Generalized Pr\"ufer variables for perturbations of Jacobi and CMV matrices

TL;DR

This paper extends the use of generalized Pr"ufer variables to Jacobi and CMV matrices, providing new tools for analyzing spectral properties under various perturbations.

Contribution

It adapts generalized Pr"ufer variables to Jacobi and CMV matrices and demonstrates their application to random and decaying oscillatory perturbations.

Findings

Effective analysis of spectral perturbations using Pr"ufer variables.

Application to random $L^2$ perturbations of Jacobi and CMV matrices.

Insights into decaying oscillatory perturbations of periodic matrices.

Abstract

Pr\"ufer variables are a standard tool in spectral theory, developed originally for perturbations of the free Schr\"odinger operator. They were generalized by Kiselev, Remling, and Simon to perturbations of an arbitrary Schr\"odinger operator. We adapt these generalized Prufer variables to the setting of Jacobi and Szeg\H{o} recursions. We present an application to random $L^2$ perturbations of Jacobi and CMV matrices, and an application to decaying oscillatory perturbations of periodic Jacobi and CMV matrices.