A Strange Vertex Condition Coming From Nowhere

TL;DR

This paper establishes the convergence of solutions to perforated domain problems to a 1D limit with a modified vertex condition, revealing how perforations influence the spectral properties of quantum graphs.

Contribution

It introduces a novel analysis of how perforations affect the limit problem and vertex conditions in quantum graphs, including the derivation of the constant  and its impact.

Findings

Norm-resolvent and spectral convergence proven

The constant  appears in the vertex condition of the limit problem

Perforations alter the spectrum of the quantum graph

Abstract

We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-\Delta u_\varepsilon = f$ on a domain $\Omega_\varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+\mu u = f$ on the interval $(0,1)$, where $\mu$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighbourhood and the vertex neighbourhood is chosen correctly, the constant $\mu$ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.