关于具有复奇异系数的正交算子唯一性问题
Uniqueness problem for accretive Schrödinger operators with complex singular coefficients
AI总结：
本文研究了一维正交算子的唯一性问题，通过分析复奇异系数的准微分表达式，得出了保证算子相等的构造性条件。
摘要：
本文研究了一维正交算子的唯一性问题，该算子与形式微分表达式 $ l[u] =-u''+qu + i[(ru)'+ru'] $ 相关联，其中系数为复值且满足 $ q=s+Q' $，$ s \in L^1_{loc}(\mathbb{R}) $，$ Q, r \in L^2_{loc}(\mathbb{R}) $。特别地，势函数 $ q $ 可以是直线上的Radon测度。通过特别选择的准导数，将表达式 $ l $ 视为准微分表达式。在 $ L^2(\mathbb{R}) $ 空间中，描述了与表达式 $ l $ 相关的最小 $ \mathrm{L}_{0} $ 和最大 $ \mathrm{L} $ 算子的域。我们找到了关于 $ \mathrm{Im}\,r $ 在 $ \pm \infty $ 附近行为的构造性条件，这些条件保证了当算子 $ \mathrm{L}_{0} $ 是 accretive 时 $ \mathrm{L}_{0}=\mathrm{L} $。
英文摘要：
This paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space $L^{2}(\mathbb{R})$. The coefficients of the expression are complex-valued and satisfy \begin{equation*} q=s+Q', \quad s \in L^1_{loc}\left(\mathbb{R}\right) \quad\text{and}\quad Q, r \in L^2_{loc}\left(\mathbb{R}\right), \end{equation*} where the derivative is understood in the sense of distributions. In particular, the potential $q$ can be a Radon measure on the line. With the help of specially selected quasi-derivatives, the expression $l$ is treated as a quasi-differential expression. The domains of the minimal $\mathrm{L}_{0}$ and maximal $\mathrm{L}$ operators associated with the expression $l$ in the space $L^{2}(\mathbb{R})$ are described. We find constructive conditions on the behaviour of $\mathrm{Im}\,r$ near $\pm \infty$ that guarantee that $\mathrm{L}_{0}=\mathrm{L}$ if the operator $\mathrm{L}_{0}$ is accretive. We prove that these conditions are sharp even in the class of differential operators with smooth real-valued coefficients. Examples are given to illustrate the main results of the paper.
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