Spectral Geometry And Inverse Scattering Theory

Author : Huaian Diao,Hongyu Liu
Publisher : Springer Nature
Page : 388 pages
File Size : 45,9 Mb
Release : 2023-10-31
Category : Mathematics
ISBN : 9783031346156

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Spectral Geometry and Inverse Scattering Theory by Huaian Diao,Hongyu Liu Pdf

Inverse scattering problems are a vital subject for both theoretical and experimental studies and remain an active field of research in applied mathematics. This book provides a detailed presentation of typical setup of inverse scattering problems for time-harmonic acoustic, electromagnetic and elastic waves. Moreover, it provides systematical and in-depth discussion on an important class of geometrical inverse scattering problems, where the inverse problem aims at recovering the shape and location of a scatterer independent of its medium properties. Readers of this book will be exposed to a unified framework for analyzing a variety of geometrical inverse scattering problems from a spectral geometric perspective. This book contains both overviews of classical results and update-to-date information on latest developments from both a practical and theoretical point of view. It can be used as an advanced graduate textbook in universities or as a reference source for researchers in acquiring the state-of-the-art results in inverse scattering theory and their potential applications.

Author : Khosrow Chadan,David Colton,William Rundell,Lassi P?iv?rinta
Publisher : SIAM
Page : 208 pages
File Size : 43,7 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0898719712

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An Introduction to Inverse Scattering and Inverse Spectral Problems by Khosrow Chadan,David Colton,William Rundell,Lassi P?iv?rinta Pdf

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.

Author : Vesselin M. Petkov,Luchezar N. Stoyanov
Publisher : John Wiley & Sons
Page : 428 pages
File Size : 48,7 Mb
Release : 2017-01-30
Category : Mathematics
ISBN : 9781119107668

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Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems by Vesselin M. Petkov,Luchezar N. Stoyanov Pdf

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.

Author : Hiroshi Isozaki,Yaroslav Kurylev
Publisher : Unknown
Page : 0 pages
File Size : 43,5 Mb
Release : 2014-06
Category : Mathematics
ISBN : 4864970211

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Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds by Hiroshi Isozaki,Yaroslav Kurylev Pdf

This manuscript is devoted to a rigorous and detailed exposition of the spectral theory and associated forward and inverse scattering problems for the Laplace-Beltrami operators on asymptotically hyperbolic manifolds. Based upon the classical stationary scattering theory in ℝn, the key point of the approach is the generalized Fourier transform, which serves as the basic tool to introduce and analyse the time-dependent wave operators and the S-matrix. The crucial role is played by the characterization of the space of the scattering solutions for the Helmholtz equations utilizing a properly defined Besov-type space. After developing the scattering theory, we describe, for some cases, the inverse scattering on the asymptotically hyperbolic manifolds by adopting, for the considered case, the boundary control method for inverse problems.The manuscript is aimed at graduate students and young mathematicians interested in spectral and scattering theories, analysis on hyperbolic manifolds and theory of inverse problems. We try to make it self-consistent and, to a large extent, not dependent on the existing treatises on these topics. To our best knowledge, it is the first comprehensive description of these theories in the context of the asymptotically hyperbolic manifolds.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

Author : Hiroshi Isozaki
Publisher : Springer Nature
Page : 130 pages
File Size : 50,7 Mb
Release : 2020-09-26
Category : Science
ISBN : 9789811581991

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Inverse Spectral and Scattering Theory by Hiroshi Isozaki Pdf

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.

Author : Hiroshi Isozaki
Publisher : American Mathematical Soc.
Page : 243 pages
File Size : 50,7 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821834213

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Inverse Problems and Spectral Theory by Hiroshi Isozaki Pdf

This volume grew out of a workshop on spectral theory of differential operators and inverse problems held at the Research Institute for Mathematical Sciences (Kyoto University). The gathering of nearly 100 participants at the conference suggests the increasing interest in this field of research. The focus of the book is on spectral theory for differential operators and related inverse problems. It includes selected topics from the following areas: electromagnetism, elasticity, the Schrodinger equation, differential geometry, and numerical analysis. The material is suitable for graduate students and researchers interested in inverse problems and their applications.

