二维广义Chebyshev-Koornwinder振子代数
The algebra of two dimensional generalized Chebyshev - Koornwinder oscillator
摘要：
在以前的工作\cite{N46,N47}作者已经定义了与两个变量的Chebyshev-Koornwinder多项式相关联的类扰动系统。我们称这个系统为广义Chebyshev-Koornwinder振子.本文研究了Chebyshev-Koornwinder振子的无限维李代数的性质。我们构造了这个代数在定义在以Steiner内摆线为界的区域上的函数的Hilbert空间$\mathcal{H}$中的精确不可约表示。函数是平方可积的Chebyshev-Koornwinder多项式的正交性措施，这些多项式形成一个正交归一化的基础上的空间$\mathcal{H}$。本文所研究的广义振子可以看作是由三个相互作用的振子组成的多玻色子量子系统的最简单的非平凡例子。
英文摘要：
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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