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

TL;DR

This paper analyzes the asymptotic behavior of eigenfunctions at the junction of shrinking tubes, providing precise normalization and describing the resulting singularity, thus advancing understanding of spectral properties in complex geometries.

Contribution

It offers the exact asymptotic behavior of normalized eigenfunctions and characterizes the singularity at the junction, improving previous convergence results.

Findings

Exact asymptotic behavior of eigenfunctions established

Description of (N-1)-order singularity at junction

Improved convergence results from prior work

Abstract

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).