Geometry of reflecting rays and inverse spectral problems
著者
書誌事項
Geometry of reflecting rays and inverse spectral problems
（Pure and applied mathematics）
Wiley, c1992
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注記
Bibliography: p. [304]-309
Includes indexes
内容説明・目次
内容説明
The behaviour of reflecting rays plays an essential role in many problems of mathematical physics. This book studies different geometric properties of reflecting rays for manifolds with smooth boundary and their applications to different inverse spectral and scattering problems. This is a developing area in which the authors have made important contributions. Results concerning the particular problems studied and which arise in several important domains of modern physics are presented. Some chapters concerning the generic properties of reflecting rays can be used for courses for graduate students.
目次
- Part 1 Preliminaries from differential topology and microlocal analysis: jets and transversality theorems
- generalized bicharacteristics
- wave front sets of distributions. Part 2 Reflecting rays: billiard ball map
- periodic rays for several convex bodies
- Poincare map
- scattering rays
- examples. Part 3 Generic properties of reflecting rays: generic properties and smooth embeddings
- elementary generic properties
- absence of tangent segments
- non-degeneracy of reflecting rays. Part 4 Bumpy metrics: Poincare map for closed geodesics
- local perturbations of smooth surfaces
- non-degeneracy and transversality
- global perturbations of smooth surfaces. Part 5 Poisson relation for manifolds with boundary: Poisson relation for convex domains
- Poisson relation for arbitrary domains. Part 6 Poisson summation formula for manifolds with boundary: global parametrix for mixed problem
- Poisson summation formula. Part 7 Inverse spectral results for generic bounded domains: planar domains
- interpolating Hamiltonians
- approximations of closed geodesics by periodic reflecting rays
- Poisson relation for generic strictly convex domains. Part 8 Poisson relation for the scattering kernel: representation of the scattering kernel
- Poisson relation for the scattering kernel. Part 9 Singularities of the scattering kernel for generic domains. Part 10 Scattering invariants for several strictly convex domains: hyperbolicity of scattering trajectories
- existence of scattering rays and asymptotic of their sojourn times
- asymptotic of the coefficients of the main singularity.
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