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arXiv · 1402.5299

The algebra of two dimensional generalized Chebyshev - Koornwinder oscillator

V. V. Borzov·E. V. Damaskinsky
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Abstract

In the previous works \cite{N46,N47} authors have defined the oscillator-like system that associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev - Koornwinder oscillator. In this paper we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev - Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space $\mathcal{H}$ of functions that are defined on a region which bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev - Koornwinder polynomials and these polynomials form an orthonormalized basis in the space $\mathcal{H}$. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.

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V. V. Borzov, E. V. Damaskinsky. 2014-03-10. The algebra of two dimensional generalized Chebyshev - Koornwinder oscillator. https://doi.org/10.1063/1.4897655

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