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

We are concerned with the inverse scattering problem for the full line Schr"odinger operator $-{\mathrm{\partial }}_x^2+q(x)$ with a steplike potential $q$ a priori known on ${\mathbb{R}}_{+}=(0,\mathrm{\infty })$. Assuming $q{\mathrm{\mid }}_{{\mathbb{R}}_{+}}$ is known and short range, we show that the unknown part $q{\mathrm{\mid }}_{{\mathbb{R}}_{-}}$ 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 $q{\mathrm{\mid }}_{{\mathbb{R}}_{+}}$ 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 $q{\mathrm{\mid }}_{{\mathbb{R}}_{-}}$ is not assumed to have any pattern of behavior at $-\mathrm{\infty }$, 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 ${\mathbb{R}}_{-}$.