Confluent and Double-Confluent Heun Equations: Convergence of Solutions in Series of Coulomb Wavefunctions

Abstract

The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test assures that the domains of convergence of these solutions include an additional singular point. Further, by using a limit proposed by Leaver, we obtain solutions for the double-confluent Heun equation (DCHE). In addition, we get solutions for the so-called Whittaker-Ince limit of the CHE and DCHE. For these four equations, new solutions are generated by transformations of variables. In the second place, for each of the above equations we obtain one-sided series solutions ($n\ge 0$) by truncating on the left the two-sided series. Finally we discuss the time dependence of the Klein-Gordon equation in two cosmological models and the spatial dependence of the Schrödinger equation to a family of quasi-exactly solvable potentials. For a subfamily of these potentials we obtain infinite-series solutions which converge and are bounded for all values of the independent variable, in opposition to a common belief}.

Throughout this version we use the tilde to denote asymptotic equality

References in corpus (1)

• Ince's limits for confluent and double-confluent Heun equations

Cited by in corpus (8)

• Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions
• Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
• Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations
• Integral relations for solutions of confluent Heun equations
• On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole
• Series solutions of confluent Heun equations in terms of incomplete Gamma-functions