najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-2-182-191najo-907Research ArticlePAPERS, PRESENTED AT THE CONFERENCEPAPERS, PRESENTED AT THE CONFERENCEAn introduction to the spectral asymptotics of a damped wave equation on metric graphsLipovsk´yJ.
Department of Physics, Faculty of Science
Rokitansk´eho 62, 500 03 Hradec Kr´alov´e
jiri.lipovsky@uhk.czUniversity of Hradec Kr´alov´eCzech Republic20151408202562182191Copyright © Lipovsk´y J., 20252025Lipovsk´y J.Lipovsk´y J.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/907
This paper summarizes the main results of [1] for the spectral asymptotics of the damped wave equation. We define the notion of a high frequency abscissa, a sequence of eigenvalues with imaginary parts going to plus or minus infinity and real parts going to some real number. We give theorems on the number of such high frequency abscissas for particular conditions on the graph. We illustrate this behavior in two particular examples.
damped wave equationspectrummetric graphsThe research was supported by the project Development of postdoc activities at the University of Hradec Kr´alov´e, CZ.1.07/2.3.00/30.0015.ReferencesFreitas P., Lipovsk´y J. Eigenvalue asymptotics for the damped wave equation on metric graphs (2013). arXiv : http://arxiv.org/abs/1307.6377.Freitas P., Lipovsk´y J. Eigenvalue asymptotics for the damped wave equation on metric graphs (2013). arXiv : http://arxiv.org/abs/1307.6377.Borisov D., Freitas P. Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation. J. Diff. Eq., 247, P. 3028-3039 (2009).Borisov D., Freitas P. Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation. J. Diff. Eq., 247, P. 3028-3039 (2009).Kuchment P. Quantum graphs: an introduction and a brief survey. Analysis on Graphs and Its Applications. Proc. Symp. Pure. Math. (Providence, RI: American Mathematical Society), P. 291-314 (2008).Kuchment P. Quantum graphs: an introduction and a brief survey. Analysis on Graphs and Its Applications. Proc. Symp. Pure. Math. (Providence, RI: American Mathematical Society), P. 291-314 (2008).Exner P., Lipovsk´y J. Resonances from perturbations of quantum graphs with rationally related edges. J. Phys. A.: Math. Theor., 43, P. 105301 (2010).Exner P., Lipovsk´y J. Resonances from perturbations of quantum graphs with rationally related edges. J. Phys. A.: Math. Theor., 43, P. 105301 (2010).Kottos T., Smilansky U. Quantum chaos on graphs. Phys. Rev. Lett., 79, P. 4794-4797 (1997).Kottos T., Smilansky U. Quantum chaos on graphs. Phys. Rev. Lett., 79, P. 4794-4797 (1997).Akkermans E., Comtet A., Desbois J., Montambaux G., Texier C. Spectral determinant on quantum graphs. Annals of Physics, 284, P. 10-51 (2000).Akkermans E., Comtet A., Desbois J., Montambaux G., Texier C. Spectral determinant on quantum graphs. Annals of Physics, 284, P. 10-51 (2000).Band R., Harrison J. M., Joyner C. H. Finite pseudo orbit expansion for spectral quantities of quantum graphs. J. Phys. A: Math. Theor., 45, P. 325204 (2012).Band R., Harrison J. M., Joyner C. H. Finite pseudo orbit expansion for spectral quantities of quantum graphs. J. Phys. A: Math. Theor., 45, P. 325204 (2012).
The authors declare that there are no conflicts of interest present.