The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra
Abbreviated Journal Title
Commun. Math. Phys.
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.
Communications in Mathematical Physics
"The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra" (2014). Faculty Bibliography 2010s. 5356.