  We deepen the study of Dirichlet eigenvalues in bounded domains where a thin
tube is attached to the boundary. As its section shrinks to a point, the
problem is spectrally stable and we quantitatively investigate the rate of
convergence of the perturbed eigenvalues. We detect the proper quantity which
sharply measures the perturbation's magnitude. It is a sort of torsional
rigidity of the tube's section relative to the domain. This allows us to
sharply describe the asymptotic behavior of the perturbed spectrum, even when
eigenvalues converge to a multiple one. The final asymptotics of eigenbranches
depend on the local behavior near the junction of eigenfunctions chosen in a
special way.
  The present techniques also apply when the perturbation of the Dirichlet
eigenvalue problem consists in prescribing homogeneous Neumann boundary
conditions on a small portion of the boundary of the domain.
