Uniqueness problem for accretive Schr\"{o}dinger operators with complex singular coefficients

Abstract

The paper studies the uniqueness problem for the one-dimensional Schr\"{o}dinger 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 quasi-differential. 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.