Spectra of graph neighborhoods and scattering

TL;DR

This paper investigates how the spectrum of the Laplace-Beltrami operator on thin Riemannian manifolds modeled on a graph behaves asymptotically as the thickness parameter tends to zero, revealing the spectral limit object as a quantum graph.

Contribution

It provides complete asymptotic expansions for eigenvalues and eigenfunctions of thin manifolds modeled on graphs, linking spectral data to scattering data and identifying the limit quantum graph.

Findings

Asymptotic eigenvalue expansions in terms of scattering data

Identification of the spectral limit as a quantum graph

Construction of approximate eigenfunctions from scattering and graph data

Abstract

Let $(G_\epsilon)_{\epsilon>0}$ be a family of '$\epsilon$-thin' Riemannian manifolds modeled on a finite metric graph $G$, for example, the $\epsilon$-neighborhood of an embedding of $G$ in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on $G_\epsilon$ as $\epsilon\to 0$, for various boundary conditions. We obtain complete asymptotic expansions for the $k$th eigenvalue and the eigenfunctions, uniformly for $k\leq C\epsilon^{-1}$, in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family $(G_\epsilon)$. Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all eigenfunctions are obtained in this way.