Series solution of a central potential problem with three-term recursion relation

TL;DR

This paper presents a simple method for solving the radial Schrödinger equation with combined Coulomb and harmonic potentials, focusing on low-lying bound states and the conditions for finite polynomial solutions.

Contribution

It introduces a novel approach to handle three-term recursion relations in the series solution of the Schrödinger equation with combined potentials.

Findings

Finite polynomial solutions occur only under specific potential relations.

The method simplifies solving for low-lying bound states.

The approach addresses the complexity of three-term recursion relations.

Abstract

The series solution of the radial part of the Schr\"odinger equation for simultaneous coulomb and harmonic potential involves three-term recursion relation and is thus difficult to solve for bound states. We have suggested a simple method to solve for low lying states. Finite polynomial solutions exist only if the coulomb and oscillator potentials are nontrivially related.