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Quantum, classical symmetries, and action-angle variables by factorization of superintegrable systems

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Abstract

The purpose of this work is to present a method based on the factorizations used in one-dimensional quantum mechanics in order to find the symmetries of quantum and classical superintegrable systems in higher dimensions. We apply this procedure to the harmonic oscillator and Kepler–Coulomb systems to show the differences with other more standard approaches. We describe in detail the basic ingredients to make explicit the parallelism of classical and quantum treatments. One of the most interesting results is the finding of action-angle variables as a natural component of the classical symmetries within this formalism.

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References

  1. J. Bertrand, Théorème relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. 77, 849–853 (1873)

    MATH  Google Scholar 

  2. L. Infeld, T. Hull, The factorization method. Rev. Mod. Phys. 23, 21–68 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. E. Schrödinger, A method of determining quantum mechanical eigenvalues and eigenfunctions; further studies on solving eigenvalue problems by factorization. Proc. Roy. Irish Acad. 46, 183–206 (1941)

    MATH  Google Scholar 

  4. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer- Verlag, Berlin, 1991)

    Book  MATH  Google Scholar 

  5. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001)

    Book  MATH  Google Scholar 

  6. D.J. Fernández, Supersymmetric quantum mechanics, Advanced summer school in physics 2009: Front. Contemp. Phys., 1287 (2010) 3-36, AIP Publishing

  7. B. Mielnik, O. Rosas-Ortiz, Factorization: little or great algorithm? J. Phys. A: Math. Gen. 37, 10007 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. S. Post, P. Winternitz, An infinite family of superintegrable deformations of the Coulomb potential. J. Phys. A: Math. Theor. 43, 222001 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. E.G. Kalnins, J.M. Kress, W. Miller Jr., Recurrence relation approach to higher order quantum superintegrability. SIGMA 7, 031 (2011)

    MathSciNet  MATH  Google Scholar 

  10. E.G. Kalnins, W. Miller Jr., Structure results for higher order symmetry algebras of 2D classical superintegrable systems. J. Nonl. Sys. App. 29–40 (2012)

  11. E. Celeghini, Ş. Kuru, J. Negro, M.A. del Olmo, A unified approach to quantum and classical TTW systems based on factorizations. Ann. Phys. 332, 27–37 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. J.A. Calzada, Ş. Kuru, J. Negro, Superintegrable Lissajous systems on the sphere. Eur. Phys. J. Plus 129, 164 (2014)

    Article  Google Scholar 

  13. A. Ballesteros, F.J. Herranz, Ş. Kuru, J. Negro, The anisotropic oscillator on curved spaces: a new exactly solvable model. Ann. Phys. 373, 399–423 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. A. Ballesteros, F.J. Herranz, Ş. Kuru, J. Negro, Factorization approach to superintegrable systems: formalism and applications. Phys. At. Nucl. 80, 389–396 (2017)

    Article  Google Scholar 

  15. Yu.N. Demkov, Symmetry group of isotropic oscillator. Soviet. Phys. JETP 36, 63–66 (1959)

    MathSciNet  MATH  Google Scholar 

  16. D.M. Fradkin, Three dimensional isotropic harmonic oscillator and SU(3). Am. J. Phys. 33, 207–211 (1965)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. D.M. Fradkin, Existence of the dynamic symmetries o(4) and su(3) for all classical central potential problems. Prog. Theor. Phys. 37, 798–812 (1967)

    Article  ADS  MATH  Google Scholar 

  18. N.W. Evans, Superintegrability in classical mechanics. Phys. Rev. A 41, 5666–5676 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  19. W. Miller Jr., S. Post, P. Winternitz, Classical and quantum superintegrability with applications. J. Phys. A: Math. Theor. 46, 423001 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. S. Rauch-Wojciechovski, Superintegrability of the Calogero–Moser system. Phys. Lett. A 95, 279–281 (1983)

    Article  MathSciNet  Google Scholar 

  21. E. Drigho-Filho, Ş. Kuru, J. Negro, L.M. Nieto, Superintegrability of the Fock–Darwin system. Ann. Phys. 383, 101–119 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. J.M. Jauch, E.L. Hill, On the problem of degeneracy in quantum mechanics. Phys. Rev. 57, 641–645 (1940)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. F. Tremblay, A.V. Turbiner, P. Winternitz, An infinite family of solvable and integrable quantum systems on a plane. J. Phys. A: Math. Theor. 42, 242001 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. F. Tremblay, A.V. Turbiner, P. Winternitz, Periodic orbits for an infinite family of classical superintegrable systems. J. Phys. A: Math. Theor. 43, 015202 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Ş. Kuru, J. Negro, O. Ragnisco, The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories. Phys. Lett. A 381, 3355–3363 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Ş. Kuru, J. Negro, Factorizations of one-dimensional classical systems. Ann. Phys. 323, 413–431 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. L. Delisle-Doray, V. Hussin, Ş. Kuru, J. Negro, Classical ladder functions for Rosen–Morse and curved Kepler–Coulomb systems. Ann. Phys. 405, 69–82 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. M.A. Rodriguez, P. Winternitz, Quantum superintegrability and exact Solvability in N dimensions. J. Math. Phys. 43, 1309–1322 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. P.E. Verrier, N.W. Evans, A new superintegrable Hamiltonian. J. Math. Phys. 49, 022902 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. A. Ballesteros, F.J. Herranz, F. Musso, The anisotropic oscillator on the 2D sphere and the hyperbolic plane. Nonlinearity 26, 971–990 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Ş. Kuru, J. Negro, M.A. del Olmo, Dynamical algebras for Pöschl-Teller Hamiltonian hierarchies. Ann. Phys. 324, 2548–2560 (2012)

