Spectra of graph neighborhoods and scattering

Details Spectra of graph neighborhoods and scattering Spectra of graph neighborhoods and scattering

2007-10-17

arxiv.org/abs/0710.3405v4

Mathematics

Spectral Theory

37 pages, 3 figures, added references, added comment at end of Section 1.2, changed comment after Theorem 30; in v4: made appe

Scientific paper

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.

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