The su(2)_α Hahn oscillator and a discrete Hahn-Fourier transform

Abstract

We define the quadratic algebra su(2)_α which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation label j, a positive integer) can be extended to representations of su(2)_α. We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)_α. It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Hahn-Fourier transform is computed explicitly. The matrix of this discrete Hahn-Fourier transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

References in corpus (1)

• The Wigner distribution function for the one-dimensional parabose oscillator

Cited by in corpus (9)

• Clifford algebras, Fourier transforms and quantum mechanics
• The Bannai-Ito algebra and some applications
• Exact solution of the position-dependent effective mass and angular frequency Schrödinger equation: harmonic oscillator model with quantized confinement parameter
• Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)
• The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
• Exact solution of the position-dependent mass Schrödinger equation with the completely positive oscillator-shaped quantum well potential
• A finite quantum oscillator model related to special sets of Racah polynomials
• Dynamical symmetry of a semiconfined harmonic oscillator model with a position-dependent effective mass
• The $\mathfrak{su}(2)$ Krawtchouk oscillator model under the $\mathcal{C}\mathcal{P}$ deformed symmetry