On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type
Pritam Ganguly
TL;DR
The work extends Rellich‑type asymptotics to exterior domains in rank‑one noncompact symmetric spaces by studying Δ_X f + (λ^2+ρ^2) f = 0. A reduction to a hypergeometric radial equation via spherical harmonic decomposition yields sharp L^p growth lower bounds in geodesic annuli, leading to nonexistence of L^p(Ω) solutions for 1 ≤ p ≤ 2 when |Im(λ)| ≤ γ_p ρ, with γ_p = 2/p − 1, and to a Liouville‑type uniqueness result in Hardy norms. The analysis hinges on Harish–Chandra expansions, the p‑dependent L^p spectrum, and the exponential volume growth, revealing genuinely non‑Euclidean spectral phenomena absent in the Euclidean setting. The results are sharp and illuminate how the geometry of X shapes eigenfunction decay, growth, and uniqueness; the paper also outlines connections to other Rellich‑type frameworks and open directions for extensions to broader noncompact spaces and groups.
Abstract
We study eigenfunctions of the Laplace-Beltrami operator $Δ_X$ in exterior domains $Ω$ of rank-one Riemannian symmetric spaces of noncompact type $X$, a class that includes all hyperbolic spaces. Extending the classical $L^2$-Rellich theorem for the Euclidean Laplacian, we investigate the asymptotic behavior and $L^p$-integrability of solutions to the Helmholtz equation \[ Δ_X f + (λ^2 + ρ^2) f = 0 \quad \text{in } Ω, \] where $λ\in \mathbb{C}\setminus i\mathbb{Z}$ and $ρ$ is the half-sum of positive roots. We obtain sharp Rellich-type quantitative $L^p$-growth estimates of~$f$ in geodesic annuli, leading to the nonexistence of $L^p(Ω)$-solutions for the optimal range $1 \leq p \leq 2$ and spectral parameters $λ$ satisfying $|Im(λ)| \leq (2/p - 1)ρ$. As a by-product of our study, we also establish a Rellich-type uniqueness theorem for eigenfunctions in terms of Hardy-type norms. Our results geometrically extend the Euclidean Rellich theorem, revealing how exponential volume growth and the dependence of the $L^p$-spectrum of $Δ_X$ on $p$ give rise to genuinely non-Euclidean spectral phenomena.