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

TL;DR

This paper develops a method to recover the unknown part of a steplike potential in a Schrödinger operator on the full line using a determinant formula involving Marchenko and Hankel operators, even without decay assumptions at minus infinity.

Contribution

It introduces a novel determinant formula for reconstructing the unknown steplike potential segment using scattering data and operator theory, extending inverse scattering techniques.

Findings

Explicit formula for the unknown potential segment using operator determinants.

Construction of the kernel of a Hankel operator from reflection coefficients.

Analysis of scattering quantities without decay assumptions at minus infinity.

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 $\Reals_-$.