Controllable subspaces and real-part observability for Schrödinger equations on $\mathbb T^d$

Abstract

We study internal controllability of Schrödinger equations on tori with controls acting only through their real parts. This real-part constraint leads to a real-linear control problem for which full null controllability may fail. We characterize the maximal null-controllable subspace and show that the uncontrollable directions are precisely given by the purely imaginary stationary modes. These results provide an explicit description of controllable and uncontrollable directions for Schrödinger equations with real-part controls. The characterization is obtained through observability inequalities with sharp stationary correction terms. For cylindrical open control regions, we prove such an estimate by a real-part compactness--uniqueness argument. For admissible measurable product control regions in the shifted free case, we establish the corresponding estimate for rough spacetime control sets. The proof combines complex-valued observability inputs with a double-spectrum analysis arising from the coupling of the forward and backward Schrödinger evolutions in the real-part observation.