Semigroup properties of Schrödinger operators with complex singular potentials
Abstract
In the complex Hilbert space L2(ℝ+) on the half-line ℝ+ = [0, ∞), we investigate non-symmetric Schrödinger operators Su = −u'' + qu, u(0) = 0, with complex-valued, highly singular distributional potentials q = s + Q' ∈ Hloc−1(ℝ+), s ∈ L1,loc(ℝ+), Q ∈ L2,loc(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 L2(ℝ+) and C-self-adjoint with respect to the complex conjugation involution C : u ↦ u̅. 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 ≥ 0 is contractive or holomorphic in a sector of the complex plane.
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BibTeX
@misc{mikhailets2026semigroup,
title = {{Semigroup properties of Schrödinger operators with complex singular potentials}},
author = {Vladimir Mikhailets and Volodymyr Molyboga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16670},
eprint = {2610.16670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16670},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Semigroup properties of Schrödinger operators with complex singular potentials
AU - Vladimir Mikhailets
AU - Volodymyr Molyboga
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16670
UR - https://arxiv.org/abs/2610.16670
AB - In the complex Hilbert space L_2(ℝ_+) on the half-line ℝ_+ = [0, ∞), we investigate non-symmetric Schrödinger operators Su = −u'' + qu, u(0) = 0, with complex-valued, highly singular distributional potentials q = s + Q' ∈ H_loc^−1(ℝ_+), s ∈ L_1,loc(ℝ_+), Q ∈ L_2,loc(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(ℝ_+) and C-self-adjoint with respect to the complex conjugation involution C : u ↦ u̅. 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 ≥ 0 is contractive or holomorphic in a sector of the complex plane.
ER -