论文标题

Robin和Neumann Eigenvalues在公制图上的差异

Differences between Robin and Neumann eigenvalues on metric graphs

论文作者

Band, Ram, Schanz, Holger, Sofer, Gilad

论文摘要

我们在公制图上考虑了Laplacian,在某些图顶点和Neumann-Kirchhoff条件下配备了Robin($δ$ -Type)顶点条件。相应的特征值称为Robin特征值,而如果在所有顶点施加了Neumann-Kirchhoff条件,则称为Neumann特征值。这些特征值对之间的差异序列称为罗宾 - 尼曼差距。我们证明,该序列的限制平均值是存在的,并且等于几何量,类似于平面域获得的几何量。此外,我们表明该序列是统一的,并提供明确的上限和下限。我们还研究了序列的可能的积累点,并将其与间隙的相关概率分布相关。为了证明我们的主要结果,我们证明了局部的Weyl定律,以及在本征函数散射幅度的第二瞬间的明确表达式。

We consider the Laplacian on a metric graph, equipped with Robin ($δ$-type) vertex condition at some of the graph vertices and Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann-Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin-Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains. Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.

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