Inter-relations between additive shape invariant superpotentials

Abstract

All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on $\hslash$, and their $\hslash$-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schrödinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus providing a pathway to interconnect all known additive shape invariant superpotentials.

arXiv comment14 pages, 1 figure

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• Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of $\hslash$
• Numerical study of the SWKB condition of novel classes of exactly solvable systems