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E-grāmata: Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems 2e 2nd Edition [Wiley Online]

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  • Formāts: 432 pages
  • Izdošanas datums: 13-Jan-2017
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 1119107687
  • ISBN-13: 9781119107682
Citas grāmatas par šo tēmu:
  • Wiley Online
  • Cena: 137,50 €*
  • * this price gives unlimited concurrent access for unlimited time
Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems 2e 2nd EditionPalielināt
  • Formāts: 432 pages
  • Izdošanas datums: 13-Jan-2017
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 1119107687
  • ISBN-13: 9781119107682
Citas grāmatas par šo tēmu:
Wiley Online E-booksMore info about Wiley Online
Rokasgrāmatas mājas lapa: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119107682

This book is a new edition of a title originally published in1992. No other book has been published that treats inverse spectral and inverse scattering results by using the so called Poisson summation formula and the related study of singularities. This book presents these in a closed and comprehensive form, and the exposition is based on a combination of different tools and results from dynamical systems, microlocal analysis, spectral and scattering theory.

The content of the first edition is still relevant, however the new edition will include several new results established after 1992; new text will comprise about a third of the content of the new edition. The main chapters in the first edition in combination with the new chapters will provide a better and more comprehensive presentation of importance for the applications inverse results. These results are obtained by modern mathematical techniques which will be presented together in order to give the readers the opportunity to completely understand them. Moreover, some basic generic properties established by the authors after the publication of the first edition establishing the wide range of applicability of the Poison relation will be presented for first time in the new edition of the book.

Preface ix
1 Preliminaries from differential topology and microlocal analysis
1(25)
1.1 Spaces of jets and transversality theorems
1(4)
1.2 Generalized bicharacteristics
5(10)
1.3 Wave front sets of distributions
15(8)
1.4 Boundary problems for the wave operator
23(2)
1.5 Notes
25(1)
2 Reflecting rays
26(31)
2.1 Billiard ball map
26(5)
2.2 Periodic rays for several convex bodies
31(9)
2.3 The Poincare map
40(9)
2.4 Scattering rays
49(7)
2.5 Notes
56(1)
3 Poisson relation for manifolds with boundary
57(25)
3.1 Traces of the fundamental solutions of and 2
58(4)
3.2 The distribution σ(t)
62(2)
3.3 Poisson relation for convex domains
64(7)
3.4 Poisson relation for arbitrary domains
71(10)
3.5 Notes
81(1)
4 Poisson summation formula for manifolds with boundary
82(36)
4.1 Global parametrix for mixed problems
82(12)
4.2 Principal symbol of FB
94(9)
4.3 Poisson summation formula
103(14)
4.4 Notes
117(1)
5 Poisson relation for the scattering kernel
118(21)
5.1 Representation of the scattering kernel
118(9)
5.2 Location of the singularities of s(t, θ, ω)
127(3)
5.3 Poisson relation for the scattering kernel
130(7)
5.4 Notes
137(2)
6 Generic properties of reflecting rays
139(34)
6.1 Generic properties of smooth embeddings
139(6)
6.2 Elementary generic properties of reflecting rays
145(10)
6.3 Absence of tangent segments
155(5)
6.4 Non-degeneracy of reflecting rays
160(12)
6.5 Notes
172(1)
7 Bumpy surfaces
173(31)
7.1 Poincare maps for closed geodesies
173(9)
7.2 Local perturbations of smooth surfaces
182(9)
7.3 Non-degeneracy and transversality
191(8)
7.4 Global perturbations of smooth surfaces
199(3)
7.5 Notes
202(2)
8 Inverse spectral results for generic bounded domains
204(38)
8.1 Planar domains
204(10)
8.2 Interpolating Hamiltonians
214(7)
8.3 Approximations of closed geodesies by periodic reflecting rays
221(14)
8.4 The Poisson relation for generic strictly convex domains
235(6)
8.5 Notes
241(1)
9 Singularities of the scattering kernel
242(22)
9.1 Singularity of the scattering kernel for a non-degenerate (ω, θ)-ray
242(10)
9.2 Singularities of the scattering kernel for generic domains
252(1)
9.3 Glancing ω-rays
253(5)
9.4 Generic domains in R3
258(5)
9.5 Notes
263(1)
10 Scattering invariants for several strictly convex domains
264(34)
10.1 Singularities of the scattering kernel for generic θ
264(9)
10.2 Hyperbolicity of scattering trajectories
273(8)
10.3 Existence of scattering rays and asymptotic of their sojourn times
281(6)
10.4 Asymptotic of the coefficients of the main singularity
287(9)
10.5 Notes
296(2)
11 Poisson relation for the scattering kernel for generic directions
298(39)
11.1 The Poisson relation for the scattering kernel
298(5)
11.2 Generalized Hamiltonian flow
303(6)
11.3 Invariance of the Hausdorff dimension
309(11)
11.4 Further regularity of the generalized Hamiltonian flow
320(5)
11.5 Proof of Proposition 11.1.2
325(11)
11.6 Notes
336(1)
12 Scattering kernel for trapping obstacles
337(14)
12.1 Scattering rays with sojourn times tending to infinity
337(6)
12.2 Scattering amplitude and the cut-off resolvent
343(4)
12.3 Estimates for the scattering amplitude
347(3)
12.4 Notes
350(1)
13 Inverse scattering by obstacles
351(45)
13.1 The scattering length spectrum and the generalized geodesic flow
351(5)
13.2 Proof of Theorem 13.1.2
356(7)
13.3 An example: star-shaped obstacles
363(2)
13.4 Tangential singularities of scattering rays I
365(3)
13.5 Tangential singularities of scattering rays II
368(6)
13.6 Reflection points of scattering rays and winding numbers
374(6)
13.7 Recovering the accessible part of an obstacle
380(5)
13.8 Proof of Proposition 13.4.2
385(9)
13.9 Notes
394(2)
References 396(9)
Topic Index 405(4)
Symbol Index 409
Vesselin Petkov, Professor Emeritus, IMB, Unversité de Bordeaux, France.

Luchezar Stoyanov, Professor, School of Mathematics and Statistics, University of Western Australia.
Permanent link: https://www.kriso.lv/db/9781119107682_pe.html
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