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

SIGMA 11 (2015), 015, 23 pages      arXiv:1405.4847      https://doi.org/10.3842/SIGMA.2015.015
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Howard S. Cohl a and Rebekah M. Palmer b
a) Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA
b) Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

Received May 20, 2014, in final form February 09, 2015; Published online February 14, 2015

Abstract
For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

Key words: fundamental solution; hypersphere; Fourier expansion; Gegenbauer expansion.

pdf (503 kb)   tex (29 kb)

References

1. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
2. Barut A.O., Inomata A., Junker G., Path integral treatment of the hydrogen atom in a curved space of constant curvature, J. Phys. A: Math. Gen. 20 (1987), 6271-6280.
3. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Vol. 67, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1954.
4. Chandrasekhar S., An introduction to the study of stellar structure, Dover Publications, Inc., New York, 1957.
5. Cohl H.S., Fundamental solution of Laplace's equation in hyperspherical geometry, SIGMA 7 (2011), 108, 14 pages, arXiv:1108.3679.
6. Cohl H.S., Kalnins E.G., Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A: Math. Theor. 45 (2012), 145206, 32 pages, arXiv:1105.0386.
7. Cohl H.S., Rau A.R.P., Tohline J.E., Browne D.A., Cazes J.E., Barnes E.I., Useful alternative to the multipole expansion of $1/r$ potentials, Phys. Rev. A 64 (2001), 052509, 5 pages, arXiv:1104.1499.
8. Cohl H.S., Tohline J.E., A compact cylindrical Green's function expansion for the solution of potential problems, Astrophys. J. 527 (1999), 86-101.
9. Folland G.B., Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995.
10. Fox L., Parker I.B., Chebyshev polynomials in numerical analysis, Oxford University Press, London - New York - Toronto, 1968.
11. Freeden W., Schreiner M., Spherical functions of mathematical geosciences: a scalar, vectorial, and tensorial setup, Advances in Geophysical and Environmental Mechanics, Vol. 67, Springer-Verlag, Berlin, 2008.
12. Genest V.X., Vinet L., Zhedanov A., The Bannai-Ito algebra and a superintegrable system with reflections on the two-sphere, J. Phys. A: Math. Theor. 47 (2014), 205202, 13 pages, arXiv:1401.1525.
13. Grosche C., Karayan Kh.H., Pogosyan G.S., Sissakian A.N., Quantum motion on the three-dimensional sphere: the ellipso-cylindrical bases, J. Phys. A: Math. Gen. 30 (1997), 1629-1657.
14. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials. II. The two- and three-dimensional sphere, Fortschr. Phys. 43 (1995), 523-563.
15. Hakobyan Ye.M., Pogosyan G.S., Sissakian A. N. V.V.I., Isotropic oscillator in the space of constant positive curvature. Interbasis expansions, Phys. Atomic Nuclei 62 (1999), 623-637, quant-ph/9710045.
16. Herranz F.J., Ballesteros Á., Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006), 010, 22 pages, math-ph/0512084.
17. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. The $n$-dimensional sphere, J. Math. Phys. 40 (1999), 1549-1573.
18. Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
19. Lee J.M., Riemannian manifolds. An introduction to curvature, Graduate Texts in Mathematics, Vol. 176, Springer-Verlag, New York, 1997.
20. Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften, Vol. 52, 3rd ed., Springer-Verlag, New York, 1966.
21. Mhaskar H.N., Narcowich F.J., Prestin J., Ward J.D., $L^p$ Bernstein estimates and approximation by spherical basis functions, Math. Comp. 79 (2010), 1647-1679, arXiv:0810.5075.
22. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
23. Narcowich F.J., Rowe S.T., Ward J.D., A novel Galerkin method for solving PDEs on the sphere using highly localized kernel bases, arXiv:1404.5263.
24. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W., NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.
25. Pogosyan G.S., Yakhno A., Lie algebra contractions and separation of variables. Three-dimensional sphere, Phys. Atomic Nuclei 72 (2009), 836-844.
26. Schrödinger E., Eigenschwingungen des sphärischen Raumes, Comment. Pontificia Acad. Sci. 2 (1938), 321-364.
27. Schrödinger E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A. 46 (1940), 9-16.
28. Wen Z.Y., Avery J., Some properties of hyperspherical harmonics, J. Math. Phys. 26 (1985), 396-403.