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

Ambre Chabert; Thibault Lefeuvre

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arXiv · 2603.12177

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

Ambre Chabert·Thibault Lefeuvre

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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 magnetic zonal states. These states resemble zonal harmonics on the sphere and equidistribute on Lagrangian tori in phase space.

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Ambre Chabert, Thibault Lefeuvre. 2026-03-12. Zonal states and improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces. https://arxiv.org/abs/2603.12177

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