Solvable potentials of shape invariance in two steps

Abstract

Within the framework of supersymmetric quantum mechanics, we obtain a class of solvable potentials of shape invariance in two steps, where the parameters a 1 and a 2 of partner potentials are related to each other by translation a 2 = a 1 + α. It is found that discontinuity at some x-points is a characteristic of the two-step superpotentials, therefore giving rise to Dirac delta-function singularities to the corresponding potentials.

Algebraic shape invariant potentials as the generalized deformed oscillator

“…In the present work, we explore the algebraic properties of shape invariance condition in arbitrary k steps, within the framework of SUSYQM. By imposing extra relations of identification ( 8) among the k superpotentials and the k remainders, respectively, we obtain the associated simplified potential algebra of shape invariance in k steps (13). This simplified version of potential algebra is found similar to that of shape invariance in one-step.…”

“…The analytical properties of the Pöschl-Teller I & II potentials in two steps have been discussed in Refs. [11,13].…”

“…In this way, the operator A † (A) as desired changes the eigenvalues of N by +(−) 1 k , respectively. We mention here that the above potential algebra (13) based on the set of generators { I, A † , A, N } is similar to that of shape invariance condition in one-step [18]- [21]. Moreover, based on our formulation, this algebra is also equivalent to the so-called generalized deformed oscillator algebra, which has been under extensive studied [23]- [25].…”

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

“…In fact, the particular form of superpotential shown in (3.29), expressed in terms of the function g(x) that satisfies the first-order differential equation (3.28), is not strange to us. Recently, both equations (3.28) and (3.29) have been developed in the study of shape invariance condition in two steps in the context of standard SUSY QM [70]. There, based on these two equations, new solvable shape invariant potentials in two steps are able to construct.…”

Two-step shape invariance in the framework of $$\mathcal{N}$$ -fold supersymmetry

“…For a quantum mechanical system with position-dependent mass the same approach was used to handle broken SUSY problem [22]. Recently a class of solvable potentials of translational SI in two steps were obtained [23,24]. It was found that discontinuity at some points was a characteristic of the two-step superpotentials, therefore giving rise to Dirac delta-function singularities in the corresponding potentials if they are considered in the whole line x ∈ (−∞, +∞).…”