On the Quasi-Exact Solvability of the Confluent Heun Equation

TL;DR

This paper demonstrates that the Confluent Heun Equation can be reduced to a quasi-exactly solvable form under certain parameters, allowing polynomial solutions linked to specific physical models like the two-center Coulomb problem.

Contribution

It establishes a connection between the CHEq and $sl(2,\mathbb{R})$-based quasi-exact solvability, providing explicit polynomial solutions for related quantum models.

Findings

Polynomial solutions for CHEq under specific parameters.

Connection to known equations like the Generalized Spheroidal and Razavy equations.

Application to quantum systems with two Coulomb centers.

Abstract

It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{\mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.