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 $-{\partial }_x^2+q(x)$ with a steplike potential $q$ a priori known on ${\mathbb{R}}_{+}=(0,\infty )$. Assuming $q{|}_{{\mathbb{R}}_{+}}$ is known and short range, we show that the unknown part $q{|}_{{\mathbb{R}}_{-}}$ 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 $q{|}_{{\mathbb{R}}_{+}}$ 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 $q{|}_{{\mathbb{R}}_{-}}$ 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 ${\mathbb{R}}_{-}$.

References in corpus (1)

• The boundary control approach to the Titchmarsh-Weyl $m-$function