{"product_id":"geometry-of-the-generalized-geodesic-flow-and-inverse-spectral-problems-isbn-9781119107668","title":"Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems","description":"\u003cp\u003eThis 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.\u003cbr\u003e\u003cbr\u003eThe 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.\u003c\/p\u003e \u003cp\u003ePreface ix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Preliminaries from differential topology and microlocal analysis 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Spaces of jets and transversality theorems 1\u003c\/p\u003e \u003cp\u003e1.2 Generalized bicharacteristics 5\u003c\/p\u003e \u003cp\u003e1.3 Wave front sets of distributions 15\u003c\/p\u003e \u003cp\u003e1.4 Boundary problems for the wave operator 23\u003c\/p\u003e \u003cp\u003e1.5 Notes 25\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Reflecting rays 26\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Billiard ball map 26\u003c\/p\u003e \u003cp\u003e2.2 Periodic rays for several convex bodies 31\u003c\/p\u003e \u003cp\u003e2.3 The Poincare map 40\u003c\/p\u003e \u003cp\u003e2.4 Scattering rays 49\u003c\/p\u003e \u003cp\u003e2.5 Notes 56\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Poisson relation for manifolds with boundary 57\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Traces of the fundamental solutions of ◻ and ◻\u003csup\u003e2\u003c\/sup\u003e 58\u003c\/p\u003e \u003cp\u003e3.2 The distribution σ(t) 62\u003c\/p\u003e \u003cp\u003e3.3 Poisson relation for convex domains 64\u003c\/p\u003e \u003cp\u003e3.4 Poisson relation for arbitrary domains 71\u003c\/p\u003e \u003cp\u003e3.5 Notes 81\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Poisson summation formula for manifolds with boundary 82\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Global parametrix for mixed problems 82\u003c\/p\u003e \u003cp\u003e4.2 Principal symbol of \u003ci\u003eF\u003csub\u003eB\u003c\/sub\u003e \u003c\/i\u003e94\u003c\/p\u003e \u003cp\u003e4.3 Poisson summation formula 103\u003c\/p\u003e \u003cp\u003e4.4 Notes 117\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Poisson relation for the scattering kernel 118\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Representation of the scattering kernel 118\u003c\/p\u003e \u003cp\u003e5.2 Location of the singularities of s(t, θ, ω) 127\u003c\/p\u003e \u003cp\u003e5.3 Poisson relation for the scattering kernel 130\u003c\/p\u003e \u003cp\u003e5.4 Notes 137\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Generic properties of reflecting rays 139\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Generic properties of smooth embeddings 139\u003c\/p\u003e \u003cp\u003e6.2 Elementary generic properties of reflecting rays 145\u003c\/p\u003e \u003cp\u003e6.3 Absence of tangent segments 155\u003c\/p\u003e \u003cp\u003e6.4 Non-degeneracy of reflecting rays 160\u003c\/p\u003e \u003cp\u003e6.5 Notes 172\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Bumpy surfaces 173\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Poincare maps for closed geodesics 173\u003c\/p\u003e \u003cp\u003e7.2 Local perturbations of smooth surfaces 182\u003c\/p\u003e \u003cp\u003e7.3 Non-degeneracy and transversality 191\u003c\/p\u003e \u003cp\u003e7.4 Global perturbations of smooth surfaces 199\u003c\/p\u003e \u003cp\u003e7.5 Notes 202\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Inverse spectral results for generic bounded domains 204\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Planar domains 204\u003c\/p\u003e \u003cp\u003e8.2 Interpolating Hamiltonians 214\u003c\/p\u003e \u003cp\u003e8.3 Approximations of closed geodesics by periodic reflecting rays 221\u003c\/p\u003e \u003cp\u003e8.4 The Poisson relation for generic strictly convex domains 235\u003c\/p\u003e \u003cp\u003e8.5 Notes 241\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Singularities of the scattering kernel 242\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Singularity