Oscillator-Morse-Coulomb mappings and algebras for constant or position-dependent mass

TL;DR

This paper unifies the analysis of radial harmonic oscillator, Morse, and Coulomb Schrödinger equations using point canonical transformations, extending algebraic structures to position-dependent mass scenarios.

Contribution

It introduces a unified algebraic framework for these quantum systems and extends deformed su(1,1) algebras to position-dependent mass cases, generalizing known relationships.

Findings

Spectrum generating su(1,1) algebra transforms into potential algebra for Morse and Coulomb.

Deformed su(1,1) algebra is extended to position-dependent mass Schrödinger equations.

The algebraic approach aligns with shape invariance methods in variable mass contexts.

Abstract

The bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the $D$-dimensional radial Coulomb Schr\"odinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schr\"odinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a $d$-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.