Dominated splittings and the spectrum of singular quasi-periodic Jacobi operators

TL;DR

This paper characterizes the resolvent set of singular quasi-periodic Jacobi operators using dominated splittings of associated cocycles, extending known results from Schrödinger operators to a broader class.

Contribution

It generalizes the characterization of the resolvent set via dominated splittings to include singular quasi-periodic Jacobi operators, expanding the scope of spectral analysis.

Findings

Resolvent set characterized by dominated splittings

Extension of Johnson's result to singular Jacobi operators

Broader applicability in spectral theory

Abstract

We prove that the resolvent set of any, possibly singular, quasi periodic Jacobi operator is characterized as the set of all energies whose associated Jacobi cocycles induce a dominated splitting. This extends a well-known result by R. A. Johnson for Schr\"odinger operators.