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M.A. Shubin, "Pseudodifferential Operators and Spectral Theory"

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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