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 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 researchers.

Table of Contents:
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.

Review :
From the reviews of the second edition: "This is the second edition of Shubin's already classical book. It provides a fairly short, highly readable nice introduction to microlocal analysis, with emphasis on its application to spectral theory ... . For anybody who holds a first course in PDO and FIO we highly recommend ... . The book is very well written, in simple and direct language. From the very basics at the beginning, the reader reaches a fairly advanced graduate level at the end." (Tibor Odor, Acta Scientiarum Mathematicarum, Vol. 73, 2007)