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, 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 2x2 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 then 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 2x2. The matrix elements of J_3 in this basis are given explicitly.

33 pages

References in corpus (4)

• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• Bispectrality of the Complementary Bannai-Ito Polynomials
• Imaginary Powers of the Dunkl Harmonic Oscillator
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)

Cited by in corpus (45)

• The Dunkl oscillator in three dimensions
• Exact Solutions of the 2D Dunkl--Klein--Gordon Equation: The Coulomb Potential and the Klein--Gordon Oscillator
• $SU(1,1)$ solution for the Dunkl oscillator in two dimensions and its coherent states
• Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions
• Algebraic approach for the one-dimensional Dirac-Dunkl oscillator
• $SU(1,1)$ solution for the Dunkl-Coulomb problem in two dimensions and its coherent states
• Dunkl-Graphene in constant magnetic field
• Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential
• The Bannai-Ito algebra and some applications
• Exact solution of two dimensional Dunkl harmonic oscillator in Non-Commutative phase-space
• On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
• Relativistic Solutions of Generalized-Dunkl Harmonic and Anharmonic Oscillators
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Landau levels for the $(2+1)$ Dunkl-Klein-Gordon oscillator
• Ideal Bose Gas and Blackbody Radiation in the Dunkl Formalism
• Rationally-extended Dunkl oscillator on the line
• Dunkl-Pauli Equation in the Presence of a Magnetic Field
• The condensation of ideal Bose gas in a gravitational field in the framework of Dunkl-statistic
• Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schrödinger equation
• Thermodynamics of bosons systems related to Dunkl differential-difference operators
• The Generalized Fokker-Planck Equation in terms of Dunkl-type Derivatives
• Dunkl-Schrödinger equation with time-dependent harmonic oscillator potential
• A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials
• The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions
• Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature
• Dunkl-Schrodinger Equation in Higher Dimension
• Time-Dependent Dunkl-Schrödinger Equation with an Angular-Dependent Potential
• The Condensation of Ideal Dunkl-Bose Gas in Power-Law Traps
• Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum
• One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
• On Dunkl-Bose-Einstein Condensation in Harmonic Traps
• The total angular momentum algebra related to the ${\mathrm{S}}_3$ Dunkl Dirac equation
• Time-dependent Dunkl-Pauli Oscillator
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Liénard Type Nonlinear Oscillators and Quantum Solvability
• Bounding the Wigner Deformation Parameter in Harmonically Trapped Bose Gases
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Spectral and Thermal Analysis of the Morse Potential within the Dunkl Formalism: Analytical Approximations and Applications
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties
• A superintegrable model with reflections on $S^3$ and the rank two Bannai-Ito algebra
• Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Orthogonal polynomials and the deformed Jordan plane