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

TL;DR

This paper investigates the uniqueness of solutions for a complex Schrödinger operator with singular coefficients, providing conditions under which the minimal and maximal operators coincide when the operator is accretive.

Contribution

It introduces a quasi-differential approach to analyze the operator with complex, singular coefficients and establishes conditions for operator equality based on the behavior of the imaginary part of r.

Findings

Conditions for operator equality when accretive

Handling of complex singular coefficients via quasi-derivatives

Characterization of minimal and maximal operators domains

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.