Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere

Abstract

We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials.

References in corpus (18)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• An infinite family of solvable and integrable quantum systems on a plane
• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Bispectrality of multivariable Racah-Wilson polynomials
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
• Properties of some families of hypergeometric orthogonal polynomials in several variables
• Superintegrability and higher order constants for classical and quantum systems
• Third order superintegrable systems separating in polar coordinates
• Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$
• Superintegrability and higher order constants for quantum systems
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Deformation Quantization of Superintegrable Systems and Nambu Mechanics
• Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics
• Superintegrable 3-body systems on the line
• Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
• Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Branes, Quantum Nambu Brackets, and the Hydrogen Atom

Cited by in corpus (31)

• Classical and Quantum Superintegrability with Applications
• Superintegrability in two dimensions and the Racah-Wilson algebra
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
• Infinite families of superintegrable systems separable in subgroup coordinates
• Symmetry algebra for the generic superintegrable system on the sphere
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
• Hahn polynomials on polyhedra and quantum integrability
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Quadratic algebra contractions and 2nd order superintegrable systems
• The Generic Superintegrable System on the 3-Sphere and the $9j$ Symbols of $\mathfrak{su}(1,1)$
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere
• Hahn polynomials for hypergeometric distribution
• (Quasi)-exact-solvability on the sphere $S^n$
• Bargmann and Barut-Girardello models for the Racah algebra
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• Gaudin model for the multinomial distribution
• Abundant Superintegrable Systems and Hessian Structures
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• A superintegrable model with reflections on $S^3$ and the rank two Bannai-Ito algebra
• Darboux transformations from the Appell-Lauricella operator
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions