Robust Observability for Schr\"odinger Equations with Rough Potentials on 2D Compact Hyperbolic Surfaces
This paper investigates the robustness of quantum observability on compact hyperbolic surfaces under $L^2$ potential perturbations. Since $L^2$ regularity is strictly subcritical in dimension two, for every non-empty open set $\Omega\subset M$ and $T>0$, the solution of $(i\partial_t+\Delta-V)u=0$ satisfies the space-time observability estimate $$ \|u_0\|_{L^2(M)}^2 \leq C\int_0^T \|e^{-it(-\Delta+V)}u_0\|_{L^2(\Omega)}^2\,dt. $$ Our results provide a quantitative confirmation that the delocalization of high-energy quantum states on negatively curved manifolds is robust against $L^2$-class microscopic scattering. The proof combines the hyperbolic dynamics of the geodesic flow with semiclassical analysis. A key ingredient is an $L^4$ spectral cluster estimate with an arbitrarily small loss, obtained by exploiting Jacobi field analysis and Bourgain--Demeter $l^2$-decoupling. This estimate allows us to obtain refined spectral localized semiclassical Strichartz estimates adapted to rough potentials and to show that the potential's contribution vanishes in the propagation of semiclassical measures. The full-support property of invariant semiclassical measures on hyperbolic surfaces then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.