Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras

Abstract

We present some algebraic models for the quantum oscillator based upon Lie superalgebras. The Hamiltonian, position and momentum operator are identified as elements of the Lie superalgebra, and then the emphasis is on the spectral analysis of these elements in Lie superalgebra representations. The first example is the Heisenberg-Weyl superalgebra sh(2|2), which is considered as a "toy model". The representation considered is the Fock representation. The position operator has a discrete spectrum in this Fock representation, and the corresponding wavefunctions are in terms of Charlier polynomials. The second example is sl(2|1), where we construct a class of discrete series representations explicitly. The spectral analysis of the position operator in these representations is an interesting problem, and gives rise to discrete position wavefunctions given in terms of Meixner polynomials. This model is more fundamental, since it contains the paraboson oscillator and the canonical oscillator as special cases.