Description

Book Synopsis
Deals with the theory of linear ordinary differential operators of arbitrary order. Unlike treatments that focus on spectral theory, this work centres on the construction of special eigen functions (generalized Jost solutions) and on the inverse problem: the problem of reconstructing the operator from minimal data associated to the special eigen functions.

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

Direct and Inverse Scattering on the Line

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    A Paperback by Richard Beals, Percy Deift, Carlos Tomei

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      Publisher: MP-AMM American Mathematical
      Publication Date: Publication Date: 30/12/2015
      ISBN13: 9781470420543, 978-1470420543
      ISBN10:

      Description

      Book Synopsis
      Deals with the theory of linear ordinary differential operators of arbitrary order. Unlike treatments that focus on spectral theory, this work centres on the construction of special eigen functions (generalized Jost solutions) and on the inverse problem: the problem of reconstructing the operator from minimal data associated to the special eigen functions.

      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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