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Direct and Inverse Scattering on the Line : Mathematical Surveys and Monographs - Richard Beals

Direct and Inverse Scattering on the Line

Mathematical Surveys and Monographs

Hardcover Published: January 1988
ISBN: 9780821815304
Number Of Pages: 209

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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-Backlund 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.

The Forward Problem: Distinguished solutions
Fundamental matrices Fundamental tensors
Behavior of fundamental tensors as $ x \rightarrow\infty$; the Functions $\Delta_k$
Behavior of fundamental tensors as $z\rightarrow\infty$
Behavior of fundamental tensors as $z\rightarrow0$
Construction of fundamental matrices
Global properties of fundamental matrices; the transition matrix $\delta$
Symmetries of fundamental matrices
The Green's function for $L$
Generic operators and scattering data
Algebraic properties of scattering data
Analytic roperties of scattering data
Scattering data for $\tilde m$; determination of $\tilde v$ from $v$
Scattering data for $L^\ast$ Generic selfadjoint operators and scattering data
The Green's function revisited
Genericity at $z=0$ Genericity at $z\ne0$
Summary of properties of scattering data
The Inverse Problem: Normalized eigenfunctions for odd order inverse data
The vanishing lemma
The Cauchy operator Equations for the inverse problem
Factorization near $z=0$ and property (20.6)
Reduction to a Fredholm equation Existence of $h^\#$
Properties of $h^\#$ Properties of $\mu^\#(x,z)$ and $\mu(x,z)$ as $z\rightarrow\infty$ and as $x\rightarrow-\infty$
Proof of the basic inverse theorem
The scalar factorization problem for $\delta$
The inverse problem at $x=+\infty$ and the bijectivity of the map $L\mapsto S(L)=(Z(L),v(L))$ The even order case The second order problem
Applications: Flows
Eigenfunction expansions and classical scattering theory
Inserting and removing poles
Matrix factorization and first order systems
Rational approximation
Some formulas
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780821815304
ISBN-10: 082181530X
Series: Mathematical Surveys and Monographs
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 209
Published: January 1988
Publisher: American Mathematical Society
Country of Publication: US
Dimensions (cm): 26.04 x 17.78  x 1.27
Weight (kg): 0.56

Earn 240 Qantas Points
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