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Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras

Elchin Jafarov (UGent) and Joris Van der Jeugt (UGent)
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Abstract
We present some algebraic models for the quantum oscillator based upon Lie superalgebras. The Hamiltonian, position and momentum operator are identified as elements of the Lie superalgebra, and then the emphasis is on the spectral analysis of these elements in Lie superalgebra representations. The first example is the Heisenberg-Weyl superalgebra sh(2 vertical bar 2), which is considered as a "toy model". The representation considered is the Fock representation. The position operator has a discrete spectrum in this Fock representation, and the corresponding wavefunctions are in terms of Charlier polynomials. The second example is sl(2 vertical bar 1), where we construct a class of discrete series representations explicitly. The spectral analysis of the position operator in these representations is an interesting problem, and gives rise to discrete position wavefunctions given in terms of Meixner polynomials. This model is more fundamental, since it contains the paraboson oscillator and the canonical oscillator as special cases.
Keywords
ALGEBRA, POLYNOMIALS, REPRESENTATIONS, orthogonal polynomials, Lie superalgebra, oscillator models

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MLA
Jafarov, Elchin, and Joris Van der Jeugt. “Quantum Oscillator Models with a Discrete Position Spectrum in the Framework of Lie Superalgebras.” Journal of Physics : Conference Series, vol. 512, IOP, 2014, doi:10.1088/1742-6596/512/1/012034.
APA
Jafarov, E., & Van der Jeugt, J. (2014). Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras. In Journal of Physics : Conference Series (Vol. 512). Bristol, UK: IOP. https://doi.org/10.1088/1742-6596/512/1/012034
Chicago author-date
Jafarov, Elchin, and Joris Van der Jeugt. 2014. “Quantum Oscillator Models with a Discrete Position Spectrum in the Framework of Lie Superalgebras.” In Journal of Physics : Conference Series. Vol. 512. Bristol, UK: IOP. https://doi.org/10.1088/1742-6596/512/1/012034.
Chicago author-date (all authors)
Jafarov, Elchin, and Joris Van der Jeugt. 2014. “Quantum Oscillator Models with a Discrete Position Spectrum in the Framework of Lie Superalgebras.” In Journal of Physics : Conference Series. Vol. 512. Bristol, UK: IOP. doi:10.1088/1742-6596/512/1/012034.
Vancouver
1.
Jafarov E, Van der Jeugt J. Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras. In: Journal of Physics : Conference Series. Bristol, UK: IOP; 2014.
IEEE
[1]
E. Jafarov and J. Van der Jeugt, “Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras,” in Journal of Physics : Conference Series, Mexico, Mexico, 2014, vol. 512.
@inproceedings{5808532,
  abstract     = {{We present some algebraic models for the quantum oscillator based upon Lie superalgebras. The Hamiltonian, position and momentum operator are identified as elements of the Lie superalgebra, and then the emphasis is on the spectral analysis of these elements in Lie superalgebra representations. The first example is the Heisenberg-Weyl superalgebra sh(2 vertical bar 2), which is considered as a "toy model". The representation considered is the Fock representation. The position operator has a discrete spectrum in this Fock representation, and the corresponding wavefunctions are in terms of Charlier polynomials. The second example is sl(2 vertical bar 1), where we construct a class of discrete series representations explicitly. The spectral analysis of the position operator in these representations is an interesting problem, and gives rise to discrete position wavefunctions given in terms of Meixner polynomials. This model is more fundamental, since it contains the paraboson oscillator and the canonical oscillator as special cases.}},
  articleno    = {{012034}},
  author       = {{Jafarov, Elchin and Van der Jeugt, Joris}},
  booktitle    = {{Journal of Physics : Conference Series}},
  issn         = {{1742-6588}},
  keywords     = {{ALGEBRA,POLYNOMIALS,REPRESENTATIONS,orthogonal polynomials,Lie superalgebra,oscillator models}},
  language     = {{eng}},
  location     = {{Mexico, Mexico}},
  pages        = {{9}},
  publisher    = {{IOP}},
  title        = {{Quantum oscillator models with a discrete position spectrum in the framework of Lie superalgebras}},
  url          = {{http://dx.doi.org/10.1088/1742-6596/512/1/012034}},
  volume       = {{512}},
  year         = {{2014}},
}

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