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THE PHYSICS TEACHER ◆ Vol. 40, March 2002174 Robert J. Dufresne Robert J. Dufresne is a Research Assistant Professor in the Department of Physics at the University of Massachusetts. He received his Ph.D. in Nuclear Theory from UMass in 1987. His research interests inc lude assessment, instructional technology, and models of cognition. He is the coordinator of the Assessing-to-Learn materials development and teacher enhancement project. Department of Physics, Box 34525 University of Massachusetts Amherst, MA 01003–4525; dufresne@physics.
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SEMICLASSICAL ANALYSIS
Maciej Zworski
Department of Mathematics
University of California, BerkeleyPREFACE
This book originated with a course I taught at UC Berkeley during the spring
of 2003, with class notes taken by my colleague Lawrence C. Evans. Various
versions of these notes have been available on-line as the Evans-Zworski
Lecture notes on semiclassical analysis and our original intention was to
use them as the basis of a coauthored book. Craig Evans’s contributions
to the current manuscript can be recognized by anybody familiar with his
popular PDE text [E]. In the end, the scope of the project and other
commitments prevented Craig Evans from participating fully in the nal
stages of the e ort, and he decided to withdraw from the responsibility of
authorship, generously allowing me to make use of the contribution he has
already made. I and my readers owe him a great debt, for this book would
never have appeared without his participation.
Semiclassical analysis provides PDE techniques based on the classical-
quantum (particle-wave) correspondence. These techniques include such well
known tools as geometric optics and the WKB approximation. Examples of
problems studied in this subject are high energy eigenvalue asymptotics or
e ective dynamics for solutions of evolution equations. From the mathemat-
ical point of view, semiclassical analysis is a branch of microlocal analysis
which, broadly speaking, applies harmonic analysis and symplectic geometry
to the study of linear and non-linear PDE.
The book is intended to be a graduate level text introducing readers
to semiclassical and microlocal methods in PDE. It is augmented in later
chapters with many specialized advanced topics. Readers are expected to
have reasonable familiarity with standard PDE theory (as recounted, for
example, in Parts I and II of [E]), as well as a basic understanding of linear
34 PREFACE
functional analysis. On occasion familiarity with di erential forms will also
prove useful.
Several excellent treatments of semiclassical analysis have appeared re-
cently. The book [D-S] by Dimassi and Sj ostrand starts with the WKB-
method, develops the general semiclassical calculus, and then provides high
tech spectral asymptotics. Martinez [M] provides a systematic development
of FBI transform techniques, with applications to microlocal exponential
estimates and to propagation estimates. This text is intended as a more
elementary, but much broader, introduction. Except for the general symbol
calculus, for which we followed Chapter 7 of [D-S], there is little overlap
with these other two texts, or with the in uential books by Hel er [ He] and
by Robert [R]. Guillemin and Sternberg [G-St] o er yet another perspec-
tive on the subject, very much complementary to that given here. Their
notes concentrate on global and functorial aspects of semiclassical analy-
sis, in particular on the theory of Fourier integral operators and on trace
formulas.
The approach to semiclassical analysis presented here is in uenced by my
long collaboration with Johannes Sj ostrand. I would like to thank him for
sharing his philosophy and insights over the years. I rst learned microlocal
analysis from Richard Melrose, Victor Guillemin and Gunther Uhlmann and
it is a pleasure to acknowledge my debt to them. Discussions of semiclassical
physics and chemistry with Stephane Nonnenmacher, Paul Brumer, William
H. Miller and Robert Littlejohn have been enjoyable and valuable. They
have added a lot to my appreciation of the subject.
I am especially grateful to Stephane Nonnenmacher, Semyon Dyatlov,
Claude Zuily, Oran Gannot, Xi Chen, Hans Christianson, Je Galkowski,
Justin Holmer, Long Jin, Gordon Lino and Steve Zelditch for their very
careful reading of the earlier versions this book and for their many valuable
comments and corrections.
My thanks also to Faye Yeager for typing the original lecture notes and
to Jonathan Dorfman for T X advice. Stephen Moye at the AMS providedE
fantastic help on deeper T X issues.E
I will maintain on my website at the UC Berkeley Mathematics De-
partment http://math.berkeley.edu/~zworski a list of errata and cor-
rections. Please let me know about any errors you nd.
I have been supported by NSF grants during the writing of this book,
most recently by NSF grant DMS-0654436.
