扩展非局部星型图上单点相互作用问题的拓扑及其特征值
Topology in One Point Interaction Problem on Extended Non-Local Star Graphs and its Eigenvalues
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中文总结 AI 辅助
该研究针对星型度量图上带非局部边界条件的Sturm-Liouville算子，通过构造特殊解与非局部特征函数，分析其逆谱问题及拓扑恢复的关键条件，证明系统在必要条件下可解并探寻非平凡非局部特征值。
中文摘要 AI 辅助
作者研究星型度量图上Sturm-Liouville算子的逆谱问题。该星型图的顶点附有m条边，这些边施加了满足适当非局部边界条件的非局部Sturm-Liouville算子。在该顶点处，作者考虑单点相互作用条件，以建模固定在图边端的度量图，该模型用于表示随时间变化的振动或通量，这些振动或通量在作为控制/调节中心的顶点处被监测。作者证明该系统在必要条件下可解，而恢复给定拓扑的网络/度量图的拓扑至关重要。分析伊始，构造固定在边端且在顶点处保持连续的特殊解，该解对应于弦根据特定频率在顶点处垂直振动的模型。非局部特征函数发挥作用，随后尝试寻找非平凡的非局部特征值。
英文摘要
The author studies the inverse spectral problem of Sturm-Liouville operator on a star-like metric graph. At the vertex of this star-like graph, there are attached $m$ edges that imposed with non-local Sturm-Liouville operator satisfying some suitable non-local boundary conditions. At the vertex, we consider one point interaction condition to model a metric graph that fixed on the end of edges of the graph. This models the vibration or flux that changes over time that monitored at the vertex which serves as certain control/regulation center. The author shows that the system is solvable under very necessary conditions. It is crucial to recover the topology of the network/metric graph which the topology is given. To begin the analysis, one constructs the special solution fixed on one end of edges while maintaining continuous at the vertex. This models a string that is vibrating vertically at the vertex according to certain frequencies. The non-local characteristic function plays a role, and then, one tries to find a non-trivial non-local eigenvalue.
