Spectra of graph neighborhoods and scattering

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 $k\le C{\epsilon }^{-1}$, 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.

37 pages, 3 figures, added references, added comment at end of Section 1.2, changed comment after Theorem 30; in v4: made appendix into a separate paper (arXiv:0711.2869), added reference, minor corrections

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