Pseudodifferential Operators and Spectral Theory

• 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.

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)