On the curvature and torsion effects in one dimensional waveguides

Abstract

Abstract.We consider the Laplace operator in a thin tube of R 3 with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies.Mathematics Subject Classification.

Location of the nodal set for thin curved tubes

“…However, it turns out that the Dirichlet Laplacian in Γ (which is in fact the Dirichlet Laplacian in I, −∆ I D , because Γ is parametrized by arc length) is not the right operator governing the spectral properties of −∆ T D in this limit. Instead, it will be clear in a moment that the right operator for this purpose is the one-dimensional Schrödinger operator 6) where the potential function is determined uniquely by the first curvature of Γ:…”

“…The geometrical meaning of our special choice for rotations (R µν ) is that we restrict to non-twisted tubes in the language of [19], which simplifies the analysis considerably. It has been noticed recently in [19,6] that other choices for the rotation may change the spectral picture, too. Namely, in view of the eigenvalue asymptotics obtained in [6], it seems to be reasonable to conjecture that a version of our Theorem 2.3 will still hold for twisted tubes, provided that v 0 is replaced by a more complicated potential, depending also on the higher curvatures of Γ and the geometry of ω.…”

Spectral analysis in thin tubes with axial heterogeneities

“…Previous mathematical analysis has been performed concerning the relation between the e¤ects of curvature, torsion, or transversal heterogeneities in wave propagation through thin tubes (see, for instance, [2], [3], [4], [6], [8], [11], [15], [16]). …”

“…Then, substituting the above expansions in (3.8), collecting power like terms in the resulting equation, and taking the terms of order 2 we conclude that j 0 should satisfy Collecting terms of order 3 , we obtain that j 1 satisfies…”

Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods

“…The asymptotics of the spectra of elliptic operators in thin domains were studied by many authors, see, for instance [2][3][4][5][6][7][8][9][10]12,14], and the references therein. There are two types of thin domains usually considered, namely, thin rods and thin plates.…”