Convergence of spectra of graph-like thin manifolds

TL;DR

This paper studies how the spectral properties of Laplace-Beltrami operators on thin, graph-like manifolds converge to those of differential operators on the limiting graph, revealing different regimes of boundary conditions.

Contribution

It characterizes the spectral convergence of graph-like manifolds to various types of Laplacians on the limiting graph, including Kirchhoff, Dirichlet, and coupled boundary conditions.

Findings

Spectral convergence to Kirchhoff Laplacian on the graph.

Different boundary conditions emerge depending on the shrinking rates.

A third regime with nontrivial vertex coupling is identified.

Abstract

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.