Direct and Inverse Scattering for the Matrix Schrödinger Equation

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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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.

• Matrix Schrodinger Equations
• General Boundary Conditions
• Marchenko Method
• Inverse Scattering
• Scattering Data
• Stokes phenomena and connection problems
• inverse problems
• Levinson's theorem
• generalized Fourier map
• Riemann-Hilbert problem
• partial differential equations

• 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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