Harmonic Analysis in Phase Space. (AM-122)

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Overview

This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.

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Editorial Reviews

Zentralblatt fur Mathematik
[This book] is a valiant attempt to present an account of [harmonic analysis in phase space], with an emphasis on the analysis-quantum mechanics and pseudodifferential operators.... The author has taken great pains to express himself clearly and ... the notation is consistent throughout.... The author should be congratulated on a very valuable addition to the library of harmonic analysis.
Zentralblatt für Mathematik

[This book] is a valiant attempt to present an account of [harmonic analysis in phase space], with an emphasis on the analysis-quantum mechanics and pseudodifferential operators.... The author has taken great pains to express himself clearly and ... the notation is consistent throughout.... The author should be congratulated on a very valuable addition to the library of harmonic analysis.
From the Publisher
"[This book] is a valiant attempt to present an account of [harmonic analysis in phase space], with an emphasis on the analysis-quantum mechanics and pseudodifferential operators.... The author has taken great pains to express himself clearly and ... the notation is consistent throughout.... The author should be congratulated on a very valuable addition to the library of harmonic analysis."—Zentralblatt für Mathematik
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Product Details

  • ISBN-13: 9780691085289
  • Publisher: Princeton University Press
  • Publication date: 3/1/1989
  • Series: Annals of Mathematics Studies Series , #12
  • Edition description: New Edition
  • Pages: 288
  • Product dimensions: 5.90 (w) x 8.90 (h) x 0.80 (d)

Table of Contents

Preface vii
Prologue: Some Matters of Notation 3
Chapter 1. The Heisenberg Group and Its Representations 9
1. Background from physics 9
Hamiltonian mechanics 10
Quantum mechanics 12
Quantization 15
2. The Heisenberg group 17
The automorphisms of the Heisenberg group 19
3. The Schrodinger representation 21
The integrated representation 23
Twisted convolution 25
The uncertainty principle 27
4. The Fourier-Wigner transform 30
Radar ambiguity functions 33
5. The Stone-von Neumann theorem 35
The group Fourier transform 37
6. The Fock-Bargmann representation 39
Some motivation and history 47
7. Hermite functions 51
8. The Wigner transform 56
9. The Laguerre connection 63
10. The nilmanifold representation 68
11. Postscripts 73
Chapter 2. Quantization and Pseudodifferential Operators 78
1. The Weyl correspondence 79
Covariance properties 83
Symbol classes 86
Miscellaneous remarks and examples 90
2. The Kohn-Nirenberg correspondence 93
3. The product formula 103
4. Basic pseudodifferential theory 111
Wave front sets 118
5. The Calderon-Vaillancourt theorems 121
6. The sharp Garding inequality 129
7. The Wick and anti-Wick correspondences 137
Chapter 3. Wave Packets and Wave Fronts 143
1. Wave packet expansions 144
2. A characterization of wave front sets 154
3. Analyticity and the FBI transform 159
4. Gabor expansions 164
Chapter 4. The Metaplectic Representation 170
1. Symplectic linear algebra 170
2. Construction of the metaplectic representation 177
The Fock model 180
3. The infinitesimal representation 185
4. Other aspects of the metaplectic representation 191
Integral formulas 191
Irreducible subspaces 194
Dependence on Planck's constant 195
The extended metaplectic representation 196
The Groenewold-van Hove theorems 197
Some applications 199
5. Gaussians and the symmetric space 200
Characterizations of Gaussians 206
6. The disc model 210
7. Variants and analogues 216
Restrictions of the metaplectic representation 216
U(n,n) as a complex symplectic group 217
The spin representation 220
Chapter 5. The Oscillator Semigroup 223
1. The Schrodinger model 223
The extended oscillator semigroup 234
2. The Hermite semigroup 236
3. Normalization and the Cayley transform 239
4. The Fock model 246
Appendix A. Gaussian Integrals and a Lemma on Determinants 256
Appendix B. Some Hilbert Space Results 260
Bibliography 265
Index 275
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