Inverse Spectral And Scattering Theory
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Inverse Spectral And Scattering Theory
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Author : Hiroshi Isozaki
language : en
Publisher: Springer Nature
Release Date : 2020-09-26
Inverse Spectral And Scattering Theory written by Hiroshi Isozaki and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-26 with Science categories.
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.
An Introduction To Inverse Scattering And Inverse Spectral Problems
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Author : Khosrow Chadan
language : en
Publisher: SIAM
Release Date : 1997-01-01
An Introduction To Inverse Scattering And Inverse Spectral Problems written by Khosrow Chadan and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-01 with Mathematics categories.
Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.
A Qualitative Approach To Inverse Scattering Theory
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Author : Fioralba Cakoni
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-28
A Qualitative Approach To Inverse Scattering Theory written by Fioralba Cakoni and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-28 with Mathematics categories.
Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006
The Inverse Problem Of Scattering Theory
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Author : Z.S. Agranovich
language : en
Publisher: Courier Dover Publications
Release Date : 2020-05-21
The Inverse Problem Of Scattering Theory written by Z.S. Agranovich and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-21 with Mathematics categories.
This monograph by two Soviet experts in mathematical physics was a major contribution to inverse scattering theory. The two-part treatment examines the boundary-value problem with and without singularities. 1963 edition.
An Introduction To Inverse Scattering And Inverse Spectral Problems
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Author : Khosrow Chadan
language : en
Publisher: SIAM
Release Date : 1997-01-01
An Introduction To Inverse Scattering And Inverse Spectral Problems written by Khosrow Chadan and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-01-01 with Mathematics categories.
Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.
Spectral Theory Of Infinite Area Hyperbolic Surfaces
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Author : David Borthwick
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-09-13
Spectral Theory Of Infinite Area Hyperbolic Surfaces written by David Borthwick and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-09-13 with Mathematics categories.
This book is a self-contained monograph on spectral theory for non-compact Riemann surfaces, focused on the infinite-volume case. By focusing on the scattering theory of hyperbolic surfaces, this work provides a compelling introductory example which will be accessible to a broad audience. The book opens with an introduction to the geometry of hyperbolic surfaces. Then a thorough development of the spectral theory of a geometrically finite hyperbolic surface of infinite volume is given. The final sections include recent developments for which no thorough account exists.
Inverse Scattering Theory And Transmission Eigenvalues
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Author : Fioralba Cakoni
language : en
Publisher: SIAM
Release Date : 2022-12-07
Inverse Scattering Theory And Transmission Eigenvalues written by Fioralba Cakoni and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-07 with Mathematics categories.
Inverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles. This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.
Inverse Acoustic And Electromagnetic Scattering Theory
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Author : David Colton
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Inverse Acoustic And Electromagnetic Scattering Theory written by David Colton and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
It has now been almost ten years since our first book on scattering theory ap peared [32]. At that time we claimed that "in recent years the development of integral equation methods for the direct scattering problem seems to be nearing completion, whereas the use of such an approach to study the inverse scattering problem has progressed to an extent that a 'state of the art' survey appears highly desirable". Since we wrote these words, the inverse scattering problem for acoustic and electromagnetic waves has grown from being a few theoreti cal considerations with limited numerical implementations to a weH developed mathematical theory with tested numerical algorithms. This maturing of the field of inverse scattering theory has been based on the realization that such problems are in general not only nonlinear but also improperly posed in the sense that the solution does not depend continuously on the measured data. This was emphasized in [32] and treated with the ideas and tools available at that time. Now, almost ten years later, these initial ideas have developed to the extent that a monograph summarizing the mathematical basis of the field seems appropriate. This book is oUf attempt to write such a monograph. The inverse scattering problem for acoustic and electromagnetic waves can broadly be divided into two classes, the inverse obstacle problem and the inverse medium problem.
Inverse Problems In Quantum Scattering Theory
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Author : K. Chadan
language : en
Publisher: Springer
Release Date : 2013-01-22
Inverse Problems In Quantum Scattering Theory written by K. Chadan and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-22 with Science categories.
Spectral And Scattering Theory For Ordinary Differential Equations
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Author : Christer Bennewitz
language : en
Publisher: Springer Nature
Release Date : 2020-10-27
Spectral And Scattering Theory For Ordinary Differential Equations written by Christer Bennewitz and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-27 with Mathematics categories.
This graduate textbook offers an introduction to the spectral theory of ordinary differential equations, focusing on Sturm–Liouville equations. Sturm–Liouville theory has applications in partial differential equations and mathematical physics. Examples include classical PDEs such as the heat and wave equations. Written by leading experts, this book provides a modern, systematic treatment of the theory. The main topics are the spectral theory and eigenfunction expansions for Sturm–Liouville equations, as well as scattering theory and inverse spectral theory. It is the first book offering a complete account of the left-definite theory for Sturm–Liouville equations. The modest prerequisites for this book are basic one-variable real analysis, linear algebra, as well as an introductory course in complex analysis. More advanced background required in some parts of the book is completely covered in the appendices. With exercises in each chapter, the book is suitable for advanced undergraduate and graduate courses, either as an introduction to spectral theory in Hilbert space, or to the spectral theory of ordinary differential equations. Advanced topics such as the left-definite theory and the Camassa–Holm equation, as well as bibliographical notes, make the book a valuable reference for experts.