Observability and Semiclassical Control for Schr\"odinger Equations on Non-compact Hyperbolic Surfaces

We study the observability of the Schr\"odinger equation on $X$, a non-compact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, functions on $X$ are identified as sections of flat Hilbert bundles over $M$. We develop a semiclassical analysis framework for such bundles and generalize the result of semiclassical control estimates in [Dyatlov and Jin, Acta Math., 220 (2018), pp. 297-339] to all flat Hilbert bundles over $M$, with uniform constants with respect to the choice of bundle. Furthermore, when the Riemannian cover $X \to M$ is a normal cover with a virtually Abelian deck transformation group $\Gamma$, we combine the uniform semiclassical control estimates on flat Hilbert bundles with the generalized Bloch theory to derive observability from any $\Gamma$-periodic open subsets of $X$. We also discuss applications of the uniform semiclassical control estimates in spectral geometry.