Convergence of spectra of graph-like thin manifolds
By Pavel Exner, Olaf Post. 2003
171 citations recorded by the source corpus.
Abstract 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.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/cc1d959fa98bface6f985858eaf57cee9a9cf975
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Homogeneous trees of second order Sturm-Liouville equations: A general theory and program (2008, background): “This usually happens when the ideal vertex corresponds to an object with some internal structure in a real world problem, see [79, 80, 140, 147, 151] for various examples.”