Semigroup properties of Schrödinger operators with complex singular potentials

Abstract

In the complex Hilbert space $L_2(\mathbb{R}_+)$ on the half-line $\mathbb{R}_+ = [0, \infty)$, we investigate non-symmetric Schrödinger operators \[ Su = -u'' + qu, \qquad u(0) = 0, \] with complex-valued, highly singular distributional potentials \[ q = s + Q' \in H_{\mathrm{loc}}^{-1}(\mathbb{R}_+), \qquad s \in L_{1,\mathrm{loc}}(\mathbb{R}_+), \quad Q \in L_{2,\mathrm{loc}}(\mathbb{R}_+), \] where the derivative $Q'$ 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.