An example of double confluent Heun equation: Schroedinger equation with supersingular plus Coulomb potential

TL;DR

This paper applies a new algorithm to solve the double confluent Heun equation in a quantum mechanics context, specifically for a particle in a combined supersingular and Coulomb potential, revealing exact solutions for specific parameters.

Contribution

It introduces an application of a recent algorithm to find solutions of the double confluent Heun equation in a quantum potential problem, highlighting cases with exact algebraic solutions.

Findings

Existence of exact algebraic solutions for specific parameter values.

Successful application of the algorithm to a complex quantum potential.

Insights into the spectral properties of the combined potential.

Abstract

A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a repulsive supersingular term, Ar(-4), plus an attractive Coulombian one, -Zr(-1). The existence of exact algebraic solutions for certain values of A is discussed.