Soft-core Coulomb potentials and Heun's differential equation

TL;DR

This paper explores the Schrödinger equation with a class of soft-core Coulomb potentials, demonstrating its reduction to Heun's differential equations and employing the Asymptotic Iteration Method to find eigenstates and eigenvalues.

Contribution

It establishes a connection between soft-core Coulomb potentials and Heun's equations, providing explicit solutions and approximation methods for eigenstates.

Findings

Reduction of Schrödinger's equation to Heun's equations for specific potentials

Explicit polynomial solutions in certain parameter regimes

Application of Asymptotic Iteration Method for eigenvalue problems

Abstract

Schroedinger's equation with the attractive potential V(r) = -Z/(r^q+ b^q)^(1/q), Z > 0, b > 0, q >= 1, is shown, for general values of the parameters Z and b, to be reducible to the confluent Heun equation in the case q=1, and to the generalized Heun equation in case q=2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases this factor becomes a polynomial. The Asymptotic Iteration Method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detail solutions for both cases are provided.