On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type

We study eigenfunctions of the Laplace--Beltrami operator $\Delta_X$ in exterior domains $\Omega$ 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 analyze the asymptotic behaviour and $L^p$-integrability of solutions to the Helmholtz equation $$\Delta_X f + (\lambda^2 + \rho^2) f = 0 \quad \text{in } \Omega,$$ where $\lambda \in \mathbb{C}\setminus i\mathbb{Z}$ and $\rho$ denotes the half-sum of positive roots. We establish sharp Rellich-type quantitative $L^p$-growth estimates in geodesic annuli, which yield the nonexistence of nontrivial $L^p(\Omega)$-solutions in the optimal range $1 \leq p \leq 2$ for spectral parameters satisfying $|\Im(\lambda)| \leq (2/p - 1)\rho$. For non-real spectral parameters, we further obtain refined Rellich-type uniqueness results under weak $L^p$-assumptions. As a by-product, we also prove a Rellich-type uniqueness theorem in terms of Hardy-type norms. Our results provide a geometric extension of the Euclidean Rellich theorem, highlighting the role of exponential volume growth and the $p$-dependence of the $L^p$-spectrum of $\Delta_X$ in producing genuinely non-Euclidean spectral phenomena.