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Direct and Inverse Scattering on the Line


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Table of Contents

  • 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

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