论文标题
在Pleijel的节点域定理上用于量子图
On Pleijel's nodal domain theorem for quantum graphs
论文作者
论文摘要
我们建立了Pleijel定理的公表示图形，内容涉及$ n $ th eigenfunction的节点域$ν_n$的渐近学，包括紧凑型公制图的广泛类函数（s），包括schrödinger，包括$ l^1 $ - potientials和vertepe $ prop的schrödinger，以及$ l^1 $ - plapex的$ - p。并且没有关于边缘长度，图形拓扑的任何假设或在顶点上特征函数的行为。 {除其他外，这些结果表征了序列$（\ frac {c {ν_n} {n}）_ {n \ in \ Mathbb n} $的积累点，始终显示出$（0,1] $的有限子集，以前是$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $拉普拉斯（Laplacian），尤其是在具有自然条件的特殊情况下，我们表明，对于具有理性依赖的边缘长度的图形，可以找到$ {ν_n} \ not \ sim {n} $的特征。
We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $ν_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $L^1$-potentials and a variety of vertex conditions as well as the $p$-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. {Among other things, these results characterise the accumulation points of the sequence $(\frac{ν_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. This} extends the previously known result that $ν_n\sim n$ \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs with rationally dependent edge lengths, one can find eigenfunctions thereon for which ${ν_n}\not\sim {n}$; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.
