Few-Body Systems

, 60:59 | Cite as

Exact Energy of the Two and Three-Body Interactions in the Trigonometric Three-Body Problem via Jack Polynomials

  • H. Rahmati
  • A. LatifiEmail author


The Hamiltonian of the three-body problem with two and three-body interactions of Calogero–Sutherland type written in an appropriate set of variables is shown to be a differential operator of Laplace–Beltrami \(H_{LB}\) type. Its eigenfunction is expressed then in terms of Jack Polynomials. In the case of identical distinguishable fermions, the only possible partition and its corresponding occupation state in Fock space is exhibited. In the co-moving frame with the center of mass, the exact explicit excitation energy due to interactions is given in terms of the parameters of the problem. The excitation energy due to interactions is expressed in such a way that it is possible to distinguish the excitation energy due to the pairwise interactions neglecting the three-body interaction as well as the excitation energy due to the pure three-body interactions neglecting the pairwise interactions. Different possible levels of excitation energy due to different type of interactions are explicitly written.



  1. 1.
    B.A. Bernevig, F.D.M. Haldane, Properties of non-abelian fractional quantum hall states at filling \(\nu =k/r\). Phys. Rev. Lett. 101, 246806 (2008). ADSCrossRefGoogle Scholar
  2. 2.
    F. Calogero, 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, 1425–1430 (1974). ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    H.H. Chen, Y.C. Lee, N.R. Pereira, Algebraic internal wave solitons and the integrable calogero–moser–sutherland n-body problem. Phys. Fluids 22, 187–188 (1979). ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Estienne, R. Santachiara, Relating jack wavefunctions to \({\text{ WA }}_{{{\rm k}}-1}\) theories. J. Phys. A Math. Theor. 42, 445209 (2009). 10.1088/1751-8113/42/44/445209ADSCrossRefGoogle Scholar
  5. 5.
    F. Lesage, V. Pasquier, D. Serban, Dynamical correlation functions in the calogero–sutherland model. Nucl. Phys. B 435, 585–603 (1995). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    N.A. Nekrasov, Seiberg-witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    C. Quesne, Exactly solvable three-particle problem with three-body interaction. Phys. Rev. A 55, 3931–3934 (1997). ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    C. Quesne, Three-Body Generalizations of the Sutherland Problem (Springer, New York, 2000), pp. 411–420. CrossRefGoogle Scholar
  9. 9.
    M. Rosenbaum, A. Turbiner, A. Capella, Solvability of the g2 integrable system. Int. J. Mod. Phys. A 13, 3885–3903 (1998). ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    K. Sogo, Eigenstates of calogero–sutherland-moser model and generalized schur functions. J. Math. Phys. 35, 2282–2296 (1994). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R.P. Stanley, Some combinatorial properties of jack symmetric functions. Adv. Math. 77, 76–115 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. Sutherland, Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971). ADSCrossRefGoogle Scholar
  13. 13.
    J. Wolfes, On the three-body linear problem with three-body interaction. J. Math. Phys. 15, 1420–1424 (1974). ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics, Faculty of SciencesQom University of TechnologyQomIran

Personalised recommendations