Few-Body Systems

, 60:28 | Cite as

The Inverse-Scattering Problem: The View of a Few-Nucleon Theorist

  • Peter U. SauerEmail author
Part of the following topical collections:
  1. Ludwig Faddeev Memorial Issue


The theoretical step from the experimental phase shifts in the partial waves of the nuclear force to a parametrization of the two-nucleon interaction is discussed. A nuclear-physics solution to the inverse-scattering problem is recalled. The procedure is only based on the assumption of Hermiticity for the underlying potential. The procedure is provided by the off-energy-shell continuation of the two-nucleon transition matrix. It is compared with the strategies of mathematicians for the same problem. Faddeev strongly influenced the mathematical side of the problem.



The author thanks the Nuclear-Theory Group of the Carnegie-Mellon University Pittsburgh, Michel Baranger, Bertrand Giraud and Susanta K. Mukhopadhay, for the pleasant collaboration years back, from which the work of Ref. [3] originated. He especially thanks Bertrand Giraud for his encouragement to this contribution, his critical reading and useful suggestions. He also thanks Arnas Deltuva and Steven Karataglidis for their critical reading of the manuscript.


  1. 1.
    M.L. Halbert, P. Paul, K.A. Snover, E.K. Warburton, Proceedings of the International Symposium on Few Particle Problems in Nuclear Physics, Dubna USSR, 1979Google Scholar
  2. 2.
    L.D. Faddeev, B. Seckler, The inverse problem in the quantum theory of scattering. J. Math. Phys. 4, 72 (1963)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Baranger, B. Giraud, S.K. Mukhopadhyay, P.U. Sauer, Off-energy-shell continuation of the two-nucleon transition matrix. Nucl. Phys. A 138, 1 (1969)ADSCrossRefGoogle Scholar
  4. 4.
    T. Hamada, I.D. Johnston, A potential model representation of two-nucleon data below 315 MeV. Nucl. Phys. A 34, 382 (1962)CrossRefGoogle Scholar
  5. 5.
    K.E. Lassila, M.H. Hull Jr., H.M. Ruppel, F.A. McDonald, G. Breit, Note on a nucleon–nucleon potential. Phys. Rev. 126, 881 (1962)ADSCrossRefGoogle Scholar
  6. 6.
    M. Baranger, Recent progress in the understanding of finite nuclei from the nucleon–nucleon interaction, in Proceedings of the International School of Physics “Enrico Fermi”, Course XL, “Nuclear Structure and Nuclear Reactions” (Academic Press, New York, 1969), p. 511Google Scholar
  7. 7.
    K.A. Brueckner, Two-body forces and nuclear saturation. III. Details of the structure of the nucleus. Phys. Rev. 97, 1353 (1955)ADSCrossRefGoogle Scholar
  8. 8.
    K.A. Brueckner, Many-body problem for strongly interacting particles. II. Linked cluster expansion. Phys. Rev. 100, 36 (1955)ADSCrossRefGoogle Scholar
  9. 9.
    J.R. Taylor, Scattering Theory (Dover Publications, Mineola, 1983), p. 138Google Scholar
  10. 10.
    L.D. Faddeev, Scattering theory for a three-particle system. Sov. Phys. JETP 12, 1014 (1961)MathSciNetGoogle Scholar
  11. 11.
    L.D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory, Academy of Sciences of the USSR Works of the Steklov Mathematical Institute, vol. 69 (1963). Israel Program for Scientific Translations, Jerusalem (1965)Google Scholar
  12. 12.
    R.V. Reid, Local phenomenological nucleon–nucleon potentials. Ann. Phys. 50, 411 (1968)ADSCrossRefGoogle Scholar
  13. 13.
    P.U. Sauer, Parametrization of the \(^1S_0\) two-nucleon transition matrix. Ann. Phys. 80, 242 (1973)ADSCrossRefGoogle Scholar
  14. 14.
    M. Haftel, Off-shell continuation of the two-nucleon transition matrix with a bound state. Phys. Rev. Lett. 25, 120 (1970)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    P.U. Sauer, Off-energy-shell continuation of the two-nucleon transition matrix in the presence of tensor forces. Nucl. Phys. A 170, 497 (1971)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    P.U. Sauer, H. Walliser, Off-shell continuation of the proton–proton transition matrix, in Proceedings of the International Conference “Few Body Problems in Nuclear and Particle Physics” (Les Presses de L’Université Laval, Quebec, 1975), p. 108Google Scholar
  17. 17.
    P.U. Sauer, J.A. Tjon, Three-nucleon calculations without the explicit use of two-nucleon potentials. Nucl. Phys. A 216, 541 (1973)ADSCrossRefGoogle Scholar
  18. 18.
    R. Machleidt, D.R. Entem, Chiral effective field theory and nuclear forces. Phys. Rep. 503, 1 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    E. Epelbaum, H. Krebs, U.-G. Meißner, Improved chiral nucleon–nucleon potential up to next-to-next-to-next-leading order. Eur. Phys J. A 51, 53 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    R.G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill Company, New York, 1966), p. 610Google Scholar
  21. 21.
    E.K. Sklyanin, L.A. Takhtadzhyan, L.D. Faddeev, Quantum inverse problem method. I. Solitons in solids. Theor. Math. Phys. 40, 688 (1979)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Insttute for Theoretical PhysicsLeibniz UniversityHannoverGermany

Personalised recommendations