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

TL;DR

This paper improves bounds on eigenfunctions of magnetic Laplacians on hyperbolic surfaces and introduces magnetic zonal states that saturate these bounds, revealing their phase space distribution.

Contribution

It provides polynomial improvements to $L^$ bounds and characterizes magnetic zonal states that saturate classical bounds on hyperbolic surfaces.

Findings

Polynomially improved $L^$ bounds for eigenfunctions.

Explicit construction of magnetic zonal states saturating bounds.

Magnetic zonal states resemble zonal harmonics and distribute on Lagrangian tori.

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 H\"ormander 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.