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arXiv 1302.6142math-phmath.MP
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平面II中的Dunkl振子：对称代数的表示

The Dunkl oscillator in the plane II : representations of the symmetry algebra

Vincent X. Genest, Mourad E. H. Ismail, Luc Vinet, Alexei Zhedanov

备注：
33 pages
发表信息：
Commun. Math. Phys. 329, 999-1029 (2014)
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摘要：

第一部分研究了平面Dunkl振子模型的超可积性、波函数和重叠系数。这里的有限维表示的对称代数的系统，称为Schwinger-Dunkl代数sd（2），进行调查。代数sd（2）有六个生成元，包括两个对合和一个中心元，可以看作是李代数u（2）的变形。其中两个对称生成元J_3和J_2分别与笛卡尔坐标和极坐标中的变量分离有关。利用抛物声波产生/湮灭算子，构造了sd（2）表示的两种基：笛卡尔基和圆基.在笛卡尔基中，算子J_3是对角的，而算子J_2以三对角的方式作用。在循环基中，算子J_2是块上三角形，所有块都是2x2，算子J_3以三对角方式作用。两个基之间的展开系数由Krawtchouk多项式给出。在一般情况下，J_2在循环基中的特征向量由Heun多项式生成，其分量用para-Krawtchouk多项式表示。在完全迷向的情况下，J_2的特征向量由小-1 Jacobi多项式或普通Jacobi多项式生成。然后考虑算子J_2是对角的基。在此基础上，Schwinger-Dunkl代数的定义关系意味着J_3以块三对角方式作用，所有块都是2x2。给出了该基中J_3的矩阵元。

英文摘要：

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra u(2). Two of the symmetry generators, J_3 and J_2, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J_3 is diagonal and the operator J_2 acts in a tridiagonal fashion. In the circular basis, the operator J_2 is block upper-triangular with all blocks 2x2 and the operator J_3 acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J_2 in the circular basis are generated by the Heun polynomials and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J_2 are generated by little -1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J_2 is diagonal is then considered. In this basis, the defining relations of the Schwinger-Dunkl algebra imply that J_3 acts in a block tridiagonal fashion with all blocks 2x2. The matrix elements of J_3 in this basis are given explicitly.

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