The $d$-dimensional softcore Coulomb potential and the generalized confluent Heun equation

TL;DR

This paper analyzes the generalized confluent Heun equation in multiple dimensions and explicitly solves the Schrödinger equation with a softcore Coulomb potential, revealing polynomial solutions and their orthogonal properties.

Contribution

It provides a comprehensive analysis of the generalized confluent Heun equation and explicitly characterizes polynomial solutions for the softcore Coulomb potential in multiple dimensions.

Findings

Explicit conditions for polynomial solutions are derived.

A three-term recurrence relation for polynomial coefficients is established.

Polynomial solutions form finite sequences of orthogonal polynomials with known properties.

Abstract

An analysis of the generalized confluent Heun equation $(\alpha_2r^2+\alpha_1r)\,y''+(\beta_2r^2+\beta_1r+\beta_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{\alpha_i, \beta_i, \varepsilon_i\}$ are real parameters, is presented. With the aid of these general results, the quasi exact solvability of the Schr\"odinger eigenproblem generated by the softcore Coulomb potential $V(r)=-e^2Z/(r+b),\, b>0$, is explicitly resolved. Necessary and sufficient conditions for polynomial solvability are given. A three-term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly. We prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials. Properties such as recurrence relations, Christoffel-Darboux formulas, and the moments of the weight function are discussed. We also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials.