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Convergence and applications of some solutions of the confluent Heun equation

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  • El-Jaick, Léa Jaccoud
  • Figueiredo, Bartolomeu D.B.

Abstract

We study the convergence of a group of solutions in series of confluent hypergeometric functions for the confluent Heun equation. These solutions are expansions in two-sided infinite series (summation from minus to plus infinity) which are interpreted as a modified version of expansions proposed by Leaver (1986). We show that the two-sided solutions yield two nonequivalent groups of one-sided series solutions (summation from zero to plus infinity). In the second place, we find that one-sided solutions of one of these groups can be used to solve an equation which describes a time-dependent two-level system of Quantum Optics. For this problem, in addition to finite-series solutions, we obtain infinite-series wavefunctions which are convergent and bounded for any value of the time t, and vanish when t goes to infinity.

Suggested Citation

  • El-Jaick, Léa Jaccoud & Figueiredo, Bartolomeu D.B., 2016. "Convergence and applications of some solutions of the confluent Heun equation," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 234-259.
    • Handle: RePEc:eee:apmaco:v:284:y:2016:i:c:p:234-259
    DOI: 10.1016/j.amc.2016.03.003
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    1. El-Jaick, Léa Jaccoud & Figueiredo, Bartolomeu D.B., 2015. "Integral relations for solutions of the confluent Heun equation," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 885-904.
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