Solutions to a generalized spheroidal wave equation: Teukolsky's equations in general relativity, and the two-center problem in molecular quantum mechanics

Journal Article · Thu May 01 04:00:00 EDT 1986 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.527130· OSTI ID:5888714

Leaver, E W

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. 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.

• Research Organization: Department of Physics and the College of Science Computer, University of Utah, Salt Lake City, Utah 84112
• OSTI ID: 5888714
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 27:5; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
ASYMPTOTIC SOLUTIONS
BLACK HOLES
COMPUTER CALCULATIONS
CONVERGENCE
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD THEORIES
FUNCTIONS
GENERAL RELATIVITY THEORY
HYPERGEOMETRIC FUNCTIONS
LAGUERRE POLYNOMIALS
PARTIAL DIFFERENTIAL EQUATIONS
POLYNOMIALS
RECURSION RELATIONS
SCHROEDINGER EQUATION
SERIES EXPANSION
SINGULARITY
WAVE EQUATIONS
WAVE FUNCTIONS