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

TL;DR

This paper studies how solutions to the semi-classical Schr"odinger equation localize on closed Riemann surfaces, especially when the potential is only bounded and not smooth, using regularization and local-to-global estimates.

Contribution

It introduces a regularization approach to handle bounded, irregular potentials and establishes a local-to-global estimate linking potential regularity to solution concentration.

Findings

Regularization technique manages non-smooth potentials.

Established local-to-global estimate for solution concentration.

Quantified influence of potential regularity on localization.

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.