Application of supersymmetric WKB method to cyclic shape invariant potentials

Abstract

We examine the accuracy and the validity of the lowest order supersymmetric WKB (SWKB) formula for cyclic shape invariant potentials (CSIPs). For period-2 CSIPs, we show analytically that the SWKB formula can yield exact eigenenergies for either all the even states or all the odd states. Such alternate exactness of the SWKB formula is due to the fact that period-2 CSIPs can also be considered as a kind of translational shape invariant potential. For CSIPs with periods greater than 2, we note that the accuracy of the SWKB formula also demonstrates similar alternating patterns. However, as a consequence of the rapid oscillations in the potential at large distances, the SWKB quantization formula fails to produce highly accurate results even in the high-energy limit.

Generation and classification of the translational shape-invariant potentials based on the analytical transfer matrix method

“…To date, several kinds of SIPs have been thoroughly studied, including translational shape invariant potentials (TSIPs), [4,5] scaling shape invariant potentials, [6] and cyclic shape invariant potentials. [7,8] However, only the TSIPs have the closed-form expressions for superpotentials and eigenfunctions.…”