Convergence of spectra of graph-like thin manifolds

Pavel Exner Olaf Post

TBD mathscidoc:1910.43028

Journal of Geometry and Physics, 54, (1), 77-115, 2005.5
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 LaplaceBeltrami 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.
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@inproceedings{pavel2005convergence,
  title={Convergence of spectra of graph-like thin manifolds},
  author={Pavel Exner, and Olaf Post},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122240352050557},
  booktitle={Journal of Geometry and Physics},
  volume={54},
  number={1},
  pages={77-115},
  year={2005},
}
Pavel Exner, and Olaf Post. Convergence of spectra of graph-like thin manifolds. 2005. Vol. 54. In Journal of Geometry and Physics. pp.77-115. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122240352050557.
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