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

SIGMA 9 (2013), 057, 28 pages      arXiv:1212.4766      https://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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References

  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  2. Askey R., An integral of Ramanujan and orthogonal polynomials, J. Indian Math. Soc. (N.S.) 51 (1987), 27-36.
  3. Bonatsos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
  4. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  5. Daskaloyannis C., Tanoudis Y., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
  6. Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055.
  7. Gao S., Wang Y., Hou B., The classification of Leonard triples of Racah type, Linear Algebra Appl. 439 (2013), 1834-1861.
  8. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.
  9. Genest V.X., Vinet L., Zhedanov A., Superintegrability in two dimensions and the Racah-Wilson algebra, arXiv:1307.5539.
  10. Granovskii Y.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  11. Granovskii Y.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. I. An oscillator, Theoret. and Math. Phys. 91 (1992), 474-480.
  12. Granovskii Y.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
  13. Granovskii Y.I., Zhedanov A.S., Lutzenko I.M., Quadratic algebra as a "hidden" symmetry of the Hartmann potential, J. Phys. A: Math. Gen. 24 (1991), 3887-3894.
  14. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510-524.
  15. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables, J. Phys. A: Math. Gen. 29 (1996), 5949-5962.
  16. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. I. 2D classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  17. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  18. Kalnins E.G., Kress J.M., Miller Jr. W., Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399-3411, arXiv:0708.3044.
  19. Kalnins E.G., Kress J.M., Miller Jr. W., Structure relations for the symmetry algebras of quantum superintegrable systems, J. Phys. Conf. Ser. 343 (2012), 012075, 12 pages.
  20. Kalnins E.G., Kress J.M., Miller Jr. W., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials, SIGMA 5 (2009), 008, 24 pages, arXiv:0901.3081.
  21. Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, math-ph/0307039.
  22. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
  23. Kalnins E.G., Miller Jr. W., Pogosyan G.S., Contractions of Lie algebras: applications to special functions and separation of variables, J. Phys. A: Math. Gen. 32 (1999), 4709-4732.
  24. Kalnins E.G., Miller Jr. W., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A: Math. Theor. 40 (2007), 11525-11538.
  25. Kalnins E.G., Miller Jr. W., Post S., Models for the 3D singular isotropic oscillator quadratic algebra, Phys. Atomic Nuclei 73 (2010), 359-366.
  26. Kalnins E.G., Miller Jr. W., Post S., Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, SIGMA 7 (2011), 051, 26 pages, arXiv:1010.3032.
  27. Kalnins E.G., Miller Jr. W., Subag E., Heinenen R., Contractions of 2nd order superintegrable systems in 2D, in preparation.
  28. Kalnins E.G., Williams G.C., Miller Jr. W., Pogosyan G.S., On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space, J. Phys. A: Math. Gen. 35 (2002), 4755-4773.
  29. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  30. Koornwinder T.H., Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials, in Orthogonal Polynomials and their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 46-72.
  31. Krall H.L., Frink O., A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100-115.
  32. Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560-566.
  33. Létourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Physics 243 (1995), 144-168.
  34. Miller Jr. W., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1977.
  35. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor., to appear, arXiv:1309.2694.
  36. Mostafazadeh A., Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, math-ph/0107001.
  37. Talman J.D., Special functions: a group theoretic approach (based on lectures by Eugene P. Wigner), W.A. Benjamin, Inc., New York - Amsterdam, 1968.
  38. Terwilliger P., The universal Askey-Wilson algebra and the equitable presentation of Uq(sl2), SIGMA 7 (2011), 099, 26 pages, arXiv:1107.3544.
  39. Weimar-Woods E., The three-dimensional real Lie algebras and their contractions, J. Math. Phys. 32 (1991), 2028-2033.
  40. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.

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