Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

TL;DR

This paper studies how eigenfunctions of the magnetic Laplacian on hyperbolic surfaces distribute in phase space across different energy levels, revealing regimes of invariant measures, quantum ergodicity, and equidistribution.

Contribution

It characterizes the semiclassical measures for magnetic Laplacians on hyperbolic surfaces across various energy regimes, including proving quantum ergodicity and equidistribution results.

Findings

In low-energy, any invariant measure of the magnetic flow can be realized as a semiclassical measure.

At the critical energy, quantum unique ergodicity holds with a quantitative convergence rate.

In high-energy, a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.

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.