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

We study eigenfunctions of the Laplace–Beltrami operator ${\mathrm{\Delta }}_X$ in exterior domains $\mathrm{\Omega }$ 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 $${\mathrm{\Delta }}_{X}f+({\lambda }^2+{\rho }^2)f=0\text{in }\mathrm{\Omega },$$ 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 $L^p(\mathrm{\Omega })$-solutions in the optimal range $1\le p\le 2$ for spectral parameters satisfying $\mathrm{\mid }\mathrm{\Im }(\lambda )\mathrm{\mid }\le (2\mathrm{/}p-1)\rho$. 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 ${\mathrm{\Delta }}_X$ in producing genuinely non-Euclidean spectral phenomena.