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

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 Schrödinger operators.

We correct an incorrect statement in Remark 1.2 (ii) of the published paper. The only other essential changes appear in the proof of Proposition 5.1 where we realized that in equation (5.5), a mere upper bound is sufficient for the proving the claim of the proposition. This implies that the main result of the paper extends to all almost periodic Jacobi operators, irrespective of the base dynamics

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• Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
• Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts
• Spectral Dimension for $\beta$-almost periodic singular Jacobi operators and the extended Harper's model