Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

TL;DR

This paper develops a spectral decimation method for analyzing the spectra of piecewise centrosymmetric Jacobi operators on graphs, linking their spectral properties to orthogonal polynomials and Laplacians.

Contribution

It introduces a novel approach to relate the spectral theory of these operators to probabilistic Laplacians via explicit constructions and orthogonal polynomials.

Findings

Spectral theory can be explicitly related to orthogonal polynomials.

Constructed new graphs and operators via edge substitution.

Provided examples of self-similar Jacobi matrices fitting the framework.

Abstract

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.