Inverse spectral and scattering theory : an introduction /
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| Author / Creator: | Isozaki, Hiroshi, author. |
|---|---|
| Imprint: | Singapore : Springer, [2020] |
| Description: | 1 online resource |
| Language: | English |
| Series: | SpringerBriefs in Mathematical Physics ; Volume 38 SpringerBriefs in mathematical physics ; v. 38. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/12607892 |
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| 100 | 1 | |a Isozaki, Hiroshi, |e author. | |
| 245 | 1 | 0 | |a Inverse spectral and scattering theory : |b an introduction / |c Hiroshi Isozaki. |
| 264 | 1 | |a Singapore : |b Springer, |c [2020] | |
| 300 | |a 1 online resource | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 338 | |a online resource |b cr |2 rdacarrier | ||
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| 490 | 1 | |a SpringerBriefs in Mathematical Physics ; |v Volume 38 | |
| 504 | |a Includes bibliographical references and index. | ||
| 588 | |a Description based on online resource; title from digital title page (viewed on October 21, 2020). | ||
| 505 | 0 | |a Intro -- Preface -- Contents -- 1 One-Dimensional Inverse Problems -- 1.1 Functional Analysis and Weyl-Stone-Titchmarsh-KodairaTheory -- 1.1.1 Self-Adjoint Extensions -- 1.1.2 Weyl Function -- 1.1.3 Eigenfunction Expansion -- 1.2 Spectral Data on the Finite Interval -- 1.3 When Eigenvalues Determine the Potential? -- 1.4 Gel'fand-Levitan Equation -- 1.5 Spectral Mapping -- 1.5.1 Analytic Functions on a Banach Space -- 1.5.2 Analytic Map Associated with the Eigenvalue Problem -- 1.5.3 Liouville Transformation -- 1.6 Isospectral Deformation -- 1.7 Scattering on the Half-Line | |
| 505 | 8 | |a 1.7.1 Generalized Sine Transform -- 1.7.2 The Core of Gel'fand-Levitan Theory -- 1.7.3 Jost Solution and Spectral Data -- 1.8 Partial Data and Local Uniqueness -- 2 Multi-Dimensional Inverse Boundary Value Problems -- 2.1 Multi-Dimensional Borg-Levinson Theorem -- 2.2 Gel'fand Problem -- 2.3 Kac Problem -- 2.3.1 Isospectral Manifolds -- 2.3.2 Spectral Invariants -- 2.4 Calderón Problem -- 2.4.1 Complex Geometrical Optics Solutions -- 2.4.2 Anisotropic Conductivity -- 2.4.3 Carleman Estimates -- 2.4.4 Two-Dimensional Problem -- 3 Multi-Dimensional Gel'fand-Levitan Theory | |
| 505 | 8 | |a 3.1 Spectra and Scattering Phenomena -- 3.2 Spectral Theory for Schrödinger Operators -- 3.3 Generalized Eigenfunctions -- 3.4 The Role of Volterra Integral Operator -- 3.5 Faddeev Theory -- 3.6 Changing Green Operators -- 3.7 Direction Dependent Green Operators -- 3.8 Inverse Scattering at a Fixed Energy -- 3.9 Multi-Dimensional Gel'fand-Levitan Theory -- 3.9.1 Justification of Volterra Properties -- 3.9.2 Gel'fand-Levitan Equation -- 3.10 ∂-Approach -- 4 Boundary Control Method -- 4.1 The Role of Spectral Theory -- 4.2 Boundary Distance Function: From R(M) to M -- 4.3 From BSD to R(M) | |
| 505 | 8 | |a 4.3.1 From BSD to the Domain of Influence -- 4.3.2 Boundary Normal Geodesic -- 4.4 From the Domain of Influence to the Topology of M -- 4.5 From R(M) to the Differentiable Structure -- 4.6 From R(M) to the Riemannian Metric -- 4.7 Wave Propagation -- 4.8 Other Spectral Data -- 4.9 Inverse Scattering -- 5 Other Topics -- 5.1 Backscattering and Fixed Angle Scattering -- 5.2 Maxwell Equation -- 5.3 Perturbed Periodic Lattices -- 5.4 Stäckel Metric -- References -- Index | |
| 520 | |a The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg-Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg's S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev-Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis. . | ||
| 650 | 0 | |a Spectral theory (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh85126408 | |
| 650 | 0 | |a Scattering (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh85118046 | |
| 650 | 0 | |a Inverse problems (Differential equations) |0 http://id.loc.gov/authorities/subjects/sh85067684 | |
| 650 | 7 | |a Inverse problems (Differential equations) |2 fast |0 (OCoLC)fst00978098 | |
| 650 | 7 | |a Scattering (Mathematics) |2 fast |0 (OCoLC)fst01106580 | |
| 650 | 7 | |a Spectral theory (Mathematics) |2 fast |0 (OCoLC)fst01129072 | |
| 655 | 0 | |a Electronic books. | |
| 655 | 4 | |a Electronic books. | |
| 776 | 0 | 8 | |c Original |z 9811581983 |z 9789811581984 |w (OCoLC)1180970723 |
| 830 | 0 | |a SpringerBriefs in mathematical physics ; |v v. 38. |0 http://id.loc.gov/authorities/names/no2014125102 | |
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