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Scattering Theory: Some Old and New Problems

  • Book
  • © 2000

Overview

Authors:
  1. Dmitri R. Yafaev
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1735)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Basic concepts

    • Dmitri R. Yafaev
    Pages 1-13

  3. Short-range interactions. asymptotic completeness

    • Dmitri R. Yafaev
    Pages 14-23

  4. Short-range interactions. Miscellaneous

    • Dmitri R. Yafaev
    Pages 24-29

  5. Long-range interactions. The scheme of smooth perturbations

    • Dmitri R. Yafaev
    Pages 30-39

  6. The generalized fourier transform

    • Dmitri R. Yafaev
    Pages 40-46

  7. Long-range matrix potentials

    • Dmitri R. Yafaev
    Pages 47-52

  8. A stationary representation

    • Dmitri R. Yafaev
    Pages 53-58

  9. The short-range case

    • Dmitri R. Yafaev
    Pages 59-66

  10. The long-range case

    • Dmitri R. Yafaev
    Pages 67-79

  11. The relative scattering matrix

    • Dmitri R. Yafaev
    Pages 80-85

  12. Setting the scattering problem

    • Dmitri R. Yafaev
    Pages 86-95

  13. Resolvent equations for three-particle systems

    • Dmitri R. Yafaev
    Pages 96-105

  14. Asymptotic completeness. A sketch of proof

    • Dmitri R. Yafaev
    Pages 106-117

  15. The scattering matrix and eigenfunctions for multiparticle systems

    • Dmitri R. Yafaev
    Pages 118-127

  16. New channels of scattering

    • Dmitri R. Yafaev
    Pages 128-136

  17. The heisenberg model

    • Dmitri R. Yafaev
    Pages 137-144

  18. Infinite obstacle scattering

    • Dmitri R. Yafaev
    Pages 145-153

  19. Back Matter

    Pages 155-169
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Keywords

About this book

Scattering theory is, roughly speaking, perturbation theory of self-adjoint operators on the (absolutely) continuous spectrum. It has its origin in mathematical problems of quantum mechanics and is intimately related to the theory of partial differential equations. Some recently solved problems, such as asymptotic completeness for the Schrödinger operator with long-range and multiparticle potentials, as well as open problems, are discussed. Potentials for which asymptotic completeness is violated are also constructed. This corresponds to a new class of asymptotic solutions of the time-dependent Schrödinger equation. Special attention is paid to the properties of the scattering matrix, which is the main observable of the theory. The book is addressed to readers interested in a deeper study of the subject.

Reviews

“This book provides a modern survey of important problems of the mathematical scattering theory. … The book fills a gap in the literature because it emphasizes the stationary approach to the scattering theory problems.” (Michael Perelmuter, zbMATH 0951.35003, 2022)

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