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

TL;DR

This work proves a geometric uncertainty principle for the hyperbolic half-plane $\mathbb{H}^2$ by linking a thick-set assumption to a high-frequency spectral projector inequality for the Laplace–Beltrami operator. The authors develop a novel rectangle-tiling and coordinate-change approach to reduce to a uniformly elliptic problem, then apply the Logunov–Malinnikova propagation of smallness and a Sobolev-injection lemma to obtain the global bound $\|\Pi_\Lambda u\|_{L^2_g(\mathbb{H}^2)} \le C e^{C\Lambda} \|\Pi_\Lambda u\|_{L^2_g(\omega)}$ for all $\Lambda\ge\tfrac14$. They also establish the converse: spectral-inequality implies thickness via heat-kernel observability and Gaussian bounds, yielding a complete equivalence between thickness and the spectral estimate on a non-compact, non-Euclidean manifold. The results extend the uncertainty principle for spectral concentration to curved geometry and non-perturbative settings, with potential implications for control theory and harmonic analysis on manifolds. The techniques blend geometric coverings, uniform-ellipticity via coordinate changes, propagation of smallness, and parabolic observability to address high-frequency spectral behavior in hyperbolic geometry.

Abstract

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--Veseli{ć}, 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.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['lemma: gradient']}
• Lemma 2.3: Spectral estimate implies observability
• Lemma 2.4: Gaussian bounds
• proof : Proof of lemma \ref{['lemma: gaussian bounds heat kernel']}
• Theorem 2: logunovmalinnikova2018qtttve_propag_smallness
• proof : Proof of Lemma \ref{['lemma: observability']}