Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces
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.
Fields of Study & Subjects
Spectral Theory in Mathematical Physics
Mathematical Dynamics and Fractals
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Journal de l’École polytechnique — Mathématiques
- Publisher
- MathDoc/Centre Mersenne
- Volume
- 13
- Pages
- 593–627
- Publication date
- 6 March 2026
- Language
- EN