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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 aremultiplied by an appropriate deformation function $h(s)> 0$. One the mainrequirements on $h(s)$ is that it has a single point of global maximum. We findthe asymptotic behaviors of the eigenvalues and weakly effective operators asthe diameters of the tubes tend to zero. It is shown that such behaviors arenot influenced by some geometric features of the tube, such as curvature,torsion and twisting, and so a huge amount of different deformed tubes areasymptotically described by the same weakly effective operator.

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