Semiclassical analysis of the magnetic Laplacian on hyperbolic surfaces

TL;DR

This work analyzes the semiclassical limits of eigenfunctions for the magnetic Laplacian on a closed hyperbolic surface with constant magnetic field, focusing on the regime $k\to\infty$ for eigenpairs $k^{-2}\Delta_k u_k=(E+\varepsilon_k)u_k$. It identifies three energy regimes with distinct quantum limits: below the critical energy $E_c=\tfrac{1}{2}B^2$ all invariant measures on the energy shell $\{p=E\}$ occur as defect measures, at $E=E_c$ the Liouville measure is the unique defect measure and a quantitative ergodic statement holds, and above $E_c$ a density-one subsequence equidistributes toward Liouville in the spirit of QUE. The proofs combine Weinstein's periodic-operator construction, magnetic Gaussian beams, and a quantitative version of QUE via horocycle ergodicity and a long-time Egorov argument that remains valid up to polynomial times near the critical energy. The paper also derives improved $L^\infty$ bounds in the critical regime and constructs magnetic zonal states that saturate Hörmander bounds, including a detailed description of their defect measures supported on a two-dimensional invariant torus. Together, these results illuminate how classical magnetic dynamics on hyperbolic surfaces governs quantum limits and provides new quantitative aspects of QUE in a curved, magnetized setting.

Abstract

The magnetic Laplacian on hyperbolic surfaces provides a rich analytic framework in which a variety of quantum phenomena emerge. The present note, written for the \emph{Proceedings of the Journées EDP 2025}, is a concise overview of the main results obtained in [arXiv:2505.08584] and work in preparation by the author with L. Charles and A. Chabert.

Theorems & Definitions (5)

• Theorem 2.1: Three semiclassical regimes
• Theorem 2.2: Polynomial convergence rate at the critical energy
• Theorem 3.1: $L^\infty$ bounds for magnetic eigenfunctions
• Proposition 3.2
• Theorem 3.3: Semiclassical defect measure of magnetic zonal states