Towards an inverse scattering theory for two-dimensional nondecaying potentials

The inverse scattering method is considered for the nonstationary Schrödinger equation with the potential $u(x_{1},x_{2})$ nondecaying in a finite number of directions in the $x$ plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the Bäcklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.