具有Miura势的薛定谔算子的逆散射，I。独特的Riccati代表和ZS-AKNS系统
Inverse scattering for Schrödinger operators with Miura potentials, I. Unique Riccati representatives and ZS-AKNS systems
摘要：
这是第一次在一系列文件的散射理论的一维薛定谔运营商与高度奇异的潜力$q\in H^{-1}(R)$。在本文中，我们研究了与正薛定谔算子相关联的Miura势$q$，该算子允许唯一的$u\in L^1(R)\cap L^2(R)$的Riccati表示$q=u'+u^2$。这样的电势具有良好定义的反射系数$r(k)$，其满足$|r(k)|<1$并且唯一地确定$u$。我们证明了散射映射$S:u\mapsto r$是实解析的，具有实解析逆。要做到这一点，我们利用一个自然的复杂化的散射映射与EST-AKNS系统。在随后的论文中，我们将考虑更大类的势，包括具有束缚态的奇异势。
英文摘要：
This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials $q\in H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrödinger operators that admit a Riccati representation $q=u'+u^2$ for a unique $u\in L^1(R)\cap L^2(R)$. Such potentials have a well-defined reflection coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We show that the scattering map $S:u\mapsto r$ is real-analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.
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。
