Resolvent bounds imply observability from measurable time sets for Schr\"odinger equations

TL;DR

This paper demonstrates that resolvent bounds for the Laplace--Beltrami operator on compact Riemannian manifolds imply observability and controllability of the Schr"odinger equation from measurable time sets, with broad applications.

Contribution

It establishes a link between resolvent bounds and observability for Schr"odinger equations, extending control results to measurable time sets under various geometric conditions.

Findings

Resolvent bounds imply observability from positive measure time sets.

Applicability includes cases satisfying the geometric control condition.

Results hold for compact surfaces of negative curvature.

Abstract

We prove that on a compact Riemannian manifold, resolvent bounds for the Laplace--Beltrami operator imply observability, and thus controllability, for the Schr\"odinger 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.