Abstract
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 \({\mathfrak{u}(2)}\). 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 2 × 2 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 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 2 × 2. The matrix elements of J 3 in this basis are given explicitly.
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Genest, V.X., Ismail, M.E.H., Vinet, L. et al. The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra. Commun. Math. Phys. 329, 999–1029 (2014). https://doi.org/10.1007/s00220-014-1915-2
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DOI: https://doi.org/10.1007/s00220-014-1915-2