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 Schr\"odinger operator $-\partial_x^2+q(x)$ with a steplike potential $q$ a priori known on $\Reals_+=(0,\infty)$. Assuming $q|{\Reals+}$ is known and short range, we show that the unknown part $q|{\Reals-}$ of $q$ can be recovered by {equation*} q|{\Reals-}(x)=-2\partial_x^2\log\det(1+(1+\mathbb{M}_x^+)^{-1}\mathbb{G}_x), {equation*} where $\mathbb{M}x^+$ is the classical Marchenko operator associated to $q|{\Reals_+}$ and $\mathbb{G}x$ is a trace class integral Hankel operator. The kernel of $\mathbb{G}_x$ is explicitly constructed in term of the difference of two suitably defined reflection coefficients. Since $q|{\Reals_-}$ is not assumed to have any pattern of behavior at $-\infty$, 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 $\Reals_-$.