论文标题
最大化较高特征值的量子树是不平衡的
Quantum trees which maximize higher eigenvalues are unbalanced
论文作者
论文摘要
研究了在给定平均边缘长度的树类中，在公制的树图上最大化Laplacian的所有特征值的等等问题。事实证明，为了进行续订，$ k $ th -th -th -thation特征值的独特最大化器是三个长度为$ 2 k -1 $，$ 1 $和$ 1 $的星形图。这补充了先前已知的结果，即所有等边星图都最大化了第一个非零特征值，并表明，对于更高的特征值的等量术问题的优化者可能会较低平衡，这一观察结果是从较高的特征值对Eigenvalues of Eigenvalue of Euclidean domains domains domains domains domains domains domains of Eigenvalue的数字结果所知。
The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th positive eigenvalue is the star graph with three edges of lengths $2 k - 1$, $1$ and $1$. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape -- an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.
