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Finite oscillator models : the Hahn oscillator

Elchin Jafarov (UGent) , Nedialka Stoilova (UGent) and Joris Van der Jeugt (UGent)
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Abstract
A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra u(2)(alpha). This algebra is a deformation of the Lie algebra u(2) extended by a parity operator, with the deformation parameter alpha. A class of irreducible unitary representations of u(2)(alpha) 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.
Keywords
SYSTEMS, QUANTIZATION, SPACE, 2-DIMENSIONAL OSCILLATOR, quantum oscillator, finite quantum physics, Hahn polynomial

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Chicago
Jafarov, Elchin, Nedialka Stoilova, and Joris Van der Jeugt. 2011. “Finite Oscillator Models : the Hahn Oscillator.” Journal of Physics A-mathematical and Theoretical 44 (26).
APA
Jafarov, Elchin, Stoilova, N., & Van der Jeugt, J. (2011). Finite oscillator models : the Hahn oscillator. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 44(26).
Vancouver
1.
Jafarov E, Stoilova N, Van der Jeugt J. Finite oscillator models : the Hahn oscillator. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 2011;44(26).
MLA
Jafarov, Elchin, Nedialka Stoilova, and Joris Van der Jeugt. “Finite Oscillator Models : the Hahn Oscillator.” JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 44.26 (2011): n. pag. Print.
@article{1266613,
  abstract     = {A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra u(2)(alpha). This algebra is a deformation of the Lie algebra u(2) extended by a parity operator, with the deformation parameter alpha. A class of irreducible unitary representations of u(2)(alpha) 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.},
  articleno    = {265203},
  author       = {Jafarov, Elchin and Stoilova, Nedialka and Van der Jeugt, Joris},
  issn         = {1751-8113},
  journal      = {JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL},
  keyword      = {SYSTEMS,QUANTIZATION,SPACE,2-DIMENSIONAL OSCILLATOR,quantum oscillator,finite quantum physics,Hahn polynomial},
  language     = {eng},
  number       = {26},
  pages        = {15},
  title        = {Finite oscillator models : the Hahn oscillator},
  url          = {http://dx.doi.org/10.1088/1751-8113/44/26/265203},
  volume       = {44},
  year         = {2011},
}

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