Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case

TL;DR

This paper analyzes the spectral behavior of quantum wave guides shrinking to a graph with Dirichlet boundary conditions, showing convergence of spectra to a decoupled graph Laplacian, including effects of curvature.

Contribution

It establishes spectral convergence for shrinking wave guides to a decoupled graph Laplacian, incorporating curvature effects and boundary condition assumptions.

Findings

Dirichlet spectrum converges to the graph Laplacian spectrum

Small vertex neighborhoods lead to decoupling of edges

Curvature introduces additional potential in the limit spectrum

Abstract

We consider a family of open sets $M_\epsilon$ which shrinks with respect to an appropriate parameter $\epsilon$ to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of $M_\epsilon$ converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature.