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arXiv · 2505.08584

Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

Laurent Charles·Thibault Lefeuvre
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Abstract

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.

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Laurent Charles, Thibault Lefeuvre. 2025-05-13. Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces. https://arxiv.org/abs/2505.08584

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