Uniqueness problem for accretive Schrödinger operators with complex singular coefficients

TL;DR

This work develops a regularized framework for 1D Schrödinger operators with complex, singular coefficients using Shin–Zettl quasi-derivatives to define maximal and minimal operators in $L^{2}(\mathbb{R})$. It derives constructive m-accretivity criteria for the minimal operator, yielding $\mathrm{L}_{0}=\mathrm{L}$ when accretive, through two main results: a growth-based condition on $\mathrm{Im}\,r$ at infinity and an interval-type criterion on $\mathrm{Im}\,r$ over sequences of intervals extending to infinity. These results extend self-adjointness-type conclusions to non-self-adjoint, singular-coefficient Schrödinger operators, including potentials that are complex Radon measures, and provide tools for spectral analysis and semigroup generation in non-Hermitian settings. The interval-type approach offers flexibility in handling localized growth patterns of the imaginary part of the coefficients and connects to classical criteria in the literature on self-adjointness for Sturm–Liouville-type operators.

Abstract

The 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 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.

Theorems & Definitions (16)

• Theorem 1
• Lemma 1: Generalized Lagrange identity Zttl1975
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['th_Properties']}
• Theorem 2
• Remark 1
• Corollary 1
• Lemma 3
• proof