有序积、$W_{\infty}$ -代数与二元奇偶正交多项式
Ordered Products, $W_{\infty}$-Algebra, and Two-Variable, Definite-Parity, Orthogonal Polynomials
摘要：
证明了$W_{\infty}$ -代数的Cartan子代数是二元定奇偶多项式空间。给出了这些多项式的显式表达式及其基本性质。也已表明，他们进行的无限维不可约表示的$su(1,1)$代数具有的频谱有界从下面。这个代数的差分算子的实现也得到。对于特定值的排序参数$s$，他们被确定与经典的正交多项式的离散变量，如迈克斯纳，迈克斯纳-Pollaczek，和阿斯基-威尔逊多项式。关于变量$s$，它们满足超几何型的二阶特征值方程。用这些多项式表示了一族势的零能量散射态。提出了将差分与微分联系起来的桥梁是勒诺努-维格纳压缩及其逆。
英文摘要：
It has been shown that the Cartan subalgebra of $W_{\infty}$- algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. Also has been shown that they carry the infinite dimensional irreducible representation of the $su(1,1)$ algebra having the spectrum bounded from below. A realization of this algebra in terms of difference operators is also obtained. For particular values of the ordering parameter $s$ they are identified with the classical orthogonal polynomials of a discrete variable, such as the Meixner, Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable $s$ they satisfy a second order eigenvalue equation of hypergeometric type. Exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials. It has been put forward that it is the İnönü-Wigner contraction and its inverse that form bridge between the difference and differential calculus.
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