Non-Relativistic Closed-Form Energy Spectrum of a Hyperbolic Molecular Potential Through the Asymptotic Iteration Method

In this study, we consider a potential expressed as a hyperbolic-sine function aiming to achieve the energy eigenvalues in a closed form, that is, as an analytical expression. Based on this, the Schrödinger equation is constructed within the framework of non-relativistic quantum mechanics and is tackled by using the Asymptotic Iteration Method. The potential in question was previously addressed in the literature. As an alternative, we obtain the complete energy spectrum in a closed form for the single-well regime of the potential function, by way of the quasi-exact solvability where the system has analytical energy eigenvalues once a certain condition is met, or a relation between the potential parameters is satisfied. This is provided by the applicability of the Asymptotic Iteration Method to both quasi-exact and numerical solutions. Thus, the effects of the potential parameters on the energy spectrum can be seen separately. We conclude that the accuracy of the obtained closed-form energy spectrum is quite high as evidenced by the strong agreement with the numerically obtained ones. Furthermore, it is seen that this consistency improves as the energy level increases. The obtained analytical expression can also be used as a simple analytical model for vibrational spectrum of molecular systems described by anharmonic single-well potentials.