Non-self-adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. II. Inverse problem

The inverse spectral and scattering problem of the Zakharov–Shabat (ZS) operator is studied. The similarity transformation between ZS operators is examined when their potentials Q(x) have the common nonvanishing asymptotic values Q± at the infinity. The Marchenko equation is derived from the Parseval equation. We give the necessary as well as the sufficient condition of the scattering data for the potential of the specified class.