Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $-\partial_x^2+q(x)$ 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 $\Reals_-$.

3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
16
The categories of this paper are: stat
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat