Analysis of the spectrum generating algebra method for obtaining energy spectra

We analyze and clarify how the SGA (spectrum generating algebra) method has been applied to different potentials. We emphasize that each energy level Eν obtained originally by Morse belongs to a different so (2,1) multiplet. The corresponding wave functions ψν are eigenfuntions of the compact generators J ₀ ^ν with the same eigenvalue k0, but with different eigenvalues qν of the Casimir operators Q. We derive a general expression for all effective potentials which have ψ λ _ν, ν+m (r) ∝ (J ₊ ^ν) mψ λ _ν, ν(r) as eigenfunctions, without using supersymmetry formalism. The different actions of SGA is further illustrated by two diagrams.