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 Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical 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 nonlinear 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 which provide a link to current research literature.
Readership: Graduate students and research mathematicians interested in semiclassical and microlocal methods in partial differential equations
Author(s): Maciej Zworski
Series: Graduate Studies in Mathematics, Vol. 138
Publisher: American Mathematical Society
Year: 2012
Language: English
Pages: C+xii+431+B
Cover
S Title
Semiclassical Analysis
Copyright
© 2012 by the American Mathematical Society
ISBN 978-0-8218-8320-4
QC174.17.D54Z96 2012 515-dc23
LCCN 2012010649
Contents
PREFACE
Chapter 1 INTRODUCTION
1.1. BASIC THEMES
1.1.1. PDE with small parameters.
1.1.2. Basic techniques
1.1.3. Microlocal analysis
1.1.4. Other directions
1.2. CLASSICAL AND QUANTUM MECHANICS
1.2.1. Observables
1.2.2. Dynamics
1.3. OVERVIEW
1.4. NOTES
Part 1 BASIC THEORY
Chapter 2 SYMPLECTIC GEOMETRY AND ANALYSIS
2.1. FLOWS
2.2. SYMPLECTIC STRUCTURE ON R^2n
2.3. SYMPLECTIC MAPPINGS
2.4. HAMILTONIAN VECTOR FIELDS
2.5. LAGRANGIAN SUBMANIFOLDS
2.6. NOTES
Chapter 3 FOURIER TRANSFORM, STATIONARY PHASE
3.1. FOURIER TRANSFORM ON S°
3.2. FOURIER TRANSFORM ON S'
3.3. SEMICLASSICAL FOURIER TRANSFORM
3.4. STATIONARY PHASE IN ONE DIMENSION
3.5. STATIONARY PHASE IN HIGHER DIMENSIONS
3.5.1. Quadratic phase function.
3.5.2. General phase function
3.5.3. Important Examples
3.6. OSCILLATORY INTEGRALS
3.7. NOTES
Chapter 4 SEMICLASSICAL QUANTIZATION
4.1. DEFINITIONS
4.1.1. Quantization rules
4.1.2. Quantization on S and S'
4.2. QUANTIZATION FORMULAS
4.2.1. Symbols depending only on x.
4.2.2. Linear symbols
4.2.3. Commutators
4.2.4. Exponentials of linear symbols
4.2.5. Exponentials of quadratic symbols
4.2.6. Conjugation by Fourier transform
4.3. COMPOSITION, ASYMPTOTIC EXPANSIONS
4.3.1. Composing symbols
4.3.2. Asymptotics
4.3.3. Transforming between different quantizations
4.3.4. Standard quantization
4.4. SYMBOL CLASSES
4.4.1. Order functions and symbol classes
4.4.2. Asymptotic series
4.4.3. Quantization
4.4.4. Semiclassical expansions in So.
4.4.5. More useful formulas
4.5. OPERATORS ON L^2
4.5.1. Symbols in S
4.5.2. Symbols in S and S.
4.6. COMPACTNESS
4.7. INVERSES, GARDING INEQUALITIES
4.7.1. Inverses
4.7.2. Garding inequalities
4.8. NOTES
Part 2 APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS
Chapter 5 SEMICLASSICAL DEFECT MEASURES
5.1. CONSTRUCTION, EXAMPLES
5.2. DEFECT MEASURES AND PDE
5.2.1. Properties of semiclassical defect measures
5.3. DAMPED WAVE EQUATION
5.3.1. Quantization and semiclassical defect measures on the torus.
5.3.2. A damped wave equation
5.3.3. Resolvent estimates
5.3.4. Energy decay
5.4. NOTES
Chapter 6 EIGENVALUES AND EIGENFUNCTIONS
6.1. THE HARMONIC OSCILLATOR
6.1.1. Eigenvalues and eigenfunctions of Po.
6.1.2. Higher dimensions, rescaling
6.1.3. Asymptotic distribution of eigenvalues
6.2. SYMBOLS AND EIGENFUNCTIONS
6.2.1. Concentration in phase space
6.2.2. Projections
6.3. SPECTRUM AND RESOLVENTS
6.4. WEYL'S LAW
6.5. NOTES
Chapter 7 ESTIMATES FOR SOLUTIONS OF PDE
7.1. CLASSICALLY FORBIDDEN REGIONS
7.2. TUNNELING
7.3. ORDER OF VANISHING
7.4. L°° ESTIMATES FOR QUASIMODES
7.4.1. Quasimodes
7.4.2. Preliminary estimates
7.4.3. Nondegeneracy, localization, and L°° bounds
7.4.4. Bounds for spectral clusters
7.5. SCHAUDER ESTIMATES
7.5.1. Littlewood-Paley decomposition
7.5.2. Holder continuity
7.5.3. Schauder estimates
7.6. NOTES
Part 3 ADVANCED THEORY AND APPLICATIONS
Chapter 8 MORE ON THE SYMBOL CALCULUS
8.1. BEALS'S THEOREM
8.2. REAL EXPONENTIATION OF OPERATORS
8.3. GENERALIZED SOBOLEV SPACES
8.3.1. Sobolev spaces compatible with symbols
8.3.2. Application: Estimates for eigenfunctions
8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, AND MICROLOCALITY
8.4.1. Tempered functions and operators, localization
8.4.2. Semiclassical wavefront sets
8.4.3. Essential support
