Discrete series representations for $\mathfrak {sl}(2|1)$, Meixner polynomials and oscillator models

Abstract

We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number β > 0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter γ. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the position operator can be continuous or infinite discrete, depending on the value of γ. We determine the position wavefunctions both in the continuous and discrete case, and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2) ⊂ sl(2|1), it can be seen why the case γ = 1 corresponds to the paraboson oscillator. Consequently, taking the values (β, γ) = (1/2, 1) in the sl(2|1) model yields the canonical oscillator.

Finite Oscillator Models Described by the Lie Superalgebra

“…These discrete series representations of sl(2|1), labelled by a positive number β > 0, have been determined recently. 11 In these representations,Ĥ has the same spectrum as the canonical oscillator. General forms forq (andp) involve one parameter γ > 0 (like our p in the finite dimensional case).…”

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

“…Motion of the non-relativistic quantum harmonic oscillator under the suddenly exposed constant external field is an exception here. The rest of the known exactly-solvable quantum harmonic oscillator models exhibit dynamical symmetries, which belong to Lie (super)algebra or q-deformation of the Heisenberg-Weyl algebra [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter

“…Other attractive examples of the quantum harmonic oscillator model leading to polynomial solutions for the wave functions of the stationary states are the non-relativistic parabose oscillator model within the non-canonical approach [5], the relativistic oscillator models [6,7], finiteand infinite-discrete harmonic oscillator models [8][9][10][11] as well as a hybrid discrete-continuous harmonic oscillator model [12].…”