Analysis of the SGA 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 $E_\nu$ obtained originally by Morse belongs to a {\em different} ${\mathfrak {so}}(2,1)$ multiplet. The corresponding wavefunctions $\Psi_\nu$ are eigenfuntions of the compact generators $J^\nu_0$ with the same eigenvalue $k_0$, but with different eigenvalues $q_\nu$ of the Casimir operators $Q$. We derive a general expression for all effective potentials which have $\Psi_{\lambda_\nu,\nu+m}(r) \propto (J_+^\nu)^m ~\Psi_{\lambda_\nu,\nu}(r)$ as eigenfunctions, without using super-symmetry formalism. The different actions of SGA is further illustrated by two diagrams.