论文标题
分段中心对称雅各比操作员的光谱拆卸图
Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs
论文作者
论文摘要
我们研究了在相关的替代图家族中定义的一类分段中心对称雅各比运算符的光谱理论。给定有限的中心对称矩阵，该矩阵被视为有限的有向路径图上的重量矩阵，而概率的laplacian在局部有限连接的图形上被视为重量矩阵，我们构造了一个新的图形和一个新的操作员，由边缘替代。我们的主要结果证明，使用某些正交多项式的分段中心对称雅各比操作员的光谱理论可以明确与概率拉普拉斯的光谱理论相关。我们的主要工具涉及所谓的光谱分解，从分形分析以及经典的Schur补体中得知。我们包括一些适合我们框架的自相似雅各比矩阵的示例。
We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we construct a new graph and a new operator by edge substitution. Our main result proves that the spectral theory of the piecewise centrosymmetric Jacobi operator can be explicitly related to the spectral theory of the probabilistic Laplacian using certain orthogonal polynomials. Our main tools involve the so-called spectral decimation, known from the analysis on fractals, and the classical Schur complement. We include several examples of self-similar Jacobi matrices that fit into our framework.
