  Consider two domains connected by a thin tube: it can be shown that,
generically, the mass of a given eigenfunction of the Dirichlet Laplacian
concentrates in only one of them. The restriction to the other domain, when
suitably normalized, develops a singularity at the junction of the tube, as the
channel section tends to zero. Our main result states that, under a
nondegeneracy condition, the normalized limiting profile has a singularity of
order N-1, where N is the space dimension. We give a precise description of the
asymptotic behavior of eigenfunctions at the singular junction, which provides
us with some important information about its sign near the tunnel entrance.
More precisely, the solution is shown to be one-sign in a neighborhood of the
singular junction. In other words, we prove that the nodal set does not enter
inside the channel.
