Below is the ascii version of the abstract for 03-533.
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Pavel Exner, Olaf Post
Convergence of spectra of graph-like thin manifolds
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 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.