Multiparticle Quantum Scattering in Constant Magnetic Fields

Christian Gerard and Izabella Laba

American Mathematical Society, Mathematical Surveys and Monographs, vol. 90, 2002.

For more information about the published version, or to purchase a copy, go to the AMS bookstore.

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

From the preface

This monograph is devoted to the spectral and scattering theory of quantum Hamiltonians describing systems of N interacting particles in an external constant magnetic field. Most of it consists of the results obtained by the authors between 1993 and 2000.

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) asymptotic completeness, which, roughly speaking, is a statement that all solutions of the Schroedinger equation under consideration must asymptotically follow certain prescribed patterns. (The precise mathematical formulation of this is given in Chapter 6.) There is a vast body of literature on this and other aspects of 2-particle scattering, see e.g., Reed and Simon, Methods of Modern Mathematical Physics vol. III, or Hormander, The analysis of Linear Partial Differential Operators vols. II, IV, for an overview. For N>2 particles, the problem becomes much more complicated. It was only in the last 20 years or so that the N-body scattering theory underwent a period of rapid development, beginning with the work of Enss in the late 1970's and culminating in the proof of N-body asymptotic completeness by Sigal-Soffer (1987) and Derezinski (1993), with significant contributions by many other authors, including Mourre, Perry-Sigal-Simon, Froese-Herbst, Graf, Isozaki, and others. We refer the reader to the monograph Scattering Theory of Classical and Quantum N-Particle Systems by J. Derezinski and C. Gerard for a more detailed account of that story and for a self-contained presentation of the results obtained in the 1980s and 90s.

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 quantum Hall effect; most of it assumes that there are no interactions between the particles save for the Pauli exclusion principle, but it is possible that at some point the scattering effects will have to be taken into account. In astrophysics, there is some evidence that strong magnetic fields exist on the surfaces of neutron stars and white dwarfs. "Quantum dots" are a prime example of quantum systems that can be significantly affected by magnetic fields of strength comparable to what can actually be achieved in existing laboratories. physicists have also been studying highly excited (Rydberg) atoms in magnetic fields, which offer an opportunity to study the phenomena of "quantum chaos". See e.g., Atoms in Strong Magnetic Fields, by H. Ruder, G. Wunner, H. Herold, and F. Geyer, Springer-Verlag 1994, for a survey of some of the recent (theoretical and experimental) work on the subject. There is also a growing interest in magnetic Hamiltonians among mathematicians and mathematical physicists. In particular, questions such as the stability of matter, eigenvalue and resonance asymptotics, and decay of eigenfunctions were recently addressed in the literature.

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 Commutator Methods and Spectral Theory of N-Body Hamiltonians by Amrein, Boutet-de-Monvel, and Georgescu. However, anyone willing to accept without proof the results of "standard" N-body scattering that we will invoke should also be able to follow all of our arguments. In fact, parts of this book (especially Chapters 1 and 2) may serve as an introduction to the N-body theory. We emphasize that no previous exposure to magnetic Schroedinger operators is required.

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".

Table of contents

  1. Fundamentals
    1. Introduction
      • One-particle Hamiltonian
      • Basic concepts of scattering theory
      • Two-particle Hamiltonians
      • N-particle Hamiltonians: an overview
    2. Hamiltonians
      • Three-dimensional systems
      • Two-dimensional systems
      • Assumptions on the Hamiltonians
      • Selfadjointness
      • Weyl quantization
      • Almost analytic extensions
      • Direct integrals
    3. 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
    4. Bound and scattering states
      • Charged systems
      • Neutral systems
  2. Geometrical methods
    1. N-body geometry
    2. Channel Hamiltonians
      • Definition
      • Channel reducing transformations
    3. Partitions of unity
      • Cylindrical partition of unity
      • Conical partition of unity
      • A few technical results
  3. The Mourre theory
    1. Local inequalities for operators
    2. Basics of the Mourre theory
    3. The strongly charged case
    4. Analytically fibered operators
    5. Two-particle neutral systems
      • Three-dimensional case: basic results
      • Three-dimensional case: the Mourre theory for H(k)
      • Two-dimensional case: basic results
    6. Dispersive channels
      • Two-dimensional case
      • Three-dimensional case
    7. Three-dimensional dispersive systems
      • The main results
      • A few technical lemmas
      • Proof of the Mourre estimate
      • Estimating the second commutator
    8. Two-dimensional dispersive systems
  4. Basic propagation estimates
    1. Preliminaries
      • Defining the dynamics
      • Wave operators
      • Time-dependent observables
    2. Asymptotic energy and maximal velocity estimates
    3. Asymptotic velocity along the field
    4. Minimal velocity estimates
      • Abstract estimates
      • Minimal velocity estimates for 3-particle systems
  5. Geometrical methods II
    1. The center of orbit observable
      • Introduction
      • Basic properties
    2. Admissible functions
      • Center of charge coordinates
      • Definition of admissible functions
      • A geometric construction
      • Construction of admissible functions
    3. Propagation estimates in the transversal direction
      • The strongly charged case
      • The general case
  6. Wave operators and scattering theory
    1. Channel identification operators
    2. Three-dimensional charged systems
      • General considerations
      • The short-range case
      • The long-range case
    3. Three-dimensional dispersive systems
    4. Two-dimensional charged systems
    5. Two-dimensional systems with neutral pairs
  7. Open problems
  8. Appendix