Direct and inverse scattering for the matrix Schrödinger equation (Q2307050)

The monograph describes the (direct and inverse) scattering problem for the matrix Schrödinger operator on a half-line with a general self-adjoint boundary condition. The problem has applications to quantum star graphs and operators on the full line. The authors define the input data set (the potential and the boundary condition) and the scattering data set (the scattering matrix and the bound state information). The direct scattering problem can then be viewed as a map from the Fadeev class of input data sets to the Marchenko class of scattering data sets. In other words, in this problem one finds the scattering matrix and bound states from the potential and the boundary condition. The inverse scattering problem is then viewed as the inverse map from the Marchenko class to the Fadeev class. Characterization of these classes using certain conditions is given in the book. Existence, uniqueness, construction, and characterisation of the problem are discussed. Unlike other publications, the authors do not solve the problem separately for the Dirichlet and non-Dirichlet boundary conditions. They describe all possible self-adjoint conditions in a similar way how they are usually described in quantum graphs. The main result of the book is establishing one-to-one correspondence between Fadeev class and Marchenko class. The monograph consists of six chapters and an appendix. The first chapter is introductory, the authors define the main aims and present an overview of the problem. In the second chapter, the direct and inverse problem are introduced mathematically, including Fadeev and Marchenko classes. Chapters 3 and 4 deal with the direct scattering problem; Chapter 5 with the inverse scattering problem. Examples are given in the sixth chapter. The necessary preliminaries are summarized in the appendix. The book has 624 pages.