Semiclassical Analysis

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Semiclassical Analysis


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     © 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


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