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Informationen zum Autor Vesselin Petkov, Professor Emeritus, IMB, Unversité de Bordeaux, France. Luchezar Stoyanov, Professor, School of Mathematics and Statistics, University of Western Australia. Klappentext 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. Zusammenfassung 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. Inhaltsverzeichnis Preface ix 1 Preliminaries from differential topology and microlocal analysis 1 1.1 Spaces of jets and transversality theorems 1 1.2 Generalized bicharacteristics 5 1.3 Wave front sets of distributions 15 1.4 Boundary problems for the wave operator 23 1.5 Notes 25 2 Reflecting rays 26 2.1 Billiard ball map 26 2.2 Periodic rays for several convex bodies 31 2.3 The Poincare map 40 2.4 Scattering rays 49 2.5 Notes 56 3 Poisson relation for manifolds with boundary 57 3.1 Traces of the fundamental solutions of ¿ and ¿2 58 3.2 The distribution ¿(t) 62 3.3 Poisson relation for convex domains 64 3.4 Poisson relation for arbitrary domains 71 3.5 Notes 81 4 Poisson summation formula for manifolds with boundary 82 4.1 Global parametrix for mixed problems 82 4.2 Principal symbol of FB 94 4.3 Poisson summation formula 103 4.4 Notes 117 5 Poisson relation for the scattering kernel 118 5.1 Representation of the scattering kernel 118 5.2 Location of the singularities of s(t, ¿, ¿) 127 5.3 Poisson relation for the scattering kernel 130 5.4 Notes 137 6 Generic properties of reflecting rays 139 6.1 Generic properties of smooth embeddings 139 6.2 Elementary generic properties of reflecting rays 145 6.3 Absence of tangent segments 155 6.4 Non-degeneracy of reflecting rays 160 6.5 Notes 172 7 Bumpy surfaces 173 7.1 Poincare maps for closed geodesics 173 7.2 Local perturbations of smooth surfaces 182 7.3 Non-degeneracy and transversality 191 7.4 Global perturbations of smooth surfaces 199 7.5 Notes 202 8 Inverse spectral results for generic bounded domains 204 8.1 Planar domains 204 8.2 Interpolating Hamiltonians 214 8.3 Approximations of closed geodesics by periodic reflecting rays 221 8.4 The Poisson relation for generic strictly convex domains 235 8.5 Notes 241 9 Singularities of the...
