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

    Authors

    V. X. Genest; M. E. H. Ismail; L. Vinet;A. Zhedanov

    Comments

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    Abbreviated Journal Title

    Commun. Math. Phys.

    Keywords

    Physics, Mathematical

    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 . 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.

    Journal Title

    Communications in Mathematical Physics

    Volume

    329

    Issue/Number

    3

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    999

    Last Page

    1029

    WOS Identifier

    WOS:000336973900009

    ISSN

    0010-3616

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