Physics of Particles and Nuclei Letters

, Volume 14, Issue 2, pp 331–335 | Cite as

Integrability of Calogero–Coulomb problems

  • Tigran Hakobyan
  • Armen NersessianEmail author
Physics of Elementary Particles and Atomic Nuclei. Theory


In this short review we describe the integrability properties of the Calogero-type perturbations of one- and two-center Coulomb problems and of the Stark–Coulomb problem. We present the explicit expressions of their constants of motion and show that these systems admit partial separation of variables.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Komarov and S. Slavyanov, “The two Coulomb centres problem at large centre separation,” J. Phys. B 1, 1066 (1968)ADSCrossRefGoogle Scholar
  2. 1a.
    Yu. Demkov and I. Komarov, “Hypergeometric partial solutions in the problem of two Coulomb centers,” Theor. Math. Phys. 38, 174 (1979).CrossRefzbMATHGoogle Scholar
  3. 2.
    F. Calogero, “Solution of a three-body problem in one dimension,” J. Math. Phys. 10 (1969); “Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials,” J. Math. Phys. 12, 419 (1971).Google Scholar
  4. 3.
    S. Wojciechowski, “Superintegrability of the Calogero-Moser system,” Phys. Lett. A 95, 279 (1983).ADSMathSciNetCrossRefGoogle Scholar
  5. 4.
    J. Moser, “Three integrable hamiltonian systems connected with isospectral deformations,” Adv. Math. 16, 197 (1975).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 5.
    M. Olshanetsky and A. Perelomov, “Classical integrable finite dimensional systems related to lie algebras,” Phys. Rep. 71, 313 (1981).ADSMathSciNetCrossRefGoogle Scholar
  7. 6.
    A. P. Polychronakos, “Physics and mathematics of Calogero particles,” J. Phys. A 39, 12793 (2006).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 7.
    A. Khare, “Exact solution of an N-body problem in one dimension,” J. Phys. A 29, L45 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 8.
    T. Hakobyan, O. Lechtenfeld, and A. Nersessian, “Superintegrability of generalized Calogero models with oscillator or Coulomb potential,” Phys. Rev. D: Part. Fields 90, 101701(R) (2014).ADSCrossRefGoogle Scholar
  10. 9.
    T. Hakobyan and A. Nersessian, “Runge-Lenz vector in Calogero-Coulomb problem,” Phys. Rev. A 92, 022111 (2015).ADSCrossRefGoogle Scholar
  11. 10.
    T. Hakobyan and A. Nersessian, “Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems,” Phys. Rev. D: Part. Fields 93, 045025 (2016).ADSMathSciNetCrossRefGoogle Scholar
  12. 11.
    M. Feigin and T. Hakobyan, “On dunkl angular momenta algebra,” J. High Energy Phys. 11, 107 (2015).ADSMathSciNetCrossRefGoogle Scholar
  13. 12.
    M. Feigin, O. Lechtenfeld, and A. Polychronakos, “The quantum angular Calogero-Moser model,” J. High Energy Phys. 1307, 162 (2013).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 13.
    T. Hakobyan, A. Nersessian, and V. Yeghikyan, “Cuboctahedric higgs oscillator from the Calogero model,” J. Phys. A 42, 205206 (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 14.
    F. Correa and O. Lechtenfeld, “The tetrahexahedric angular Calogero model,” J. High Energy Phys. 10, 191 (2015).ADSMathSciNetCrossRefGoogle Scholar
  16. 15.
    T. Hakobyan, D. Karakhanyan, and O. Lechtenfeld, “The structure of invariants in conformal mechanics,” Nucl. Phys. B 886, 399 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 15a.
    T. Hakobyan, S. Krivonos, O. Lechtenfeld, and A. Nersessian, “Hidden symmetries of integrable conformal mechanical systems,” Phys. Lett. A 374, 801 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 15b.
    T. Hakobyan, O. Lechtenfeld, A. Nersessian, and A. Saghatelian, “Invariants of the spherical sector in conformal mechanics,” J. Phys. A 44, 055205 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 15c.
    O. Lechtenfeld, A. Nersessian, and V. Yeghikyan, “Action-angle variables for dihedral systems on the circle,” Phys. Lett. A 374, 4647 (2010).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 16.
    C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Trans. Am. Math. Soc. 311, 167 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 17.
    M. Feigin, “Intertwining relations for the spherical parts of generalized Calogero operators,” Theor. Math. Phys. 135, 497 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 18.
    V. X. Genest, A. Lapointe, and L. Vinet, “The Dunkl-Coulomb problem in the plane,” Phys. Lett. A 379, 923 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 19.
    T. Hakobyan, O. Lechtenfeld, and A. Nersessian, “The spherical sector of the Calogero model as a reduced matrix model,” Nucl. Phys. B 858, 250 (2012).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 20.
    P. J. Redmond, “Generalization of the Runge-Lenz vector in the presence of an electric field,” Phys. Rev. B 133, 1352 (1964).ADSMathSciNetCrossRefGoogle Scholar
  25. 21.
    K. Helfrich, “Constants of motion for separable oneparticle problems with cylinder symmetry,” Theor. Chim. Acta (Berl.) 24, 271 (1972).CrossRefGoogle Scholar
  26. 22.
    L. Landau and E. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1989; Pergamon, Oxford, 1977).Google Scholar
  27. 23.
    C. Coulson and A. Joseph, “A constant of the motion for the two-centre Kepler problem,” Int. J. Quantum Chem. 1, 337 (1967).ADSCrossRefGoogle Scholar
  28. 24.
    H. A. Erikson and E. L. Hill, “A note about one-electron states of diatomic molecules,” Phys. Rev. 76, 29 (1949).ADSCrossRefzbMATHGoogle Scholar
  29. 25.
    S. P. Alliluev and A. V. Matveenko, “Symmetry group of the hydrogen molecular ion (A system with separable variables),” Sov. Phys. JETP 24, 1260 (1967).ADSzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Yerevan State UniversityYerevanArmenia
  2. 2.Tomsk Polytechnic UniversityTomskRussia

Personalised recommendations