Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems

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.

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 σ(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 F_B 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, θ, ω) 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 scattering kernel 242

9.1 Singularity of the scattering kernel for a non-degenerate (ω, θ)-ray 242

9.2 Singularities of the scattering kernel for generic domains 252

9.3 Glancing ω-rays 253

9.4 Generic domains in ℝ³ 258

9.5 Notes 263

10 Scattering invariants for several strictly convex domains 264

10.1 Singularities of the scattering kernel for generic θ 264

10.2 Hyperbolicity of scattering trajectories 273

10.3 Existence of scattering rays and asymptotic of their sojourn times 281

10.4 Asymptotic of the coefficients of the main singularity 287

10.5 Notes 296

11 Poisson relation for the scattering kernel for generic directions 298

11.1 The Poisson relation for the scattering kernel 298

11.2 Generalized Hamiltonian flow 303

11.3 Invariance of the Hausdorff dimension 309

11.4 Further regularity of the generalized Hamiltonian flow 320

11.5 Proof of Proposition 11.1.2 325

11.6 Notes 336

12 Scattering kernel for trapping obstacles 337

12.1 Scattering rays with sojourn times tending to infinity 337

12.2 Scattering amplitude and the cut-off resolvent 343

12.3 Estimates for the scattering amplitude 347

12.4 Notes 350

13 Inverse scattering by obstacles 351

13.1 The scattering length spectrum and the generalized geodesic flow 351

13.2 Proof of Theorem 13.1.2 356

13.3 An example: star-shaped obstacles 363

13.4 Tangential singularities of scattering rays I 365

13.5 Tangential singularities of scattering rays II 368

13.6 Reflection points of scattering rays and winding numbers 374

13.7 Recovering the accessible part of an obstacle 380

13.8 Proof of Proposition 13.4.2 385

13.9 Notes 394

References 396

Topic Index 405

Symbol Index 409