Abstract
We consider the problem of determining the asymptotics for the number of points moving along a metric graph. This problem is motivated by the problem of the evolution of wave packets, which at the initial moment of time are localized in a small neighborhood of one point. It turns out that the number of points, as a function of time, allows a polynomial approximation. This polynomial is expressed via Barnes’ multiple Bernoulli polynomials, which are related to the problem of counting the number of lattice points in expanding simplexes. In this paper we give explicit formulas for the first two terms of the expansion for the counting function of the number of moving points. The leading term was found earlier and depends only on the number of vertices, the number of edges and the lengths of the edges. The second term in the expansion shows what happens to the graph when one or two edges are removed. In particular, whether it breaks up into several connected components or not. In this paper, examples of the calculation of the leading and second terms are given.
Similar content being viewed by others
References
Barvinok, A., Integer Points in Polyhedra, EMS Zurich Lectures in Adv. Math., vol. 9, Zárich: European Mathematical Society, 2008.
Lando, S.K., Introduction to Discrete Mathematics, Moscow: MCCME, 2012 (Russian).
Ramírez Alfonsín, J.L., The Diophantine Frobenius Problem, Oxford Lecture Ser. Math. Appl., vol. 30, New York: Oxford Univ. Press, 2005.
Chernyshev, V. L. and Shafarevich, A. I., Statistics of Gaussian Packets onMetric and Decorated Graphs, Philos. Trans. A Math. Phys. Eng. Sci., 2014, vol. 372, no. 2007, 20130145, 11 pp.
Chernyshev, V. L. and Tolchennikov, A. A., Correction to the Leading Term of Asymptotics in the Problem of Counting the Number of Points Moving on a Metric Tree, Russ. J. Math. Phys., 2017, vol. 24, no. 3, pp. 290–298.
Lehmer, D.H., The Lattice Points of an n-Dimensional Tetrahedron, Duke Math. J., 1940, vol. 7, no. 1, pp. 341–353.
Spencer, D.C., The Lattice Points of Tetrahedra, J. Math. Phys. Mass. Inst. Tech., 1942, vol. 21, pp. 189–197.
Barnes,E.W., The Theory of the Double Gamma Function, Trans. Cambridge Philos. Soc., 1901, vol. 196, nos. 274–286, pp. 265–387.
Chernyshev, V. L. and Tolchennikov, A. A., Asymptotic Estimate for the Counting Problems Corresponding to the Dynamical System on Some Decorated Graphs, Ergodic Theory Dynam. Systems, 2017, 12 pp.
Beukers, F., The Lattice-Points of n-Dimensional Tetrahedra, Indag. Math., 1975, vol. 37, no. 5, pp. 365–372.
Chernyshev, V. L., Time-Dependent Schr¨odinger Equation: Statistics of the Distribution of Gaussian Packets on a Metric Graph, Proc. Steklov Inst. Math., 2010, vol. 270, no. 1, pp. 246–262; see also: Tr. Mat. Inst. Steklova, 2010, vol. 270, pp. 249–265.
Berkolaiko, G., Quantum Star Graphs and Related Systems, PhD Thesis, Bristol, Univ. of Bristol, 2000, 135 pp.
Chernyshev, V. L. and Tolchennikov, A.A., How the Permutation of Edges of a Metric Graph Affects the Number of Points Moving along the Edges, arXiv:1410.5015 (2014).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chernyshev, V.L., Tolchennikov, A.A. The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph. Regul. Chaot. Dyn. 22, 937–948 (2017). https://doi.org/10.1134/S1560354717080032
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354717080032