Convergence of spectra of graph-like thin manifolds

Details Convergence of spectra of graph-like thin manifolds Convergence of spectra of graph-like thin manifolds

2003-12-10

arxiv.org/abs/math-ph/0312028v2

J. Geom. Phys. 54 (2005), 77-115

Physics

Mathematical Physics

38 pages, 6 figures (small changes, Sec 9 extended)

Scientific paper

10.1016/j.geomphys.2004.08.003

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 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.

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