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Title: Generalized spheroidal wave equation and limiting cases

Abstract

We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called Whittaker-Ince limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the Whittaker-Hill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindemann-Stieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Pressmore » (1945)] is associated with two expansions in series of Bessel functions. We also discuss a set of solutions in series of hypergeometric and confluent hypergeometric functions for the GSWE and use their Leaver limits to obtain infinite-series solutions for the Schroedinger equation with an asymmetric double-Morse potential. Finally, the possibility of extending the solutions of the GSWE to the general Heun equation is briefly discussed.« less

Authors:
 [1]
  1. Instituto de Cosmologia, Relatividade e Astrofisica (ICRA-BR), Centro Brasileiro de Pesquisas Fisicas (CBPF), Rua Dr. Xavier Sigaud, CEP 22290-180, Rio de Janeiro 150 (Brazil)
Publication Date:
OSTI Identifier:
20929619
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2406057; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMMETRY; BESSEL FUNCTIONS; GEOMETRY; HYPERGEOMETRIC FUNCTIONS; MATHEMATICAL SOLUTIONS; MATHIEU EQUATION; MORSE POTENTIAL; SCHROEDINGER EQUATION; SINGULARITY

Citation Formats

Bonorino Figueiredo, B. D. Generalized spheroidal wave equation and limiting cases. United States: N. p., 2007. Web. doi:10.1063/1.2406057.
Bonorino Figueiredo, B. D. Generalized spheroidal wave equation and limiting cases. United States. doi:10.1063/1.2406057.
Bonorino Figueiredo, B. D. Mon . "Generalized spheroidal wave equation and limiting cases". United States. doi:10.1063/1.2406057.
@article{osti_20929619,
title = {Generalized spheroidal wave equation and limiting cases},
author = {Bonorino Figueiredo, B. D.},
abstractNote = {We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called Whittaker-Ince limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the Whittaker-Hill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindemann-Stieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press (1945)] is associated with two expansions in series of Bessel functions. We also discuss a set of solutions in series of hypergeometric and confluent hypergeometric functions for the GSWE and use their Leaver limits to obtain infinite-series solutions for the Schroedinger equation with an asymmetric double-Morse potential. Finally, the possibility of extending the solutions of the GSWE to the general Heun equation is briefly discussed.},
doi = {10.1063/1.2406057},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}
  • The differential equation, x(x-x/sub 0/)(d/sup 2/y/dx/sup 2/)+(B/sub 1/+B/sub 2/x) (dy/dx)+(..omega../sup 2/x(x-x/sub 0/) -(2eta..omega..(x-x/sub 0/)+B/sub 3/)y = 0, arises both in the quantum scattering theory of nonrelativistic electrons from polar molecules and ions, and, in the guise of Teukolsky's equations, in the theory of radiation processes involving black holes. This article discusses analytic representations of solutions to this equation. Previous results of Hylleraas (E. Hylleraas, Z. Phys. 71, 739 (1931)), Jaffe (G. Jaffe, Z. Phys. 87, 535 (1934)), Baber and Hasse (W. G. Baber and H. R. Hasse, Proc. Cambridge Philos. Soc. 25, 564 (1935)), and Chu and Stratton (L. J.more » Chu and J. A. Stratton, J. Math. Phys. (Cambridge, Mass.) 20, 3 (1941)) are reviewed, and a rigorous proof is given for the convergence of Stratton's spherical Bessel function expansion for the ordinary spheroidal wave functions. An integral is derived that relates the eigensolutions of Hylleraas to those of Jaffe. The integral relation is shown to give an integral equation for the scalar field quasinormal modes of black holes, and to lead to irregular second solutions to the equation. New representations of the general solutions are presented as series of Coulomb wave functions and confluent hypergeometric functions. The Coulomb wave-function expansion may be regarded as a generalization of Stratton's representation for ordinary spheroidal wave functions, and has been fully implemented and tested on a digital computer.« less
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