We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
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July 2013
Research Article|
July 24 2013
Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras
Ian Marquette
Ian Marquette
a)
School of Mathematics and Physics,
The University of Queensland
, Brisbane, QLD 4072, Australia
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Electronic mail: [email protected]
J. Math. Phys. 54, 071702 (2013)
Article history
Received:
April 26 2013
Accepted:
July 06 2013
Citation
Ian Marquette; Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras. J. Math. Phys. 1 July 2013; 54 (7): 071702. https://doi.org/10.1063/1.4816086
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