Semigroup properties of Schrödinger operators with complex singular potentials

In the complex Hilbert space $L_2({\mathbb{R}}_{+})$ on the half-line ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$, we investigate non-symmetric Schrödinger operators $$Su=-u^{\mathrm{\prime }\mathrm{\prime }}+qu,u(0)=0,$$ with complex-valued, highly singular distributional potentials $$q=s+Q^{\mathrm{\prime }}\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}({\mathbb{R}}_{+}),s\in L_{1,\mathrm{l}\mathrm{o}\mathrm{c}}({\mathbb{R}}_{+}),Q\in L_{2,\mathrm{l}\mathrm{o}\mathrm{c}}({\mathbb{R}}_{+}),$$ where the derivative $Q^{\mathrm{\prime }}$ is understood in the sense of Schwartz distributions. A rigorous operator-theoretic setting is established within the framework of Shin–Zettl quasi-derivatives. We prove that every accretive operator $S$ of this class is automatically $m$-accretive in $L_2({\mathbb{R}}_{+})$ and $C$-self-adjoint with respect to the complex conjugation involution $C:u\mapsto \bar{u}$. In particular, no additional boundary conditions at infinity are needed, meaning that a Weyl limit-point type situation always prevails for accretive realizations. Constructive local conditions on the coefficients $s$ and $Q$ are found under which the generated strongly continuous semigroup $\{e^{-tS}{\}}_{t\ge 0}$ is contractive or holomorphic in a sector of the complex plane.