具有对称性量子图上Schro¨dinger算子的分解及谱条件

Decomposition of the Schro¨dinger Operator on SymmetricQuantum Graphs and Spectral Conditions

摘要: 该文分别考虑了 n 次中心对称量子图与笛卡尔积量子图. 根据循环群的不可约表示及两类对称图上平方可积函数空间的分解定理, 给出了定义在两对称结构量子图上 Schrödinger 算子的分解定理. 进一步考虑了一种特殊的 n 次中心对称量子图: 循环图. 在满足 δ 耦合顶点条件下给出了其上定义的 Lapace 算子的谱条件, 为进一步研究两类量子图的谱估计问题打下基础.

Abstract: This paper considers n subcenter-symmetric quantum graphs and Cartesian product graphs. According to the irreducible representations of cyclic groups and the de- composition theorems of the square-integrable function space on these two types of symmetric graphs, it establishes decomposition theorems for the Schrödinger oper- ators defined on the two symmetric quantum graphs. Furthermore, it considers a special class of n subcenter-symmetric quantum graphs: cyclic graphs, and establishes the spectral conditions of the Laplace operator under δ coupling vertex conditions, which lays a foundation for the further study of spectral estimates for these two types of quantum graphs.