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

Laplace Operator in Networks of Thin Fibers: Spectrum Near the Threshold