The quantum angular Calogero-Moser model

  • Mikhail Feigin
  • Olaf LechtenfeldEmail author
  • Alexios P. Polychronakos


The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of \( {{\mathbb{R}}^n} \) to the ‘angular Calogero-Moser model’ on the sphere S n−1.We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a ‘relative angular Calogero-Moser model’, which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.


Integrable Equations in Physics Integrable Field Theories 


  1. [1]
    F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    M. Olshanetsky and A. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    M. Olshanetsky and A. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    Calogero-Moser system Scholarpedia webpage,
  6. [6]
    T. Hakobyan, A. Nersessian and V. Yeghikyan, Cuboctahedric Higgs oscillator from the Calogero model, J. Phys. A 42 (2009) 205206 [arXiv:0808.0430] [INSPIRE].MathSciNetADSGoogle Scholar
  7. [7]
    T. Hakobyan, S. Krivonos, O. Lechtenfeld and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374 (2010) 801 [arXiv:0908.3290] [INSPIRE].MathSciNetADSGoogle Scholar
  8. [8]
    T. Hakobyan, O. Lechtenfeld, A. Nersessian and A. Saghatelian, Invariants of the spherical sector in conformal mechanics, J. Phys. A 44 (2011) 055205 [arXiv:1008.2912] [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    T. Hakobyan, O. Lechtenfeld and A. Nersessian, The spherical sector of the Calogero model as a reduced matrix model, Nucl. Phys. B 858 (2012) 250 [arXiv:1110.5352] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    O. Lechtenfeld, A. Nersessian and V. Yeghikyan, Action-angle variables for dihedral systems on the circle, Phys. Lett. A 374 (2010) 4647 [arXiv:1005.0464] [INSPIRE].MathSciNetADSGoogle Scholar
  11. [11]
    T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian and V. Yeghikyan, Action-angle variables and novel superintegrable systems, Phys. Part. Nuclei 43 (2012) 577 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M.V. Feigin, Intertwining relations for spherical parts of generalized Calogero operators, Theor. Math. Phys. 135 (2003) 497.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J.A. Minahan and A.P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993) 265 [hep-th/9206046] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    F. Calogero and C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials, J. Math. Phys. 15 (1974) 1425 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    P.W. Higgs, Dynamical symmetries in a spherical geometry. 1, J. Phys. A 12 (1979) 309 [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    H.I. Leemon, Dynamical symmetries in a spherical geometry. 2, J. Phys. A 12 (1979) 489 [INSPIRE].MathSciNetADSGoogle Scholar
  18. [18]
    A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. [19]
    L. Brink, T. Hansson and M.A. Vasiliev, Explicit solution to the N body Calogero problem, Phys. Lett. B 286 (1992) 109 [hep-th/9206049] [INSPIRE].MathSciNetADSGoogle Scholar
  20. [20]
    C.F. Dunkl and Y. Hu, Orthogonal polynomials of several variables, Cambridge University Press, Cambridge U.K. (2001).zbMATHCrossRefGoogle Scholar
  21. [21]
    J.F. Van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Commun. Math. Phys. 188 (1997) 467 [q-alg/9609032].ADSzbMATHCrossRefGoogle Scholar
  22. [22]
    G.J. Heckman, A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, W. Barker and P. Sally eds., Progr. Math. 101 (1991) 181, Birkhäuser, (1991).Google Scholar
  23. [23]
    C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994) 237.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge U.K. (1990).zbMATHCrossRefGoogle Scholar
  25. [25]
    J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974) 1420 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    M. Olshanetsky, Wave functions of quantum integrable systems, Theor. Math. Phys. 57 (1983) 1048 [Teor. Mat. Fiz. 57 (1983) 148] [INSPIRE].
  27. [27]
    A. Golynski, diploma paper, Moscow State University, Moscow Russia (1999).Google Scholar
  28. [28]
    A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 [hep-th/9210109] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Mikhail Feigin
    • 1
  • Olaf Lechtenfeld
    • 2
    Email author
  • Alexios P. Polychronakos
    • 3
    • 4
  1. 1.School of Mathematics and StatisticsUniversity of GlasgowGlasgowU.K.
  2. 2.Institut für Theoretische Physik and Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany
  3. 3.CCPP, Department of PhysicsNYUNew YorkU.S.A.
  4. 4.Physics DepartmentThe City College of the CUNYNew YorkU.S.A.

Personalised recommendations