New Wine in Old Bottle: An Improved Perspective on the Greene-Aldrich Approximation for the Generalized Pöschl-Teller Potential

The generalized Pöschl–Teller potential is well-known and is widely applied in molecular physics, chemistry, and nonlinear physics. In the past, when the vintage Greene–Aldrich approximation scheme for the centrifugal potential of the generalized Pöschl–Teller potential was examined, the consensus was that it would break down or become unreliable for large values of the potential parameter α. However, this study challenges that perspective by reviewing the problem in a new light and proposing a new and improved Greene–Aldrich approximation that mitigates the breakdown effect. Additionally, it is demonstrated that the energy eigenvalues produced for short and long-range potentials are in considerably closer agreement with those obtained using the numerical integration method than with the results of earlier research.