Energy spectrum of a generalized Scarf potential using the asymptotic iteration method and the tridiagonal representation approach

Abstract

The well-known trigonometric Scarf potential is generalized by adding a sinusoidal term and then treated using the Asymptotic Iteration Method (AIM) and the Tridiagonal Representation Approach (TRA). The energy spectrum of the associated bound states is computed. For the AIM, we have improved convergence of the quantization condition that terminates the iterations asymptotically. This is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results (plateau of convergence). We have shown that with increasing iteration, this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point. The value of this point (or points) may depend on the physical parameters. The numerical results have been compared favorably with those resulting from the TRA.