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SIGMA, 2011, том 7, 051, 26 стр.
(Mi sigma609)
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Эта публикация цитируется в 25 научных статьях (всего в 25 статьях)
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the $3$-Sphere
Ernie G. Kalninsa, Willard Miller Jr.b, Sarah Postc
a Department of Mathematics, University of Waikato, Hamilton, New Zealand
b School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
c Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128 succ. Centre-Ville, Montréal (QC) H3C 3J7, Canada
Аннотация:
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.
Ключевые слова:
superintegrability; quadratic algebras; multivariable Wilson polynomials; multivariable Racah polynomials
DOI:
https://doi.org/10.3842/SIGMA.2011.051
 Полный текст:
PDF файл (467 kB)
Полный текст:
http://emis.mi.ras.ru/journals/SIGMA/2011/051/
Список литературы:
PDF файл
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Реферативные базы данных:
  
ArXiv:
1010.3032
Тип публикации:
Статья
MSC: 81R12; 33C45
Поступила: 31 января 2011 г.; в окончательном варианте 23 мая 2011 г.; опубликована 30 мая 2011 г.
Язык публикации: английский
Образец цитирования:
Ernie G. Kalnins, Willard Miller Jr., Sarah Post, “Two-Variable Wilson Polynomials and the Generic Superintegrable System on the $3$-Sphere”, SIGMA, 7 (2011), 051, 26 pp.
Цитирование в формате AMSBIB
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\paper Two-Variable Wilson Polynomials and the Generic Superintegrable System on the $3$-Sphere
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\papernumber 051
\totalpages 26
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Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Эта публикация цитируется в следующих статьяx:
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Ernie G. Kalnins, Willard Miller Jr., “Structure theory for extended Kepler–Coulomb 3D classical superintegrable systems”, SIGMA, 8 (2012), 034, 25 pp.
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Levesque D., Post S., Winternitz P., “Infinite Families of Superintegrable Systems Separable in Subgroup Coordinates”, J. Phys. A-Math. Theor., 45:46 (2012), 465204
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Kalnins E.G. Kress J.M. Miller Jr. W., “Structure Relations for the Symmetry Algebras of Quantum Superintegrable Systems”, 7th International Conference on Quantum Theory and Symmetries (Qts7), Journal of Physics Conference Series, 343, IOP Publishing Ltd, 2012, 012075
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Kalnins E.G., Kress J.M., Miller Jr. W., “Extended Kepler-Coulomb Quantum Superintegrable Systems in Three Dimensions”, J. Phys. A-Math. Theor., 46:8 (2013), 085206
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Ernest G. Kalnins, Willard Miller Jr., Sarah Post, “Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials”, SIGMA, 9 (2013), 057, 28 pp.
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Genest V.X. Vinet L. Zhedanov A., “The Multivariate Krawtchouk Polynomials as Matrix Elements of the Rotation Group Representations on Oscillator States”, J. Phys. A-Math. Theor., 46:50 (2013), 505203
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Genest V.X. Vinet L. Zhedanov A., “The Singular and the 2:1 Anisotropic Dunkl Oscillators in the Plane”, J. Phys. A-Math. Theor., 46:32 (2013), 325201
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Celeghini E. Kuru S. Negro J. del Olmo M.A., “A Unified Approach to Quantum and Classical TTW Systems Based on Factorizations”, Ann. Phys., 332 (2013), 27–37
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Genest V.X., Vinet L., Zhedanov A., “Superintegrability in Two Dimensions and the Racah-Wilson Algebra”, Lett. Math. Phys., 104:8 (2014), 931–952
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Genest V.X. Vinet L. Zhedanov A., “The Bannai–Ito Algebra and a Superintegrable System with Reflections on the Two-Sphere”, J. Phys. A-Math. Theor., 47:20 (2014), 205202
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Vincent X. Genest, Luc Vinet, “The Generic Superintegrable System on the $3$-Sphere and the ${9j}$ Symbols of ${\mathfrak{su}(1,1)}$”, SIGMA, 10 (2014), 108, 28 pp.
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Genest V.X., Vinet L., “the Multivariate Hahn Polynomials and the Singular Oscillatorle”, J. Phys. A-Math. Theor., 47:45 (2014), 455201
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Kalnins E.G. Miller Jr. W., “Quadratic Algebra Contractions and Second-Order Superintegrable Systems”, Anal. Appl., 12:5, SI (2014), 583–612
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Genest V.X., Vinet L., Zhedanov A., “the Racah Algebra and Superintegrable Models”, 8Th International Symposium on Quantum Theory and Symmetries (Qts8), Journal of Physics Conference Series, 512, IOP Publishing Ltd, 2014, 012011
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Miller Jr. Willard, “the Theory of Contractions of 2D 2Nd Order Quantum Superintegrable Systems and Its Relation To the Askey Scheme For Hypergeometric Orthogonal Polynomials”, 8Th International Symposium on Quantum Theory and Symmetries (Qts8), Journal of Physics Conference Series, 512, IOP Publishing Ltd, 2014, 012012
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Miller Jr. Willard, Turbiner A.V., “(Quasi)-Exact-Solvability on the Sphere S-N”, J. Math. Phys., 56:2 (2015), 023501
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Sarah Post, “Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$”, SIGMA, 11 (2015), 057, 17 pp.
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Miller Jr. W. Li Q., “Wilson Polynomials/Functions and Intertwining Operators For the Generic Quantum Superintegrable System on the 2-Sphere”, Xxxth International Colloquium on Group Theoretical Methods in Physics (Icgtmp) (Group30), Journal of Physics Conference Series, 597, IOP Publishing Ltd, 2015, 012059
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De Bie, H.; Genest, V. X.; Lemay, J.-M.; Vinet, L., “A superintegrable model with reflections on S3 and the rank two Bannai–Ito algebra”, Acta Polytechnica, 56:3 (2016), 166-172
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Kalnins, E.; Miller, W., Jr.; Subag, E., “Laplace equations, conformal superintegrability and bocher contractions”, Acta Polytechnica, 56:3 (2016), 214-223.
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De Bie H. Genest V.X. Lemay J.-M. Luc Vinet, “A superintegrable model with reflections on S ^{ n ?1 } and the higher rank Bannai–Ito algebra”, J. Phys. A-Math. Theor., 50:19 (2017), 195202
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Iliev P., “The Generic Quantum Superintegrable System on the Sphere and Racah Operators”, Lett. Math. Phys., 107:11 (2017), 2029–2045
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Iliev P., “Symmetry Algebra For the Generic Superintegrable System on the Sphere”, J. High Energy Phys., 2018, no. 2, 044
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De Bie H. Genest V.X. van de Vijver W. Vinet L., “A Higher Rank Racah Algebra and the Z(2)(N) Laplace–Dunkl Operator”, J. Phys. A-Math. Theor., 51:2 (2018), 025203
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De Bie H., Iliev P., Vinet L., “Bargmann and Barut-Girardello Models For the Racah Algebra”, J. Math. Phys., 60:1 (2019), 011701
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