Zonal states and improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces

Abstract

We establish polynomially improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces in the critical energy regime. We also show that, below the critical energy, the Hörmander bound is saturated by explicit eigenstates, which we call \emph{magnetic zonal states}. These states resemble zonal harmonics on the sphere and equidistribute on Lagrangian tori in phase space.

Figures (2)

• Figure 1: A crossing of $\gamma_E$ with a singularity of order $2$.
• Figure 2: Left hand-side: two $T_E$-periodic magnetic trajectories inside the geodesic disk $\mathbb{D}(i,R_E)$ are depicted in the Poincaré hyperbolic disk model. The point $y_1$ in the interior of the disk has two preimages under the map $\Psi$ but the point $y_0 \in \partial \mathbb{D}(i,R_E)$ has only one. Right-hand side: zooming in near the boundary $\partial \mathbb{D}(i,R_E)$. The various parameters introduced in the proof of Proposition \ref{['proposition:pushforward']} are depicted.

Theorems & Definitions (32)

• Theorem 1.1: $L^\infty$ bounds of magnetic eigenfunctions
• Theorem 1.2: Semiclassical defect measure of magnetic zonal states
• Theorem 1.3: $L^p$ bounds of magnetic eigenfunctions
• Definition 2.1: Twisted semiclassical $\Psi$DOs
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof