Opening Gaps in the Spectrum of Strictly Ergodic Jacobi and CMV Matrices

TL;DR

This paper proves that for generic continuous and analytic sampling functions, all spectral gaps predicted by the Gap Labelling Theorem are open in certain classes of Jacobi and CMV matrices, advancing understanding of their spectral properties.

Contribution

It establishes the generic openness of spectral gaps for Jacobi and CMV matrices with continuous and analytic sampling functions, respectively, providing new mechanisms for gap opening.

Findings

All predicted spectral gaps are open for generic continuous sampling functions.

Analytic resonance tongue boundaries facilitate gap opening in the subcritical region.

The results apply to both Jacobi and CMV matrices, broadening spectral theory insights.

Abstract

We prove that dynamically defined Jacobi and CMV matrices associated with generic continuous sampling functions have all gaps predicted by the Gap Labelling Theorem open. We also give a mechanism for generic gap opening for quasi-periodic analytic sampling functions in the subcritical region following from the analyticity of resonance tongue boundaries for both Jacobi and CMV matrices.