Algebraic shape invariant potentials as the generalized deformed oscillator

Abstract

Within the framework of supersymmetric quantum mechanics, we study the simplified version of potential algebra of shape invariance condition in k steps, where k is an arbitrary positive integer. The associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built-in Z k -grading structure. The algebraic realization of shape invariance condition in k steps is therefore formulated by the method of Z k -graded deformed oscillator. Based on this formulation, we explicitly construct the general algebraic properties for shape invariant potentials in k steps, in which the parameters of partner potentials are related to each other by translation a 1 = a 0 + δ. The obtained results include the cyclic shape invariant potentials of period k as a special case. *

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

“…where H ± are a pair of type A 2-fold SUSY Hamiltonians, H i1 is (one of) its intermediate Hamiltonians, and ψ ± P are second-order parafermions satisfying 25) we see that the triple (H P , Q ± P ) satisfies the second-order paraSUSY relations in [39] (Q ± P ) 2 = 0,…”

“…The translational SI potentials were shown to possess a potential algebra involving three generators of angular momentum type. The potential algebra for the case of SI in k steps (k being an arbitrary positive integer) was found [25] to be equivalent to the generalized deformed oscillator algebra that had a built-in Z 4 grading structure and was constructed in terms of the generators of the deformed harmonic oscillator (I, A, A † , N) as well as the grading generator T of the cyclic group of order k. The obtained potentials included the cyclic SI potentials of period k as a special case.…”

Shape Invariant Potentials in Second-Order Supersymmetric Quantum Mechanics

“…Additionally, the shape invariant potentials constructed in the framework of 2-SUSY QM are similar to those shape invariant potentials in two steps discussed in the context of standard SUSY QM. That is, there must be an analogous potential algebra underlying the 2-SUSY shape invariant potentials, thus providing an alternative way of getting the energy eigenvalues by algebraic methods [34,46,69,71].…”