Long-Time Dispersion and Threshold Resonances for the Landau Hamiltonian on the Hyperbolic Plane

Abstract

A uniform magnetic field completely suppresses long-time dispersion for the Euclidean Landau Hamiltonian. We show that negative curvature fundamentally changes this behavior. For the Landau Hamiltonian with arbitrary field strength $B\in\mathbb R$ on the hyperbolic plane $\mathbb H$, we establish sharp dispersive estimates for the Schr\"odinger evolution on the absolutely continuous spectral subspace. The decay is $|t|^{-1}$ at short times and, strikingly, improves to $|t|^{-3/2}$ at large times whenever $B\notin\frac12+\mathbb Z$. At half-integer fields, we identify a threshold resonance and isolate its leading contribution, which yields the slower but sharp rate $|t|^{-1/2}$. Thus the long-time dynamics exhibit a resonance-driven dichotomy with no analogue in the flat constant-field problem. Our analysis is based on an explicit representation of the magnetic Schr\"odinger kernel together with a precise separation of the Landau levels and threshold profile. As a further consequence, for every nonthreshold field we obtain global-in-time Strichartz estimates on the absolutely continuous spectral subspace throughout the full hyperbolic admissible region.