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We show the existence and orthogonality of wave operators naturally associated to a compatible Laplacian on a complete manifold with a corner of codimension 2. In fact, we prove asymptotic completeness i.e. that the image of these wave operators is equal to the space of absolutely continuous states of the compatible Laplacian. We achieve this last result using time dependent methods coming from many-body Schr\"odinger equations., Comment: We prove asymptotic completeness for compatible Laplacians on complete manifolds with corners of codimension 2. This version adds references and corrects typos. appears in Rev.colomb.mat. vol.49 no.1 Bogot\'a jan./jun. 2015

We apply Mourre theory to compatible Laplacians on manifolds with corners of codimension 2 in order to prove absence of singular spectrum, that non-threshold eigenvalues have finite multiplicity and could accumulate only at thresholds or infinity. It turns out that we need Mourre estimates on manifolds with cylindrical ends where the results are both expected and consequences of more general theorems. In any case we also provide a description, interesting in its own, of Mourre theory in such context that makes our text complete and suggests generalizations to higher order codimension corners. We use theorems of functional analysis that are suitable for these geometric applications., Comment: This paper has been withdrawn by the author in order to correct reference [3] respect to the $C^1(A)$-property

The analytic dilation method was originally used in the context of many body Schr\"odinger operators. In this paper we adapt it to the context of compatible Laplacians on complete manifolds with corners of codimension two. As in the original setting of application we show that the method allows us to: First, meromorphically extend the matrix elements associated to analytic vectors. Second, to prove absence of singular spectrum. Third, to find a discrete set that contains the accumulation points of the pure point spectrum, and finally, it provides a theory of quantum resonances. Apart from these results, we win also a deeper understanding of the essential spectrum of compatible Laplacians on complete manifolds with corners of codimension 2.