Spectral And Scattering Theory
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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.
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.
Spectral And Scattering Theory
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Author : Alexander G. Ramm
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Spectral And Scattering Theory written by Alexander G. Ramm 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 2013-06-29 with Mathematics categories.
Proceedings of Sessions from the First Congress of the International Society for Analysis, Applications and Computing held in Newark, Delaware, June, 2-, 1997
Quantum Scattering And Spectral Theory
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Author : D. B. Pearson
language : en
Publisher:
Release Date : 1988
Quantum Scattering And Spectral Theory written by D. B. Pearson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.
FROM THE PREFACE: This book deals with the foundations of the quantum theory of scattering. Scattering theory may be regarded either as a branch of mathematical physics or, increasingly, as a branch of mathematics worthy of independent study in its own right. The importance of spectral analysis to the theory is central; every modern text on scattering theory makes reference to the methods and ideas of spectral analysis, and conversely any comprehensive treatment of spectral theory will refer to methods and ideas drawn from applications to quantum theory, and to quantum scattering in particular. Much of the material in this volume, while relating to important aspects of the theory, is new or is presented for the first time in book form.
Spectral Methods In Quantum Field Theory
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Author : Noah Graham
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-05-08
Spectral Methods In Quantum Field Theory written by Noah Graham 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 2009-05-08 with Science categories.
In this monograph we apply scattering theory methods to calculations in quantum ?eld theory, with a particular focus on properties of the quantum vacuum. These methods will provide e?cient and reliable solutions to a - riety of problems in quantum ?eld theory. Our approach will also elucidate in a concrete context many of the subtleties of quantum ?eld theory, such as divergences, regularization, and renormalization, by connecting them to more familiar results in quantum mechanics. We will use tools of scattering theory to characterize the spectrum of energyeigenstatesinapotentialbackground,hencethetermspectralmethods. This mode spectrum comprises both discrete bound states and a continuum of scattering states. We develop a powerful formalism that parameterizes the e?ects of the continuum by the density of states, which we compute from scattering data. Summing the zero-point energies of these modes gives the energy of the quantum vacuum, which is one of the central quantities we study.Althoughthemostcommonlystudiedbackgroundpotentialsarisefrom static soliton solutions to the classical equations of motion, these methods are not limited to such cases.
Spectral And Scattering Theory
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Author : Mitsuru Ikawa
language : en
Publisher:
Release Date : 1994
Spectral And Scattering Theory written by Mitsuru Ikawa and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with categories.
Multiple Scattering Theory
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Author : J. S. Faulkner
language : en
Publisher:
Release Date : 2018
Multiple Scattering Theory written by J. S. Faulkner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Energy-band theory of solids categories.
"In 1947, it was discovered that multiple scattering theory can be used to solve the Schrèodinger equation for the stationary states of electrons in a solid. Written by experts in the field, Dr. J S Faulkner, G M Stocks, and Yang Wang, this book collates the results of numerous studies in the field of multiple scattering theory and provides a comprehensive, systematic approach to MSTs." -- Prové de l'editor.
Scattering Theory For Diffraction Gratings
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Author : Calvin H. Wilcox
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Scattering Theory For Diffraction Gratings written by Calvin H. Wilcox 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 2012-12-06 with Mathematics categories.
The scattering of acoustic and electromagnetic waves by periodic sur faces plays a role in many areas of applied physics and engineering. Opti cal diffraction gratings date from the nineteenth century and are still widely used by spectroscopists. More recently, diffraction gratings have been used as coupling devices for optical waveguides. Trains of surface waves on the oceans are natural diffraction gratings which influence the scattering of electromagnetic waves and underwater sound. Similarly, the surface of a crystal acts as a diffraction grating for the scattering of atomic beams. This list of natural and artificial diffraction gratings could easily be extended. The purpose of this monograph is to develop from first principles a theory of the scattering of acoustic and electromagnetic waves by periodic surfaces. In physical terms, the scattering of both time-harmonic and transient fields is analyzed. The corresponding mathematical model leads to the study of boundary value problems for the Helmholtz and d'Alembert wave equations in plane domains bounded by periodic curves. In the formal ism adopted here these problems are intimately related to the spectral analysis of the Laplace operator, acting in a Hilbert space of functions defined in the domain adjacent to the grating.
Spectral And Scattering Theory For Second Order Partial Differential Operators
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Author : Kiyoshi Mochizuki
language : en
Publisher: CRC Press
Release Date : 2017-06-01
Spectral And Scattering Theory For Second Order Partial Differential Operators written by Kiyoshi Mochizuki and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-01 with Mathematics categories.
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.