Finite oscillator models: the Hahn oscillator

Abstract

A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra u(2) α . This algebra is a deformation of the Lie algebra u(2) extended by a parity operator, with deformation parameter α. A class of irreducible unitary representations of u(2) α is constructed. In the finite oscillator model, the (discrete) spectrum of the position operator is determined, and the position wavefunctions are shown to be dual Hahn polynomials. Plots of these discrete wavefunctions display interesting properties, similar to those of the parabose oscillator. We show indeed that in the limit, when the dimension of the representations goes to infinity, the discrete wavefunctions tend to the continuous wavefunctions of the parabose oscillator.

Deformed su(1,1) Algebra as a Model for Quantum Oscillators

“…Note that for a = 1/2 the paraboson wave functions reduce to the canonical oscillator wave functions, as the corresponding Laguerre polynomials become Hermite polynomials [16,Appendix]. So therefore, one can say that under the limit (a, c) → (1/2, +∞) the new oscillator models introduced in this paper reduce to the canonical quantum oscillator.…”

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

“…We shall recall some formulas for the paraboson oscillator (see [18] or the appendix of [19]). In terms of the momentum and position operatorp andq, the Hamiltonian of the paraboson oscillator is given bŷ…”

“…From the action ofĤ 0 and from the explicit action of the commutator [q,p] on the basis vectors |β, n , it is clear that for β = 1/2 the paraboson oscillator yields the canonical oscillator. The position wavefunctions for the paraboson oscillator can then be determined by constructing the formal eigenvectors ofq (see [19]) or by different techniques [31]. To see that the Lie superalgebra generated by the paraboson operators is indeed osp(1|2), let…”

“…Since these representations are finite-dimensional, one is dealing with "finite oscillator models", of potential use in optical image processing [6]. Recently, this model has been extended by introducing an additional parameter in the algebra, thus leading to finite paraboson oscillator models [19,20]. In [22], the Lie algebra su(2) was extended to the Lie superalgebra sl(2|1), yielding another interesting finite oscillator model with appealing position spectra and remarkable discrete wavefunctions.…”

“…For each x belonging to the spectrum ofq, the corresponding formal eigenvector is given by (21). Since essentially only γ 2 plays a role, and since γ = 0 is a redundant case, we shall in the rest of the paper deal with γ > 0.…”