Introduction to ill-posed aspects of nuclear scattering

  • Pierre C. Sabatier
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 236)


This pedagogical survey is presented following a suggestion of the Workshop Committee. It is tried to present in a simple way the main ill-posed aspects of inverse problems arising in nuclear scattering. One also explains on some examples how they are dealt with or how they are circumvented.


Inverse Problem Inverse Scattering Born Approximation Compressive Mapping Fixed Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    For more details, and references, see K. Chadan & P.C. Sabatier “Inverse Problems in Quantum Scattering Theory” Springer-Verlag, New York Heidelberg Berlin 1977.Google Scholar
  2. (2).
    Ref (1) page 146 and ff.Google Scholar
  3. (3).
    Ref (1) page 64Google Scholar
  4. (4)a.
    For recent references on the three-dimensional inverse problem see R.G. Newton “The Marchenko and Gel'fand Levitan Methods in the Inverse Scattering Problem in one and three dimensions” in “Conference on Inverse Scattering: Theory and Application” J. Bee Bednar et al. ed. SIAM Philadelphia (1983) and (same author)Google Scholar
  5. (4)b.
    A Faddeev — Marchenko method for Inverse Scattering in three dimensions. Inverse Problems 2 (1985). A study of the ill-posed aspects of the inverse problem at fixed energy in the class of truncated potentials by Y. Loubatières will be published soon.Google Scholar
  6. (5).
    P.C. Sabatier: Well-posed Questions and Exploration of the Space of Parameters in Linear and Non Linear Inversion. In “Inverse Problems of Acoustic and Elastic Waves” F. Santosa et al. Ed. SIAM Philadelphia 1984.Google Scholar
  7. (6).
    V. Glaser and A. Martin, H. Grosse, W. Thirring: A family of optimal conditions for the absence of bound states in a potential. In Studies in mathematical Physics (eds. E.H. Lieb, B. Simon, A.S. Wightman) Princeton U.P. 1976. See also B. Simon same ref. also: A. Martin and P.C. Sabatier: Impedance, zero energy wave function, and bound states. J. Math. Phys. 18, 1623–1626 (1977).Google Scholar
  8. (7).
    See for example P.E. Hodgson: Nuclear Reactions and Nuclear Structure — Clarenton Press, Oxford, 1971. For a recent monograph on applications of inversion theory, particulary in nuclear scattering, see Zachariev B.N. and Suzko A.A. “Potentials and quantum scattering direct and inverse problems”, to be published by Energoatomisdat (Moscow) in 1985).Google Scholar
  9. (8).
    A recent example is given by A.A. Ioannides and R.S. Mackintosh: A method for S-matrix to Potential Inversion at Fixed Energy. Nucl. Phys. A 438, 354 (1985).Google Scholar
  10. (9).
    For a review and references see: P.C. Sabatier: “Rational Reflection Coefficients in One-Dimensional Inverse Scattering and Applications. In “Conference on Inverse Scattering: Theory and Application” J. Bee Bednar et al. eds. SIAM, Philadelphia 1983.Google Scholar
  11. (10)a.
    R. Lipperheide and H. Fiedeldey: Inverse Problem for Potential Scattering at fixed Energy I: Z. Phys. A 286, 45–46 (1978) and id. II: Z. Phys. A 301, 81–89 (1981). R. Lipperheide, S. Sofianos and H. Fiedeldey Potential Inversion for scattering at fixed energy: Phys. Rev. C 26, 770–772 (1982).Google Scholar
  12. (10)b.
    H. Burger, L.J. Allen, H. Fiedeldy, S.A. Sofianos and R. Lipperheide: Potentials obtained by inversion of e — He atomic scattering data: Physics Lett. 97 A, 39–41 (1983). B.V. Rudyak, A.a; Surko and B.N. Zachariev: Exactly solvable models (Crum-Krein Transformation in the λ2, E, plane). Phys. Scripts 29, 515–517, (1984).Google Scholar
  13. (11).
    A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi: Higher transcendental functions. Mac Graw Hill Ed. 1953.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Pierre C. Sabatier
    • 1
  1. 1.Département de Physique MathématiqueUniversité des Sciences et Techniques du LanguedocMontpellier cedexFrance

Personalised recommendations