广义图$p$ -Laplacian的节点域计数
Nodal domain count for the generalized graph $p$-Laplacian
摘要：
受线性薛定谔算子的启发，我们考虑了离散图上的广义$p$ -Laplacian算子，并给出了新的结果，刻画了该算子的几个谱性质，特别注意了其本征函数的节点域计数.与一维连续$p$ -Laplacian一样，证明了森林上离散广义$p$ -Laplacian的变分谱是全谱.此外，我们展示了如何转移外尔的拉普拉斯算子的非线性情况下的不等式，并证明了新的上限和下界的节点域的数目的广义$p$ -拉普拉斯算子的每一个特征函数的一般图，包括变分特征对。特别是，当应用到线性情况下$p=2$，除了恢复众所周知的功能，新的结果提供了新的性质的线性薛定谔算子。
英文摘要：
Inspired by the linear Schrödinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schrödinger operator.
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