Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

TL;DR

This work analyzes semiclassical defect measures of magnetic Laplacians on closed hyperbolic surfaces under a constant magnetic field, revealing a trichotomy of dynamical regimes tied to the energy relative to the critical value Ec = B^2/2. The authors develop a twisted semiclassical calculus and employ Weinstein averaging to construct eigenstates concentrating on periodic magnetic orbits forEc below Ec, establish Quantum Unique Ergodicity with a polynomial rate at Ec, and prove density-one equidistribution for Ec below the upper energy band. A key part of the analysis connects the classical magnetic flow with quantum limits, using long-time Egorov arguments and the unique ergodicity of the horocyclic flow to obtain quantitative rates. The results illuminate how classical dynamics control quantum limits in the magnetic setting and lay groundwork for extending to variable curvature in a follow-up work.

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.

Theorems & Definitions (41)

• Theorem 1.1: The three classical/quantum regimes
• Theorem 1.2: Polynomial rate of convergence in the critical regime
• Theorem 1.3: Perturbation by a potential
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Proposition 2.5
• proof