Direct and Inverse Scattering on the Line
American Mathematical Society (Publisher)
Will be published approx. on 30. December 2015
Book
Paperback/Softback
209 pages
978-1-4704-2054-3 (ISBN)
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-Baecklund 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 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-Baecklund 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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 229 mm
Width: 152 mm
Weight
415 gr
ISBN-13
978-1-4704-2054-3 (9781470420543)
Copyright in bibliographic data is held by Nielsen Book Services Limited or its licensors: all rights reserved.
Schweitzer Classification
Content
Part I. The Forward Problem
Distinguished solutions
Fundamental matrices
Fundamental tensors
Behavior of fundamental tensors as |x|??; the Functions ?k
Behavior of fundamental tensors as z??
Behavior of fundamental tensors as z?0
Construction of fundamental matrices
Global properties of fundamental matrices; the transition matrix ?
Symmetries of fundamental matrices
The Green's function for L
Generic operators and scattering data
Algebraic properties of scattering data
Analytic properties of scattering data
Scattering data for m~; determination of v~ from v
Scattering data for L?
Generic selfadjoint operators and scattering data
The Green's function revisited
Genericity at z=0
Genericity at z?0
Summary of properties of scattering data
Part II. 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 ?#(x,z) and ?(x,z) as z?? and as x???
Proof of the basic inverse theorem
The scalar factorization problem for ?
The inverse problem at x=+? and the bijectivity of the map L?S(L)=(Z(L),v(L))
The even order case
The second order problem
Part III. Applications
Flows
Eigenfunction expansions and classical scattering theory
Inserting and removing poles
Matrix factorization and first order systems
Appendix A. Rational approximation
Appendix B. Some formulas
Distinguished solutions
Fundamental matrices
Fundamental tensors
Behavior of fundamental tensors as |x|??; the Functions ?k
Behavior of fundamental tensors as z??
Behavior of fundamental tensors as z?0
Construction of fundamental matrices
Global properties of fundamental matrices; the transition matrix ?
Symmetries of fundamental matrices
The Green's function for L
Generic operators and scattering data
Algebraic properties of scattering data
Analytic properties of scattering data
Scattering data for m~; determination of v~ from v
Scattering data for L?
Generic selfadjoint operators and scattering data
The Green's function revisited
Genericity at z=0
Genericity at z?0
Summary of properties of scattering data
Part II. 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 ?#(x,z) and ?(x,z) as z?? and as x???
Proof of the basic inverse theorem
The scalar factorization problem for ?
The inverse problem at x=+? and the bijectivity of the map L?S(L)=(Z(L),v(L))
The even order case
The second order problem
Part III. Applications
Flows
Eigenfunction expansions and classical scattering theory
Inserting and removing poles
Matrix factorization and first order systems
Appendix A. Rational approximation
Appendix B. Some formulas