The Matrix Schrödinger Equation and the Characterization of the Scattering Data

Abstract

In this chapter we describe the basic ingredients of the direct and inverse scattering problems for the matrix Schrödinger equation on the half line with the general self-adjoint boundary condition. We show how the analysis of star graphs and the Schrödinger scattering problem on the full line can be reduced to the study of the matrix Schrödinger equation on the half line with some appropriate self-adjoint boundary conditions. To analyze the direct and inverse problems on the half line, we introduce the input data set consisting of a potential and two constant boundary matrices describing the boundary condition. We define the Faddeev class of input data sets by imposing some appropriate restrictions on the input data sets. We introduce the scattering data set consisting of a scattering matrix and the bound-state information. We define the Marchenko class of scattering data sets by imposing some appropriate restrictions on the scattering data sets. The unique solutions to the direct and inverse scattering problems are achieved by establishing a one-to-one correspondence between the Faddeev class of input data sets and the Marchenko class of scattering data sets. Various equivalent descriptions of the Marchenko class are introduced. Such equivalent descriptions allow us to present various different but equivalent results for the characterization of the scattering data in the solution to the inverse problem. For the reader’s convenience various equivalent characterization theorems are stated but their proofs are postponed until Chap. 5.