On some solutions to generalized spheroidal wave equations and applications

Abstract

Expansions in series of Coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations (GSWEs) are analysed and written together in pairs. Each pair consists of a solution in series of hypergeometric functions and another in series of Coulomb wavefunctions and has the same recurrence relations for the series coefficients, but the solutions inside it present different radii of convergence. Expansions without a phase parameter are derived by truncating the series with a phase parameter. For the Whittaker-Hill equation, solutions are found by treating that equation as a particular case of GSWE while, for the confluent GSWE, solutions, with and without a phase parameter, are given as pairs of series of Coulomb wavefunctions. Amongst the applications there are equations for the time dependence of Dirac test fields in some nonflat Friedmann-Robertson-Walker spacetimes, the radial Schrödinger equation for an electron in the field of two Coulombian centres and the Schrödinger equation for the Razavy-type quasi-exactly solvable potentials. For these problems it is possible to find wavefunctions in terms of infinite series, regular and convergent over the entire range of the independent variable, by matching expansions belonging to one or more of the above pairs. The infinite-series solutions for the Razavy-type potentials, in addition to the polynomial ones, suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi-classical methods of approximation.