Racah多项式与$\mathfrak{su}(1,1)$的再耦合方案
Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
摘要：
再耦合计划的四个副本的$\mathfrak{su}(1,1)$，一般的超可积系统的3球，和二元Racah多项式之间的连接被确定。Racah多项式被表示为在不同球坐标系中分离的本征函数之间的连接系数，并且等价地表示为张量积表示的不同不可约分解。作为该模型的结果，给出了二次代数${\rm QR}(3)$的一个推广。它示出，这个代数关闭只包含一个额外的移位算子，超出了二元Racah多项式，其多项式本征函数的本征值算子。变量和度之间的对偶性，因此双谱的多项式，被解释在独立的解决方案的膨胀系数。
英文摘要：
The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.
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