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SIGMA, 2011, том 7, 051, 26 стр. (Mi sigma609)  

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   HTML файл

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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\by Ernie G.~Kalnins, Willard Miller Jr., Sarah Post
\paper Two-Variable Wilson Polynomials and the Generic Superintegrable System on the $3$-Sphere
\jour SIGMA
\yr 2011
\vol 7
\papernumber 051
\totalpages 26
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\crossref{https://doi.org/10.3842/SIGMA.2011.051}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855243905}


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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:
    1. Ernie G. Kalnins, Willard Miller Jr., “Structure theory for extended Kepler–Coulomb 3D classical superintegrable systems”, SIGMA, 8 (2012), 034, 25 pp.  mathnet  crossref  mathscinet
    2. Levesque D., Post S., Winternitz P., “Infinite Families of Superintegrable Systems Separable in Subgroup Coordinates”, J. Phys. A-Math. Theor., 45:46 (2012), 465204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. 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  crossref  isi  scopus
    4. 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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. 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.  mathnet  crossref  mathscinet
    6. 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  crossref  mathscinet  zmath  isi  elib  scopus
    7. 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  crossref  mathscinet  zmath  isi  elib  scopus
    8. 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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. Genest V.X., Vinet L., Zhedanov A., “Superintegrability in Two Dimensions and the Racah-Wilson Algebra”, Lett. Math. Phys., 104:8 (2014), 931–952  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. 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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. 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.  mathnet  crossref
    12. Genest V.X., Vinet L., “the Multivariate Hahn Polynomials and the Singular Oscillatorle”, J. Phys. A-Math. Theor., 47:45 (2014), 455201  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Kalnins E.G. Miller Jr. W., “Quadratic Algebra Contractions and Second-Order Superintegrable Systems”, Anal. Appl., 12:5, SI (2014), 583–612  crossref  mathscinet  zmath  isi  scopus
    14. 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  crossref  isi  scopus
    15. 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  crossref  isi  scopus
    16. Miller Jr. Willard, Turbiner A.V., “(Quasi)-Exact-Solvability on the Sphere S-N”, J. Math. Phys., 56:2 (2015), 023501  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Sarah Post, “Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$”, SIGMA, 11 (2015), 057, 17 pp.  mathnet  crossref  mathscinet
    18. 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  crossref  isi  scopus
    19. 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  crossref  scopus
    20. Kalnins, E.; Miller, W., Jr.; Subag, E., “Laplace equations, conformal superintegrability and bocher contractions”, Acta Polytechnica, 56:3 (2016), 214-223.  crossref  scopus
    21. 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  crossref  mathscinet  zmath  isi  scopus
    22. Iliev P., “The Generic Quantum Superintegrable System on the Sphere and Racah Operators”, Lett. Math. Phys., 107:11 (2017), 2029–2045  crossref  mathscinet  zmath  isi  scopus
    23. Iliev P., “Symmetry Algebra For the Generic Superintegrable System on the Sphere”, J. High Energy Phys., 2018, no. 2, 044  crossref  mathscinet  isi  scopus
    24. 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  crossref  mathscinet  zmath  isi  scopus
    25. De Bie H., Iliev P., Vinet L., “Bargmann and Barut-Girardello Models For the Racah Algebra”, J. Math. Phys., 60:1 (2019), 011701  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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