The algebra of two dimensional generalized Chebyshev - Koornwinder oscillator

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.

27 pages, 3 figures, submited