Quantum scattering theory was the main area of my graduate and postgraduate research. My Ph.D. thesis problem was scattering theory for quantum particles in a constant magnetic field. After writing a couple of modest papers on my own (1993-94), I was lucky to strike a collaboration with Christian Gerard. In our three joint papers (1994-96), we were able to make significant progress on the problem and prove asymptotic completeness for large classes of N-particle systems.

Most papers in scattering theory refer to a chain of earlier results by various authors. It can be very difficult, even for an expert, to track down the entire argument. The literature is not always reliable - significant gaps and errors have been found in published work. For this reason, Christian and I decided to write up our results - those already published as well as those we were still producing - in a self-contained "review paper". The review paper eventually became a book. It was completed in the summer of 2000, more than four years after its conception.

From the preface

Table of contents

Quantum scattering theory is the subfield of quantum mechanics that deals with the large-time asymptotics of the solutions of the Schroedinger equation and with the structure of the continuous spectrum of the corresponding Schroedinger operator. One of its main problems is to prove (or disprove)

Our work was largely inspired by these developments: we set out to extend the new results on asymptotic completeness to the case of N-body systems in a constant magnetic field. Such systems are of considerable interest in quantum physics. There is a large body of research on the

The purpose of this book is twofold. Firstly, in Chapter 1 we provide a general introduction to the spectral theory of N-body magnetic Hamiltonians, aimed at a wider audience of mathematical physicists. Secondly, we present a proof of asymptotic completeness for large classes of magnetic Hamiltonians, namely for generic 3-body systems and for N-body systems whose all proper subsystems have nonzero total electric charge. The proof requires much more than simply applying the known methods in a slightly different situation. This book focuses on the new methods and techniques that are specific to the magnetic case. In particular, this includes an extension of the Mourre theory to "dispersive" Hamiltonians with a rather complicated structure (Chapter 3), and a geometrical analysis of the propagation of charged systems (Chapter 5). Our goal was to give a clear and reasonably self-contained presentation of the subject and to provide a solid foundation for further research.

The book is addressed mostly to researchers and graduate students in mathematical physics. We do expect the reader to be familiar with quantum mechanics, functional analysis, and modern PDE theory (especially with pseudodifferential calculus). A background in N-body scattering and abstract Mourre theory will be useful, but not indispensable. To the readers who wish to acquire such background we recommend the monograph by Derezinski and Gerard (mentioned above), and

Some of the results presented here were first published in a series of research articles (1994-96). However, much of the material, including all of our results in the 2-dimensional case and a large part of the geometrical analysis of Chapter 5, is published here for the first time. The Mourre theory for magnetic Hamiltonians (Chapter 3) has been completely reworked and rewritten, especially in the case we call ``dispersive".

- Fundamentals
- Introduction
- One-particle Hamiltonian
- Basic concepts of scattering theory
- Two-particle Hamiltonians
- N-particle Hamiltonians: an overview

- Hamiltonians
- Three-dimensional systems
- Two-dimensional systems
- Assumptions on the Hamiltonians
- Selfadjointness
- Weyl quantization
- Almost analytic extensions
- Direct integrals

- Reducing transformations
- Center of mass coordinates
- Magnetic translations
- Pseudomomentum of the center of mass
- Neutral systems
- Charged systems
- Cutoffs in the pseudomomentum
- Details of calculations
- One-particle magnetic Hamiltonians
- A neutral pair

- Bound and scattering states
- Charged systems
- Neutral systems

- Introduction
- Geometrical methods
- N-body geometry
- Channel Hamiltonians
- Definition
- Channel reducing transformations

- Partitions of unity
- Cylindrical partition of unity
- Conical partition of unity
- A few technical results

- The Mourre theory
- Local inequalities for operators
- Basics of the Mourre theory
- The strongly charged case
- Analytically fibered operators
- Two-particle neutral systems
- Three-dimensional case: basic results
- Three-dimensional case: the Mourre theory for H(k)
- Two-dimensional case: basic results

- Dispersive channels
- Two-dimensional case
- Three-dimensional case

- Three-dimensional dispersive systems
- The main results
- A few technical lemmas
- Proof of the Mourre estimate
- Estimating the second commutator

- Two-dimensional dispersive systems

- Basic propagation estimates
- Preliminaries
- Defining the dynamics
- Wave operators
- Time-dependent observables

- Asymptotic energy and maximal velocity estimates
- Asymptotic velocity along the field
- Minimal velocity estimates
- Abstract estimates
- Minimal velocity estimates for 3-particle systems

- Preliminaries
- Geometrical methods II
- The center of orbit observable
- Introduction
- Basic properties

- Admissible functions
- Center of charge coordinates
- Definition of admissible functions
- A geometric construction
- Construction of admissible functions

- Propagation estimates in the transversal direction
- The strongly charged case
- The general case

- The center of orbit observable
- Wave operators and scattering theory
- Channel identification operators
- Three-dimensional charged systems
- General considerations
- The short-range case
- The long-range case

- Three-dimensional dispersive systems
- Two-dimensional charged systems
- Two-dimensional systems with neutral pairs

- Open problems
- Appendix