Semi-Classical Localization of the Schr\"odinger Resolvent on Closed Riemann Surfaces

Abstract

This paper investigates the localization properties of solutions to the semi-classical Schr\"odinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising from irregular potentials, specifically those that are merely bounded. We employ a regularization technique to manage the potential's lack of smoothness and establish a local-to-global estimate. This result provides a quantitative measure of how the local regularity of the potential influences the global concentration of the solution, thereby bridging the gap between smooth and non-continuous regimes in semi-classical analysis.