Energy spectrum of a generalized Scarf potential using the Asymptotic Iteration Method and the Tridiagonal Representation Approach

TL;DR

This paper extends the trigonometric Scarf potential with a sinusoidal term and computes its energy spectrum using AIM and TRA, improving convergence and comparing results for bound states.

Contribution

It introduces a generalized Scarf potential and enhances the AIM convergence technique, providing a comparative analysis with TRA for bound state energies.

Findings

AIM convergence is improved by identifying stable initial value ranges.

The energy spectra computed by AIM and TRA show good agreement.

Optimal iteration numbers depend on physical parameters.

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 are 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.