Multiparticle quantum scattering in constant magnetic fields (Q2772079)

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 operator and the commentator method are repeated. The 3-body dispersive case is considered in detail. New proofs are given.NEWLINENEWLINEPropagation estimates allow to analyse the evolution of the perturbed system. Basics on propagation estimates concerning \(N\)-body systems without magnetic fields are given in chapter 4. In particular, they study estimates for the asymptotic energy, minimal and maximal velocities, and the asymptotic velocity along the field. These considerations are extended to systems in magnetic fields (chapter 5), i.e. to the geometrical analysis of the propagation of charged systems.NEWLINENEWLINEThe final results in the book are the proof of asymptotic completeness for several \(N\)-body systems in a constant magnetic field (chapter 6). The proofs are now well prepared by the previous analysis on the Mourre theory and on the propagation estimates. The asymptotic completeness is proved for three-dimensional strongly charged systems including long-range and short-range potentials, for 3-body dispersive systems in \(\mathbb{R}^3\), for two-dimensional charged systems, and for two-dimensional 3-body systems with at least one neutral pair.NEWLINENEWLINE Finally, a list of open problems is given for further directions of research in this field. One problem is to prove the asymptotic completeness for general \(N\)-body systems in a constant magnetic field.