Spectral And Scattering Theory For Ordinary Differential Equations
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Author |
: Christer Bennewitz |
Publisher |
: Springer Nature |
Total Pages |
: 379 |
Release |
: 2020-10-27 |
ISBN-10 |
: 9783030590888 |
ISBN-13 |
: 3030590887 |
Rating |
: 4/5 (88 Downloads) |
Synopsis Spectral and Scattering Theory for Ordinary Differential Equations by : Christer Bennewitz
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 |
: |
Publisher |
: Elsevier |
Total Pages |
: 219 |
Release |
: 2011-09-21 |
ISBN-10 |
: 9780080871240 |
ISBN-13 |
: 0080871240 |
Rating |
: 4/5 (40 Downloads) |
Synopsis Spectral Theory and Asymptotics of Differential Equations by :
Spectral Theory and Asymptotics of Differential Equations
Author |
: W.N. Everitt |
Publisher |
: Springer |
Total Pages |
: 338 |
Release |
: 2006-11-15 |
ISBN-10 |
: 9783540374442 |
ISBN-13 |
: 3540374442 |
Rating |
: 4/5 (42 Downloads) |
Synopsis Spectral Theory and Differential Equations by : W.N. Everitt
Author |
: Hiroshi Isozaki |
Publisher |
: Springer Nature |
Total Pages |
: 130 |
Release |
: 2020-09-26 |
ISBN-10 |
: 9789811581991 |
ISBN-13 |
: 9811581991 |
Rating |
: 4/5 (91 Downloads) |
Synopsis Inverse Spectral and Scattering Theory by : Hiroshi Isozaki
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 |
: Richard Beals |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 226 |
Release |
: 2015-03-02 |
ISBN-10 |
: 9781470420543 |
ISBN-13 |
: 1470420546 |
Rating |
: 4/5 (43 Downloads) |
Synopsis Direct and Inverse Scattering on the Line by : Richard Beals
This book deals with the theory of linear ordinary differential operators of arbitrary order. Unlike treatments that focus on spectral theory, this work centers on the construction of special eigenfunctions (generalized Jost solutions) and on the inverse problem: the problem of reconstructing the operator from minimal data associated to the special eigenfunctions. In the second order case this program includes spectral theory and is equivalent to quantum mechanical scattering theory; the essential analysis involves only the bounded eigenfunctions. For higher order operators, bounded eigenfunctions are again sufficient for spectral theory and quantum scattering theory, but they are far from sufficient for a successful inverse theory. The authors give a complete and self-contained theory of the inverse problem for an ordinary differential operator of any order. The theory provides a linearization for the associated nonlinear evolution equations, including KdV and Boussinesq. The authors also discuss Darboux-Bäcklund transformations, related first-order systems and their evolutions, and applications to spectral theory and quantum mechanical scattering theory. Among the book's most significant contributions are a new construction of normalized eigenfunctions and the first complete treatment of the self-adjoint inverse problem in order greater than two. In addition, the authors present the first analytic treatment of the corresponding flows, including a detailed description of the phase space for Boussinesq and other equations. The book is intended for mathematicians, physicists, and engineers in the area of soliton equations, as well as those interested in the analytical aspects of inverse scattering or in the general theory of linear ordinary differential operators. This book is likely to be a valuable resource to many. Required background consists of a basic knowledge of complex variable theory, the theory of ordinary differential equations, linear algebra, and functional analysis. The authors have attempted to make the book sufficiently complete and self-contained to make it accessible to a graduate student having no prior knowledge of scattering or inverse scattering theory. The book may therefore be suitable for a graduate textbook or as background reading in a seminar.
Author |
: Yulia E. Karpeshina |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 410 |
Release |
: 2003 |
ISBN-10 |
: 9780821832967 |
ISBN-13 |
: 0821832964 |
Rating |
: 4/5 (67 Downloads) |
Synopsis Advances in Differential Equations and Mathematical Physics by : Yulia E. Karpeshina
This volume presents the proceedings of the 9th International Conference on Differential Equations and Mathematical Physics. It contains 29 research and survey papers contributed by conference participants. The conference provided researchers a forum to present and discuss their recent results in a broad range of areas encompassing the theory of differential equations and their applications in mathematical physics. Papers in this volume represent some of the most interesting results and the major areas of research that were covered, including spectral theory with applications to non-relativistic and relativistic quantum mechanics, including time-dependent and random potential, resonances, many body systems, pseudodifferential operators and quantum dynamics, inverse spectral and scattering problems, the theory of linear and nonlinear partial differential equations with applications in fluid dynamics, conservation laws and numerical simulations, as well as equilibrium and nonequilibrium statistical mechanics. The volume is intended for graduate students and researchers interested in mathematical physics.
Author |
: Kiyoshi Mochizuki |
Publisher |
: CRC Press |
Total Pages |
: 131 |
Release |
: 2017-06-01 |
ISBN-10 |
: 9781351648943 |
ISBN-13 |
: 1351648942 |
Rating |
: 4/5 (43 Downloads) |
Synopsis Spectral and Scattering Theory for Second Order Partial Differential Operators by : Kiyoshi Mochizuki
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 |
: Alexander G. Ramm |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 220 |
Release |
: 1998-04-30 |
ISBN-10 |
: 0306458292 |
ISBN-13 |
: 9780306458293 |
Rating |
: 4/5 (92 Downloads) |
Synopsis Spectral and Scattering Theory by : Alexander G. Ramm
Proceedings of Sessions from the First Congress of the International Society for Analysis, Applications and Computing held in Newark, Delaware, June, 2-, 1997
Author |
: Ian W. Knowles |
Publisher |
: Springer |
Total Pages |
: 517 |
Release |
: 2006-11-14 |
ISBN-10 |
: 9783540479833 |
ISBN-13 |
: 354047983X |
Rating |
: 4/5 (33 Downloads) |
Synopsis Differential Equations and Mathematical Physics by : Ian W. Knowles
The meeting in Birmingham, Alabama, provided a forum for the discussion of recent developments in the theory of ordinary and partial differential equations, both linear and non-linear, with particular reference to work relating to the equations of mathematical physics. The meeting was attended by about 250 mathematicians from 22 countries. The papers in this volume all involve new research material, with at least outline proofs; some papers also contain survey material. Topics covered include: Schrödinger theory, scattering and inverse scattering, fluid mechanics (including conservative systems and inertial manifold theory attractors), elasticity, non-linear waves, and feedback control theory.
Author |
: Keiji Yajima |
Publisher |
: Amer Mathematical Society |
Total Pages |
: 322 |
Release |
: 1994 |
ISBN-10 |
: 4314101075 |
ISBN-13 |
: 9784314101073 |
Rating |
: 4/5 (75 Downloads) |
Synopsis Spectral and Scattering Theory and Applications by : Keiji Yajima
This work contains the proceedings from a conference on Spectral and Scattering Theory, held in July 1992 at Tokyo Institute of Technology, in celebration of the 60th birthday of ShigeToshi Kuroda. It is a guide to recent results in spectral and scattering theory and applications to linear and nonlinear equations. Among the application areas covered are Schrodinger and wave equations, Boltzmann and MHD equations, and elliptic and parabolic equations. Abstract spectral theory is also discussed. It is aimed at mathematicians and graduate students in operator theory, partial differential equations, mathematical physics, and applied mathematics, in addition to physicists and chemists working in such areas as atomic or molecular physics.