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

TL;DR

This paper develops a semiclassical analysis framework for Schr"odinger equations on non-compact hyperbolic surfaces, establishing uniform control estimates and observability results using generalized Bloch theory and flat Hilbert bundles.

Contribution

It generalizes semiclassical control estimates to all flat Hilbert bundles over hyperbolic surfaces and derives observability from periodic subsets in non-compact covers.

Findings

Established uniform semiclassical control estimates for flat Hilbert bundles.

Derived observability results from periodic open subsets of non-compact hyperbolic surfaces.

Applied results to spectral geometry problems.

Abstract

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.