Maciej ZworskiContents
Preface 3
Chapter 1. Introduction 11
x1.1. Basic themes 11
x1.2. Classical and quantum mechanics 13
x1.3. Overview 15
x1.4. Notes 18
Part 1. BASIC THEORY
Chapter 2. Symplectic geometry and analysis 21
x2.1. Flows 21
2nx2.2. Symplectic structure onR 22
x2.3. mappings 24
x2.4. Hamiltonian vector elds 28
x2.5. Lagrangian submanifolds 31
x2.6. Notes 34
Chapter 3. Fourier transform, stationary phase 35
x3.1. Fourier transform onS 35
0x3.2. Fourier onS 43
x3.3. Semiclassical Fourier transform 46
x3.4. Stationary phase in one dimension 48
x3.5. phase in higher dimensions 54
x3.6. Oscillatory integrals 60
56 Contents
x3.7. Notes 62
Chapter 4. Semiclassical quantization 63
x4.1. De nitions 64
x4.2. Quantization formulas 67
x4.3. Composition, asymptotic expansions 73
x4.4. Symbol classes 80
2x4.5. Operators on L 89
x4.6. Compactness 95
x4.7. Inverses, Garding inequalities 98
x4.8. Notes 104
Part 2. APPLICATIONS TO PARTIAL DIFFERENTIAL
EQUATIONS
Chapter 5. Semiclassical defect measures 107
x5.1. Construction, examples 107
x5.2. Defect measures and PDE 112
x5.3. Damped wave equation 114
x5.4. Notes 125
Chapter 6. Eigenvalues and eigenfunctions 127
x6.1. The harmonic oscillator 127
x6.2. Symbols and eigenfunctions 132
x6.3. Spectrum and resolvents 137
x6.4. Weyl’s Law 140
x6.5. Notes 144
Chapter 7. Estimates for solutions of PDE 145
x7.1. Classically forbidden regions 146
x7.2. Tunneling 149
x7.3. Order of vanishing 154
1x7.4. L estimates for quasimodes 158
x7.5. Schauder estimates 164
x7.6. Notes 173
Part 3. ADVANCED THEORY AND APPLICATIONS
Chapter 8. More on the symbol calculus 177Contents 7
x8.1. Beals’s Theorem 177
x8.2. Real exponentiation of operators 183
x8.3. Generalized Sobolev spaces 188
x8.4. Wavefront sets, essential support, microlocality 193
x8.5. Notes 202
Chapter 9. Changing variables 203
x9.1. Invariance, half-densities 203
x9.2. Changing symbols 209
x9.3. Invariant symbol classes 212
x9.4. Notes 222
Chapter 10. Fourier integral operators 223
x10.1. Operator dynamics 224
x10.2. An integral representation formula 230
x10.3. Strichartz estimates 239
px10.4. L estimates for quasimodes 244
x10.5. Notes 248
Chapter 11. Quantum and classical dynamics 249
x11.1. Egorov’s Theorem 249
x11.2. Quantizing symplectic mappings 255
x11.3. Quantizing linear symplectic mappings 261
x11.4. Egorov’s Theorem for longer times 268
x11.5. Notes 275
Chapter 12. Normal forms 277
x12.1. Overview 277
x12.2. Normal forms: real symbols 279
x12.3. Propagation of singularities 283
x12.4. Normal forms: complex symbols 286
x12.5. Quasimodes, pseudospectra 290
x12.6. Notes 293
Chapter 13. The FBI transform 295
x13.1. Motivation 295
x13.2. Complex analysis 297
x13.3. FBI transform and Bergman kernels 3068 Contents
x13.4. Quantization and Toeplitz operators 315
x13.5. Applications 325
x13.6. Notes 339
Part 4. SEMICLASSICAL ANALYSIS ON MANIFOLDS
Chapter 14. Manifolds 343
x14.1. De nitions, examples 343
x14.2. Pseudodi erential operators on manifolds 349
x14.3. Schr odinger operators on manifolds 358
x14.4. Notes 366
Chapter 15. Quantum ergodicity 367
x15.1. Classical ergodicity 368
x15.2. A weak Egorov Theorem 370
x15.3. Weyl’s Law generalized 372
x15.4. Quantum ergodic theorems 374
x15.5. Notes 381
Part 5. APPENDICES
Appendix A. Notation 385
xA.1. Basic notation 385
xA.2. Functions, di erentiation 387
xA.3. Operators 389
xA.4. Estimates 390
xA.5. Symbol classes 391
Appendix B. Di erential forms 393
xB.1. De nitions 393
xB.2. Push-forwards and pull-backs 396
xB.3. Poincare’s Lemma 398
xB.4. Di erential forms on manifolds 399
Appendix C. Functional analysis 401
xC.1. Operator theory 401
xC.2. Spectral theory 405
xC.3. Trace class operators 413
Appendix D. Fredholm theory 417Contents 9
xD.1. Grushin problems 417
xD.2. Fredholm operators 418
xD.3. Meromorphic continuation 420
Bibliography 423
Index 429