{ "id": "1302.6142", "version": "v1", "published": "2013-02-25T16:39:37.000Z", "updated": "2013-02-25T16:39:37.000Z", "title": "The Dunkl oscillator in the plane II : representations of the symmetry algebra", "authors": [ "Vincent X. Genest", "Mourad E. H. Ismail", "Luc Vinet", "Alexei Zhedanov" ], "comment": "33 pages", "journal": "Commun. Math. Phys. 329, 999-1029 (2014)", "categories": [ "math-ph", "math.MP" ], "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.", "revisions": [ { "version": "v1", "updated": "2013-02-25T16:39:37.000Z" } ], "analyses": { "keywords": [ "symmetry algebra", "representations", "tridiagonal fashion", "circular basis", "blocks 2x2" ], "tags": [ "journal article" ], "publication": { "publisher": "Springer", "journal": "Communications in Mathematical Physics", "doi": "10.1007/s00220-014-1915-2", "year": 2014, "month": "Aug", "volume": 329, "number": 3, "pages": 999 }, "note": { "typesetting": "TeX", "pages": 33, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2014CMaPh.329..999G" } } }