Direct And Inverse Scattering On The Line

Author : Richard Beals
Publisher : American Mathematical Soc.
ISBN 13 : 1470420546
Total Pages : 226 pages
Book Rating : 4.43/5 ( download)

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Download or read book Direct and Inverse Scattering on the Line written by Richard Beals and published by American Mathematical Soc.. This book was released on 2015-03-02 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Richard Beals
Publisher :
ISBN 13 : 9781470412555
Total Pages : 225 pages
Book Rating : 4.51/5 ( download)

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Download or read book Direct and Inverse Scattering on the Line written by Richard Beals and published by . This book was released on 2014-06-29 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Deals with the theory of linear ordinary differential operators of arbitrary order. This book centers on the construction of special eigenfunctions and on the inverse problem. It is suitable for mathematicians, physicists, and engineers in the area of soliton equations, as well as those interested in the analytical aspects of inverse scattering.

Author : Tuncay Aktosun
Publisher : Springer Nature
ISBN 13 : 3030384314
Total Pages : 631 pages
Book Rating : 4.19/5 ( download)

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Download or read book Direct and Inverse Scattering for the Matrix Schrödinger Equation written by Tuncay Aktosun and published by Springer Nature. This book was released on 2020-05-19 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: Authored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.

Author : Richard Beals, Percy Deift, Carlos Tomei
Publisher : American Mathematical Soc.
ISBN 13 : 9780821874837
Total Pages : 228 pages
Book Rating : 4.37/5 ( download)

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Download or read book Direct and Inverse Scattering on the Line written by Richard Beals, Percy Deift, Carlos Tomei and published by American Mathematical Soc.. This book was released on with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Deals with the theory of linear ordinary differential operators of arbitrary order. This book centers on the construction of special eigenfunctions and on the inverse problem. It is suitable for mathematicians, physicists, and engineers in the area of soliton equations, as well as those interested in the analytical aspects of inverse scattering.

Author : University of Minnesota. Institute for Mathematics and Its Applications
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.43/5 ( download)

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Download or read book On the Direct and Inverse Scattering for the Matrix Schroedinger Equation on the Line written by University of Minnesota. Institute for Mathematics and Its Applications and published by . This book was released on 1999 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Ramazan Ercan
Publisher :
ISBN 13 :
Total Pages : 204 pages
Book Rating : 4.49/5 ( download)

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Download or read book Scattering and Inverse Scattering on the Line for a First-order System with Energy-dependent Potentials written by Ramazan Ercan and published by . This book was released on 2019 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: A first-order system of two linear ordinary differential equations is analyzed. The linear system contains a spectral parameter, and it has two coefficients that are functions of the spatial variable x. Those two functions act as potentials in the linear system and they also linearly contain the spectral parameter ʎ, and hence they are referred to as energy-dependent potentials. Such a linear system arises in the solution to a pair of integrable nonlinear partial differential equations (known as the derivative nonlinear Schr ̈odinger equations) via the so-called inverse scattering transform method.The direct and inverse problems for the corresponding first-order linear system with energy-dependent potentials are investigated. In the direct problem, when the two potentials belong to the Schwartz class, the properties of the corresponding scattering coefficients and so-called bound-state data are derived. In the inverse problem, the two potentials are recovered from the scattering data set consisting of the scattering coefficients and bound-state data. The solutions to the direct and inverse problems are achieved by relating the scattering data and the potentials in the energy-dependent system to those in a pair of first-order system with energy independent potentials. An alternate solution to the inverse problem is given by formulating a linear integral equation (referred to as the alternate Marchenko integral equation), and the energy-dependent potentials are recovered with the help of the solution to the alternate Marchenko equation.

Author : Vladislav V. Kravchenko
Publisher : Birkhäuser
ISBN 13 : 9783030478483
Total Pages : 154 pages
Book Rating : 4.83/5 ( download)

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Download or read book Direct and Inverse Sturm-Liouville Problems written by Vladislav V. Kravchenko and published by Birkhäuser. This book was released on 2020-08-18 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the most recent developments in the theory and practice of direct and inverse Sturm-Liouville problems on finite and infinite intervals. A universal approach for practical solving of direct and inverse spectral and scattering problems is presented, based on the notion of transmutation (transformation) operators and their efficient construction. Analytical representations for solutions of Sturm-Liouville equations as well as for the integral kernels of the transmutation operators are derived in the form of functional series revealing interesting special features and lending themselves to direct and simple numerical solution of a wide variety of problems. The book is written for undergraduate and graduate students, as well as for mathematicians, physicists and engineers interested in direct and inverse spectral problems.

Author : E. R. Pike
Publisher : Elsevier
ISBN 13 : 0080540732
Total Pages : 1831 pages
Book Rating : 4.33/5 ( download)

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Download or read book Scattering, Two-Volume Set written by E. R. Pike and published by Elsevier. This book was released on 2001-10-09 with total page 1831 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scattering is the collision of two objects that results in a change of trajectory and energy. For example, in particle physics, such as electrons, photons, or neutrons are "scattered off" of a target specimen, resulting in a different energy and direction. In the field of electromagnetism, scattering is the random diffusion of electromagnetic radiation from air masses is an aid in the long-range sending of radio signals over geographic obstacles such as mountains. This type of scattering, applied to the field of acoustics, is the spreading of sound in many directions due to irregularities in the transmission medium. Volume I of Scattering will be devoted to basic theoretical ideas, approximation methods, numerical techniques and mathematical modeling. Volume II will be concerned with basic experimental techniques, technological practices, and comparisons with relevant theoretical work including seismology, medical applications, meteorological phenomena and astronomy. This reference will be used by researchers and graduate students in physics, applied physics, biophysics, chemical physics, medical physics, acoustics, geosciences, optics, mathematics, and engineering. This is the first encyclopedic-range work on the topic of scattering theory in quantum mechanics, elastodynamics, acoustics, and electromagnetics. It serves as a comprehensive interdisciplinary presentation of scattering and inverse scattering theory and applications in a wide range of scientific fields, with an emphasis, and details, up-to-date developments. Scattering also places an emphasis on the problems that are still in active current research. The first interdisciplinary reference source on scattering to gather all world expertise in this technique Covers the major aspects of scattering in a common language, helping to widening the knowledge of researchers across disciplines The list of editors, associate editors and contributors reads like an international Who's Who in the interdisciplinary field of scattering

Author : University of Minnesota. Institute for Mathematics and Its Applications
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.56/5 ( download)

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Download or read book Direct and Inverse Scattering for Selfadjoint Hamiltonian Systems on the Line written by University of Minnesota. Institute for Mathematics and Its Applications and published by . This book was released on 1999 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Percy Deift
Publisher :
ISBN 13 :
Total Pages : 334 pages
Book Rating : 4.39/5 ( download)

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Download or read book Inverse Scattering on the Line written by Percy Deift and published by . This book was released on 1978 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: