论文标题
部分可观测时空混沌系统的无模型预测
Morse theory for discrete magnetic operators and nodal count distribution for graphs
论文作者
论文摘要
给定有限连接的图形$ g $ n $ vertices的离散schrödinger运营商$ h $，节点计数$ ϕ（h，k）$表示$ k $ th eigenVector更改符号的边数。 {\ em Signing} $ h $ of $ h $是通过更改$ h $的一些偏外条目的符号构建的任何真实对称矩阵，其节点计数是根据签名定义的。 $ h $的签名集中在自然定义的圆环$ \ mathbb {t} _h $``磁扰动''的$ h $。G。berkolaiko的$ h $。g。berkolaiko发现，每个签名$ h'$ h $ of $ h $ of $ h $ a $ h $都是每个eigenvalue $λ_k：\ mathbb {摩尔斯指数等于节点盈余。与平面链接的配置空间相关。一般而言，我们猜想，鼻孔盈余分布以CLT的方式收敛到高斯，因为第一个$ G $的Betti数量是Infinity。
Given a discrete Schrödinger operator $h$ on a finite connected graph $G$ of $n$ vertices, the nodal count $ϕ(h,k)$ denotes the number of edges on which the $k$-th eigenvector changes sign. A {\em signing} $h'$ of $h$ is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of $h$, and its nodal count is defined according to the signing. The set of signings of $h$ lie in a naturally defined torus $\mathbb{T}_h$ of ``magnetic perturbations" of $h$. G. Berkolaiko discovered that every signing $h'$ of $h$ is a critical point of every eigenvalue $λ_k:\mathbb{T}_h \to \mathbb{R}$, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if $h_α \in \mathbb{T}_h$ is a critical point of $λ_k$ and the eigenvector vanishes at a single vertex $v$ of degree $d$, then the critical point lies in a nondegenerate critical submanifold of dimension $d+n-4$, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of $ϕ(h',k)-(k-1)$, averaged over all signings $h'$ of $h$. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of $G$ goes to infinity.
