Author | Shubin, Mikhail A. author |
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Title | Pseudodifferential Operators and Spectral Theory [electronic resource] / by Mikhail A. Shubin |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2001 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-3-642-56579-3 |
Descript | XII, 288 p. online resource |
I. Foundations of ?DO Theory -- ยง 1. Oscillatory Integrals -- ยง 2. Fourier Integral Operators (Preliminaries) -- ยง 3. The Algebra of Pseudodifferential Operators and Their Symbols -- ยง 4. Change of Variables and Pseudodifferential Operators on Manifolds -- ยง 5. Hypoellipticity and Ellipticity -- ยง 6. Theorems on Boundedness and Compactness of Pseudodifferential Operators -- ยง 7. The Sobolev Spaces -- ยง 8. The Fredholm Property, Index and Spectrum -- II. Complex Powers of Elliptic Operators -- ยง 9. Pseudodifferential Operators with Parameter. The Resolvent -- ยง 10. Definition and Basic Properties of the Complex Powers of an Elliptic Operator -- ยง 11. The Structure of the Complex Powers of an Elliptic Operator -- ยง 12. Analytic Continuation of the Kernels of Complex Powers -- ยง 13. The ?-Function of an Elliptic Operator and Formal Asymptotic Behaviour of the Spectrum -- ยง 14. The Tauberian Theorem of Ikehara -- ยง 15. Asymptotic Behaviour of the Spectral Function and the Eigenvalues (Rough Theorem) -- III. Asymptotic Behaviour of the Spectral Function -- ยง 16. Formulation of the Hormander Theorem and Comments -- ยง 17. Non-linear First Order Equations -- ยง 18. The Action of a Pseudodifferential Operator on an Exponent -- ยง 19. Phase Functions Defining the Class of Pseudodifferential Operators -- ยง 20. The Operator exp(โ it A) -- ยง 2l. Precise Formulation and Proof of the Hormander Theorem -- ยง 22. The Laplace Operator on the Sphere -- IV. Pseudodifferential Operators in ?n -- ยง 23. An Algebra of Pseudodifferential Operators in ?n -- ยง 24. The Anti-Wick Symbol. Theorems on Boundedness and Compactness -- ยง 25. Hypoellipticity and Parametrix. Sobolev Spaces. The Fredholm Property -- ยง 26. Essential Self-Adjointness. Discreteness of the Spectrum -- ยง 27. Trace and Trace Class Norm -- ยง 28. The Approximate Spectral Projection -- ยง 29. Operators with Parameter -- ยง 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