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The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator

### Abstract

In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys.175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is 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 of functions that are defined on a region which is 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 . 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.

© 2014 AIP Publishing LLC

Received 27 April 2014
Accepted 25 September 2014
Published online 17 October 2014

Acknowledgments:
The authors are grateful to Petr Kulish and Vladimir Lyakhovsky for the valuable discussions. The work of EVD was done under the partial support of the RFBR Grant No. 12-01-00207.

Article outline:

I. INTRODUCTION
II. PRELIMINARIES
A. The ChK-polynomials
B. ChK-oscillator
1. Generalized sectorial oscillator
2. Generalized radial oscillator
3. Generalized boundary oscillator
III. ALGEBRA OF GENERALIZED SECTORIAL OSCILLATOR
IV. ALGEBRA OF GENERALIZED RADIAL OSCILLATOR
V. ALGEBRA OF GENERALIZED BOUNDARY OSCILLATOR
VI. ALGEBRA OF GENERALIZED CHEBYSHEV-KOORNWINDER OSCILLATOR
VII. INVESTIGATION OF ALGEBRA
A. Commutation relations between generators of the algebra
B. Center of the algebra
C. Maximal Abelian subalgebra of the algebra
D. Subalgebras and ideals in : Representation in the form of semidirect sum
E. Proof simplicity of the ideal
VIII. CONCLUSION

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2014-10-17

2016-10-23

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