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

Singularity of eigenfunctions at the junction of shrinking tubes, Part II