Spectra of graph neighborhoods and scattering

Let $(G_{ϵ}{)}_{ϵ>0}$ be a family of ‘$ϵ$-thin’ Riemannian manifolds modeled on a finite metric graph $G$, for example, the $ϵ$-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 $G_{ϵ}$ as $ϵ\to 0$, for various boundary conditions. We obtain complete asymptotic expansions for the $k$th eigenvalue and the eigenfunctions, uniformly for $k\le C{ϵ}^{-1}$, 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 $(G_{ϵ})$. 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.