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


SIGMA 7 (2011), 108, 14 pages      arXiv:1108.3679      http://dx.doi.org/10.3842/SIGMA.2011.108

Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Howard S. Cohl a, b
a) Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland, USA
b) Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand

Received August 18, 2011, in final form November 22, 2011; Published online November 29, 2011; Misprints are corrected January 29, 2012

Abstract
Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hypersphere SRd with R>0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the associated Legendre function of the second kind on the cut (Ferrers function of the second kind) with degree and order given by d/2−1 and 1−d/2 respectively, with real argument between plus and minus one.

Key words: hyperspherical geometry; fundamental solution; Laplace's equation; separation of variables; Ferrers functions.

pdf (370 Kb)   tex (18 Kb)       [previous version:  pdf (370 kb)   tex (18 kb)]

References

  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  2. Berakdar J., Concepts of highly excited electronic systems, Wiley-VCH, New York, 2003.
  3. Bers L., John F., Schechter M. (Editors), Partial differential equations, Interscience Publishers, New York, 1964.
  4. Cooper J.W., Fano U., Prats F., Classification of two-electron excitation levels of helium, Phys. Rev. Lett. 10 (1963), 518-521.
  5. Delves L.M., Tertiary and general-order collisions. II, Nuclear Phys. 20 (1960), 275-308.
  6. Doob J.L., Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften, Vol. 262, Springer-Verlag, New York, 1984.
  7. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
  8. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
  9. Fano U., Correlations of two excited electrons, Rep. Progr. Phys. 46 (1983), 97-165.
  10. Fock V., On the Schrödinger equation of the helium atom. I, Norske Vid. Selsk. Forhdl. 31 (1958), no. 22, 7 pages.
    Fock V., On the Schrödinger equation of the helium atom. II, Norske Vid. Selsk. Forhdl. 31 (1958), no. 23, 8 pages.
  11. Folland G.B., Introduction to partial differential equations, Mathematical Notes, Princeton University Press, Princeton, N.J., 1976.
  12. Gilbarg D., Trudinger N.S., Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1983.
  13. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.
  14. Grigor'yan A.A., Existence of the Green's function on a manifold, Russ. Math. Surv. 38 (1983), no. 1, 190-191.
  15. Grigor'yan A.A., The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds, Math. USSR Sb. 56 (1987), 349-358.
  16. Grigor'yan A.A., Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, Vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.
  17. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.
  18. 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.
  19. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and the separation of variables: interbase expansions, J. Phys. A: Math. Gen. 34 (2001), 521-554.
  20. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contraction and interbases expansions on n-sphere, in Quantum Theory and Symmetries (Kraków, 2001), World Sci. Publ., River Edge, NJ, 2002, 389-395.
  21. Kalnins E.G., Miller W. Jr., Pogosyan G.S., The Coulomb-oscillator relation on n-dimensional spheres and hyperboloids, Phys. Atomic Nuclei 65 (2002), 1086-1094.
  22. Lee J.M., Riemannian manifolds, Graduate Texts in Mathematics, Vol. 176, Springer-Verlag, New York, 1997.
  23. Leemon H.I., Dynamical symmetries in a spherical geometry. II, J. Phys. A: Math. Gen. 12 (1979), 489-501.
  24. Lin C.D., Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995), 1-83.
  25. Olevski M.N., Triorthogonal systems in spaces of constant curvature in which the equation Δ2uu=0 allows a complete separation of variables, Mat. Sbornik N.S. 27 (1950), 379-426.
  26. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010.
  27. Oprea J., Differential geometry and its applications, 2nd ed., Classroom Resource Materials Series, Mathematical Association of America, Washington, DC, 2007.
  28. Pack R.T., Parker G.A., Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. Theory, J. Chem. Phys. 87 (1987), 3888-3921.
  29. Schrödinger E., Eigenschwingungen des sphärischen Raumes, Comment. Pontificia Acad. Sci. 2 (1938), 321-364.
  30. Schrödinger E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A 46 (1940), 9-16.
  31. Smith F.T., Generalized angular momentum in many-body collisions, Phys. Rev. 120 (1960), 1058-1069.
  32. Takeuchi M., Modern spherical functions, Translations of Mathematical Monographs, Vol. 135, American Mathematical Society, Providence, RI, 1994.
  33. Thurston W.P., Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997.
  34. Timofeev A.F., Integration of functions, OGIZ, Moscow - Leningrad, 1948 (in Russian).
  35. Vilenkin N.Ja., Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968.
  36. Vinitski S.I., Mardoyan L.G., Pogosyan G.S., Sissakian A.N., Strizh T.A., Hydrogen atom in curved space. Expansion in free solutions on a three-dimensional sphere, Phys. Atomic Nuclei 56 (1993), 321-327.
  37. Zernike F., Brinkman H.C., Hypersphärische Funktionen und die in sphärische Bereichen orthogonalen Polynome, Proc. Akad. Wet. Amsterdam 38 (1935), 161-170.
  38. Zhukov M.V., Danilin B.V., Fedorov D.V., Bang J.M., Thompson I.J., Vaagen J.S., Bound state properties of Borromean halo nuclei: 6He and 11Li, Phys. Rep. 231 (1993), 151-199.

Previous article   Next article   Contents of Volume 7 (2011)