Author : David Borthwick
Publisher : Birkhäuser
Page : 471 pages
File Size : 44,6 Mb
Release : 2016-07-12
Category : Mathematics
ISBN : 9783319338774

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Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick Pdf

This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)

Author : Jingzhi Li,Hongyu Liu
Publisher : Springer Nature
Page : 373 pages
File Size : 47,6 Mb
Release : 2023-09-07
Category : Science
ISBN : 9789819937721

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Numerical Methods for Inverse Scattering Problems by Jingzhi Li,Hongyu Liu Pdf

This book highlights the latest developments on the numerical methods for inverse scattering problems associated with acoustic, electromagnetic, and elastic waves. Inverse scattering problems are concerned with identifying unknown or inaccessible objects by wave probing data, which makes possible many industrial and engineering applications including radar and sonar, medical imaging, nondestructive testing, remote sensing, and geophysical exploration. The mathematical study of inverse scattering problems is an active field of research. This book presents a comprehensive and unified mathematical treatment of various inverse scattering problems mainly from a numerical reconstruction perspective. It highlights the collaborative research outputs by the two groups of the authors yet surveys and reviews many existing results by global researchers in the literature. The book consists of three parts respectively corresponding to the studies on acoustic, electromagnetic, and elastic scattering problems. In each part, the authors start with in-depth theoretical and computational treatments of the forward scattering problems and then discuss various numerical reconstruction schemes for the associated inverse scattering problems in different scenarios of practical interest. In addition, the authors provide an overview of the existing results in the literature by other researchers. This book can serve as a handy reference for researchers or practitioners who are working on or implementing inverse scattering methods. It can also serve as a graduate textbook for research students who are interested in working on numerical algorithms for inverse scattering problems.

Author : K. Chadan,P. C. Sabatier
Publisher : Springer Science & Business Media
Page : 364 pages
File Size : 52,7 Mb
Release : 2013-04-18
Category : Science
ISBN : 9783662121252

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Inverse Problems in Quantum Scattering Theory by K. Chadan,P. C. Sabatier Pdf

Author : Isozaki Hiroshi
Publisher : World Scientific
Page : 300 pages
File Size : 46,7 Mb
Release : 1998-08-15
Category : Science
ISBN : 9789813232716

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Maxwell Equation by Isozaki Hiroshi Pdf

How can one determine the physical properties of the medium or the geometrical properties of the domain by observing electromagnetic waves? To answer this fundamental problem in mathematics and physics, this book leads the reader to the frontier of inverse scattering theory for electromagnetism. The first three chapters, written comprehensively, can be used as a textbook for undergraduate students. Beginning with elementary vector calculus, this book provides fundamental results for wave equations and Helmholtz equations, and summarizes the potential theory. It also explains the cohomology theory in an easy and straightforward way, which is an essential part of electromagnetism related to geometry. It then describes the scattering theory for the Maxwell equation by the time-dependent method and also by the stationary method in a concise, but almost self-contained manner. Based on these preliminary results, the book proceeds to the inverse problem for the Maxwell equation. The chapters for the potential theory and elementary cohomology theory are good introduction to graduate students. The results in the last chapter on the inverse scattering for the medium and the determination of Betti numbers are new, and will give a current scope for the inverse spectral problem on non-compact manifolds. It will be useful for young researchers who are interested in this field and trying to find new problems.

Author : Alexander G. Ramm
Publisher : Springer Science & Business Media
Page : 207 pages
File Size : 47,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781489915528

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Spectral and Scattering Theory by Alexander G. Ramm Pdf

Proceedings of Sessions from the First Congress of the International Society for Analysis, Applications and Computing held in Newark, Delaware, June, 2-, 1997

Author : Bertero
Publisher : CRC Press
Page : 454 pages
File Size : 48,8 Mb
Release : 1992-02-27
Category : Technology & Engineering
ISBN : 0750301430

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Inverse Problems in Scattering and Imaging by Bertero Pdf

Inverse Problems in Scattering and Imaging is a collection of lectures from a NATO Advanced Research Workshop that integrates the expertise of physicists and mathematicians in different areas with a common interest in inverse problems. Covering a range of subjects from new developments on the applied mathematics/mathematical physics side to many areas of application, the book achieves a blend of research, review, and tutorial contributions. It is of interest to researchers in the areas of applied mathematics and mathematical physics as well as those working in areas where inverse problems can be applied.

Author : J. L. Bona,Roy Choudhury,David Kaup,Ams-IMS-Siam Joint Summer Research Conference on the Legacy of Inverse Scattering Transfor
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 45,9 Mb
Release : 2002
Category : Science
ISBN : 9780821831618

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The Legacy of the Inverse Scattering Transform in Applied Mathematics by J. L. Bona,Roy Choudhury,David Kaup,Ams-IMS-Siam Joint Summer Research Conference on the Legacy of Inverse Scattering Transfor Pdf

Swift progress and new applications characterize the area of solitons and the inverse scattering transform. There are rapid developments in current nonlinear optical technology: Larger intensities are more available; pulse widths are smaller; relaxation times and damping rates are less significant. In keeping with these advancements, exactly integrable soliton equations, such as $3$-wave resonant interactions and second harmonic generation, are becoming more and more relevant in experimental applications. Techniques are now being developed for using these interactions to frequency convert high intensity sources into frequency regimes where there are no lasers. Other experiments involve using these interactions to develop intense variable frequency sources, opening up even more possibilities. This volume contains new developments and state-of-the-art research arising from the conference on the ""Legacy of the Inverse Scattering Transform"" held at Mount Holyoke College (South Hadley, MA). Unique to this volume is the opening section, ""Reviews"". This part of the book provides reviews of major research results in the inverse scattering transform (IST), on the application of IST to classical problems in differential geometry, on algebraic and analytic aspects of soliton-type equations, on a new method for studying boundary value problems for integrable partial differential equations (PDEs) in two dimensions, on chaos in PDEs, on advances in multi-soliton complexes, and on a unified approach to integrable systems via Painleve analysis. This conference provided a forum for general exposition and discussion of recent developments in nonlinear waves and related areas with potential applications to other fields. The book will be of interest to graduate students and researchers interested in mathematics, physics, and engineering.

Author : Fioralba Cakoni,David Colton,Houssem Haddar
Publisher : SIAM
Page : 203 pages
File Size : 46,5 Mb
Release : 2016-10-28
Category : Mathematics
ISBN : 9781611974461

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Inverse Scattering Theory and Transmission Eigenvalues by Fioralba Cakoni,David Colton,Houssem Haddar Pdf

Inverse scattering theory is a major theme of applied mathematics, and it has 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 particular problems in the development of efficient inversion algorithms. Although linearized models continue to play an important role in many applications, an increased need to focus on problems in which multiple scattering effects cannot be ignored has led to a central role for nonlinearity, and the possibility of collecting large amounts of data over limited regions of space means that the ill-posed nature of the inverse scattering problem has become a problem of central importance.? Initial efforts to address the nonlinear and the ill-posed nature of the inverse scattering problem focused on nonlinear optimization methods. While efficient in many situations, strong a priori information is necessary for their implementation. This problem led to a qualitative approach to inverse scattering theory in which the amount of a priori information is drastically reduced, although at the expense of only obtaining limited information about the values of the constitutive parameters. This qualitative approach (the linear sampling method, the factorization method, the theory of transmission eigenvalues, etc.) is the theme of Inverse Scattering Theory and Transmission Eigenvalues.? The authors begin with a basic introduction to the theory, then proceed to more recent developments, including a detailed discussion of the transmission eigenvalue problem; present the new generalized linear sampling method in addition to the well-known linear sampling and factorization methods; and in order to achieve clarification of presentation, focus on the inverse scattering problem for scalar homogeneous media.?

Author : J. Bee Bednar
Publisher : SIAM
Page : 310 pages
File Size : 47,9 Mb
Release : 1983-01-01
Category : Mathematics
ISBN : 0898711908

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Conference on Inverse Scattering--Theory and Application by J. Bee Bednar Pdf