    Article  ADS  MATH  Google Scholar 

  32. S. Garneau-Desroches, V. Hussin, Ladder operators and coherent states for the Rosen–Morse system and its rational extensions. J. Phys. A: Math. Theor. 54, 475201 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Ş. Kuru, J. Negro, Classical spectrum generating algebra of the Kepler–Coulomb system and action-angle variables. Phys. Lett. A 376, 260–264 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Ş. Kuru, J. Negro, Spectrum generating algebras’ of classical systems: the Kepler–Coulomb potential. J. Phys: Conf. Ser. 343, 012063 (2012)

    MATH  Google Scholar 

  35. D.C. Fernández, J. Negro, M.A. del Olmo, Group approach to the factorization of the radial oscillator equation. Ann. Phys. 252, 386–412 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. M. Gadella, J. Negro, L.M. Nieto, G.P. Pronko, M. Santander, Spectrum generating algebras for the free motion in S3. J. Math. Phys. 52, 063509 (2016)

    Article  ADS  MATH  Google Scholar 

  37. J.-P. Amiet, S. Weigert, Commensurate harmonic oscillators: classical symmetries. J. Math. Phys. 43, 4110–4126 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. F. Calogero, Isochronous Systems, 2008, Oxford: OUP, (Oxford)

  39. T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian, V. Yeghikyana, Integrable generalizations of oscillator and Coulomb systems via action-angle variables. Phys. Lett. A 376, 679–686 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics, 3rd edn. (Addison-Wesley, New York, 2001)

    MATH  Google Scholar 

  41. R. Campoamor-Stursberg, M. Gadella, Ş. Kuru, J. Negro, Action-angle variables, ladder operators and coherent states. Phys. Lett. A 376, 2515–2521 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. J.A. Calzada, Ş. Kuru, J. Negro, Superintegrable Lissajous systems on the sphere. Eur. Phys. J. Plus 129, 1–15 (2014)

    Article  Google Scholar 

  43. S. Cruz y Cruz, Ş. Kuru, J. Negro, Classical motion and coherent states for Pöschl-Teller potentials. Phys. Lett. A 372, 1391–1405 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. S. Post, P. Winternitz, A nonseparable quantum superintegrable system in 2D real Euclidean space. J. Phys. A: Math. Theor. 44, 162001 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. I. Marquette, J. Zhang, Y.-Z. Zhang, Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces. J. Phys. A: Math. Theor. 56, 355201 (2023)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. E.G. Kalnins, W. Miller Jr., Structure results for higher order symmetry algebras of 2D classical superintegrable systems. J. Nonl. Sys. App. 29–40 (2012)

  47. C. Gonera, On the superintegrability of TTW model. Phys. Lett. A 376, 2341–2343 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. D. Lévesque, S. Post, P. Winternitz, Infinite families of superintegrable systems separable in subgroup coordinates. J. Phys. A: Math. Theor. 45, 465204 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. E.G. Kalnins, J.M. Kress, W. Miller Jr., Superintegrability and higher order integrals for quantum systems. J. Phys. A: Math. Theor. 43, 265205 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. I. Yurdusen, O.O. Tuncer, P. Winternitz, Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion. J. Phys. A: Math. Theor. 54, 305201 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  51. F.J. Herranz, A. Ballesteros, Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature. SIGMA 2, 010 (2006)

    MathSciNet  MATH  Google Scholar 

  52. I. Marquette, C. Quesne, New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. J. Math. Phys. 54, 042102 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. I. Marquette, C. Quesne, Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions. J. Math. Phys. 56, 062102 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was supported by Spanish MCIN with funding from European Union NextGenerationEU (PRTRC17.I1) and Consejeria de Educacion from JCyL through QCAYLE project, as well as MCIN project PID2020-113406GB-I00. Ş. K. thanks Ankara University and the warm hospitality of the Department of Theoretical Physics of the University of Valladolid, where most part of this work has been carried out.

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Authors and Affiliations

  1. Department of Physics, Faculty of Science, Ankara University, 06100, Ankara, Turkey

    Şengül Kuru

  2. Departamento de Física Teórica, Atómica y Óptica, and IMUVA, Universidad de Valladolid, 47011, Valladolid, Spain

    Javier Negro & Sergio Salamanca

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  1. Şengül Kuru
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  2. Javier Negro
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Correspondence to Javier Negro.

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Kuru, Ş., Negro, J. & Salamanca, S. Quantum, classical symmetries, and action-angle variables by factorization of superintegrable systems. Eur. Phys. J. Plus 138, 931 (2023). https://doi.org/10.1140/epjp/s13360-023-04524-x

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