Methods of modern mathematical physics: III Scattering theory

Scattering theory is the study of an interacting system on a scale of time and/or distance which is large compared to the scale of the interaction itself. As such, it is the most effective means, sometimes the only means, to study microscopic nature. To understand the importance of scattering theory, consider the variety of ways in which it arises. First, there are various phenomena in nature (like the blue of the sky) which are the result of scattering. In order to understand the phenomenon (and to identify it as the result of scattering) one must understand the underlying dynamics and its scattering theory. Second, one often wants to use the scattering of waves or particles whose dynamics on knows to determine the structure and position of small or inaccessible objects. For example, in x-ray crystallography (which led to the discovery of DNA), tomography, and the detection of underwater objects by sonar, the underlying dynamics is well understood. What one would like to construct are correspondences that link, via the dynamics, the position, shape, and internal structure of the object to the scattering data. Ideally, the correspondence should be an explicit formula which allows one to reconstruct, at least approximately, the object from the scattering data. The main test of any proposed particle dynamics is whether one can construct for the dynamics a scattering theory that predicts the observed experimental data. Scattering theory was not always so central the physics. Even thought the Coulomb cross section could have been computed by Newton, had he bothered to ask the right question, its calculation is generally attributed to Rutherford more than two hundred years later. Of course, Rutherford's calculation was in connection with the first experiment in nuclear physics. Title page Introduction Contents of Other Volumes XI: SCATTERING THEORY 1. An overview of scattering phenomena 2. Classical particle scattering 3. The basic principles of scattering in Hilbert space Appendix 1 Stationary phase methods Appendix 2 Trace ideal properties of f(x)g(-i∇) Appendix 3 A general invariance principle for wave operators 4. Quantum scattering I: Two-body case 5. Quantum scattering II: N-body case 6. Quantum scattering III: Eigenfunction expansions Appendix Introduction to eigenfunction expansions by the auxiliary space method 7. Quantum scattering IV: Dispersion relations 8. Quantum scattering V: Central potentials A. Reduction of the S-matrix by symmetries B. The partial wave expansion and its convergence C. Phase shifts and their connection to the Schrödinger equation D. The variable phase equation E. Jost functions and Lev;nson"s theorem F. Analyticity of the partial wave amplitude for generalized Yukawa potentials G. The Kohn variational principle Appendix 1 Legendre polynomials and spherical Bessel functions Appendix 2 Jost solutions for oscillatory potentials Appendix 3 Jost solutions and the fundamental problems of scattering theory 9. Long-range potentials 10. Optical and acoustical scattering I: Schrödinger operator methods Appendix Trace class properties of Green's functions 11. Optical and acoustical scattering II: The Lax-Phillips method Appendix The twisting trick 12. The linear Boltzmann equation 13. Nonlinear wave equations Appendix Conserved currents 14. Spin wave scattering 15. Quantum field scattering I: The external field 16. Quantum field scattering II: The Haag-Ruelle theory 17. Phase space analysis of scattering and spectral theory Appendix The RAGE theorem Notes Notes on scattering theory on C*-algebras Problems MATERIAL PREPRINTED FROM VOLUME IV XIII.6 The absence of singular continuous spectrum I: General theory XIII.7 The absence of singular continuous spectrum II: Smooth perturbations A. Weakly coupled quantum systems B. Positive commutators and repulsive potentials C. Local smoothness and wave operators for repulsive potentials XIII.8 The absence of singular continuous spectrum III: Weighted L2 spaces Notes Problems List of Symbols Index The simplest system with which to illustrate the ideas of scattering theory is the classical mechanics of a single particle moving in an external force field F(r).