Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems (eBook)

Vesselin M. Petkov, Luchezar N. Stoyanov (Autoren)

eBook Download: PDF

2016 | 2. Auflage
John Wiley & Sons (Verlag)
978-1-119-10767-5 (ISBN)

Infos

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.

Vesselin Petkov, Professor Emeritus, IMB, Unversité de Bordeaux, France.

Luchezar Stoyanov, Professor, School of Mathematics and Statistics, University of Western Australia.

Inhalt

Cover 1
Title Page 5
Copyright 6
Contents 7
Preface 11
Chapter 1 Preliminaries from differential topology and microlocal analysis 19
1.1 Spaces of jets and transversality theorems 19
1.2 Generalized bicharacteristics 23
1.3 Wave front sets of distributions 33
1.4 Boundary problems for the wave operator 41
1.5 Notes 43
Chapter 2 Reflecting rays 44
2.1 Billiard ball map 44
2.2 Periodic rays for several convex bodies 49
2.3 The Poincaré map 58
2.4 Scattering rays 67
2.5 Notes 74
Chapter 3 Poisson relation for manifolds with boundary 75
3.1 Traces of the fundamental solutions of and 2 76
3.2 The distribution ?(t) 80
3.3 Poisson relation for convex domains 82
3.4 Poisson relation for arbitrary domains 89
3.5 Notes 99
Chapter 4 Poisson summation formula for manifolds with boundary 100
4.1 Global parametrix for mixed problems 100
4.2 Principal symbol of FB 112
4.3 Poisson summation formula 121
4.4 Notes 135
Chapter 5 Poisson relation for the scattering kernel 136
5.1 Representation of the scattering kernel 136
5.2 Location of the singularities of s(t, ?, ?) 145
5.3 Poisson relation for the scattering kernel 148
5.4 Notes 155
Chapter 6 Generic properties of reflecting rays 157
6.1 Generic properties of smooth embeddings 157
6.2 Elementary generic properties of reflecting rays 163
6.3 Absence of tangent segments 173
6.4 Non-degeneracy of reflecting rays 178
6.5 Notes 190
Chapter 7 Bumpy surfaces 191
7.1 Poincaré maps for closed geodesics 191
7.2 Local perturbations of smooth surfaces 200
7.3 Non-degeneracy and transversality 209
7.4 Global perturbations of smooth surfaces 217
7.5 Notes 220
Chapter 8 Inverse spectral results for generic bounded domains 222
8.1 Planar domains 222
8.2 Interpolating Hamiltonians 232
8.3 Approximations of closed geodesics by periodic reflecting rays 239
8.4 The Poisson relation for generic strictly convex domains 253
8.5 Notes 259
Chapter 9 Singularities of the scattering kernel 260
9.1 Singularity of the scattering kernel for a non-degenerate (?, ?)-ray 260
9.2 Singularities of the scattering kernel for generic domains 270
9.3 Glancing ?-rays 271
9.4 Generic domains in R3 276
9.5 Notes 281
Chapter 10 Scattering invariants for several strictly convex domains 282
10.1 Singularities of the scattering kernel for generic ? 282
10.2 Hyperbolicity of scattering trajectories 291
10.3 Existence of scattering rays and asymptotic of their sojourn times 299
10.4 Asymptotic of the coefficients of the main singularity 305
10.5 Notes 314
Chapter 11 Poisson relation for the scattering kernel for generic directions 316
11.1 The Poisson relation for the scattering kernel 316
11.2 Generalized Hamiltonian flow 321
11.3 Invariance of the Hausdorff dimension 327
11.4 Further regularity of the generalized Hamiltonian flow 338
11.5 Proof of Proposition 11.1.2 343
11.6 Notes 354
Chapter 12 Scattering kernel for trapping obstacles 355
12.1 Scattering rays with sojourn times tending to infinity 355
12.2 Scattering amplitude and the cut-off resolvent 361
12.3 Estimates for the scattering amplitude 365
12.4 Notes 368
Chapter 13 Inverse scattering by obstacles 369
13.1 The scattering length spectrum and the generalized geodesic flow 369
13.2 Proof of Theorem 13.1.2 374
13.3 An example: star-shaped obstacles 381
13.4 Tangential singularities of scattering rays I 383
13.5 Tangential singularities of scattering rays II 386
13.6 Reflection points of scattering rays and winding numbers 392
13.7 Recovering the accessible part of an obstacle 398
13.8 Proof of Proposition 13.4.2 403
13.9 Notes 412
References 414
Topic Index 423
Symbol Index 427
EULA 429

Rezensionen

"Thus, while solving the usual inverse problems is the rst main concern, a second
one is the study of generic properties, supported by the bumpy surfaces theorem.
Understanding scattering and sojourn times is the third main topic, and the fourth is
obtaining inverse scattering results. The material added for this edition focuses on the
latter two" Boris Hasselblatt on behalf of Mathematical Reviews, October 2017

Produkt-Details

• Erscheint lt. Verlag: 7.11.2016
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Applied Mathematics in Science • generalized geodesics • generic properties of periodic geodesics • Geodätischer Fluss • Geodätischer Fluss • Geometrie • Geometrie u. Topologie • Geometry & Topology • inverse scattering problems • inverse spectral problems • length spectrum • Mathematics • Mathematics Special Topics • Mathematik • Mathematik in den Naturwissenschaften • periodic geodesics • Poisson relation • reflecting rays • scattering amplitude • scattering rays • Spezialthemen Mathematik
• ISBN-10: 1-119-10767-9 / 1119107679
• ISBN-13: 978-1-119-10767-5 / 9781119107675