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Generalized spheroidal wave equation and limiting cases

B. D. Bonorino Figueiredo
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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. Another procedure, called Whittaker-Ince limit, 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 by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindmann-Stieltjes solutions for the Mathieu equation 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 Schrödinger 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.

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B. D. Bonorino Figueiredo. 2006-12-12. Generalized spheroidal wave equation and limiting cases. https://arxiv.org/abs/math-ph/0611048

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