The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

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 . 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 x 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 x 2. The matrix elements of J (3) in this basis are given explicitly.