Fourier integral operators

J.J. Duistermaat0817638210, 9780817681074, 9780817638214, 0817681078, 3764338210, 9783764338213

This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods. Familiarity with analysis (distributions and Fourier transformation) and differential geometry is useful. Additionally, this book is designed for a one-semester introductory course on Fourier integral operators aimed at a broad audience.

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Table of contents :
Contents……Page 3
Preface……Page 4
0. Introduction……Page 7
1.1. Distribution densities on manifolds……Page 14
1.2. The method of stationary phase……Page 16
1.3. The wave front set of a distribution……Page 21
2.1. Symbols……Page 29
2.2. Distributions defined by oscillatory integrals……Page 34
2.3. Oscillatory integrals with nondegenerate phase functions……Page 36
2.4. Fourier integral operators (local theory)……Page 43
2.5. Pseudodifferential operators in R^n……Page 47
3.1. Vector fields……Page 51
3.2. Differential forms……Page 55
3.3. The canonical 1- and 2-form in T*{X)……Page 62
3.4. Symplectic vector spaces……Page 64
3.5. Symplectic differential geometry……Page 75
3.6. Lagrangian manifolds……Page 81
3.7. Conic Lagrangian manifolds……Page 88
3.8. Classical mechanics and variational calculus……Page 92
4.1. Invariant definition of the principal symbol……Page 97
4.2. Global theory of Fourier integral operators……Page 103
4.3. Products with vanishing principal symbol……Page 110
4.4. L2-continuity……Page 113
5.1. The Cauchy problem for strictly hyperbolic differential operators with C^infty coefficients……Page 119
5.2. Oscillatory asymptotic solutions. Caustics……Page 133
References……Page 144