Abstract
We discuss resonances in quantum graphs and their more general analogs having ‘edges’ of different dimensions. Since the notion of resonance may mean different things, we show that the two most common definitions, scattering and resolvent resonances, are equivalent in this case. We analyze the high-energy behavior of resonances in quantum graphs and show that it may deviate from the standard Weyl law prediction; we derive a criterion which shows when such a thing happens. We also investigate influence of magnetic fields on graph resonances and show that they are field configurations which remove all ‘true’ resonances from such systems.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adami R, Teta A (1998) On the Aharonov-Bohm Hamiltonian. Lett Math Phys 43:43–54
Aizenman M, Sims R, Warzel S (2006) Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun Math Phys 264:371–389
Berkolaiko G, Kuchment P (2013) Introduction to quantum graphs. American Mathematical Society, Providence
Bolte J, Endres S (2009) The trace formula for quantum graphs with general self-adjoint boundary conditions. Ann H Poincaré 10:189–223
Brüning J, Geyler V (2003) Scattering on compact manifolds with infinitely thin horns. J Math Phys 44:371–405
Brüning J, Exner P, Geyler VA (2003) Large gaps in point-coupled periodic systems of manifolds. J Phys A Math Gen 36:4875–4890
Cattaneo C (1997) The spectrum of the continuous Laplacian on a graph. Monatsh Math 124:215–235
Dabrowski L, Šťovíček P (1998) Aharonov-Bohm effect with δ-type interaction. J Math Phys 36:47–62
Davies EB, Pushnitski A (2011) Non-Weyl resonance asymptotics for quantum graphs. Anal PDE 4:729–756
Davies EB, Exner P, Lipovský J (2010) Non-Weyl asymptotics for quantum graphs with general coupling conditions. J Phys A Math Theor 43:474013
Exner P (1996) Contact interactions on graph superlattices. J Phys A Math Gen 29:87–102
Exner P (1997) A duality between Schrödinger operators on graphs and certain Jacobi matrices. Ann Inst H Poincaré Phys Théor 66:359–371
Exner P, Lipovský J (2010) Resonances from perturbations of quantum graphs with rationally related edges. J Phys A Math Theor 43:105301
Exner P, Lipovský J (2011) Non-Weyl resonance asymptotics for quantum graphs in a magnetic field. Phys Lett A 375:805–807
Exner P, Lipovský J (2013, to appear) Resonances on hedgehog manifolds. Acta Polytech. arXiv:1302.5269
Exner P, Post O (2013, to appear) Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds. Commun Math Phys. arXiv:1205.5129
Exner P, Šeba P (1987) Quantum motion on a halfline connected to a plane. J Math Phys 28:386–391; erratum p 2254
Exner P, Tater M, Vaněk D (2001) A single-mode quantum transport in serial-structure geometric scatterers. J Math Phys 42:4050–4078
Exner P, Helm M, Stollmann P (2007) Localization on a quantum graph with a random potential on the edges. Rev Math Phys 19:923–939
Grieser D (2008) Spectra of graph neighborhoods and scattering. Proc Lond Math Soc 97:718–752
Gutkin B, Smilansky U (2001) Can one hear the shape of a graph? J Phys A Math Gen 34:6061–6068
Hislop P, Post O (2009) Anderson localization for radial tree-like random quantum graphs. Wave Random Media 19:216–261
Kiselev A (1997) Some examples in one-dimensional ‘geometric’ scattering on manifolds. J Math Anal Appl 212:263–280
Kottos T, Smilansky U (1999) Periodic orbit theory and spectral statistics for quantum graphs. Ann Phys 274:76–124
Kuchment P (2004) Quantum graphs: I. Some basic structures. Waves Random Media 14:S107–S128
Langer RE (1931) On the zeros of exponential sums and integrals. Bull Am Math Soc 37:213–239
Pankrashkin K (2012) Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures. J Math Anal Appl 396:640–655
Post O (2011) Spectral analysis on graph-like spaces. Lecture notes in mathematics, vol 2039. Springer, Berlin
Ruedenberg K, Scherr CW (1953) Free-electron network model for conjugated systems, I. Theory J Chem Phys 21:1565–1581
Schenker JH, Aizenman M (2000) The creation of spectral gaps by graph decoration. Lett Math Phys 53:253–262
Acknowledgements
The research the results of which are reported here has been supported by the Czech Science Foundation within the project P203/11/0701.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Exner, P. (2014). Resonances in Quantum Networks and Their Generalizations. In: Matrasulov, D., Stanley, H. (eds) Nonlinear Phenomena in Complex Systems: From Nano to Macro Scale. NATO Science for Peace and Security Series C: Environmental Security. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8704-8_12
Download citation
DOI: https://doi.org/10.1007/978-94-017-8704-8_12
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-8703-1
Online ISBN: 978-94-017-8704-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)