论文标题
恢复量子图的欧拉特征的具体方法
Concrete method for recovering the Euler characteristic of quantum graphs
论文作者
论文摘要
痕量公式在光谱几何形状，尤其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，该结果将欧拉（Euler）特征的$ $ $公制图与其标准拉普拉斯（Laplacian）的光谱联系起来。这些想法也被证明是适用的，即使在实验环境中，只能测量量子图的物理实现的有限数量的特征值。 在目前的工作中，我们分析了足够的假设，以确保成功恢复$χ$。我们还研究了如何提高方法的效率，尤其是如何最大程度地减少所需的特征值的数量。最后，我们将发现与数值示例进行了比较 - 令人惊讶的是，只有几十个特征值就足够了。
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.
