Physics of Atomic Nuclei

, Volume 80, Issue 3, pp 520–528 | Cite as

Dunkl operator, integrability, and pairwise scattering in rational Calogero model

  • David Karakhanyan
Elementary Particles and Fields Theory


The integrability of the Calogero model can be expressed as zero curvature condition using Dunkl operators. The corresponding flat connections are non-local gauge transformations, which map the Calogero wave functions to symmetrized wave functions of the set of N free particles, i.e. it relates the corresponding scattering matrices to each other. The integrability of the Calogero model implies that any k-particle scattering is reduced to successive pairwise scatterings. The consistency condition of this requirement is expressed by the analog of the Yang–Baxter relation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Z. N. C. Ha, Nucl. Phys. B 435, 604 (1995).ADSCrossRefGoogle Scholar
  2. 2.
    F. D. M. Haldane, in Correlation Effects in Low- Dimensional Electron Systems, Ed. by A. Okiji and N. Kawakami (Springer, 1995), p. 3.Google Scholar
  3. 3.
    A. P. Polychronakos, Phys. Rev. Lett. 69, 703 (1992)ADSMathSciNetCrossRefGoogle Scholar
  4. 3a.
    A. P. Polychronakos, J. Phys. A 39, 12793 (2006).ADSMathSciNetCrossRefGoogle Scholar
  5. 4.
    G. J. Heckman, in Harmonic Analysis on Reductive Groups, Ed. by W. H. Barker and P. J. Sally, Progress in Mathematics}, Vol. 101 (Birkhäuser, 1991), p. 181.Google Scholar
  6. 5.
    F. Calogero J. Math. Phys. 1021911969; 124191971 ;Lett. NuovoCimento13 4111975Google Scholar
  7. 6.
    A. M. Perelomov, Teor. Mat. Fiz. 6, 364 (1971) [Theor. Math. Phys. 6, 285 (1971)].Google Scholar
  8. 7.
    J. Moser, Adv. Math. 16, 197 (1975).ADSCrossRefGoogle Scholar
  9. 8.
    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1993).Google Scholar
  10. 9.
    E. K. Sklyanin, Algebra and Analysis 6, 227 (1994).Google Scholar
  11. 10.
    S. Wojciechowski, Phys. Lett. A 95, 279 (1983).ADSMathSciNetCrossRefGoogle Scholar
  12. 11.
    Ch. F. Dunkl, Trans. Amer. Math. Soc. 311, 167 (1989).MathSciNetCrossRefGoogle Scholar
  13. 12.
    Ch. F. Dunkl, Math. Z. 197, 33 (1988); in Invariant Theory and Tableaux, Ed. by D. Stanton (Springer 1990), p. 107; Canad. J. Math. 43, 1213 (1991).MathSciNetCrossRefGoogle Scholar
  14. 13.
    G. J. Heckman, Dunkl Operators. Seminaire Bourbaki (Asterisque, 1997).zbMATHGoogle Scholar
  15. 14.
    G. V. Efimov, The Non-Local interactions (Nauka, Moskow, 1977) (in Russian).Google Scholar
  16. 15.
    N. Yonezawa and I. Tsutsui, J. Math. Phys. 47, 012104 (2006).ADSMathSciNetCrossRefGoogle Scholar
  17. 16.
    L. Fehér, I. Tsutsui, and T. Fülöp, Nucl. Phys. B 715, 713 (2005).ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Yerevan Physics InstituteYerevanArmenia

Personalised recommendations