Propagation of Waves in Networks of Thin Fibers
This chapter contains a simplified and improved version of the results obtained by the authors earlier.Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one-dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations (ODEs) on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODEs on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph.
Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it.
KeywordsLution Reso Verse
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- [ExSe89a]Exner, P., Šeba, P.: Electrons in semiconductor microstructures: a challenge to operator theorists, in Schrödinger Operators, Standard and Nonstandard, World Scientific, Singapore (1989), 79-100.Google Scholar
- [ExWe01]Exner, P., Weidl, T.: Lieb-Thirring inequalities on trapped modes in quantum wires, in Proceedings of the XIII International Congress on Mathematical Physics, International Press, Boston (2001), 437-443.Google Scholar
- [Fr96]Freidlin, M.: Markov Processes and Differential Equations: Asymptotic Problems, Birkhäuser, Basel (1996).Google Scholar
- [KuZe03]Kuchment, P., Zeng, H.: Asymptotics of spectra of Neumann Laplacians in thin domains, in Advances in Differential Equations and Mathematical Physics, Karpeshina, Yu. et al. (eds.), American Mathematical Society, Providence, RI (2003), 199-213.Google Scholar
- [MoVa06]Molchanov, S., Vainberg, B.: Transition from a network of thin fibers to quantum graph: an explicitly solvable model, in Contemporary Mathematics, American Mathematical Society, Providence, RI (2006), 227-240.Google Scholar
- [MoVa08]Molchanov, S., Vainberg, B.: Laplace operator in networks of thin fibers: spectrum near the threshold, in Stochastic Analysis in Mathematical Physics, World Scientific, Hackensack, NJ (2008), 69-93.Google Scholar
- [PaRo03]Pavlov, B., Robert, K.: Resonance optical switch: calculation of resonance eigenvalues, in Waves in Periodic and Random Media, American Mathematical Society, Providence, RI (2003), 141-169.Google Scholar
- [RuSc01]Rubinstein, J., Schatzman, M.: Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum. Arch. Rational Mech. Anal., 160, 293-306 (2001).Google Scholar