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Author: Christer Bennewitz
Publisher: Springer Nature
ISBN: 3030590887
Category : Mathematics
Languages : en
Pages : 384
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Book Description
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: Christer Bennewitz
Publisher: Springer Nature
ISBN: 3030590887
Category : Mathematics
Languages : en
Pages : 384
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Book Description
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: Khosrow Chadan
Publisher: SIAM
ISBN: 0898713870
Category : Mathematics
Languages : en
Pages : 206
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Book Description
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: Michael Ruzhansky
Publisher: Chapman & Hall/CRC
ISBN: 9781138360716
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suragan.
Author: P.A. Kuchment
Publisher: Birkhäuser
ISBN: 3034885733
Category : Science
Languages : en
Pages : 363
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Book Description
Linear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389]. They arise in many physical and technical applications [177, 178, 272]. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for integration of nonlinear differential equations. This has led to significant progress in this traditional area [27, 71, 72, 111 119, 250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393, 403, 404]. At the same time, many theoretical and applied problems lead to periodic partial differential equations. We can mention, for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107, 123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417], hydrodynamics [179, 180], elasticity theory [395], the theory of guided waves [87-89, 208, 300], homogenization theory [29, 41, 348], direct and inverse scattering [175, 206, 216, 314, 388, 406-408], parametric resonance theory [122, 178], and spectral theory and spectral geometry [103 105, 381, 382, 389]. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations. The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267].
Author: J.A. Nohel
Publisher: Springer Science & Business Media
ISBN: 9780817638627
Category : Mathematics
Languages : en
Pages : 584
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Book Description
The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential & integral equations, harmonic, complex & stochas tic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in 1960 played a central role in the development of the modern theory of dynami cal systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifi cally by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed dif ferential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gel'fand-Levitan method to the inverse scattering problem for the Schrodinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann Hypothesis. Levinson's papers are typically tightly crafted and masterpieces of brevity and clarity. It is our hope that the publication of these selected papers will bring his mathematical ideas to the attention of the larger mathematical community.
Author: Fritz Gesztesy
Publisher: American Mathematical Soc.
ISBN: 9780821842492
Category : Mathematics
Languages : en
Pages : 472
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Book Description
This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to aparticular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random andErgodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensiveintroduction into an area covered in this volume.
Author: Nikolai Chernov
Publisher: American Mathematical Soc.
ISBN: 0821838407
Category : Mathematics
Languages : en
Pages : 354
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Book Description
Surveys topics in differential equations that are associated with mathematical physics. This book includes such topics as asymptotic formulas for the ground-state energy of fermionic gas, $J$-self adjoint Dirac operators, and spectral theory of Schrodinger operators. It is suitable for mathematicians and physicists.
Author: Michael Stephen Patrick Eastham
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 148
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Book Description
Author: Ian W. Knowles
Publisher: Springer
ISBN: 354047983X
Category : Mathematics
Languages : en
Pages : 517
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Book Description
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: Richard Beals
Publisher: American Mathematical Soc.
ISBN: 1470420546
Category : Mathematics
Languages : en
Pages : 226
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Book Description
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.
