  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.
