Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convergence of spectra of graph-like thin manifolds   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.

1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