量子图上的均匀高频局域化
Uniform High-Frequency Localization on Quantum Graphs
AI总结：
该研究通过最小L²质量定义量子图拉普拉斯算子的高频局域化常数，提出奇异完备锥方法，证明其在低复杂度类中有效但存在反例，揭示其不总能决定固定图上的高频变分问题。
AI中文摘要：
我们通过本征函数必须放置在预定可测观测集中的最小$L^2$质量，研究紧度量图上拉普拉斯算子的均匀高频局域化。我们首先将此渐近局域化常数与可达边强度集上线性泛函的最小值等同；在一般的标准基尔霍夫设定下，该集合由正则世俗流形的正则高斯像控制。因此，原始环和外部到外部的路径决定正性阈值，但通常不决定正数值。我们引入一个边界感知的奇异完备锥，并证明对于树、单环图和圈秩为二的闭图，它包含所有正则世俗边能量向量。然后我们构造一个具有两个狄利克雷叶的圈秩二图，对于该图此完备原理失效。一个精确的有理分离器，结合经过验证的Krawczyk包络，产生一个位于每个边界兼容奇异扇区之外的非奇异标量世俗态。正径向导数恒等式和紧轨道闭包中的递推将此局部分离转换为单个固定度量上的精确高频本征序列。对于合适的可测观测集，真实高频局域化常数$C_\infty(\omega;\ell)$及其奇异完备对应物$C_{\mathrm{sing}}(\omega;\ell)$满足$$C_\infty(\omega;\ell)<1/2<C_{\mathrm{sing}}(\omega;\ell)$$。因此，奇异完备在几个低复杂度类中捕获定量局域化几何，但通常不决定固定量子图上的高频变分问题。
英文摘要：
We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least $L^2$-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant $C_\infty(ω;\ell)$ and its singular-completion counterpart $C_{\mathrm{sing}}(ω;\ell)$ satisfy $$C_\infty(ω;\ell)<1/2<C_{\mathrm{sing}}(ω;\ell)$$. Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.
