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

Exponential Decay of Eigenfunctions and Accumulation of Eigenvalues on Manifolds with Axial Analytic Asymptotically Cylindrical Ends