Inverse Problems in Quantum Scattering Theory

1. Front Matter

   Pages i-xxxi

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2. Some Results from Scattering Theory

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 1-19

3. Bound States—Eigenfunction Expansions

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 20-39

4. The Gel’fand-Levitan-Jost-Kohn Method

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 40-58

5. Applications of the Gel’fand-Levitan Equation

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 59-75

6. The Marchenko Method

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 76-85

7. Examples

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 86-94

8. Special Classes of Potentials

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 95-111

9. Nonlocal Separable Interactions

   • K. Chadan, P. C. Sabatier, R. G. Newton

   Pages 112-127

10. Miscellaneous Approaches to the Inverse Problems at Fixed l

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 128-154

11. Scattering Amplitudes from Elastic Cross Sections

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 155-181

12. Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 182-194

13. Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 195-213

14. Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 214-234

15. The Three-Dimensional Inverse Problem

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 235-274

16. Miscellaneous Approaches to Inverse Problems at Fixed Energy

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 275-289

17. Approximate Methods

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 290-322

18. Inverse Problems in One Dimension

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 323-388

19. Problems Connected with Discrete Spectra

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 389-415

20. Numerical Problem

    • K. Chadan, P. C. Sabatier, R. G. Newton

    Pages 416-439

The normal business of physicists may be schematically thought of as predic­ ting the motions of particles on the basis of known forces, or the propagation of radiation on the basis of a known constitution of matter. The inverse problem is to conclude what the forces or constitutions are on the basis of the observed motion. A large part of our sensory contact with the world around us depends on an intuitive solution of such an inverse problem: We infer the shape, size, and surface texture of external objects from their scattering and absorption of light as detected by our eyes. When we use scattering experiments to learn the size or shape of particles, or the forces they exert upon each other, the nature of the problem is similar, if more refined. The kinematics, the equations of motion, are usually assumed to be known. It is the forces that are sought, and how they vary from point to point. As with so many other physical ideas, the first one we know of to have touched upon the kind of inverse problem discussed in this book was Lord Rayleigh (1877). In the course of describing the vibrations of strings of variable density he briefly discusses the possibility of inferring the density distribution from the frequencies of vibration. This passage may be regarded as a precursor of the mathematical study of the inverse spectral problem some seventy years later.

• Inverses Streuproblem
• Potential
• calculus
• inverse scattering problem
• inverse scattering theory
• numerical methods
• scattering theory

• Laboratoire de Physique Théorique, Université de Paris-Sud, Orsay, France

  K. Chadan

• Université des Sciences et Techniques du Languedoc, Montpellier Cedex, France

  P. C. Sabatier

• Department of Physics, Indiana University, Bloomington, USA

  R. G. Newton

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