  In continuation with the paper arXiv:1202.4414, we investigate the asymptotic
behavior of weighted eigenfunctions in two half-spaces connected by a thin
tube. We provide several improvements about some convergences stated in
arXiv:1202.4414; most of all, we provide the exact asymptotic behavior of the
implicit normalization for solutions given in arXiv:1202.4414 and thus describe
the (N-1)-order singularity developed at a junction of the tube (where N is the
space dimension).
