细纤维网络中的波传播
Propagation of Waves in Networks of Thin Fibers
摘要：
该文件包含一个简化和改进的版本，作者早先获得的结果。讨论了当波导厚度为零，波导收缩为一维图形时，波在分支细波导网络中的传播。它表明，渐近地可以描述的传播波，频谱和预解式的限制图上的常微分方程的解决方案。图的顶点对应于波导的接合点。为了唯一地确定图上的常微分方程的解，需要知道图的顶点上的粘合条件（GC）。与关于该主题的其他出版物不同，我们考虑当谱参数大于阈值时的情况，即，波的传播可以在网络的圆柱形部分中进行。我们表明，在这种情况下，GC可以表示在与个别路口的散射矩阵。结果被扩展到低于阈值和它周围的频谱参数的值。
英文摘要：
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.
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