Inverse Scattering on the Half Line for the Matrix Schr?dinger Equation

摘要

The matrix Schr¨odinger equation is considered on the half line withthe general selfadjoint boundary condition at the origin described by twoboundary matrices satisfying certain appropriate conditions. It is assumedthat the matrix potential is integrable, is selfadjoint, and has a finite firstmoment. The corresponding scattering data set is constructed, and suchscattering data sets are characterized by providing a set of necessary andsufficient conditions assuring the existence and uniqueness of the one-toonecorrespondence between the scattering data set and the input data setcontaining the potential and boundary matrices. The work presented hereprovides a generalization of the classic result by Agranovich and Marchenkofrom the Dirichlet boundary condition to the general selfadjoint boundarycondition.