Title

The Dunkl Oscillator In The Plane Ii: Representations Of The Symmetry Algebra

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 u(2). Two of the symmetry generators, J3 and J2, 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 J3 is diagonal and the operator J2 acts in a tridiagonal fashion. In the circular basis, the operator J2 is block upper-triangular with all blocks 2 × 2 and the operator J3 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 J2 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 J2 are generated by little -1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J2 is diagonal is considered. In this basis, the defining relations of the Schwinger-Dunkl algebra imply that J3 acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J3 in this basis are given explicitly. © 2014 Springer-Verlag Berlin Heidelberg.

Publication Date

1-1-2014

Publication Title

Communications in Mathematical Physics

Volume

329

Issue

3

Number of Pages

999-1029

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00220-014-1915-2

Socpus ID

84901851629 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84901851629

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