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

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

<p><strong>Vesselin Petkov</strong>, Professor Emeritus, IMB, Unversité de Bordeaux, France. <p><strong>Luchezar Stoyanov</strong>, Professor, School of Mathematics and Statistics, University of Western Australia.

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