发表机构
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City（越南国立大学胡志明市大学城数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对满足基尔霍夫条件与狄利克雷-诺伊曼赋值的有限紧度量图，确定了确保算术疤痕可达性的准则，结合谱结果与GGCC的图论性质，证明有理独立度量下GGCC是自由薛定谔方程任意正时间精确可控性的充要条件。
AI 中文摘要
我们研究有限紧度量图上自由薛定谔方程的内部精确可控性，该图在内部顶点处满足基尔霍夫条件，在外部顶点处满足固定的狄利克雷-诺伊曼赋值。针对给定的原始环或外-外路径，我们确定了一个算术准则，该准则确保对应疤痕可被固定目标度量的精确高频本征函数访问。该准则通过将久期集的正则疤痕层与目标长度向量的轨道闭包环面相交来表述，且对于超出有理独立类的共振度量也可成立。在轨道闭包流满足固有丢番图条件下，该构造是定量的：沿精确本征序列，给定支撑外的质量呈多项式衰减，当该支撑未被控制时，会产生有限频率观测的多项式退化律。有理独立度量具有完全的相位轨道闭包，因此可使每个原始阻碍都可访问。结合该谱结果与图几何控制条件（GGCC）的已知图论刻画及充分性，我们证明，对于每个有理独立（因此对勒贝格几乎每个）度量，GGCC是自由情形下任意正时间精确可控性的充要条件。
英文摘要
We study internal exact controllability of the free Schrödinger equation on finite compact metric graphs with Kirchhoff conditions at interior vertices and a fixed Dirichlet-Neumann assignment at exterior vertices. For a prescribed primitive cycle or exterior-to-exterior path, we identify an arithmetic criterion ensuring that the corresponding scar is accessible by exact high-frequency eigenfunctions of a fixed target metric. The criterion is formulated by intersecting the regular scar stratum of the secular set with the orbit-closure torus of the target length vector, and it can hold for resonant metrics beyond the rationally independent class. Under an intrinsic Diophantine condition on the orbit-closure flow, the construction is quantitative: along an exact eigensequence the mass outside the prescribed support decays polynomially, yielding a polynomial degeneration law for finite-frequency observation when that support is uncontrolled. Rationally independent metrics have full phase-orbit closure and therefore make every primitive obstruction accessible. Combining this spectral result with the known graph-theoretic characterization and sufficiency of the Graph Geometric Control Condition (GGCC), we prove that, for every rationally independent and hence for Lebesgue-almost every metric, GGCC is necessary and sufficient for exact controllability at every positive time in the free setting.
Comments20 pages, 1 figure. Submitted
