C$_λ$-extended Oscillator Algebras: Theory and Applications to (Variants) of Supersymmetric Quantum Mechanics

Abstract

C$_λ$-extended oscillator algebras, where C$_λ$ is the cyclic group of order $λ$, are introduced and realized as generalized deformed oscillator algebras. For $λ=2$, they reduce to the well-known Calogero-Vasiliev algebra. For higher $λ$ values, they are shown to provide in their bosonic Fock space representation some interesting applications to supersymmetric quantum mechanics and some variants thereof: an algebraic realization of supersymmetric quantum mechanics for cyclic shape invariant potentials of period $λ$, a bosonization of parasupersymmetric quantum mechanics of order $p = λ-1$, and, for $λ=3$, a bosonization of pseudosupersymmetric quantum mechanics and orthosupersymmetric quantum mechanics of order two.