Generalized Morse Potential: Symmetry and Satellite Potentials

We study in detail the bound state spectrum of the generalized Morse potential~(GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schr\"odinger equation with the Laplace equation on the hyperboloid and the Schr\"odinger equation for the P\"oschl-Teller potential, we explain the exact solvability of the problem by an $so(2,2)$ symmetry algebra, and obtain an explicit realization of the latter as $su(1,1) \oplus su(1,1)$. We prove that some of the $so(2,2)$ generators connect among themselves wave functions belonging to different GMP's (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, $so(2,2)$ belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.