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

TL;DR

This paper explores solutions to a special limit of the confluent Heun equation, deriving new solution forms, analyzing their convergence, and connecting these to Schrödinger equations for specific potentials.

Contribution

It introduces Ince's limits for confluent and double-confluent Heun equations, providing explicit solutions and linking them to quantum mechanical potentials.

Findings

Solutions expressed as hypergeometric and Bessel series

Convergence properties of the solutions

Reduction of Schrödinger equations to Heun forms

Abstract

We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable $z$, while the other is given by a series of modified Bessel functions and converges for $|z|>|z_{0}|$, where $z_{0}$ denotes a regular singularity. For short, the preceding limit is called Ince's limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when $z_{0}$ tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schr\"odinger equation for inverse fourth and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince's limit, respectively.