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

TL;DR

This paper analyzes the spectral behavior of the Dirichlet Laplacian in thin, deformed tubes in three-dimensional space, revealing that eigenvalue asymptotics are primarily governed by the deformation function rather than geometric features.

Contribution

It introduces a new asymptotic analysis of the Laplacian in deformed thin tubes, showing that the eigenvalue behavior is dominated by the deformation function rather than geometric complexities.

Findings

Eigenvalues exhibit specific asymptotic behavior as tube diameter shrinks.

The weakly effective operator is determined mainly by the deformation function.

Geometric features like curvature and torsion do not influence the asymptotics.

Abstract

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.