#### Title

The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients

#### Abbreviated Journal Title

J. Phys. A-Math. Theor.

#### Keywords

CONFORMALLY FLAT SPACES; DEFORMED HEISENBERG ALGEBRA; CLASSICAL; STRUCTURE-THEORY; QUADRATIC ALGEBRAS; HERMITE-POLYNOMIALS; STACKEL; TRANSFORM; CURVED SPACES; SYSTEMS; OPERATORS; DYNAMICS; Physics, Multidisciplinary; Physics, Mathematical

#### Abstract

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u( 2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual - 1 Hahn polynomials. The connection with the Clebsch-Gordan problem of the sl(-1) (2) algebra is explained.

#### Journal Title

Journal of Physics a-Mathematical and Theoretical

#### Volume

46

#### Issue/Number

14

#### Publication Date

1-1-2013

#### Document Type

Article

#### Language

English

#### First Page

21

#### WOS Identifier

#### ISSN

1751-8113

#### Recommended Citation

"The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients" (2013). *Faculty Bibliography 2010s*. 4009.

http://stars.library.ucf.edu/facultybib2010/4009

## Comments

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