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

Abstract

The differential equation, x(x−x0)(d2y/dx2)+(B1+B2x) (dy/dx)+[ω2x(x−x0) −[2ηω(x−x0)+B3]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)], Jaffé [G. Jaffé, Z. Phys. 87, 535 (1934)], Baber and Hassé [W. G. Baber and H. R. Hassé, 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 Jaffé. 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. Both solutions given by the new algorithms are analytic in the variable x and the parameters B1, B2, B3, ω, x0, and η, and are uniformly convergent on any interval bounded away from x0. They are the first representations for generalized spheroidal wave functions that allow the direct evaluation of asymptotic magnitude and phase.