A new comb geometric control condition suffices for observability of Schrödinger equations in Euclidean space and is equivalent for fractional cases under uniform continuity of observations.
Observability and Semiclassical Control for Schr\"odinger Equations on Non-compact Hyperbolic Surfaces
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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.
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Observability of Schr\"odinger equations in Euclidean space
A new comb geometric control condition suffices for observability of Schrödinger equations in Euclidean space and is equivalent for fractional cases under uniform continuity of observations.