阶跃势部分已知时逆散射过程中的行列式公式
On the determinant formula in the inverse scattering procedure with a partially known steplike potential
摘要：
本文研究了具有阶跃势$q$的全线薛定谔算子$-\partial_x^2+q(x)$的逆散射问题。假设$q|_{\Reals_+}$是已知的并且是短距离的，我们证明了$q$的未知部分$q|_{\Reals_-}$可以通过{equation*} q恢复|_{ \Reals_- }（x）=-2 \partial_x^2 \log \det（1+（1+ \mathbb{M}_x^+）^{-1} \mathbb{G}_x），{等式 *}其中$\mathbb{M}_x^+$是与$q|_{\Reals_+}$相关联的经典Marchenko算子，而$\mathbb{G}_x$是迹类积分Hankel算子。$\mathbb{G}_x$的核是根据两个适当定义的反射系数之差来构造的。由于$q|_{\Reals_-}$在$-\infty$上不具有任何行为模式，因此定义和分析散射量成为一个严重的问题。我们的分析是基于与$\Reals_-$相关的Titchmarsh-Weyl $m$函数的一些微妙性质。
英文摘要：
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 $\Reals_+=(0,\infty)$. 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 $\Reals_-$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
