Spectral And Scattering Theory For Second Order Partial Differential Operators

Author : Kiyoshi Mochizuki
Publisher : CRC Press
ISBN 13 : 1351648942
Total Pages : 131 pages
Book Rating : 4.43/5 ( download)

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Download or read book Spectral and Scattering Theory for Second Order Partial Differential Operators written by Kiyoshi Mochizuki and published by CRC Press. This book was released on 2017-06-01 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is intended for students of graduate and postgraduate level, researchers in mathematical sciences as well as those who want to apply the spectral theory of second order differential operators in exterior domains to their own field. In the first half of this book, the classical results of spectral and scattering theory: the selfadjointness, essential spectrum, absolute continuity of the continuous spectrum, spectral representations, short-range and long-range scattering are summarized. In the second half, recent results: scattering of Schrodinger operators on a star graph, uniform resolvent estimates, smoothing properties and Strichartz estimates, and some applications are discussed.

Author : Kiyoshi Mochizuki
Publisher : CRC Press
ISBN 13 : 1498756034
Total Pages : 232 pages
Book Rating : 4.37/5 ( download)

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Download or read book Spectral and Scattering Theory for Second Order Partial Differential Operators written by Kiyoshi Mochizuki and published by CRC Press. This book was released on 2017-06-01 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is intended for students of graduate and postgraduate level, researchers in mathematical sciences as well as those who want to apply the spectral theory of second order differential operators in exterior domains to their own field. In the first half of this book, the classical results of spectral and scattering theory: the selfadjointness, essential spectrum, absolute continuity of the continuous spectrum, spectral representations, short-range and long-range scattering are summarized. In the second half, recent results: scattering of Schrodinger operators on a star graph, uniform resolvent estimates, smoothing properties and Strichartz estimates, and some applications are discussed.

Author : Hiroshi Isozaki
Publisher : Springer Nature
ISBN 13 : 9811581991
Total Pages : 130 pages
Book Rating : 4.91/5 ( download)

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Download or read book Inverse Spectral and Scattering Theory written by Hiroshi Isozaki and published by Springer Nature. This book was released on 2020-09-26 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : I.W. Knowles
Publisher : Elsevier
ISBN 13 : 9780080871660
Total Pages : 383 pages
Book Rating : 4.66/5 ( download)

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Download or read book Spectral Theory of Differential Operators written by I.W. Knowles and published by Elsevier. This book was released on 1981-01-01 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral Theory of Differential Operators

Author :
Publisher :
ISBN 13 : 9780387546773
Total Pages : pages
Book Rating : 4.74/5 ( download)

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Download or read book Partial Differential Equations written by and published by . This book was released on 1991 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Christer Bennewitz
Publisher : Springer Nature
ISBN 13 : 3030590887
Total Pages : 379 pages
Book Rating : 4.88/5 ( download)

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Download or read book Spectral and Scattering Theory for Ordinary Differential Equations written by Christer Bennewitz and published by Springer Nature. This book was released on 2020-10-27 with total page 379 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Michael Ruzhansky
Publisher : Chapman & Hall/CRC
ISBN 13 : 9781138360716
Total Pages : 0 pages
Book Rating : 4.16/5 ( download)

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Download or read book Spectral Geometry of Partial Differential Operators written by Michael Ruzhansky and published by Chapman & Hall/CRC. This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suragan.

Author : Alexander Komech
Publisher : John Wiley & Sons
ISBN 13 : 1118382889
Total Pages : 236 pages
Book Rating : 4.82/5 ( download)

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Download or read book Dispersion Decay and Scattering Theory written by Alexander Komech and published by John Wiley & Sons. This book was released on 2014-08-21 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: A simplified, yet rigorous treatment of scattering theory methods and their applications Dispersion Decay and Scattering Theory provides thorough, easy-to-understand guidance on the application of scattering theory methods to modern problems in mathematics, quantum physics, and mathematical physics. Introducing spectral methods with applications to dispersion time-decay and scattering theory, this book presents, for the first time, the Agmon-Jensen-Kato spectral theory for the Schr?dinger equation, extending the theory to the Klein-Gordon equation. The dispersion decay plays a crucial role in the modern application to asymptotic stability of solitons of nonlinear Schr?dinger and Klein-Gordon equations. The authors clearly explain the fundamental concepts and formulas of the Schr?dinger operators, discuss the basic properties of the Schr?dinger equation, and offer in-depth coverage of Agmon-Jensen-Kato theory of the dispersion decay in the weighted Sobolev norms. The book also details the application of dispersion decay to scattering and spectral theories, the scattering cross section, and the weighted energy decay for 3D Klein-Gordon and wave equations. Complete streamlined proofs for key areas of the Agmon-Jensen-Kato approach, such as the high-energy decay of the resolvent and the limiting absorption principle are also included. Dispersion Decay and Scattering Theory is a suitable book for courses on scattering theory, partial differential equations, and functional analysis at the graduate level. The book also serves as an excellent resource for researchers, professionals, and academics in the fields of mathematics, mathematical physics, and quantum physics who would like to better understand scattering theory and partial differential equations and gain problem-solving skills in diverse areas, from high-energy physics to wave propagation and hydrodynamics.

Author : Vladimir Georgiev
Publisher : Springer Nature
ISBN 13 : 3030582159
Total Pages : 317 pages
Book Rating : 4.59/5 ( download)

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Download or read book Advances in Harmonic Analysis and Partial Differential Equations written by Vladimir Georgiev and published by Springer Nature. This book was released on 2020-11-07 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book originates from the session "Harmonic Analysis and Partial Differential Equations" held at the 12th ISAAC Congress in Aveiro, and provides a quick overview over recent advances in partial differential equations with a particular focus on the interplay between tools from harmonic analysis, functional inequalities and variational characterisations of solutions to particular non-linear PDEs. It can serve as a useful source of information to mathematicians, scientists and engineers. The volume contains contributions of authors from a variety of countries on a wide range of active research areas covering different aspects of partial differential equations interacting with harmonic analysis and provides a state-of-the-art overview over ongoing research in the field. It shows original research in full detail allowing researchers as well as students to grasp new aspects and broaden their understanding of the area.

Author : David Eric Edmunds
Publisher :
ISBN 13 :
Total Pages : 604 pages
Book Rating : 4.33/5 ( download)

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Download or read book Spectral Theory and Differential Operators written by David Eric Edmunds and published by . This book was released on 1987 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive and long-awaited volume provides an up-to-date account of those parts of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. For the first time it brings together recent results in essential spectra, measures of non-compactness, entropy numbers, approximation numbers, eigenvalues, and the relationships among these concepts. The authors illustrate abstract theory with results for embedding maps between Sobolev spaces. Strong emphasis is placed on application to boundary-value problems for general second-order linear elliptic equations in an arbitrary domain in Rn. The book introduces some key eastern European work, never before available in English translation.