Uniqueness problem for accretive Schrödinger operators with complex singular coefficients

Abstract

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.

19 pages, extended version