The Magnetic Zone

Eigenfunction bounds on Riemannian manifolds quantify how concentrated an eigenfunction can be. The Hormander bound gives the worst case, but for specific geometries — negatively curved manifolds — the bounds can be improved. Adding a magnetic field (a connection on a line bundle) changes the Laplacian and potentially changes the bound.

On hyperbolic surfaces, the magnetic Laplacian admits polynomial improvements to the L-infinity bound in the critical energy regime (arXiv:2603.12177). Below critical energy, the Hormander bound is sharp — achieved by “magnetic zonal states” that generalize zonal harmonics on spheres. These states concentrate on Lagrangian tori in phase space.

The structural observation: the optimizers of the eigenfunction bound are not pathological constructions but natural geometric objects — magnetic generalizations of the classical zonal harmonics. The states that maximize concentration are the ones aligned with the geometric symmetry of the underlying surface. Above critical energy, this alignment breaks and the bound improves. The critical energy is the threshold where the magnetic field’s geometric effect — confining eigenfunctions to tori — loses its grip. Below: concentration wins, bound is sharp. Above: dispersion wins, bound improves. The magnetic field creates a confinement that the eigenfunction eventually escapes.