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Pseudodifferential Operators and Spectral Theory


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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 Trace Class 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.

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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)

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