Convergence of spectra of graph-like thin manifolds

0032293 - ÚJF 2006 RIV NL eng J - Journal Article
Exner, Pavel - Post, O.
Convergence of spectra of graph-like thin manifolds.
[Konvergence spekter tenkych variet typu grafu.]
Journal of Geometry and Physics. Roč. 54, č. 1 (2005), s. 77-115. ISSN 0393-0440. E-ISSN 1879-1662
R&D Projects: GA AV ČR IAA1048101
Institutional research plan: CEZ:AV0Z1048901
Keywords : branched quantum wave guides * convergence of eigenvalues * singular limit * Laplace-Beltrami operator
Subject RIV: BE - Theoretical Physics
Impact factor: 0.607, year: 2005

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.

Uvažujeme množinu kompaktních variet, jež kolabuje vzhledem ke vhodnému parametru na graf. Hlavní výsledek je, že spektrum Laplace-Beltramiho operátoru konverguje ke spektru (diferenciélniho)Laplaciánu na grafu s Kirchhoffovými hraničními podmínkami ve vrcholech.
Permanent Link: http://hdl.handle.net/11104/0132842