The direct and inverse scattering problems for the one-dimensional perturbed Hill operator

The problem of scattering by a one-dimensional periodic lattice $p(x)$ with impurity potential $q(x)$ is considered. A stationary scattering matrix is constructed on the basis of the asymptotics of the scattered waves, its properties are studied, and it is shown to coincide with the nonstationary scattering operator defined in the usual way in the quasimomentum representation of the unperturbed operator $H_0$. The inverse scattering problem is also solved, i.e., the problem of recovering $q(x)$ on the basis of $p(x)$ and the scattering data. Here we follow the scheme going back to the well-known work of V. A. Marchenko and L. D. Faddeev. However, solution of the inverse problem in the presence of a periodic lattice required considerable modification of classical arguments. The theory of so-called “global” quasimomentum serves as analytic basis. Conditions on the scattering data are found which are necessary with a finite second moment and sufficient in order that there exist a unique impurity potential with given scattering characteristics and a finite first moment.
Bibliography: 28 titles.