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

TL;DR

This paper extends Rellich-type asymptotic results for eigenfunctions of the Laplace--Beltrami operator on rank-one symmetric spaces, revealing new spectral phenomena influenced by exponential volume growth.

Contribution

It establishes sharp Rellich-type estimates and nonexistence results for eigenfunctions on noncompact symmetric spaces, generalizing Euclidean theorems to a geometric setting.

Findings

Nontrivial L^p solutions do not exist in the optimal range for certain spectral parameters.

Refined uniqueness results are obtained for non-real spectral parameters.

A Rellich-type theorem in Hardy norms is proved.

Abstract

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.