Robust Observability for Schrödinger Equations with Singular Potentials on 2D Compact Hyperbolic Surfaces

This paper investigates the robustness of quantum observability on compact hyperbolic surfaces subject to $L^q$ potential perturbations with $2<q\leq \infty$. Since $L^q$ regularity is strictly subcritical in dimension two, for every non-empty open set $Ω\subset M$ and $T>0$, the solution of $(i\partial_t+Δ-V)u=0$ satisfies the space-time observability estimate $\|u_0\|_{L^2(M)}^2 \leq C\int_0^T \|e^{-it(-Δ+V)}u_0\|_{L^2(Ω)}^2\,dt.$ A cornerstone of our analysis is Sogge's spectral cluster estimate, which allows us to obtain refined spectrally 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 associated with the eigenfunctions then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.