论文标题
部分可观测时空混沌系统的无模型预测
Nodal domain count for the generalized graph $p$-Laplacian
论文作者
论文摘要
受线性schrödinger运算符的启发，我们在离散图上考虑了一个广义的$ p $ laplacian运算符，并提出了新的结果，这些结果表征了该操作员的多个光谱属性，并特别注意其特征函数的节点域计数。就像一维连续的$ p $ -laplacian一样，我们证明了森林上离散的$ p $ laplacian的变化频谱是整个频谱。此外，我们展示了如何将Laplacian操作员Weyl的不平等转移到非线性情况下，并在包括多种特征Pairs在内的通用图上的广义$ p $ -laplacian的每个特征函数的节点域的数量证明了新的上和下限。特别是，当应用于线性案例$ p = 2 $，除了恢复知名功能外，新的结果还提供了线性schrödinger运算符的新属性。
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.
