论文标题
具有奇异性的微分系统的重建公式
Reconstruction formula for differential systems with a singularity
论文作者
论文摘要
我们的研究涉及奇异差分系统散射理论的某些方面$ y'-x^{ - 1} ay-q（x）y =ρby，\ x> 0 $ at $ n \ times n $ n $矩阵$ a，b，q（x），x（x），x \ in（in（0，\ infty）$，$ a，b $ as a，b $和$ $ comport a spemalter as a spemalter。当$ q（\ cdot）$平滑而$ q（0）= 0 $时，我们专注于重要的特殊情况，并以某些特殊的轮廓积分的形式得出了一种表达这种$ q（\ cdot）$的公式，其中可以用weyl -typer -type solutions编写内核。这种类型的公式在逆散射问题的建设性解决方案中起着重要作用：使用这种公式，其中以前从所谓的主方程中找到了右侧的术语，提供了解决方案过程的最后一步。为了获得上述重建公式，我们首先建立Weyl-类型解决方案的渐近扩展为$ρ\ to \ to \ infty $，带有$ o \ left（ρ^{ - 1} \ right）$ rate rate retusterderderderderderderderderderderderders。
Our studies concern some aspects of scattering theory of the singular differential systems $ y'-x^{-1}Ay-q(x)y=ρBy, \ x>0 $ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $ρ$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl - type solutions of the considered differential system. Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation, provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula we establish first the asymptotical expansions for the Weyl - type solutions as $ρ\to\infty$ with $o\left(ρ^{-1}\right)$ rate remainder estimate.
