Abstract
We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.
Highlights
Solvable (ES) Schrödinger equations (SE) allow us to understand some physical phenomena and to test some approximation schemes
Except for the PT and deformed radial Coulomb (DC) potentials, the deformed superpotential is written in terms of two combinations of parameters, the first one a being changed into a + hand the second one b remaining constant when going to the partner
We have shown that the approach of Gangopadhyaya, Mallow, and their coworkers of shape invariant (SI) potentials in conventional Supersymmetric quantum mechanics (SUSYQM) can be extended to DSI ones in deformed SUSYQM (DSUSYQM) and we have illustrated our results by considering several examples taken from [39,40,41]
Summary
Solvable (ES) Schrödinger equations (SE) allow us to understand some physical phenomena and to test some approximation schemes. In the case of conventional SUSYQM, Gangopadhyaya, Mallow, and their coworkers proposed an interesting approach to SI potentials, consisting in replacing the SI condition, which is a difference-differential equation, by an infinite set of partial differential equations The latter is obtained by expanding the superpotential in powers of hand expressing that the coefficient of each power must separately vanish [42]. They encountered a pathway for going from those superpotentials of [1] corresponding to SE that can be reduced to the confluent hypergeometric equation to those related to SE connected with the hypergeometric equation [46] It is the purpose of the present paper to propose an extension of the approach of Gangopadhyaya, Mallow, and their coworkers to the case of DSI potentials in DSUSYQM, both without and with explicit dependence of the superpotential on h.
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