An infinite family of superintegrable Hamiltonians with reflection in the plane

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly solvable. The angular part of the wave function is expressed in terms of little -1 Jacobi polynomials. The spectra exhibit "accidental" degeneracies. The superintegrability of the model is proved using the recurrence relation approach. The (higher-order) constants of motion are constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page

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

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.Comment: 25 pages; Added references; Added appendix on anti-Hermicity of the Dunkl derivativ

The Dunkl oscillator in the plane II : representations of the symmetry algebra

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.Comment: 33 page

Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah-Wilson algebra associated to the generic Racah/Wilson polynomials. A degenerate case leading to the Hahn algebra is also discussed.Comment: 14 pages; Section 3 revise

The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)

The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from a q-> -1 limit of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators including an involution, it is also a q-> -1 limit of the quantum algebra sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of the -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.Comment: 15 pages, Some minor changes from version #