vi CONTENTS
§3.5. Stationary phase in higher dimensions 46
§3.6. Oscillatory integrals 52
§3.7. Notes 54
Chapter 4. Semiclassical quantization 55
§4.1. Definitions 56
§4.2. Quantization formulas 59
§4.3. Composition, asymptotic expansions 65
§4.4. Symbol classes 72
§4.5. Operators on
L2
81
§4.6. Compactness 87
§4.7. Inverses, arding inequalities 90
§4.8. Notes 96
Part 2. APPLICATIONS TO PARTIAL DIFFERENTIAL
EQUATIONS
Chapter 5. Semiclassical defect measures 99
§5.1. Construction, examples 99
§5.2. Defect measures and PDE 104
§5.3. Damped wave equation 106
§5.4. Notes 117
Chapter 6. Eigenvalues and eigenfunctions 119
§6.1. The harmonic oscillator 119
§6.2. Symbols and eigenfunctions 124
§6.3. Spectrum and resolvents 129
§6.4. Weyl’s Law 132
§6.5. Notes 137
Chapter 7. Estimates for solutions of PDE 139
§7.1. Classically forbidden regions 140
§7.2. Tunneling 143
§7.3. Order of vanishing 148
§7.4.
L∞
estimates for quasimodes 152
§7.5. Schauder estimates 158
§7.6. Notes 167
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