Spectral And Scattering Theory For Second Order Partial Differential Operators PDF Download
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Author | : Kiyoshi Mochizuki |
Publisher | : CRC Press |
Total Pages | : 131 |
Release | : 2017-06-01 |
Genre | : Mathematics |
ISBN | : 1351648942 |
Download Spectral and Scattering Theory for Second Order Partial Differential Operators Book in PDF, ePub and Kindle
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 |
Total Pages | : 232 |
Release | : 2017-06-01 |
Genre | : Mathematics |
ISBN | : 1498756034 |
Download Spectral and Scattering Theory for Second Order Partial Differential Operators Book in PDF, ePub and Kindle
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 |
Total Pages | : 130 |
Release | : 2020-09-26 |
Genre | : Science |
ISBN | : 9811581991 |
Download Inverse Spectral and Scattering Theory Book in PDF, ePub and Kindle
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 |
Total Pages | : 383 |
Release | : 1981-01-01 |
Genre | : Mathematics |
ISBN | : 9780080871660 |
Download Spectral Theory of Differential Operators Book in PDF, ePub and Kindle
Spectral Theory of Differential Operators
Author | : |
Publisher | : |
Total Pages | : |
Release | : 1991 |
Genre | : Differential equations, Partial |
ISBN | : 9780387546773 |
Download Partial Differential Equations Book in PDF, ePub and Kindle
Author | : Dmitri R. Yafaev |
Publisher | : Springer |
Total Pages | : 185 |
Release | : 2007-05-06 |
Genre | : Mathematics |
ISBN | : 3540451706 |
Download Scattering Theory: Some Old and New Problems Book in PDF, ePub and Kindle
Scattering theory is, roughly speaking, perturbation theory of self-adjoint operators on the (absolutely) continuous spectrum. It has its origin in mathematical problems of quantum mechanics and is intimately related to the theory of partial differential equations. Some recently solved problems, such as asymptotic completeness for the Schrödinger operator with long-range and multiparticle potentials, as well as open problems, are discussed. Potentials for which asymptotic completeness is violated are also constructed. This corresponds to a new class of asymptotic solutions of the time-dependent Schrödinger equation. Special attention is paid to the properties of the scattering matrix, which is the main observable of the theory. The book is addressed to readers interested in a deeper study of the subject.
Author | : Michael Ruzhansky |
Publisher | : Chapman & Hall/CRC |
Total Pages | : 0 |
Release | : 2020 |
Genre | : Mathematics |
ISBN | : 9781138360716 |
Download Spectral Geometry of Partial Differential Operators Book in PDF, ePub and Kindle
Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suragan.
Author | : Alexander Komech |
Publisher | : John Wiley & Sons |
Total Pages | : 236 |
Release | : 2014-08-21 |
Genre | : Mathematics |
ISBN | : 1118382889 |
Download Dispersion Decay and Scattering Theory Book in PDF, ePub and Kindle
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 |
Total Pages | : 317 |
Release | : 2020-11-07 |
Genre | : Mathematics |
ISBN | : 3030582159 |
Download Advances in Harmonic Analysis and Partial Differential Equations Book in PDF, ePub and Kindle
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 | : Christer Bennewitz |
Publisher | : Springer Nature |
Total Pages | : 379 |
Release | : 2020-10-27 |
Genre | : Mathematics |
ISBN | : 3030590887 |
Download Spectral and Scattering Theory for Ordinary Differential Equations Book in PDF, ePub and Kindle
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.