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The Resource Geometry of the generalized geodesic flow and inverse spectral problems, Vesselin M. Petkov, Luchezar N. Stoyanov, (electronic book)

Geometry of the generalized geodesic flow and inverse spectral problems, Vesselin M. Petkov, Luchezar N. Stoyanov, (electronic book)

Label
Geometry of the generalized geodesic flow and inverse spectral problems
Title
Geometry of the generalized geodesic flow and inverse spectral problems
Statement of responsibility
Vesselin M. Petkov, Luchezar N. Stoyanov
Creator
Contributor
Author
Subject
Language
eng
Summary
Revised and expanded edition of: Geometry of reflecting rays and inverse spectral problems (Chichester, England; New York: Wiley, c1992).
Member of
Cataloging source
  • StDuBDS
  • StDuBDS
http://library.link/vocab/creatorName
Petkov, Vesselin
Dewey number
515.7222
Index
no index present
LC call number
QA320
LC item number
.P435 2016
Literary form
non fiction
http://library.link/vocab/relatedWorkOrContributorDate
1954-
http://library.link/vocab/relatedWorkOrContributorName
Stoyanov, Luchezar N.
http://library.link/vocab/subjectName
  • Spectral theory (Mathematics)
  • Inverse problems (Differential equations)
  • Geometry, Differential
  • Geodesic flows
  • Flows (Differentiable dynamical systems)
Summary expansion
This book is a new edition of a titleoriginally 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 thesein 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 thefirst editionis still relevant, however the new edition will include several new results established after 1992; new text willcomprise 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 whichwill 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
Target audience
specialized
Label
Geometry of the generalized geodesic flow and inverse spectral problems, Vesselin M. Petkov, Luchezar N. Stoyanov, (electronic book)
Instantiates
Publication
Carrier category
online resource
Carrier MARC source
rdacarrier
Content category
text
Content type MARC source
rdacontent
Contents
<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 &#9723; and &#9723;<sup>2</sup> 58</p> <p>3.2 The distribution &sigma;(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&nbsp;<i>F<sub>B</sub> </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, &theta;, &omega;) 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 (&omega;, &theta;)-ray 242</p> <p>9.2 Singularities of the scattering kernel for generic domains 252</p> <p>9.3 Glancing &omega;-rays 253</p> <p>9.4 Generic domains in&nbsp;&#8477;<sup>3</sup> 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 &theta; 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>
Control code
AH30712540
Edition
Second edition.
Extent
400 pages
Form of item
electronic
Governing access note
After 5 minutes Preview, click on &#x32;Request Access&#x33;, fill in a form with your details. If triggered, the book will be loaned and tied to the one user for 1 week, during which time users can read or download as they choose. 4th user request triggers auto-purchase
Isbn
9781119107668
Media category
computer
Media MARC source
rdamedia
Specific material designation
remote
Label
Geometry of the generalized geodesic flow and inverse spectral problems, Vesselin M. Petkov, Luchezar N. Stoyanov, (electronic book)
Publication
Carrier category
online resource
Carrier MARC source
rdacarrier
Content category
text
Content type MARC source
rdacontent
Contents
<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 &#9723; and &#9723;<sup>2</sup> 58</p> <p>3.2 The distribution &sigma;(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&nbsp;<i>F<sub>B</sub> </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, &theta;, &omega;) 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 (&omega;, &theta;)-ray 242</p> <p>9.2 Singularities of the scattering kernel for generic domains 252</p> <p>9.3 Glancing &omega;-rays 253</p> <p>9.4 Generic domains in&nbsp;&#8477;<sup>3</sup> 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 &theta; 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>
Control code
AH30712540
Edition
Second edition.
Extent
400 pages
Form of item
electronic
Governing access note
After 5 minutes Preview, click on &#x32;Request Access&#x33;, fill in a form with your details. If triggered, the book will be loaned and tied to the one user for 1 week, during which time users can read or download as they choose. 4th user request triggers auto-purchase
Isbn
9781119107668
Media category
computer
Media MARC source
rdamedia
Specific material designation
remote

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