Scattering Theory: Some Old and New Problems

Dmitri R. Yafaev (auth.)9783540675877, 3540675876

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.

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Table of contents :
Basic concepts….Pages 1-13
Short-range interactions. asymptotic completeness….Pages 14-23
Short-range interactions. Miscellaneous….Pages 24-29
Long-range interactions. The scheme of smooth perturbations….Pages 30-39
The generalized fourier transform….Pages 40-46
Long-range matrix potentials….Pages 47-52
A stationary representation….Pages 53-58
The short-range case….Pages 59-66
The long-range case….Pages 67-79
The relative scattering matrix….Pages 80-85
Setting the scattering problem….Pages 86-95
Resolvent equations for three-particle systems….Pages 96-105
Asymptotic completeness. A sketch of proof….Pages 106-117
The scattering matrix and eigenfunctions for multiparticle systems….Pages 118-127
New channels of scattering….Pages 128-136
The heisenberg model….Pages 137-144
Infinite obstacle scattering….Pages 145-153