## Propagation of Waves in Networks of Thin Fibers (902)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Molchanov902propagationof,

author = {S. Molchanov and B. Vainberg},

title = {Propagation of Waves in Networks of Thin Fibers},

year = {902}

}

### OpenURL

### Abstract

The paper 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 on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE 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. 1

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Citation Context ...imple. In order to determine the solutions of ODE on Γ uniquely, one needs to know the GC on the vertices of Γ. The form of the GC in the general situation was discovered quite recently in our papers =-=[21]-=--[23]. It turned out that they can be expressed in terms of scattering matrices for problems of the wave propagation through individual junctions of Ωε. These GC hold in all the cases: in the bulk of ... |

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Citation Context ...cross-sections of the channels. 20 n≥0Obviously, if the Neumann boundary condition is imposed on ∂Ωε, then λ0 = 0, and the ground state at λ = 0 exists and equal to a constant. It was shown in [21], =-=[22]-=- that the ground state at λ = λ0 does not exist for generic domains Ωε in the case of other boundary conditions on ∂Ωε. In particular, it does not exist if there are eigenvalues of Hε below λ0 or if t... |

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