量子图上的拉普拉斯算子
Laplace operators on quantum graphs
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中文总结 AI 辅助
研究在算子系统框架下引入量子图的拉普拉斯算子，通过确定特定内积定义其量子类似物，与替代定义比较并研究相关拉普拉斯算子，将谱图理论扩展到算子代数设置，迈向量子图谱理论。
中文摘要 AI 辅助
我们在算子系统框架中引入了一类与量子图相关的拉普拉斯算子。为此，我们研究了与舒尔积和转置相关的量子图的结构性质。我们的主要创新是在矩阵空间中确定了一种特定的内积，相对于该内积，量子图基础算子系统上的投影是正交的。这使我们能够定义经典关联算子和相关拉普拉斯算子的量子类似物。我们将此构造与最近提出的替代定义进行比较，并研究了几个量子图族的所得拉普拉斯算子。这些结果将谱图理论的基本构造扩展到算子代数设置，朝着量子图的谱理论迈出了一步，该理论在揭示与量子信息理论和非交换几何的新联系的同时推广了经典图论概念。
英文摘要
We introduce a class of Laplace operators associated with quantum graphs in the operator-system framework. To this end, we investigate structural properties of quantum graphs related to the Schur product and transposition. Our main innovation is the identification of a specific inner product on the space of matrices with respect to which the projection onto the operator system underlying a quantum graph is orthogonal. This enables us to define a quantum analogue of the classical incidence operator and the associated Laplace operator. We compare this construction with a recently proposed alternative definition and examine the resulting Laplace operators for several families of quantum graphs. These results extend fundamental constructions from spectral graph theory to the operator-algebraic setting, providing a step toward a spectral theory of quantum graphs that generalizes classical graph-theoretic concepts while revealing new connections with quantum information theory and noncommutative geometry.
