Existence and nonexistence results for eigenfunctions of the Laplacian in unbounded domains of H^n

We investigate, for the Laplacian operator, the existence and nonexistence of eigenfunctions of eigenvalue between zero and the first eigenvalue of the hyperbolic space H^n, for unbounded domains of H^n. If a domain is contained in a horoball, we prove that there is no positive bounded eigenfunction that vanishes on the boundary. However, if the asymptotic boundary of a domain contains an open set of the asymptotic boundary of H^n, there is a solution that converges to 0 at infinity and can be extended continuously to the asymptotic boundary. In particular, this result holds for hyperballs.