  In this paper we continue our study of the Laplacian on manifolds with axial
analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using
the complex scaling method and the Phragm\'{e}n-Lindel\"{o}f principle we prove
exponential decay of the eigenfunctions corresponding to the non-threshold
eigenvalues of the Laplacian on functions. In the case of a manifold with
(non-compact) boundary it is either the Dirichlet Laplacian or the Neumann
Laplacian. We show that the rate of exponential decay of an eigenfunction is
prescribed by the distance from the corresponding eigenvalue to the next
threshold. Under our assumptions on the behaviour of the metric at infinity
accumulation of isolated and embedded eigenvalues occur. The results on decay
of eigenfunctions combined with the compactness argument due to Perry imply
that the eigenvalues can accumulate only at thresholds and only from below. The
eigenvalues are of finite multiplicity.
