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Direct and Inverse Scattering on the Line
Richard Beals, Percy Deift, and Carlos Tomei

Mathematical Surveys and Monographs
1988; 209 pp; hardcover
Volume: 28
ISBN-10: 0-8218-1530-X
ISBN-13: 978-0-8218-1530-4
List Price: US$80
Member Price: US$64
Order Code: SURV/28
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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-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.

Table of Contents

Part I. 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 properties 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
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 \(\mu^\#(x,z)\) 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
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
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