Ławniczak Michał, Sawicki Adam, Białous Małgorzata, Sirko Leszek
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668, Warsaw, Poland.
Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668, Warsaw, Poland.
Sci Rep. 2021 Jan 15;11(1):1575. doi: 10.1038/s41598-020-80950-6.
We identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for [Formula: see text]. The theoretical predictions are confirmed experimentally using [Formula: see text] units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the [Formula: see text] scattering matrices [Formula: see text] of the systems to 2n diagonal elements, while the old measures of isoscattering require all [Formula: see text] entries. The studied problem generalizes a famous question of Mark Kac "Can one hear the shape of a drum?", originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.
我们识别并研究具有(n)个单元和(2n)条无限外部引线的串联量子图的等散射弦。我们深入了解了设计大型图和网络的原理，对于这些图和网络，在([公式：见正文])的情况下等散射特性得以保留。使用([公式：见正文])单元、四引线微波网络对理论预测进行了实验验证。在实验和数学方法中，我们的工作超越了先前的结果，通过证明使用迹函数可以解决到目前为止尚未解决的问题，即具有许多外部引线的开放复杂图和网络的散射特性是否与其形状唯一相关。迹函数的应用将系统所需的([公式：见正文])散射矩阵([公式：见正文])的输入数量减少到(2n)个对角元素，而旧的等散射度量需要所有([公式：见正文])个输入。所研究的问题将马克·卡茨（Mark Kac）著名的问题“能听到鼓的形状吗？”从最初在等谱无耗散系统的情况下进行了推广，扩展到开放图和网络的无限弦的情况。
