Spectra of graph neighborhoods and scattering

Proceedings of the London Mathematical SocietyVolume 97, Issue 3 p. 718-752 Articles Spectra of graph neighborhoods and scattering Daniel Grieser, Corresponding Author Daniel Grieser [email protected] Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, Oldenburg, D-26111 Germany[email protected]Search for more papers by this author Daniel Grieser, Corresponding Author Daniel Grieser [email protected] Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, Oldenburg, D-26111 Germany[email protected]Search for more papers by this author First published: 01 May 2008 https://doi.org/10.1112/plms/pdn020Citations: 67 2000 Mathematics Subject Classification 58J50 35P99 (Primary) 47A55 81Q10 (Secondary). AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract Let (Gε)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 Gε, as ε→0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k⩽Cε−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 (Gε). 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. Citing Literature Volume97, Issue3November 2008Pages 718-752 RelatedInformation