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

Elchin Jafarov (UGent) , Nedialka Stoilova (UGent) and Joris Van der Jeugt (UGent)
Author
Organization
Abstract
We define the quadratic algebra su(2)(alpha) 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)(alpha). We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)(alpha). 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 Fourier-Hahn transform is computed explicitly. The matrix of this discrete Fourier-Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.
Keywords
Hahn polynomials, Lie algebra deformation, Finite oscillator models

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## Citation

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Chicago
Jafarov, Elchin, Nedialka Stoilova, and Joris Van der Jeugt. 2011. “The Su(2)α Hahn Oscillator and a Discrete Fourier-Hahn Transform.” Journal of Physics A-mathematical and Theoretical 44 (35).
APA
Jafarov, Elchin, Stoilova, N., & Van der Jeugt, J. (2011). The su(2)α Hahn oscillator and a discrete Fourier-Hahn transform. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 44(35).
Vancouver
1.
Jafarov E, Stoilova N, Van der Jeugt J. The su(2)α Hahn oscillator and a discrete Fourier-Hahn transform. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 2011;44(35).
MLA
Jafarov, Elchin, Nedialka Stoilova, and Joris Van der Jeugt. “The Su(2)α Hahn Oscillator and a Discrete Fourier-Hahn Transform.” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 44.35 (2011): n. pag. Print.
@article{1917306,
abstract     = {We define the quadratic algebra su(2)(alpha) 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)(alpha). We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)(alpha). 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 Fourier-Hahn transform is computed explicitly. The matrix of this discrete Fourier-Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.},
articleno    = {355205},
author       = {Jafarov, Elchin and Stoilova, Nedialka and Van der Jeugt, Joris},
issn         = {1751-8113},
journal      = {JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL},
language     = {eng},
number       = {35},
pages        = {18},
title        = {The su(2)\ensuremath{\alpha} Hahn oscillator and a discrete Fourier-Hahn transform},
url          = {http://dx.doi.org/10.1088/1751-8113/44/35/355205},
volume       = {44},
year         = {2011},
}


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