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Geometry of the generalized geodesic flow and inverse spectral problems

Author: Vesselin Petkov; Luchezar N Stoyanov
Publisher: Chichester, West Sussex : Wiley, 2017.
Edition/Format:   Print book : English : Second editionView all editions and formats
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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  Read more...

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Document Type: Book
All Authors / Contributors: Vesselin Petkov; Luchezar N Stoyanov
ISBN: 9781119107668 1119107660
OCLC Number: 950004318
Notes: Revised and expanded edition of: Geometry of reflecting rays and inverse spectral problems (Chichester, England ; New York : Wiley, c1992).
Description: xv, 410 pages : illustrations ; 24 cm
Contents: Preface ix1 Preliminaries from differential topology and microlocal analysis 11.1 Spaces of jets and transversality theorems 11.2 Generalized bicharacteristics 51.3 Wave front sets of distributions 151.4 Boundary problems for the wave operator 231.5 Notes 252 Reflecting rays 262.1 Billiard ball map 262.2 Periodic rays for several convex bodies 312.3 The Poincare map 402.4 Scattering rays 492.5 Notes 563 Poisson relation for manifolds with boundary 573.1 Traces of the fundamental solutions of and 2 583.2 The distribution (t) 623.3 Poisson relation for convex domains 643.4 Poisson relation for arbitrary domains 713.5 Notes 814 Poisson summation formula for manifolds with boundary 824.1 Global parametrix for mixed problems 824.2 Principal symbol of FB 944.3 Poisson summation formula 1034.4 Notes 1175 Poisson relation for the scattering kernel 1185.1 Representation of the scattering kernel 1185.2 Location of the singularities of s(t, , ) 1275.3 Poisson relation for the scattering kernel 1305.4 Notes 1376 Generic properties of reflecting rays 1396.1 Generic properties of smooth embeddings 1396.2 Elementary generic properties of reflecting rays 1456.3 Absence of tangent segments 1556.4 Non-degeneracy of reflecting rays 1606.5 Notes 1727 Bumpy surfaces 1737.1 Poincare maps for closed geodesics 1737.2 Local perturbations of smooth surfaces 1827.3 Non-degeneracy and transversality 1917.4 Global perturbations of smooth surfaces 1997.5 Notes 2028 Inverse spectral results for generic bounded domains 2048.1 Planar domains 2048.2 Interpolating Hamiltonians 2148.3 Approximations of closed geodesics by periodic reflecting rays 2218.4 The Poisson relation for generic strictly convex domains 2358.5 Notes 2419 Singularities of the scattering kernel 2429.1 Singularity of the scattering kernel for a non-degenerate ( , )-ray 2429.2 Singularities of the scattering kernel for generic domains 2529.3 Glancing -rays 2539.4 Generic domains in 3 2589.5 Notes 26310 Scattering invariants for several strictly convex domains 26410.1 Singularities of the scattering kernel for generic 26410.2 Hyperbolicity of scattering trajectories 27310.3 Existence of scattering rays and asymptotic of their sojourn times 28110.4 Asymptotic of the coefficients of the main singularity 28710.5 Notes 29611 Poisson relation for the scattering kernel for generic directions 29811.1 The Poisson relation for the scattering kernel 29811.2 Generalized Hamiltonian flow 30311.3 Invariance of the Hausdorff dimension 30911.4 Further regularity of the generalized Hamiltonian flow 32011.5 Proof of Proposition 11.1.2 32511.6 Notes 33612 Scattering kernel for trapping obstacles 33712.1 Scattering rays with sojourn times tending to infinity 33712.2 Scattering amplitude and the cut-off resolvent 34312.3 Estimates for the scattering amplitude 34712.4 Notes 35013 Inverse scattering by obstacles 35113.1 The scattering length spectrum and the generalized geodesic flow 35113.2 Proof of Theorem 13.1.2 35613.3 An example: star-shaped obstacles 36313.4 Tangential singularities of scattering rays I 36513.5 Tangential singularities of scattering rays II 36813.6 Reflection points of scattering rays and winding numbers 37413.7 Recovering the accessible part of an obstacle 38013.8 Proof of Proposition 13.4.2 38513.9 Notes 394References 396Topic Index 405Symbol Index 409
Other Titles: Geometry of reflecting rays and inverse spectral problems
Responsibility: Vesselin M. Petkov, University o Bordeaux, France, Luchezar N. Stoyanov, University of Western Australia, Australia.

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"Thus, while solving the usual inverse problems is the rst main concern, a secondone is the study of generic properties, supported by the bumpy surfaces theorem.Understanding scattering and sojourn Read more...

 
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