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Working paper

Asymptotic estimate for the number of Gaussian packets on three decorated graphs

arxiv.org. math. Cornell University, 2014. No. 1403.0263.
V.L. Chernyshev, Tolchennikov A.
We study a topological space obtained from a graph by replacing vertices with smooth Riemannian manifolds, i.e. a decorated graph. We construct a semiclassical asymptotics of the solutions of Cauchy problem for a time-dependent Schroedinger equation on a decorated graph with a localized initial function. The main term of our asymptotic solution at an arbitrary finite time is the sum of Gaussian packets and generalized Gaussian packets. We study the number of such packets as time goes to infinity. We prove asymptotic estimations for this number for the following decorated graphs: cylinder with a segment, two dimensional torus with a segment, three dimensional torus with a segment.