On the solvability of confluent Heun equation and associated orthogonal polynomials

TL;DR

This paper investigates the conditions under which confluent Heun equations admit polynomial solutions, explores their orthogonal polynomial properties, and applies these findings to analyze the quasi-exact solvability of hyperbolic potentials in quantum mechanics.

Contribution

It provides necessary and sufficient conditions for polynomial solutions of confluent Heun equations and constructs associated orthogonal polynomials, including their properties and applications to hyperbolic potentials.

Findings

Derived conditions for polynomial solutions of confluent Heun equations

Constructed orthogonal polynomials and analyzed their properties

Applied results to quasi-exactly solvable hyperbolic potentials

Abstract

The present paper analyze the constraints on the confluent Heun type-equation, $(a_{3,1}r^2+a_{3,2}r)y"+(a_{2,0}r^2+a_{2,1}r+a_{2,2})y'-(\tau_{1,0}r+\tau_{1,1})y=0,$ where $|a_{3,1}|^2+|a_{3,2}|^2\neq 0, $ and $a_{i,j},i=3,2,1, j=0,1,2$ are real parameters, to admit polynomial solutions. The necessary and sufficient conditions for the existence of these polynomials are given. A three-term recurrence relation is provided to generate the polynomial solutions explicitly. We, then, prove that these polynomial solutions are a source of finite sequences of orthogonal polynomials. Several properties, such as the recurrence relation, Christoffel-Darboux formulas and the moments of the weight function, are discussed. We also show a factorization property of these orthogonal polynomials that allow for the construction of other sequences of orthogonal polynomials. For illustration, we examines the quasi- exactly solvability of the $(p,q)$-hyperbolic potential $V(r)=-V_0\sinh^p(r)/\cosh^q(r), V_0>0, p\geq 0, q>p$. The associated orthogonal polynomials generated by the solutions of the Schr\"odinger equation with the $(4,6)$-hyperbolic potential are constructed.