8.4.4. Wavefront sets of localized functions.
8.4.5. Microlocality.
8.5. NOTES
Chapter 9 CHANGING VARIABLES
9.1. INVARIANCE, HALF-DENSITIES
9.1.1. Motivation, definitions
9.1.2. Operators on half-densities
9.1.3. Quantization and half-densities
9.2. CHANGING SYMBOLS
9.2.1. Changing variables and changing symbols
9.3. INVARIANT SYMBOL CLASSES
9.3.1. Classical symbols
9.3.2. Symbol calculus for S^m
9.3.3. Changing variables for S"
9.3.4. Sharp Garding inequality again
9.3.5. Beals's Theorem again
9.4. NOTES
Chapter 10 FOURIER INTEGRAL OPERATORS
10.1. OPERATOR DYNAMICS
10.1.1. Symbols in S.
10.1.2. Time-independent, elliptic symbols
10.1.3. Time-dependent elliptic symbols
10.2. AN INTEGRAL REPRESENTATION FORMULA
10.2.1. A microlocal representation
10.2.2. Construction of the phase function.
10.2.3. Construction of the amplitude
10.3. STRICHARTZ ESTIMATES
10.3.1. Strichartz estimates.
10.4. L^p ESTIMATES FOR QUASIMODES
10.4.1. Nondegeneracy, localization, and Lp bounds
10.4.2. Bounds for spectral clusters
10.5. NOTES
Chapter 11 QUANTUM AND CLASSICAL DYNAMICS
11.1. EGOROV'S THEOREM
11.2. QUANTIZING SYMPLECTIC MAPPINGS
11.2.1. More on symplectic matrices
11.2.2. Deformation of symplectomorphisms
11.2.3. Locally quantizing symplectomorphisms
11.2.4. Microlocal reformulation
11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS
11.3.1. Quantizing J.
11.3.2. Quantizing linear symplectic mappings
11.3.3. An explicit formula
11.4. EGOROV'S THEOREM FOR LONGER TIMES
11.4.1. Estimates for flows.
11.4.2. Egorov's Theorem for long times
11.5. NOTES
Chapter 12 NORMAL FORMS
12.1. OVERVIEW
12.2. NORMAL FORMS: REAL SYMBOLS
12.2.1. More symplectic geometry
12.2.2. Symbols of real principal type
12.2.3. L2 estimates and principal type
12.3. PROPAGATION OF SINGULARITIES
12.3.1. Propagation of wavefront sets
12.4. NORMAL FORMS: COMPLEX SYMBOLS
12.5. QUASIMODES, PSEUDOSPECTRA
12.5.1. Quasimodes and eigenvalues
12.5.2. Quasimodes for nonnormal operators
12.6. NOTES
Chapter 13 THE FBI TRANSFORM
13.1. MOTIVATION
13.2. COMPLEX ANALYSIS
13.2.1. Complex differential forms.
13.2.2. Quadratic forms
13.2.3. Symplectic geometry
13.2.4. Plurisubharmonic functions
13.3. FBI TRANSFORMS AND BERGMAN KERNELS
13.4. QUANTIZATION AND TOEPLITZ OPERATORS
13.5. APPLICATIONS
13.5.1. Approximation by multiplication
13.5.2. Characterization of WFh
13.5.3. Sobolev spaces
13.5.4. Positive forms in several complex variables
13.6. NOTES
Part 4 SEMICLASSICAL ANALYSIS ON MANIFOLDS
Chapter 14 MANIFOLDS
14.1. DEFINITIONS, EXAMPLES
14.1.1. Manifolds
14.1.2. Vector bundles
14.1.3. Riemannian manifolds
14.2. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS
14.2.1. Differential operators on manifolds
14.2.2. Pseudodifferential operators on manifolds.
14.2.3. Symbols of pseudodifferential operators
14.2.4. Properties of pseudodifferential operators on manifolds
14.2.5. Pseudodifferential operators and half-densities
14.2.6. PDE on manifolds
14.3. SCHRODINGER OPERATORS ON MANIFOLDS
14.3.1. Spectral theory
14.3.2. A functional calculus.
14.3.3. Trace class operators
14.3.4. Weyl's Law for compact manifolds
14.4. NOTES
Chapter 15 QUANTUM ERGODICITY
15.1. CLASSICAL ERGODICITY
15.2. A WEAK EGOROV THEOREM
15.3. WEYL'S LAW GENERALIZED
15.4. QUANTUM ERGODIC THEOREMS
15.5. NOTES
Part 5 APPENDICES
Appendix A NOTATION
A.1. BASIC NOTATION
A.2. FUNCTIONS, DIFFERENTIATION
A.3. OPERATORS
A.4. ESTIMATES
A.4.1. Use of constants.
A.4.2. Order estimates
A.5. SYMBOL CLASSES
Appendix B DIFFERENTIAL FORMS
B.1. DEFINITIONS
B.2. PUSH-FORWARDS AND PULL-BACKS
B.3. POINCARE'S LEMMA
B.4. DIFFERENTIAL FORMS ON MANIFOLDS
Appendix C FUNCTIONAL ANALYSIS
C.1. OPERATOR THEORY
C.1.1. Operators on distributions
C.1.2. Operators and inverses
C.2. SPECTRAL THEORY
C.2.1. Spectral theory for bounded operators
C.2.2. Spectral theory for unbounded operators.
C.2.3. Minimax formulas
C.3. TRACE CLASS OPERATORS
Appendix D FREDHOLM THEORY
D.1. GRUSHIN PROBLEMS
D.2. FREDHOLM OPERATORS
D.3. MEROMORPHIC CONTINUATION
NOTES FOR THE APPENDICES
Bibliography
Index
Back Cover
