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

Ambre Chabert; Thibault Lefeuvre

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

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arxiv 2603.12177 v2 pith:PPKJCHVR submitted 2026-03-12 math.AP math.DGmath.SP

Ambre Chabert , Thibault Lefeuvre This is my paper · ORCID

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keywords magneticstateszonalboundscriticaleigenfunctionsenergyhyperbolic

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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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