Algebraic Description of Shape Invariance Revisited

TL;DR

This paper revisits the algebraic approach to shape invariance in quantum mechanics, demonstrating how specific models can be solved using representation theory when parameters are quantized.

Contribution

It provides a detailed algebraic analysis of shape invariance for several Kepler and Rosen-Morse problems, emphasizing the role of nonlinear algebraic systems and parameter quantization.

Findings

Bound states are solvable via representation theory with quantized parameters.

Algebraic systems are constructed for specific quantum models.

Shape invariance can be revisited through nonlinear algebraic frameworks.

Abstract

We revisit the algebraic description of shape invariance method in one-dimensional quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space, the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen-Morse potential problem. Following the prescription given by Gangopadhyaya et al., we first introduce certain nonlinear algebraic systems. We then show that, if the model parameters are appropriately quantized, the bound-state problems can be solved solely by means of representation theory.