Nota de FísicaPublicado
Generalized spheroidal wave equation and limiting cases
Dados da publicação
- Autores
- —
- Tipo
- Nota de Física
- Status
- Publicado
- Data de publicação
- 05/12/2006
- Descrição
- 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 Schr\"odinger 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.
- Ano
- 2006
- Vínculo com tese
- Não
- Vínculo com bolsa
- Não