of the scattering kernel for a non-degenerate (ω, θ)-ray 242\u003c\/p\u003e \u003cp\u003e9.2 Singularities of the scattering kernel for generic domains 252\u003c\/p\u003e \u003cp\u003e9.3 Glancing ω-rays 253\u003c\/p\u003e \u003cp\u003e9.4 Generic domains in ℝ\u003csup\u003e3\u003c\/sup\u003e 258\u003c\/p\u003e \u003cp\u003e9.5 Notes 263\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Scattering invariants for several strictly convex domains 264\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Singularities of the scattering kernel for generic θ 264\u003c\/p\u003e \u003cp\u003e10.2 Hyperbolicity of scattering trajectories 273\u003c\/p\u003e \u003cp\u003e10.3 Existence of scattering rays and asymptotic of their sojourn times 281\u003c\/p\u003e \u003cp\u003e10.4 Asymptotic of the coefficients of the main singularity 287\u003c\/p\u003e \u003cp\u003e10.5 Notes 296\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Poisson relation for the scattering kernel for generic directions 298\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 The Poisson relation for the scattering kernel 298\u003c\/p\u003e \u003cp\u003e11.2 Generalized Hamiltonian flow 303\u003c\/p\u003e \u003cp\u003e11.3 Invariance of the Hausdorff dimension 309\u003c\/p\u003e \u003cp\u003e11.4 Further regularity of the generalized Hamiltonian flow 320\u003c\/p\u003e \u003cp\u003e11.5 Proof of Proposition 11.1.2 325\u003c\/p\u003e \u003cp\u003e11.6 Notes 336\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Scattering kernel for trapping obstacles 337\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Scattering rays with sojourn times tending to infinity 337\u003c\/p\u003e \u003cp\u003e12.2 Scattering amplitude and the cut-off resolvent 343\u003c\/p\u003e \u003cp\u003e12.3 Estimates for the scattering amplitude 347\u003c\/p\u003e \u003cp\u003e12.4 Notes 350\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Inverse scattering by obstacles 351\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 The scattering length spectrum and the generalized geodesic flow 351\u003c\/p\u003e \u003cp\u003e13.2 Proof of Theorem 13.1.2 356\u003c\/p\u003e \u003cp\u003e13.3 An example: star-shaped obstacles 363\u003c\/p\u003e \u003cp\u003e13.4 Tangential singularities of scattering rays I 365\u003c\/p\u003e \u003cp\u003e13.5 Tangential singularities of scattering rays II 368\u003c\/p\u003e \u003cp\u003e13.6 Reflection points of scattering rays and winding numbers 374\u003c\/p\u003e \u003cp\u003e13.7 Recovering the accessible part of an obstacle 380\u003c\/p\u003e \u003cp\u003e13.8 Proof of Proposition 13.4.2 385\u003c\/p\u003e \u003cp\u003e13.9 Notes 394\u003c\/p\u003e \u003cp\u003eReferences 396\u003c\/p\u003e \u003cp\u003eTopic Index 405\u003c\/p\u003e \u003cp\u003eSymbol Index 409\u003c\/p\u003e \"Thus, while solving the usual inverse problems is the rst main concern, a second\u003cbr\u003eone is the study of generic properties, supported by the bumpy surfaces theorem.\u003cbr\u003eUnderstanding scattering and sojourn times is the third main topic, and the fourth is\u003cbr\u003eobtaining inverse scattering results. The material added for this edition focuses on the\u003cbr\u003elatter two\" \u003cb\u003eBoris Hasselblatt on behalf of Mathematical Reviews, October 2017\u003c\/b\u003e\u003cbr\u003e  \u003cp\u003e\u003cstrong\u003eVesselin Petkov\u003c\/strong\u003e, Professor Emeritus, IMB, Unversité de Bordeaux, France. \u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eLuchezar Stoyanov\u003c\/strong\u003e, Professor, School of Mathematics and Statistics, University of Western Australia.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989283750117,"sku":"NP9781119107668","price":151.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119107668.jpg?v=1761783511","url":"https:\/\/k12savings.com\/es\/products\/geometry-of-the-generalized-geodesic-flow-and-inverse-spectral-problems-isbn-9781119107668","provider":"K12savings","version":"1.0","type":"link"}