Physics – Mathematical Physics
Scientific paper
2011-07-17
Physics
Mathematical Physics
Scientific paper
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 $\Reals_-$.
Bastille Odile
Rybkin Alexei
No associations
LandOfFree
On the determinant formula in the inverse scattering procedure with a partially known steplike potential does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the determinant formula in the inverse scattering procedure with a partially known steplike potential, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the determinant formula in the inverse scattering procedure with a partially known steplike potential will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-478610