Symplectic Methods in Harmonic Analysis and in Mathematical Physics

  • Maurice A. de Gosson

Part of the Pseudo-Differential Operators book series (PDO, volume 7)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Symplectic Mechanics

    1. Front Matter
      Pages 1-1
    2. Maurice A. de Gosson
      Pages 3-17
    3. Maurice A. de Gosson
      Pages 19-30
    4. Maurice A. de Gosson
      Pages 31-40
    5. Maurice A. de Gosson
      Pages 41-50
    6. Maurice A. de Gosson
      Pages 51-63
    7. Maurice A. de Gosson
      Pages 65-76
  3. Harmonic Analysis in Symplectic Spaces

    1. Front Matter
      Pages 77-77
    2. Maurice A. de Gosson
      Pages 79-90
    3. Maurice A. de Gosson
      Pages 91-116
    4. Maurice A. de Gosson
      Pages 117-136
    5. Maurice A. de Gosson
      Pages 137-161
    6. Maurice A. de Gosson
      Pages 163-183
    7. Maurice A. de Gosson
      Pages 185-203
    8. Maurice A. de Gosson
      Pages 205-219
  4. Pseudo-differential Operators and Function Spaces

    1. Front Matter
      Pages 221-221
    2. Maurice A. de Gosson
      Pages 223-244
  5. Applications

    1. Front Matter
      Pages 245-245
    2. Maurice A. de Gosson
      Pages 247-260

About this book

Introduction

The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space is introduced and studied, where the main role is played by “Bopp operators” (also called “Landau operators” in the literature). This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors.

This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Keywords

Bopp operator deformation quantization pseudo-differential operator

Authors and affiliations

  • Maurice A. de Gosson
    • 1
  1. 1.Fak. MathematikUniversität WienWienAustria

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-7643-9992-4
  • Copyright Information Springer Basel AG 2011
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-7643-9991-7
  • Online ISBN 978-3-7643-9992-4
  • About this book