相空间中的调和分析 [Harmonic Analysis in Phase Space] 下载 mobi epub pdf 电子书 2024

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The main point of Chapter 2 is the development of the Weyl calculusof pseudodifferential operators.

内容简介

The phrase "harmonic analysis in phase space" is a concise if somewhatinadequate name for the area of analysis on Rn that involves the Heisenberggroup，quantization，the Weyl operational calculus，the metaplectic representa-tion，wave packets，and related concepts： it is meant to suggest analysis on theconfiguration space Rn done by working in the phase space Rn x Rn. The ideasthat fall under this rubric have originated in several different fidds——Fourieranalysis，partial differential equations，mathematical physics，representationtheory，and number theory，among others.

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目录

Preface
Prologue. Some Matters of Notation
CHAPTER 1.
THE HEISENBERG GROUP AND ITS REPRESENTATIONS
1. Background from physics
Hamiltonian mechanics， 10. Quantum mechanics， 12. Quantization， 15.
2. The Heisenberg group
The automorphisms of the Heisenberg group， 19.
3. The SchrSdinger representation
The integrated representation， 23. Twisted convolution， 25.
The uncertainty principle， 27.
4. The Fourier-Wigner transform
Radar ambiguity functions， 33.
5. The Stone-von Neumann theorem
The group Fourier transform， 37.
6. The Fock-Bargmann representation
Some motivation and history， 47.
7. Hermite functions
8. The Wigner transform
9. The Laguerre connection
10. The nilmanifold representation
11. Postscripts

CHAPTER 2.
QUANTIZATION AND PSEUDODIFFERENTIAL OPERATORS
1. The Weyl correspondence
Covariance properties， 83. Symbol classes， 86. Miscellaneous remarks
and examples， 90.
2. The Kohn-Nirenberg correspondence
3. The product formula
4. Basic pseudodifferential theory
Wave front sets， 118.
5. The CalderSn-Vaillancourt theorems
6. The sharp Garding inequality
7. The Wick and anti-Wick correspondences

CHAPTER 3.
WAVE PACKETS AND WAVE FRONTS
1. Wave packet expansions
2. A characterization of wave front sets
3. Analyticity and the FBI transform
4. Gabor expansions

CHAPPTER 4.
THE METAPLECTIC REPRESENTATION
1. Symplectic linear algebra
2. Construction of the metaplectic representation
The Fock model， 180.
3. The infinitesimal representation
4. Other aspects of the metaplectic representation
Integral formulas， 191. Irreducible subspaces， 194. Dependence on
Plancks constant， 195. The extended metaplectic representation， 196.
The Groenewold-van Hove theorems， 197. Some applications， 199.
5. Gaussians and the symmetric space
Characterizations of Gaussians， 206.
6. The disc model
7. Variants and analogues
Restrictions of the metaplectic representation， 216. U(n，n) as a complex
symplectic group， 217. The spin representation， 220.

CHAPTER 5.
THE OSCILLATOR SEMIGROUP
1. The SchrSdinger model
The extended oscillator semigroup， 234.
2. The Hermite semigroup
3. Normalization and the Cayley transform
4. The Fock model
Appendix A. Gaussian Integrals and a Lemma on Determinants
Appendix B. Some Hilbert Space Results
Bibliography
Index

前言/序言

　　The phrase "harmonic analysis in phase space" is a concise if somewhatinadequate name for the area of analysis on Rn that involves the Heisenberggroup， quantization， the Weyl operational calculus， the metaplectic representa-tion， wave packets， and related concepts： it is meant to suggest analysis on theconfiguration space Rn done by working in the phase space Rn x Rn. The ideasthat fall under this rubric have originated in several different fidds——Fourieranalysis， partial differential equations， mathematical physics， representationtheory， and number theory， among others. As a result， although these ideas areindividually well known to workers in such fields， their close kinship and thecross-fertilization they can provide have often been insufficiently appreciated.One of the principal objectives of this monograph is to give a coherent accountof this material， comprising not just an efficient tour of the major avenues butalso an exploration of some picturesque byways.
　　Here is a brief guide to the main features of the book.　Readers shouldbegin by perusing the Prologue and perhaps refreshing their knowledge aboutGaussian integrals by glancing at Appendix A.
　　Chapter I is devoted to the description of the representations of the Heisen-berg group and various integral transforms and special functions associated tothem， with motivation from physics. The material in the first eight sectionsis the foundation for all that follows， although readers who wish to proceedquickly to pseudodifferential operators can skip Sections 1.5-1.7.
　　The main point of Chapter 2 is the development of the Weyl calculusof pseudodifferential operators. As a tool for studying differential equations，the Weyl calculus is essentially equivalent to the standard Kohn-Nirenbergcalculus——in fact， this equivalence is the principal result of Section 2.2——but itis somewhat more elegant and more natural from the point of view of harmonicanalysis. Its close connection with the Heisenberg group yields some insightswhich are useful in the proofs of the Calder6n-Vaillancourt （0，0） estimate andthe sharp Grding inequality in Sections 2.5 and 2.6 and in the argumentsof Section 3.1. Since my aim is to provide a reasonably accessible introduc-tion rather than to develop a general theory （in contrast to H6rmander [70]），I mainly restrict attention to the standard symbol classes S.

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