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SIGMA, 2010, том 6, 066, 23 страниц (Mi sigma523)  

Эта публикация цитируется в 25 научных статьях (всего в 25 статьях)

Tools for Verifying Classical and Quantum Superintegrability

Ernest G. Kalninsa, Jonathan M. Kressb, Willard Miller Jr.c

a Department of Mathematics, University of Waikato, Hamilton, New Zealand
b School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
c School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Аннотация: Recently many new classes of integrable systems in $n$ dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of $2n-1$ symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for $n=2$ but cases where $n>2$ are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stäckel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler–Coulomb system is quantum superintegrable for all rational values of a parameter $k$ in the potential.

Ключевые слова: superintegrability; hidden algebras; quadratic algebras

DOI: https://doi.org/10.3842/SIGMA.2010.066

Полный текст: PDF файл (300 kB)
Полный текст: http://emis.mi.ras.ru/journals/SIGMA/2010/066/
Список литературы: PDF файл   HTML файл

Реферативные базы данных:

ArXiv: 1006.0864
Тип публикации: Статья
MSC: 20C99; 20C35; 22E70
Поступила: 4 июня 2010 г.; в окончательном варианте 6 августа 2010 г.; опубликована 18 августа 2010 г.
Язык публикации: английский

Образец цитирования: Ernest G. Kalnins, Jonathan M. Kress, Willard Miller Jr., “Tools for Verifying Classical and Quantum Superintegrability”, SIGMA, 6 (2010), 066, 23 pp.

Цитирование в формате AMSBIB
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\paper Tools for Verifying Classical and Quantum Superintegrability
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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, Jonathan M. Kress, Willard Miller Jr., “A Recurrence Relation Approach to Higher Order Quantum Superintegrability”, SIGMA, 7 (2011), 031, 24 pp.  mathnet  crossref  mathscinet
    2. Christiane Quesne, “Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System in $D$ Dimensions”, SIGMA, 7 (2011), 035, 21 pp.  mathnet  crossref  mathscinet
    3. Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “First Integrals of Extended Hamiltonians in $n+1$ Dimensions Generated by Powers of an Operator”, SIGMA, 7 (2011), 038, 12 pp.  mathnet  crossref  mathscinet
    4. Chanu C., Degiovanni L., Rastelli G., “Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries”, Regular & Chaotic Dynamics, 16:5 (2011), 496–503  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. 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
    6. Claudia M. Chanu, Luca Degiovanni, Giovanni Rastelli, “Superintegrable extensions of superintegrable systems”, SIGMA, 8 (2012), 070, 12 pp.  mathnet  crossref  mathscinet
    7. 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
    8. Chanu C. Degiovanni L. Rastelli G., “Generalizations of a Method for Constructing First Integrals of a Class of Natural Hamiltonians and Some Remarks About Quantization”, 7th International Conference on Quantum Theory and Symmetries (Qts7), Journal of Physics Conference Series, 343, IOP Publishing Ltd, 2012, 012101  crossref  isi  scopus
    9. 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
    10. Ayadi, V., Feher, L., Gorbe, T.F., “Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals”, Journal of Geometry and Symmetry in Physics, 27 (2012), 27–44  crossref  mathscinet  zmath
    11. 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
    12. Blaszak M., Domanski Z., Sergyeyev A., Szablikowski B.M., “Integrable Quantum Stackel Systems”, Phys. Lett. A, 377:38 (2013), 2564–2572  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Miller Jr. Willard Post S. Winternitz P., “Classical and Quantum Superintegrability with Applications”, J. Phys. A-Math. Theor., 46:42 (2013), 423001  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Riglioni D., “Classical and Quantum Higher Order Superintegrable Systems From Coalgebra Symmetry”, J. Phys. A-Math. Theor., 46:26 (2013), 265207  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. 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
    16. Rastelli G., “Extensions of Natural Hamiltonians”, 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 (Ic-Msquare 2013), Journal of Physics Conference Series, 490, ed. Vagenas E. Vlachos D., IOP Publishing Ltd, 2014, 012088  crossref  isi  scopus
    17. Chanu C.M. Degiovanni L. Rastelli G., “The Tremblay–Turbiner–Winternitz System as Extended Hamiltonian”, J. Math. Phys., 55:12 (2014), 122701  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    18. Chanu C.M. Degiovanni L. Rastelli G., “Extensions of Hamiltonian Systems Dependent on a Rational Parameter”, J. Math. Phys., 55:12 (2014), 122703  crossref  mathscinet  zmath  adsnasa  isi  scopus
    19. Calzada J.A. Kuru S. Negro J., “Superintegrable Lissajous Systems on the Sphere”, Eur. Phys. J. Plus, 129:8 (2014), 164  crossref  adsnasa  isi  elib  scopus
    20. Gonera C., Kaszubska M., “Superintegrable Systems on Spaces of Constant Curvature”, Ann. Phys., 346 (2014), 91–102  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    21. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Superintegrable Generalizations of the Kepler and Hook Problems”, Regul. Chaotic Dyn., 19:3 (2014), 415–434  mathnet  crossref  mathscinet  zmath
    22. Yuxuan Chen, Ernie G. Kalnins, Qiushi Li, Willard Miller Jr., “Examples of Complete Solvability of 2D Classical Superintegrable Systems”, SIGMA, 11 (2015), 088, 51 pp.  mathnet  crossref
    23. Claudia Maria Chanu, Luca Degiovanni, Giovanni Rastelli, “Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post–Winternitz System”, SIGMA, 11 (2015), 094, 9 pp.  mathnet  crossref
    24. Chanu C.M. Degiovanni L. Rastelli G., “Warped Product of Hamiltonians and Extensions of Hamiltonian Systems”, Xxxth International Colloquium on Group Theoretical Methods in Physics (Icgtmp) (Group30), Journal of Physics Conference Series, 597, IOP Publishing Ltd, 2015, 012024  crossref  isi  scopus
    25. Chanu C.M. Rastelli G., “Extended Hamiltonians and Shift, Ladder Functions and Operators”, Ann. Phys., 386 (2017), 254–274  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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