Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 057, 28 pages      arXiv:1212.4766      http://dx.doi.org/10.3842/SIGMA.2013.057

Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

Ernest G. Kalnins a, Willard Miller Jr. b and Sarah Post c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA
c) Department of Mathematics, U. Hawai'i at Manoa, Honolulu, HI, 96822, USA

Received May 29, 2013, in final form September 26, 2013; Published online October 02, 2013

Abstract
We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-Inönü method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2) and so(3). By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems, and using Wigner's idea of ''saving'' a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

Key words: Askey scheme; hypergeometric orthogonal polynomials; quadratic algebras.

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