On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

Details On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

2011-03-15

arxiv.org/abs/1103.2934v1

Physics

Mathematical Physics

Scientific paper

10.1016/j.jmaa.2011.03.022

We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.

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