  Consider two domains connected by a thin tube: it can be shown that the
resolvent of the Dirichlet Laplacian is continuous with respect to the channel
section parameter. This in particular implies the continuity of isolated simple
eigenvalues and the corresponding eigenfunctions with respect to domain
perturbation. Under an explicit nondegeneracy condition, we improve this
information providing a sharp control of the rate of convergence of the
eigenvalues and eigenfunctions in the perturbed domain to the relative
eigenvalue and eigenfunction in the limit domain. As an application, we prove
that, again under an explicit nondegeneracy con- dition, the case of resonant
domains features polinomial splitting of the two eigenvalues and a clear
bifurcation of eigenfunctions.
