Spectral estimate for the Laplace–Beltrami operator on the hyperbolic half-plane

The purpose of this note is to investigate the concentration properties of spectral projectors on manifolds. This question has been intensively studied (by Logvinenko–Sereda, Nazarov, Jerison–Lebeau, Kovrizhkin, Egidi–Seelmann–Veselić, Burq–Moyano, among others) in connection with the uncertainty principle. We provide the first high-frequency results in a geometric setting which is neither Euclidean nor a perturbation of Euclidean. Namely, we prove the natural (and optimal) uncertainty principle for the spectral projector on the hyperbolic half-plane.