Provides an introduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics. Written for a wide audience of mathematicians, be they interested students or researchers. Softcover.
Author(s): M.A. Shubin, S.I. Andersson
Series: Springer Series in Soviet Mathematics
Edition: 2ed.
Publisher: Springer
Year: 2001
Language: English
Pages: 302
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface to the Second Edition......Page 6
Preface to the Russian Edition......Page 8
Preface to the English Edition......Page 11
Table of Contents......Page 12
1. Oscillatory Integrals......Page 14
2. Fourier Integral Operators (Preliminaries)......Page 23
3. The Algebra of Pseudodifferential Operators and Their Symbols......Page 29
4. Change of Variables and Pseudodifferential Operators on Manifolds......Page 44
5. Hypoellipticity and Ellipticity......Page 51
6. Theorems on Boundedness and Compactness of Pseudodifferential Operators......Page 59
7. The Sobolev Spaces......Page 65
8. The Fredholm Property, Index and Spectrum......Page 78
9. Pseudodifferential Operators with Parameter. The Resolvent......Page 90
10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator......Page 100
11. The Structure of the Complex Powers of an Elliptic Operator......Page 107
12. Analytic Continuation of the Kernels of Complex Powers......Page 115
13. The C-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum......Page 125
14. The Tauberian Theorem of Ikehara......Page 133
15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem)......Page 141
16. Formulation of the Hormander Theorem and Comments......Page 146
17. Non-linear First Order Equations......Page 147
18. The Action of a Pseudodifferential Operator on an Exponent......Page 154
19. Phase Functions Defining the Class of Pseudodifferential Operators......Page 160
20. The Operator exp(-itA)......Page 163
21. Precise Formulation and Proof of the Hormander Theorem......Page 169
22. The Laplace Operator on the Sphere......Page 177
23. An Algebra of Pseudodifferential Operators in IR^n.......Page 188
24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness......Page 199
25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property......Page 206
26. Essential Self-Adjointness. Discreteness of the Spectrum......Page 210
27. Trace and Trace Class Norm......Page 215
28. The Approximate Spectral Projection......Page 219
29. Operators with Parameter......Page 228
30. Asymptotic Behaviour of the Eigenvalues......Page 236
Appendix 1. Wave Fronts and Propagation of Singularities......Page 242
Appendix 2. Quasiclassical Asymptotics of Eigenvalues......Page 253
Appendix 3. Hilbert-Schmidt and Trace Class Operators......Page 270
A Short Guide to the Literature......Page 282
Bibliography......Page 288
Index of Notation......Page 298
Subject Index......Page 300
