Pseudodifferential Operators and Spectral Theory

I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.

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, Hörmander asymptotics of the spectral function and eigenvalues, and methods of approximate spectral projection. Exercises and problems are included to help the reader master the essential techniques. The book is written for a wide audience of mathematicians, be they interested students or researcher.

Second editon of Shubin's classic from 1987 as softcover

I. Foundations of ?DO Theory.1. Oscillatory Integrals.2. Fourier Integral Operators (Preliminaries).3. The Algebra of Pseudodifferential Operators and Their Symbols.4. Change of Variables and Pseudodifferential Operators on Manifolds.5. Hypoellipticity and Ellipticity.6. Theorems on Boundedness and Compactness of Pseudodifferential Operators.7. The Sobolev Spaces.8. The Fredholm Property, Index and Spectrum.- II. Complex Powers of Elliptic Operators.9. Pseudodifferential Operators with Parameter. The Resolvent.10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator.11. The Structure of the Complex Powers of an Elliptic Operator.12. Analytic Continuation of the Kernels of Complex Powers.13. The ?-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum.14. The Tauberian Theorem of Ikehara.15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem).- III. Asymptotic Behaviour of the Spectral Function.16. Formulation of the Hormander Theorem and Comments.17. Non-linear First Order Equations.18. The Action of a Pseudodifferential Operator on an Exponent.19. Phase Functions Defining the Class of Pseudodifferential Operators.20. The Operator exp(- it A).2l. Precise Formulation and Proof of the Hormander Theorem.22. The Laplace Operator on the Sphere.- IV. Pseudodifferential Operators in ?n.23. An Algebra of Pseudodifferential Operators in ?n.24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness.25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property.26. Essential Self-Adjointness. Discreteness of the Spectrum.27. Trace and TraceClass Norm.28. The Approximate Spectral Projection.29. Operators with Parameter.30. Asymptotic Behaviour ofthe Eigenvalues.- Appendix 1. Wave Fronts and Propagation of Singularities.- Appendix 2. Quasiclassical Asymptotics of Eigenvalues.- Appendix 3. Hilbert-Schmidt and Trace Class Operators.- A Short Guide to the Literature.- Index of Notation.