Characteristic properties of scattering data for discontinuous Schrödinger equations

In this paper, we discuss the inverse scattering problem to recover the potential from the scattering data of a class of Schrödinger equations with a nonlinear spectral parameter in the boundary condition. It turns out that for real-valued potential function $q(x)$, the scattering data is defined as in the non-self-adjoint case: the scattering function, the nonreal singular values, and normalization polynomials. Characteristic properties of the spectral data are investigated. The solution of the problem is constructed by using the Gelfand–Levitan–Marchenko procedure. The uniqueness of the algorithm for the potential with given scattering data is proved.