On the determinant formula in the inverse scattering procedure with a partially known steplike potential

Abstract

We are concerned with the inverse scattering problem for the full line Schrödinger operator −∂2x + q(x) with a steplike potential q a priori known on . Assuming is known and short range, we show that the unknown part of q can be recovered bywhere is the classical Marchenko operator associated with and is a trace class integral Hankel operator. The kernel of is explicitly constructed in terms of the difference of two suitably defined reflection coefficients. Since is not assumed to have any pattern of behavior at −∞, defining and analyzing scattering quantities becomes a serious issue. Our analysis is based upon some subtle properties of the Titchmarsh–Weyl m-function associated with .