Scattering Theory: Some Old and New Problems

ISBN-13: 9783540675877 / Angielski / Miękka / 2000 / 176 str.

D. R. Yafaev; Dmitri R. Yafaev

Opis

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 Schrodinger 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 Schrodinger 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.

Recenzje

"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)

Spis treści

Part 1. The Schroedinger operator of two-particle systems 1. Basic notions 2. Short-range interactions. Asymptotic completeness 3. Short-range interactions. Miscellaneous 4. Long-range interactions. The scheme of smooth perturbations 5. The generalized Fourier transform 6. Long-range matrix potentials Part 2. The scattering matrix 7. A stationary representarion 8. The short-range case 9. The long-range cas 10. The relative scattering matrix Part 3. The multiparticle Schroedinger operator and related problems 11. Setting the scattering problem 12. Resolvent equations 13. Asymptotic completeness. A sketch of proof 14. The scattering matrix for multiparticle systems 15. New channels of scattering 16. The Heisenberg model 17. Infinite obstacle scattering