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

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.

25 pages; Added references; Added appendix on anti-Hermicity of the Dunkl derivative

References in corpus (3)

• Families of superintegrable Hamiltonians constructed from exceptional polynomials
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• Orthogonality of Hermite polynomials in superspace and Mehler type formulae

Cited by in corpus (43)

• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• 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
• Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential
• Thermal Properties of Relativistic Dunkl Oscillators
• 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
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Exact solution of two dimensional Dunkl harmonic oscillator in Non-Commutative phase-space
• Relativistic Solutions of Generalized-Dunkl Harmonic and Anharmonic Oscillators
• On the Path Integral Formulation of Wigner-Dunkl Quantum Mechanics
• Landau levels for the $(2+1)$ Dunkl-Klein-Gordon oscillator
• Ideal Bose Gas and Blackbody Radiation in the Dunkl Formalism
• A Laplace-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Rationally-extended Dunkl oscillator on the line
• 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 algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions
• Dunkl-Schrodinger Equation in Higher Dimension
• Time-Dependent Dunkl-Schrödinger Equation with an Angular-Dependent Potential
• One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
• A Path Integral Treatment of Time-dependent Dunkl Quantum Mechanics
• The total angular momentum algebra related to the ${\mathrm{S}}_3$ Dunkl Dirac equation
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• 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
• Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Free boson realization of the Dunkl intertwining operator in one dimension
• 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
• Impact of the non-canonical approach to the exact solution of the ideal one-dimensional electron gas confined with an anisotropic quantum wire of oscillator-shaped profile
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties
• 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