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Direct and Inverse Scattering for the Matrix Schrödinger Equation

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

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Abstract

In this introductory chapter the goals of the monograph are described, the contents of the remaining chapters and Appendix A are outlined, and the relevant general references in the literature are mentioned. The direct and inverse scattering problems for the matrix Schrödinger equation on the half line with the general self-adjoint boundary condition are viewed as two mappings, and it is indicated how these two mappings become inverses of each other by specifying their domains appropriately. The traditional definition of the scattering matrix in one way with the Dirichlet boundary condition and in a different way with a non-Dirichlet boundary condition is criticized, and it is indicated that such a practice makes it impossible to formulate a well-posed inverse scattering problem unless the boundary condition is already known and the Dirichlet and non-Dirichlet boundary conditions are not mixed. It is emphasized that the recovery of the boundary condition should be a part of the solution to the inverse scattering problem rather than a part of the inverse scattering data.

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Authors and Affiliations

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

    Tuncay Aktosun

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

    Ricardo Weder

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Aktosun, T., Weder, R. (2021). Introduction. In: Direct and Inverse Scattering for the Matrix Schrödinger Equation. Applied Mathematical Sciences, vol 203. Springer, Cham. https://doi.org/10.1007/978-3-030-38431-9_1

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