Semiclassical localization of Schr\"odinger's eigenfunctions

This article addresses the microlocalization of eigenfunctions for the semiclassical Schr\"odinger operator $-h^2\Delta+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the spatial concentration of these eigenfunctions, extending classical results, typically restricted to smooth potentials, to the more general case where the potential is merely bounded. Our main result provides an explicit exponential bound for the $L^2$-norm of eigenfunctions on the entire surface in terms of their $L^2$-norm on an arbitrary open subset with an exponential weight of $Ch^{-1}\log(h)^2$. This bound improves upon previous estimates for non-smooth potentials that was an exponential weight of $Ch^{-4/3}$. Our proof is based on a recent approach of the Landis conjecture develop by Logunov, Malinnikova, Nadirashvili and Nazarov (2025).