Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains

Abstract

We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are semi-infinite straight strips. We make the waveguide collapse onto a graph by squeezing the edge regions to half-lines and the vertex region to a point. In a setting in which the ratio between the width of the waveguide and the longitudinal extension of the vertex region goes to zero, we prove the convergence of the operator to a selfadjoint realization of the Laplacian on a two edged graph. In the limit operator, the boundary conditions in the vertex depend on the spectral properties of an effective one dimensional Hamiltonian associated to the vertex region.

Planar waveguide with “twisted” boundary conditions: Small width

“…The models similar to our were studied in [1], [14], [15], [16], [29]. The first four papers are devoted to the model of a thin bent waveguide.…”

“…Here the waveguide was three-dimensional and the nontrivial geometry came from localized twisting which was scaled together with the width as the bending in [1], [14], [15], [16]. The operator in [1], [14], [16], [29] was the Dirichlet Laplacian, while in [15] it was the Robin Laplacian. The main result of [1], [14], [15], [16], [29] is the convergence theorems.…”

“…The operator in [1], [14], [16], [29] was the Dirichlet Laplacian, while in [15] it was the Robin Laplacian. The main result of [1], [14], [15], [16], [29] is the convergence theorems. Namely, the effective operator was found and the uniform resolvent convergence was proven.…”

“…Namely, the effective operator was found and the uniform resolvent convergence was proven. In [16] the estimates for the rate of convergence were established under some additional restrictions for the resolvent's domain. It was also shown in [1], [14], [15], [16] that the effective operator can involve nontrivial boundary condition instead of bending, if certain one-dimensional operators possesses eigenvalue or resonance at zero.…”

“…The main difference of our model in comparison with [1], [14], [15], [16] is that instead of bending we consider a special "twisted" combination of Dirichlet and Neumann boundary conditions. It must be said that we have borrowed the idea of such combination from [17].…”