Multiparticle quantum scattering in constant magnetic fields
type: Item
instance of: scholarly article
zbMATH Open document ID: 1044.81123
author: Christian Gérard
author: Izabella Łaba
publication date: 2002-02-18T00:00:00Z
Mathematics Subject Classification ID: 81U10
Mathematics Subject Classification ID: 81-01
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 34L25
zbMATH DE Number: 1706716
zbMATH Keywords: spectral theory of magnetic \(N\)-body Hamiltonians
zbMATH Keywords: special \(N\)-body systems
zbMATH Keywords: time-dependent quantum mechanical scattering
zbMATH Keywords: large-time behaviour of solutions
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: asymptotic completeness
zbMATH Keywords: generic 3-body systems
zbMATH Keywords: textbook
zbMATH Keywords: channel Hamiltonian
zbMATH Keywords: Mourre theory
zbMATH Keywords: conjugate operator
zbMATH Keywords: commentator method
zbMATH Keywords: propagation estimates
zbMATH Keywords: asymptotic energy
zbMATH Keywords: minimal and maximal velocities
zbMATH Keywords: asymptotic velocity
zbMATH Keywords: propagation of charged systems
zbMATH Keywords: three-dimensional strongly charged systems
zbMATH Keywords: long-range and short-range potentials
zbMATH Keywords: 3-body dispersive systems
zbMATH Keywords: two-dimensional charged systems
zbMATH Keywords: open problems
MaRDI profile type: Publication
title: Multiparticle quantum scattering in constant magnetic fields
published in: Mathematical Surveys and Monographs
review text: The time-dependent quantum mechanical scattering deals with the large-time behaviour of solutions of the Schrödinger equation. From the mathematical point of view it is mainly a tool for the perturbation theory of the continuous spectram of self-adjoint operators. \(N\)-body scattering is well understood for scalar (electrostatic) potentials. In the presence of magnetic fields the situation becomes much more complicated. The book explains the present status in this field and provides a solid foundation for further research. The objectives are to give a general introduction to the spectral theory of magnetic \(N\)-body Hamiltonians and to prove the asymptotic completeness for generic 3-body systems and for special \(N\)-body systems.NEWLINENEWLINE The book is useful for researchers and graduate students who are already familiar with the \(N\)-body Schrödinger systems without magnetic fields. Starting from this level it is a good textbook, it is understandable and well written. This is also due to the fact that the authors are known in this field and experts in mathematical scattering theory.NEWLINENEWLINE The main concept for the scattering of \(N\)-particle magnetic Hamiltonian is described (chapter 1). Several aspects winch are known from the usual Schrödinger systems are different, e.g. the cluster decomposition and the charges of the clusters, the centre of mass motion the definition of bound and scattering states. The denotation for describing. \(N\)-body-systems is alwavs technical. It needed more than 20 years to find elegant geometrical methods. The channel Hamiltonian and the associated partitions of unity are explained in chapter 2.NEWLINENEWLINE One main part of the book is devoted to the Mourre theory (chapter 3), an important tool for the following results on asymptotic completeness. A detailed list on the literature concerning the Mourre estimates is given. The known definitions and arguments of the Mourre theory, the abstract framework of the conjugate…
reviewed by: Michael Demuth
description: scientific article; zbMATH DE number 1706716