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, 1238Phys. 27, (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 ≥ 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.

Integral relations for solutions of the confluent Heun equation

“…For the present case the transformation T 1 is ineffective and, so, from U J 1 ; U L 1 we can obtain only 8 pairs of solutions by composition of the transformations (18); to each pair corresponds a kernel generated by the transformations (20). For example, taking U J 2 ðzÞ ¼ T 2 U J 1 ðzÞ and U L 2 ðzÞ ¼ T 2 U L 1 ðzÞ, we find…”

“…For the version (1), in the following we reobtain these kernels by solving Eq. (15) and use the transformations (20) to generate groups of kernels closed under such transformations.…”

“…where C 2 is a constant, G ð4Þ 4 ðz; tÞ is the kernel indicated in (54), and the transformation R 2 is given in (20); then,…”

“…Observe that from the pair U baber 1 ; U 1 we can obtain 16 pairs of solutions by using the four transformations (18) and composition of them: to each pair corresponds a kernel which is obtained by using the transformations (20).…”

“…However, more important is the fact that the solutions exhibit the above behaviour predicted by the normal or subnormal Thomé solutions, and fact that the Whittaker-Ince limit (3) may generate solutions to the RCHE. In effect, most of the known solutions for the RCHE [18][19][20] has been obtained from solutions of the CHE by means of the limit (3). On account of this, and for the sake of brevity, we restrict ourselves to integral relations among solutions of the CHE.…”