Pseudodifferential Operators and Spectral Theory
| Date: 21 April 2011, 16:16
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This is the second edition of Shubin's classical book. It provides an introduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics. The applications discussed include complex powers of elliptic operators, Hormander asymptotics of the spectral function and eigenvalues, and method of approximate spectral projection. Exercises and problems are included to help the reader master the essential techniques. This book is written for a wide audience of mathematicians, be they interested students or researchers.
Table of Contents
Chapter I. Foundations of PDO Theory 1
1. Oscillatory Integrals 1
2. Fourier Integral Operators (Preliminaries) 10
3. The Algebra of Pseudodifferential Operators and Their Symbols 16
4. Change of Variables and Pseudodifferential Operators on Manifolds 31
5. Hypoellipticity and Ellipticity 38
6. Theorems on Boundedness and Compactness of Pseudodifferential Operators 46
7. The Sobolev Spaces 52
8. The Fredholm Property, Index and Spectrum 65
Chapter II. Complex Powers of Elliptic Operators 77
9. Pseudodifferential Operators with Parameter. The Resolvent 77
10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator 87
11. The Structure of the Complex Powers of an Elliptic Operator 94
12. Analytic Continuation of the Kernels of Complex Powers 102
13. The C-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum 112
14. The Tauberian Theorem of Ikehara 120
15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) 128
Chapter III. Asymptotic Behaviour of the Spectral Function 133
16. Formulation of the Hormander Theorem and Comments 133
17. Non-linear First Order Equations 134
18. The Action of a Pseudodifferential Operator on an Exponent 141
19. Phase Functions Defining the Class of Pseudodifferential Operators 147
20. The Operator exp(-itA) 150
21. Precise Formulation and Proof of the Hormander Theorem 156
22. The Laplace Operator on the Sphere 164
Chapter IV. Pseudodifferential Operators in IR^n 175
23. An Algebra of Pseudodifferential Operators in IR^n. 175
24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness 186
25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property 193
26. Essential Self-Adjointness. Discreteness of the Spectrum 197
27. Trace and Trace Class Norm 202
28. The Approximate Spectral Projection 206
29. Operators with Parameter 215
30. Asymptotic Behaviour of the Eigenvalues 223
Appendix 1. Wave Fronts and Propagation of Singularities 229
Appendix 2. Quasiclassical Asymptotics of Eigenvalues 240
Appendix 3. Hilbert-Schmidt and Trace Class Operators 257
A Short Guide to the Literature 269
Bibliography 275
Index of Notation 285
Subject Index 287
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