Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910941575118848 |
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| author | Charles, Laurent Lefeuvre, Thibault |
| author_facet | Charles, Laurent Lefeuvre, Thibault |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08584 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces Charles, Laurent Lefeuvre, Thibault Analysis of PDEs Differential Geometry Dynamical Systems 58J51, 35Q60, 37A25 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. |
| title | Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces |
| topic | Analysis of PDEs Differential Geometry Dynamical Systems 58J51, 35Q60, 37A25 |
| url | https://arxiv.org/abs/2505.08584 |