From the generalized Morse potential to a unified treatment of the $D$-dimensional singular harmonic oscillator and singular Coulomb potentials

TL;DR

This paper presents a unified approach to solving bound states of singular harmonic oscillator and Coulomb potentials in arbitrary dimensions by mapping them to the generalized Morse potential, revealing new insights and special cases.

Contribution

It introduces a simple method to derive solutions for singular potentials in any dimension from the generalized Morse potential, unifying various cases and clarifying boundary conditions.

Findings

Derived bound-state solutions in arbitrary dimensions.

Unified treatment of singular harmonic oscillator and Coulomb potentials.

Clarified conditions for bound states and degeneracies.

Abstract

Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence of bound states in a pure inversely quadratic potential.