Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature
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 HᵈR and hyperspherical SᵈR geometry, which represent Riemannian manifolds with positive constant and negative constant sect...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209867 |
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| Cite this: | Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature / H.S. Cohl, T.H. Dang, T.M. Dunster // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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Cohl, H.S. Dang, T.H. Dunster, T.M. 2025-11-28T09:33:12Z 2018 Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature / H.S. Cohl, T.H. Dang, T.M. Dunster // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 31C12; 32Q45; 33C05; 33C45; 35A08; 35J05; 42A16 arXiv: 1803.07149 https://nasplib.isofts.kiev.ua/handle/123456789/209867 https://doi.org/10.3842/SIGMA.2018.136 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 HᵈR and hyperspherical SᵈR 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 HᵈR, (−Δ+β²) on SᵈR, and present two candidate fundamental solutions for (−Δ−β²) on SᵈR. 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. We thank George Pogosyan and Loyal Durand for valuable discussions, and the referees for many helpful suggestions. T.M.D. acknowledges support from Ministerio de Economía y Competitividad, Spain, project MTM2015-67142-P (MINECO/FEDER, UE). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature |
| spellingShingle |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature Cohl, H.S. Dang, T.H. Dunster, T.M. |
| title_short |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature |
| title_full |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature |
| title_fullStr |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature |
| title_full_unstemmed |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature |
| title_sort |
fundamental solutions and gegenbauer expansions of helmholtz operators in riemannian spaces of constant curvature |
| author |
Cohl, H.S. Dang, T.H. Dunster, T.M. |
| author_facet |
Cohl, H.S. Dang, T.H. Dunster, T.M. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 HᵈR and hyperspherical SᵈR 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 HᵈR, (−Δ+β²) on SᵈR, and present two candidate fundamental solutions for (−Δ−β²) on SᵈR. 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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209867 |
| citation_txt |
Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature / H.S. Cohl, T.H. Dang, T.M. Dunster // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
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| first_indexed |
2025-12-03T07:02:14Z |
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2025-12-03T07:02:14Z |
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