  Our talk at Lisbon SAMP conference was based mainly on our recent results
(published in Comm. Math. Phys.) on small diameter asymptotics for solutions of
the Helmgoltz equation in networks of thin fibers. The present paper contains a
detailed review of these results under some assumptions which make them much
more transparent. It also contains several new theorems on the structure of the
spectrum near the threshold. small diameter asymptotics of the resolvent, and
solutions of the evolution equation.
