Graph-like models for thin waveguides with Robin boundary conditions

TL;DR

This paper studies the behavior of the Robin Laplacian in thin waveguides as their width shrinks, showing convergence to a graph model and analyzing conditions for decoupling or coupling at the edges.

Contribution

It establishes the convergence of Robin Laplacians on thin waveguides to graph Laplacians and characterizes when decoupling or coupling occurs based on resonances.

Findings

Robin Laplacian converges to a graph Laplacian as width shrinks

Decoupling conditions are generic, coupling occurs at resonances

Resonances at spectrum thresholds influence boundary conditions

Abstract

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while exceptionally a coupling can occur. The non decoupling conditions are related to the existence of resonances at the thresholds of the continuum spectrum.