Algebraic Realization of Supersymmetric Quantum Mechanics for Cyclic Shape Invariant Potentials

We study in detail the spectrum of the bosonic oscillator Hamiltonian associated with the C3-extended oscillator algebra A α0α1 , where C3 denotes a cyclic group of order three, and classify the various types of spectra in terms of the algebra parameters α0, α1. 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 A α0α1 algebras, and of Pauli spin matrices. Extension to period-λ spectra in terms of Cλ-extended oscillator algebras is outlined.