恢复量子图的欧拉示性数的具体方法
Concrete method for recovering the Euler characteristic of quantum graphs
摘要：
迹公式在谱几何，特别是量子图的研究中起着核心作用。我们工作的基础是Kurasov的结果，它将度量图的欧拉示性数$χ$与它们的标准Laplacian的谱联系起来。这些想法被证明是适用的，即使在实验环境中，只有有限数量的特征值从一个物理实现的量子图可以测量。在目前的工作中，我们分析了充分的假设，保证成功恢复$χ$。我们还研究了如何提高效率的方法，特别是如何尽量减少所需的特征值的数量。最后，我们比较我们的研究结果与数值例子-令人惊讶的是，只有几十个特征值就足够了。
英文摘要：
Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic $χ$ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured. In the present work we analyse sufficient hypotheses which guarantee the successful recovery of $χ$. We also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required. Finally, we compare our findings with numerical examples---surprisingly, just a few dozens of eigenvalues can be enough.
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