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 investigate the asymptotic behavior 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 $\rho$ 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 $L^p(\Omega)$-solutions for the optimal range $1 \leq p \leq 2$ and spectral parameters $\lambda$ satisfying $|Im(\lambda)| \leq (2/p - 1)\rho$. 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 $\Delta_X$ on $p$ give rise to genuinely non-Euclidean spectral phenomena.
Categories: math.AP math.CA