Title: Algebraic Description of Shape Invariance Revisited,
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.