Resolvent bounds imply observability from measurable time sets for Schrödinger equations
Nicolas Burq, Hui Zhu
TL;DR
This work shows that resolvent bounds for the Laplace–Beltrami operator on a compact manifold imply observability and hence controllability of the Schrödinger equation from measurable time sets. The authors develop a three-step framework: (i) a semiclassical observability bound for high-frequency components that is uniform in short time windows, (ii) a weak observability bound obtained via Littlewood–Paley theory and temporal techniques, and (iii) a strong observability result achieved through a Bardos–Lebeau–Rauch uniqueness-compactness argument together with a unique continuation lemma. The approach generalizes existing results from tori to arbitrary compact manifolds and yields corollaries under the geometric control condition and for negatively curved surfaces, linking resolvent growth to observability from time sets of positive measure. This broadens the applicability of observability and controllability results to measurable time subsets, offering new avenues for Schrödinger control in geometric settings.
Abstract
We prove that on a compact Riemannian manifold, resolvent bounds for the Laplace--Beltrami operator imply observability, and thus controllability, for the Schrödinger propagator from time sets of positive Lebesgue measure. Applications include almost all cases where observability and controllability hold from time intervals, particularly when the geometric control condition is satisfied or when the manifold is a compact surface of negative curvature.