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We develop the complex scaling for a manifold with an asymptoticallycylindrical end under an assumption on the analyticity of the metric withrespect to the axial coordinate of the end. We allow for arbitrarily slowconvergence of the metric to its limit at infinity, and prove a variant of theAguilar-Balslev-Combes theorem for the Laplacian $\Delta$ on functions. In thecase of a manifold with (noncompact) boundary it is either the Dirichlet or theNeumann Laplacian. We introduce resonances as the discrete non-real eigenvaluesof non-selfadjoint operators, obtained as deformations of the Laplacian bymeans of the complex scaling. The resonances are identified with the poles ofthe resolvent matrix elements $((\Delta-\mu)^{-1}F, G)$ meromorphiccontinuation in $\mu$ across the essential spectrum of $\Delta$, where $F$ and$G$ are elements of an explicitly given set of analytic vectors. The Laplacianhas no singular continuous spectrum, the eigenvalues can accumulate only atthresholds.

Aguilar-Balslev-Combes theorem for the Laplacian on a manifold with an axial analytic asymptotically cylindrical end