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

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 of 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.

Planar waveguide with “twisted” boundary conditions: Small width

“…In these works the most attention was paid to the case of Neumann problems and the behavior of the spectrum was studied. Similar studies but for Dirichlet problems were made in [9], [10], [11], [13], [23], [24], [27], [32], [33], [34], [37], [39], [40]. The uniform resolvent convergence for Dirichlet Laplacian in a thin bounded two-dimensional domain was established in [27], while the multidimensional case was treated in [10].…”

The adiabatic limit of Schrödinger operators on fibre bundles

“…This means that the typical energy scale of H a must be ε 2 , which is generally only the case in λ 0 ≡ const (see also the discussion of small energies in Section 4.2). De Oliveira and Verri [14] treat the situation where λ 0 has a unique, non-degenerate minimum and this scaling is of order ε. We see an advantage of our approach in the fact that a priori we do not place any restrictions on the behaviour of λ, and that we can treat also bands different from the ground state.…”

Absolute Continuity and Band Gaps of the Spectrum of the Dirichlet Laplacian in Periodic Waveguides

“…The main point in this theorem is that β ε → 1 uniformily as ε → 0. Its proof is quite similar to the proof of Theorem 3.1 in [8] and will not be presented here.…”