Abstract
We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (“fat graph”) when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.
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- 1.
Report given by G. F. D. A. at the Conference “Mathematical Technology of Networks – QGraphs 2013”, ZIF Bielefeld.
- 2.
For this work, A.M. was partially supported by a 2013–2014 “CAS-LMU Research in Residence Fellowship” at the Center for Advanced Studies Munich, by a 2014–2015 “INdAM grant Progetto Giovani”, and by the 2014–2017 MIUR-FIR grant “Cond-Math: Condensed Matter and Mathematical Physics”. A.M. gratefully acknowledges also the support of a visiting research fellowship at the International Center for Mathematical Research CIRM, Trento, and G.D. gratefully acknowledges the kind hospitality of the Center for Advanced Studies Munich, where this work was partially carried on.
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Dell’Antonio, G., Michelangeli, A. (2015). Dynamics on a Graph as the Limit of the Dynamics on a “Fat Graph”. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_5
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DOI: https://doi.org/10.1007/978-3-319-16619-3_5
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