Pseudodifferential Operators and Spectral Theory
Published on 27. February 1987
Book
Hardback
X, 278 pages
978-3-540-13621-7 (ISBN)
Description
The theory of pseudo differential operators (abbreviated PD~) is compara tively young; in its modern form it was created in the mid-sixties. The progress achieved with its help, however, has been so essential that without PD~ it would indeed be difficult to picture modern analysis and mathematical physics. PD~ are of particular importance in the study of elliptic equations. Even the simplest operations on elliptic operators (e. g. taking the inverse or the square root) lead out of the class of differential operators but will, under reasonable assumptions, preserve the class of PD~. A significant role is played by PD~ in the index theory for elliptic operators, where PD~ are needed to extend the class of possible deformations of an operator. PD~ appear naturally in the reduction to the boundary for any elliptic boundary problem. In this way, PD~ arise not as an end-in-themselves, but as a powerful and natural tool for the study of partial differential operators (first and foremost elliptic and hypo elliptic ones). In many cases, PD~ allow us not only to establish new theorems but also to have a fresh look at old ones and thereby obtain simpler and more transparent formulations of already known facts. This is, for instance, the case in the theory of Sobolev spaces. A natural generalization of PD~ are the Fourier integral operators (abbreviatedFIO), the first version ofwhich was the Maslov canonical operator.
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M.A. Shubin
Pseudodifferential Operators and Spectral Theory
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Content
I. Foundations of PDO 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 Hörmander 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(-itA).- §21. Precise Formulation and Proof of the Hörmander Theorem.- §22. The Laplace Operator on the Sphere.- IV. Pseudodifferential Operators in ?n.- §23. The Algebra of Pseudodifferential Operators in ?n.- § 24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness.- §25. Hypoellipticity and the 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 Operator.- §29. Operators with Parameter.- §30. Asymptotic Behaviour of the Eigenvalues.- Appendix 1. Wave Fronts and Propagation of Singularities.- Appendix 2. Quasiclassical Asymptotic Behaviour of Eigenvalues.- Appendix 3. Hilbert-Schmidt and Trace Class Operators.- A Short Guide to the Literature.- Index of Notation.