Inverse Spectral and Scattering Theory

  • K. Chadan
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 23/1981)


In these lectures, we shall try to give a résume of the techniques which have been devised for solving various spectral and scattering inverse problems. The invention of these techniques goes back to the 50’s, and is mainly due to Gel’fand and Levitan, and Marchenko, with important contributions by Jost and Kohn, Faddeev, Newton and Sabatier, Regge, Loeffel, Martin, Cornille, Gasymov and Levitan,… All the references are given at the end.


Integral Equation Phase Shift Transmission Coefficient Schrodinger Equation Scattering Theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


For general scattering theory, see

  1. R.G. Newton, Scattering Theory of Waves and Particles (Mc Graw Hill, 1966).Google Scholar
  2. V. de Alfaro and T. Regge, Potential Scattering, John Wiley and Sons, 1965.MATHGoogle Scholar
  3. W.O. Amrein, J.M. Jauch, and K.B. Sinha, Scattering Theory in Quantum Mechanics, Benjamin,1977.MATHGoogle Scholar
  4. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. III, Academic Press, 1979.MATHGoogle Scholar
  5. H.M. Nussenzveig, Causality and Dispersion Relations, Academic Press, 1972.Google Scholar
  6. B. Simon, Quantum Mechanics for Hamiltonians defined as quadratic forms, Princeton University Press, 1971.MATHGoogle Scholar

For the study of inverse problems, see

  1. Z.S. Agranovich and V.A. Marchenko, The Inverse Problem of Scattering Theory, English translation, Gordon and Breach, 1963.MATHGoogle Scholar
  2. K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer Verlag, 1977. Russian translation, MIR, 1980. This book contains an almost complete list of references to original works of Gel’fand and Levitan, Marchenko, Jost and Kohn, Faddeev, Newton, Sabatier, Regge, Loeffel, Martin, Cronille,… -Google Scholar
  3. P. Deift and E. Trubowitz, Inverse Scattering on the Line, Comm. Pure. Appl. Math. 32 (1979) 121.MathSciNetMATHCrossRefGoogle Scholar
  4. This paper gives a method different from the original method of Faddeev.Google Scholar
  5. R.G. Newton, Inverse scattering, I, One Dimension, J. Math. Phys. (1980) 493;Google Scholar
  6. R.G. Newton, Three Dimensions, ibid. 21 (1980) 1698. See also his recent preprint.Google Scholar

For the case of discrete spectrum, including confining potentials, see

  1. M. Levitan and M.G. Gasimov, Determination of a differential equation by two of its spectra, Russian Math. Survey (1964) 1.Google Scholar
  2. H. Grosse and A. Martin, Theory of the inverse problem for confining potentials, Nucl. Phys. B148 (1979) 413.Google Scholar
  3. Quigg, J.L. Rosner and H.B. Thacker, Phys. Rev. D18 (1978) 274; ibid. 287.ADSGoogle Scholar

For applications to Q.F.T., see

  1. K. Sklyanin and L.D. Faddeev, Sov. Phys. Dokl. 23 (1978) 902ADSGoogle Scholar
  2. J. Honerkamp, P. Weber and A. Wiesler, Nucl. Phys. B152 266.Google Scholar
  3. H. Grosse, Phys. Lett. B86 (1979) 287.MathSciNetADSGoogle Scholar
  4. H.B. Thacker, Rev. Mod. Phys. 53 (1981) 256.MathSciNetADSCrossRefGoogle Scholar
  5. These papers, especially the last one, contain full references to the works of Zakharov and Shabat, Faddeev, Manakov, Sklyanin, Thacker, Kulish, etc..Google Scholar

For applications to nonlinear partial differential equations, see

  1. C.S. Gardner, M.J. Greene, M.O. Kruskal and R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095.ADSMATHCrossRefGoogle Scholar
  2. A.G. Scott, F.Y.F. Chu and D.M. McLaughlin, Proc. IEEE 61 (1973) 1443.Google Scholar
  3. F. Calogero (Ed.), Nonlinear evolution equations solvable by the spectral transform (Pitman, London, 1978).MATHGoogle Scholar
  4. P.C. Sabatier (Ed.), Problèmes Inverses. Evolutions non linéaires (C.N.R.S., Paris, 1979).Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • K. Chadan
    • 1
  1. 1.Lab. de Phys. Théor. et Hautes EnergiesUniv. de Paris XIOrsayFrance

Personalised recommendations