An algorithm to obtain global solutions of the double confluent Heun equation

TL;DR

This paper introduces a method to find global solutions of the double confluent Heun equation by using asymptotic expansions and Wronskian quotients, demonstrated through a quantum mechanics example.

Contribution

It presents a novel procedure for constructing solutions with specified behaviors at singular points of the double confluent Heun equation.

Findings

Connection factors expressed as Wronskian quotients

Method validated with Schrödinger equation example

Feasibility demonstrated through asymptotic expansion techniques

Abstract

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schroedinger equation with a quasi-exactly-solvable potential.