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Statistics of Gaussian packets on metric and decorated graphs. (English) Zbl 1354.35122

The authors consider the Cauchy problem for a time-dependent Schrödinger equation on metric and decorated graphs with a localized initial function. An algorithm for constructing quantization rules in the case of metric graphs appears in [the authors, Math. Notes 82, No. 4, 542–554 (2007; Zbl 1160.47022); translation from Mat. Zametki 82, No. 4, 606–620 (2007)], and semiclassical asymptotics for a solution of the time-dependent Schrödinger equation with narrow packet initial conditions localized in a small neighborhood of a certain point on a metric graph was studied in [the authors, Russ. J. Math. Phys. 15, No. 1, 25–34 (2008; Zbl 1175.81115)]. Of particular importance in this paper, the authors study the statistical properties of the associated solutions (in terms of Gaussian packets) on decorated graphs and prove that under certain conditions the number of packets grows polynomially in time.

MSC:

35Q40 PDEs in connection with quantum mechanics
35F05 Linear first-order PDEs
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35B40 Asymptotic behavior of solutions to PDEs
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