Dispersive estimates 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∈ℝ on the hyperbolic plane ℍ, we establish sharp dispersive estimates for the Schrödinger 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∉1/2+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 Schrödinger 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.
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BibTeX
@misc{guo2026dispersive,
title = {{Dispersive estimates for the Landau Hamiltonian on the hyperbolic plane}},
author = {Huanqing Guo and Haoran Wang and Junyong Zhang and Jiqiang Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12999},
eprint = {2609.12999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12999},
}
RIS
TY - UNPB
TI - Dispersive estimates for the Landau Hamiltonian on the hyperbolic plane
AU - Huanqing Guo
AU - Haoran Wang
AU - Junyong Zhang
AU - Jiqiang Zheng
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12999
UR - https://arxiv.org/abs/2609.12999
AB - 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∈ℝ on the hyperbolic plane ℍ, we establish sharp dispersive estimates for the Schrödinger 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∉1/2+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 Schrödinger 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.
ER -