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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
References
Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Texts and Monographs in Physics. Springer, New York (1988)
Albeverio, S., Høegh-Krohn, R.: Point interactions as limits of short range interactions. J. Oper. Theory 6, 313–339 (1981)
Albeverio, S., Pankrashkin, K.: A remark on Krein’s resolvent formula and boundary conditions. J. Phys. A 38, 4859–4864 (2005)
Amovilli, C., Leys, F., March, N.: Electronic energy spectrum of two-dimensional solids and a chain of C atoms from a quantum network model. J. Math. Chem. 36, 93–112 (2004)
Brüning, J., Geyler, V.A.: Scattering on compact manifolds with infinitely thin horns. J. Math. Phys. 44, 371–405 (2003)
Dell’Antonio, G., Costa, E.: Effective Schrödinger dynamics on ε-thin Dirichlet waveguides via quantum graphs: I. Star-shaped graphs. J. Phys. A 43, 474014, 23 (2010)
Dell’Antonio, G., Michelangeli, A.: Schrödinger operators on half-line with shrinking potentials at the origin, SISSA preprint 6/2015/MATE (2015) http://urania.sissa.it/xmlui/handle/1963/34439
Dell’Antonio, G., Tenuta, L.: Quantum graphs as holonomic constraints. J. Math. Phys. 47, 072102, 21 (2006)
Grieser, D.: Spectra of graph neighborhoods and scattering. Proc. Lond. Math. Soc. 97, 718–752 (2008)
Harmer, M.: Hermitian symplectic geometry and extension theory. J. Phys. A 33, 9193–9203 (2000)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics, Springer, Berlin (1995). Reprint of the 1980 edition
Konno, R., Kuroda, S.T.: On the finiteness of perturbed eigenvalues. J. Fac. Sci. Univ. Tokyo Sect. I 13 55–63 (1966)
Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32, 595–630 (1999)
Kostrykin, V., Schrader, R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In: Quantum Graphs and Their Applications. Contemporary Mathematics, vol. 415, pp. 201–225. American Mathematical Society, Providence, RI (2006)
Kuroda, S.T., Nagatani, H.: Resolvent formulas of general type and its application to point interactions. J. Evol. Equ. 1, 421–440 (2001). Dedicated to the memory of Tosio Kato
Post, O.: Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case. J. Phys. A 38, 4917–4931 (2005)
Post, O.: Spectral convergence of quasi-one-dimensional spaces. Ann. Henri Poincaré 7, 933–973 (2006)
Saito, Y.: The limiting equation for Neumann Laplacians on shrinking domains. Electron. J. Differ. Equ. 31, 25 pp. (2000) (electronic)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-16619-3_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16618-6
Online ISBN: 978-3-319-16619-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)