#### Title

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

#### 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

#### ISSN

0010-3616

#### Recommended Citation

"The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra" (2014). *Faculty Bibliography 2010s*. 5356.

https://stars.library.ucf.edu/facultybib2010/5356

## Comments

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