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Direct and Inverse Scattering for the Matrix Schrödinger Equation

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  • © 2021

Overview

Authors:
  1. Tuncay Aktosun

    1. Department of Mathematics, University of Texas at Arlington, Arlington, USA

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  2. Ricardo Weder

    1. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico

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  • Presents a complete and detailed matrix Marchenko method with general boundary conditions
  • Illustrates a comprehensive treatment of scattering theory through explicit examples
  • Indicates how the inverse problem should be posed and reveals how the existing formulation is problematic unless the boundary condition is specified as part of the scattering data
  • Investigates existence, uniqueness, and construction aspects of scattering and inverse scattering problems

Part of the book series: Applied Mathematical Sciences (AMS, volume 203)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Tuncay Aktosun, Ricardo Weder
    Pages 1-17

  3. The Matrix Schrödinger Equation and the Characterization of the Scattering Data

    • Tuncay Aktosun, Ricardo Weder
    Pages 19-47

  4. Direct Scattering I

    • Tuncay Aktosun, Ricardo Weder
    Pages 49-260

  5. Direct Scattering II

    • Tuncay Aktosun, Ricardo Weder
    Pages 261-337

  6. Inverse Scattering

    • Tuncay Aktosun, Ricardo Weder
    Pages 339-484

  7. Some Explicit Examples

    • Tuncay Aktosun, Ricardo Weder
    Pages 485-543

  8. Back Matter

    Pages 545-624
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Keywords

About this book

Authored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. 


The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.


Authors and Affiliations

  • Department of Mathematics, University of Texas at Arlington, Arlington, USA

    Tuncay Aktosun

  • Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Ricardo Weder

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