Inverse Scattering on the Half Line for the Matrix Schrödinger Equation

Tuncay Aktosun

University of Texas at Arlington, Arlington, TX 76019-0408, USA
E-mail: aktosun@uta.edu

Ricardo Weder

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional
Autónoma de México, Apartado Postal 20-126, IIMAS-UNAM, México DF 01000,México
E-mail: weder@unam.mx

Received March 1, 2018.

Dedicated to Professor V.A. Marchenko
for his 95th birthday

Abstract

The matrix Schrödinger equation is considered on the half line with
the general selfadjoint boundary condition at the origin described by two
boundary matrices satisfying certain appropriate conditions. It is assumed
that the matrix potential is integrable, is selfadjoint, and has a finite first
moment. The corresponding scattering data set is constructed, and such
scattering data sets are characterized by providing a set of necessary and
sufficient conditions assuring the existence and uniqueness of the one-toone correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classic result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition.