Coupling in the singular limit of thin quantum waveguides

TL;DR

This paper studies how smooth quantum waveguides can be approximated by quantum graphs in the singular limit, revealing conditions for decoupling or coupling of the waveguide components.

Contribution

It provides a rigorous analysis of the convergence of the Laplacian in thin quantum waveguides to a quantum graph, identifying conditions for decoupling or coupling effects.

Findings

Decoupling occurs in the generic case with Dirichlet boundary conditions.

Coupling arises for specific curve families, leading to point interactions.

The analysis characterizes the limit operators as the waveguide width shrinks.

Abstract

We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.