #msc:32Q45 #msc:33C05 #msc:33C45 #msc:35A08 #msc:35J05 #msc:42A16
paper · pdf · doi:10.3842/sigma.2018.136
published as SIGMA 14 (2018), 136, 45 pages · This article recollects results published in arXiv:1201.4406, arXiv:1105.0386 and applies them in a new context
arxiv created 2018/12/31 · arxiv updated 2019/01/01
We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators (-Δ±β2) in d-dimensional, R-radius hyperbolic \mathbf HRd and hyperspherical \mathbf SRd 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 (-Δ± β2) on \mathbf HRd, (-Δ+β2) on \mathbf SRd, and present two candidate fundamental solutions for (-Δ-β2) on \mathbf SRd. 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.