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Spherical Calogero model with oscillator/Coulomb potential: Quantum case

Francisco Correa1,2,*, Tigran Hakobyan3,4,†, Olaf Lechtenfeld1,‡, and Armen Nersessian3,4,§

  • 1Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany
  • 2Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile
  • 3Yerevan State University, 1 Alex Manoogian Street, Yerevan 0025, Armenia
  • 4Tomsk Polytechnic University, Lenin Avenue 30, 634050 Tomsk, Russia

  • *francisco.correa@itp.uni-hannover.de
  • †tigran.hakobyan@ysu.am
  • ‡lechtenf@itp.uni-hannover.de
  • §arnerses@ysu.am
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Phys. Rev. D 93, 125009 – Published 7 June, 2016

DOI: https://doi.org/10.1103/PhysRevD.93.125009

Abstract

We consider the quantum mechanics of Calogero models in an oscillator or Coulomb potential on the N-dimensional sphere. Their Hamiltonians are obtained by an appropriate Dunkl deformation of the oscillator/Coulomb system on the sphere and its restriction to (Coxeter reflection) symmetric wave functions. By the same method we also find the symmetry generators and compute their algebras.

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Spherical Calogero model with oscillator/Coulomb potential: Classical case

Francisco Correa, Tigran Hakobyan, Olaf Lechtenfeld, and Armen Nersessian
Phys. Rev. D 93, 125008 (2016)

Article Text

References (26)

  1. F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. (N.Y.) 10, 2191 (1969); Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, 12, 419 (1971).
  2. M. A. Olshanetsky and A. M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rep. 71, 313 (1981); Quantum integrable systems related to Lie algebras, 94, 313 (1983).
  3. J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16, 197 (1975).
  4. S. Wojciechowski, Superintegrability of the Calogero-Moser system, Phys. Lett. 95A, 279 (1983).
  5. D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appl. Math. 31, 481 (1978).
  6. A. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69, 703 (1992).
  7. L. Brink, T. Hansson, and M. Vasiliev, Explicit solution to the N-body Calogero problem, Phys. Lett. B 286, 109 (1992).
  8. A. P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39, 12793 (2006).
  9. C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc. 311, 167 (1989).
  10. N. Gurappa and P. K. Panigrahi, Equivalence of the Calogero-Sutherland model to free harmonic oscillators, Phys. Rev. B 59, R2490 (1999); A. Galajinsky, O. Lechtenfeld, and K. Polovnikov, Calogero models and nonlocal conformal transformations, Phys. Lett. B 643, 221 (2006).
  11. T. Hakobyan, O. Lechtenfeld, and A. Nersessian, Superintegrability of generalized Calogero models with oscillator or Coulomb potential, Phys. Rev. D 90, 101701(R) (2014).
  12. M. Adler, Some finite dimensional integrable systems and their scattering behavior, Commun. Math. Phys. 55, 195 (1977).
  13. A. Khare, Exact solution of an N-body problem in one dimension, J. Phys. A 29, L45 (1996); Reply to comment Exact solution of an N-body problem in one dimension, 29, 6459 (1996); P. K. Ghosh and A. Khare, Relationship between the energy eigenstates of Calogero-Sutherland models with oscillator and Coulomb-like potentials, 32, 2129 (1999).
  14. V. X. Genest, A. Lapointe, and L. Vinet, The Dunkl-Coulomb problem in the plane, Phys. Lett. A 379, 923 (2015).
  15. T. Hakobyan and A. Nersessian, Runge-Lenz vector in Calogero-Coulomb problem, Phys. Rev. A 92, 022111 (2015).
  16. T. Hakobyan and A. Nersessian, Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems, Phys. Rev. D 93, 045025 (2016).
  17. E. Schrödinger, A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. R. Irish Acad., Sect. A 46, 9 (1940).
  18. P. W. Higgs, Dynamical symmetries in a spherical geometry: I, J. Phys. A 12, 309 (1979); H. I. Leemon, Dynamical symmetries in a spherical geometry: II, 12, 489 (1979).
  19. F. Correa, T. Hakobyan, O. Lechtenfeld, and A. Nersessian, Spherical Calogero model with oscillator/Coulomb potential: Classical case, Phys. Rev. D 93, 125008 (2016).
  20. V. B. Kuznetsov, Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218, 212 (1996).
  21. M. Feigin, Intertwining relations for the spherical parts of generalized Calogero operators, Theor. Math. Phys. 135, 497 (2003).
  22. M. Feigin and T. Hakobyan, On Dunkl angular momenta algebra, J. High Energy Phys. 11 (2015) 107.
  23. T. Hakobyan, A. Nersessian, and V. Yeghikyan, The cuboctahedric Higgs oscillator from the rational Calogero model, J. Phys. A 42, 205206 (2009); T. Hakobyan, O. Lechtenfeld, A. Nersessian, and A. Saghatelian, Invariants of the spherical sector in conformal mechanics, 44, 055205 (2011); T. Hakobyan, S. Krivonos, O. Lechtenfeld, and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374, 801 (2010); T. Hakobyan, D. Karakhanyan, and O. Lechtenfeld, The structure of invariants in conformal mechanics, Nucl. Phys. B886, 399 (2014).
  24. M. Feigin, O. Lechtenfeld, and A. Polychronakos, The quantum angular Calogero-Moser model, J. High Energy Phys. 07 (2013) 162; F. Correa and O. Lechtenfeld, The tetrahexahedric angular Calogero model, 10 (2015) 191.
  25. T. Hakobyan, O. Lechtenfeld, and A. Nersessian, The spherical sector of the Calogero model as a reduced matrix model, Nucl. Phys. B858, 250 (2012).
  26. D. M. Fradkin, Existence of the dynamic symmetries O4 and SU3 for all classical central potential problems, Prog. Theor. Phys. 37, 798 (1967); Three-dimensional isotropic harmonic oscillator and SU3, Am. J. Phys. 33, 207 (1965).

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Phys. Rev. D 93, 125009– Published 7 June, 2016
  • Received 9 April 2016

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