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Proceedings of the London Mathematical Society Advance Access published online on May 1, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn020
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© 2008 London Mathematical Society

Spectra of graph neighborhoods and scattering

Daniel Grieser

Institut für Mathematik
Carl von Ossietzky Universität Oldenburg
D-26111 Oldenburg
Germany

Received 20 November 2007.

Let (G{varepsilon}){varepsilon}>0 be a family of ‘{varepsilon}-thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the {varepsilon}-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{varepsilon}, as {varepsilon}->0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k≤C{varepsilon}–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{varepsilon}). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and the use of a priori estimates to show that all eigenfunctions are obtained in this way.

2000 Mathematics Subject Classification 58J50 35P99 (Primary) 47A55 81Q10 (Secondary).


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