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

Sébastien Campagne

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arXiv · 2601.02274

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

Sébastien Campagne

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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.

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Sébastien Campagne. 2026-01-05. Semi-Classical Localization of the Schr\"odinger Resolvent on Closed Riemann Surfaces. https://arxiv.org/abs/2601.02274

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