Semiclassical analysis:
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| 100 | 1 | |a Zworski, Maciej |d 1963- |e Verfasser |0 (DE-588)102713596X |4 aut | |
| 245 | 1 | 0 | |a Semiclassical analysis |c Maciej Zworski |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2012 | |
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| 490 | 1 | |a Graduate studies in mathematics |v 138 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: Semiclassical analysis
Autor: Zworski, Maciej
Jahr: 2012
Contents
Preface
xi
Chapter 1. Introduction 1
§1.1. Basic themes 1
§1.2. Classical and quantum mechanics 3
§1.3. Overview 5
§1.4. Notes 9
Part 1. BASIC THEORY
Chapter 2. Symplectic geometry and analysis 13
§2.1. Flows 13
§2.2. Symplectic structure on R2n 14
§2.3. Symplectic mappings 16
§2.4. Hamiltonian vector fields 20
§2.5. Lagrangian submanifolds 23
§2.6. Notes 26
Chapter 3. Fourier transform, stationary phase 27
§3.1. Fourier transform on 5? 27
§3.2. Fourier transform on 5f 35
§3.3. Semiclassical Fourier transform 38
§3.4. Stationary phase in one dimension 40
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 I? 81
§4.6. Compactness 87
§4.7. Inverses, Gärding 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. WeyPs 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
CONTENTS vii
Part 3. ADVANCED THEORY AND APPLICATIONS
Chapter 8. More on the symbol calculus 171
§8.1. Beals s Theorem 171
§8.2. Real exponentiation of Operators 177
§8.3. Generalized Sobolev Spaces 182
§8.4. Wavefront sets, essential support, and microlocality 187
§8.5. Notes 196
Chapter 9. Changing variables 197
§9.1. Invariance, half-densities 197
§9.2. Changing symbols 203
§9.3. Invariant symbol classes 206
§9.4. Notes 217
Chapter 10. Fourier integral Operators 219
§10.1. Operator dynamics 220
§10.2. An integral representation formula 226
§10.3. Strichartz estimates 235
§10.4. LP estimates for quasimodes 240
§10.5. Notes 244
Chapter 11. Quantum and classical dynamics 245
§11.1. Egorov s Theorem 245
§11.2. Quantizing symplectic mappings 251
§11.3. Quantizing linear symplectic mappings 257
§11.4. Egorov s Theorem for longer times 264
§11.5. Notes 271
Chapter 12. Normal forms 273
§12.1. Overview 273
§12.2. Normal forms: real symbols 275
§12.3. Propagation of singularities 279
§12.4. Normal forms: complex symbols 282
§12.5. Quasimodes, pseudospectra 286
§12.6. Notes 289
viii______________________________________________________CONTENTS
Chapter 13. The FBI transform 291
§13.1. Motivation 291
§13.2. Complex analysis 293
§13.3. FBI transforms and Bergman kerneis 302
§13.4. Quantization and Toeplitz Operators 311
§13.5. Applications 321
§13.6. Notes 336
Part 4. SEMICLASSICAL ANALYSIS ON MANIFOLDS
Chapter 14. Manifolds 339
§14.1. Defmitions, examples 339
§14.2. Pseudodifferential Operators on manifolds 345
§14.3. Schrödinger Operators on manifolds 354
§14.4. Notes 362
Chapter 15. Quantum ergodicity 365
§15.1. Classical ergodicity 366
§15.2. A weak Egorov Theorem 368
§15.3. Weyl s Law generalized 370
§15.4. Quantum ergodic theorems 372
§15.5. Notes 379
Part 5. APPENDICES
Appendix A. Notation 383
§A.l. Basic notation 383
§A.2. Functions, differentiation 385
§A.3. Operators 387
§A.4. Estimates 388
§A.5. Symbol classes 389
Appendix B. Differential forms 391
§B.l. Defmitions 391
§B.2. Push-forwards and pull-backs 394
§B.3. Poincare s Lemma 396
§B.4. Differential forms on manifolds 397
CONTENTS_______________________________________________________ix
Appendix C. Functional analysis 399
§C.l. Operator theory 399
§C.2. Spectral theory 403
§C3. Trace class Operators 411
Appendix D. Fredholm theory 415
§D.l. Grushin problems 415
§D.2. Fredholm Operators 416
§D.3. Meromorphic continuation 418
Bibliography 421
Index 427
|
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| spelling | Zworski, Maciej 1963- Verfasser (DE-588)102713596X aut Semiclassical analysis Maciej Zworski Providence, RI American Mathematical Society 2012 XII, 431 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 138 Includes bibliographical references and index Mathematik Quantentheorie Quantum theory Mathematics Differential equations, Partial Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics msc Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators msc Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions msc Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods msc Quantentheorie (DE-588)4047992-4 gnd rswk-swf Quasiklassische Näherung (DE-588)4296820-3 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Quantentheorie (DE-588)4047992-4 s Quasiklassische Näherung (DE-588)4296820-3 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Graduate studies in mathematics 138 (DE-604)BV009739289 138 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025237684&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zworski, Maciej 1963- Semiclassical analysis Graduate studies in mathematics Mathematik Quantentheorie Quantum theory Mathematics Differential equations, Partial Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics msc Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators msc Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions msc Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods msc Quantentheorie (DE-588)4047992-4 gnd Quasiklassische Näherung (DE-588)4296820-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4047992-4 (DE-588)4296820-3 (DE-588)4044779-0 |
| title | Semiclassical analysis |
| title_auth | Semiclassical analysis |
| title_exact_search | Semiclassical analysis |
| title_full | Semiclassical analysis Maciej Zworski |
| title_fullStr | Semiclassical analysis Maciej Zworski |
| title_full_unstemmed | Semiclassical analysis Maciej Zworski |
| title_short | Semiclassical analysis |
| title_sort | semiclassical analysis |
| topic | Mathematik Quantentheorie Quantum theory Mathematics Differential equations, Partial Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics msc Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators msc Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators msc Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions msc Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods msc Quantentheorie (DE-588)4047992-4 gnd Quasiklassische Näherung (DE-588)4296820-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Mathematik Quantentheorie Quantum theory Mathematics Differential equations, Partial Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods Quasiklassische Näherung Partielle Differentialgleichung |
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