Algebraic Shape Invariant Potentials as the Generalized Deformed Oscillator

TL;DR

This paper explores the algebraic structure of shape invariant potentials in supersymmetric quantum mechanics, revealing their equivalence to a generalized deformed oscillator algebra with Z_k-grading, and constructs explicit algebraic properties for these potentials.

Contribution

It introduces a novel algebraic formulation of shape invariance in k steps using Z_k-graded deformed oscillators, extending the understanding of cyclic shape invariant potentials.

Findings

Potential algebra is equivalent to generalized deformed oscillator algebra with Z_k-grading.

Explicit algebraic properties for shape invariant potentials in k steps are constructed.

Cyclic shape invariant potentials of period k are derived as a special case.

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 + \delta. The obtained results include the cyclic shape invariant potentials of period k as a special case.