Algebraic Realization of Supersymmetric Quantum Mechanics for Cyclic Shape Invariant Potentials

TL;DR

This paper explores the spectrum of a bosonic oscillator linked to a cyclic extended algebra, classifies its spectral types, and demonstrates how supersymmetric Hamiltonians can be constructed using these algebraic structures, with potential extensions to higher periods.

Contribution

It introduces a novel algebraic framework for realizing supersymmetric quantum mechanics in systems with cyclic shape invariant potentials, extending to general cyclic groups.

Findings

Identified spectra with periodically spaced levels similar to cyclic shape invariant potentials.

Established a realization of supersymmetric Hamiltonians using extended oscillator algebras and spin matrices.

Outlined the extension to spectra with arbitrary cyclic periods.

Abstract

We study in detail the spectrum of the bosonic oscillator Hamiltonian associated with the $C_3$-extended oscillator algebra \algthree, where $C_3$ denotes a cyclic group of order three, and classify the various types of spectra in terms of the algebra parameters $\alpha_0, \alpha_1$. In such a classification, we identify those spectra having an infinite number of periodically spaced levels, similar to those of cyclic shape invariant potentials of period three. We prove that the hierarchy of supersymmetric Hamiltonians and supercharges, corresponding to the latter, can be realized in terms of some appropriately chosen \algthree algebras, and of Pauli spin matrices. Extension to period-$\lambda$ spectra in terms of $C_{\lambda}$-extended oscillator algebras is outlined.