On the correspondence between domination and the spectrum of Jacobi operators

Abstract

In this paper, we first develop a notion of dominated splitting for M ( 2 , C ) \mathbb {M}(2,\mathbb {C}) -sequences and show it is a stable property under ‖ ⋅ ‖ ∞ \|\cdot \|_\infty -perturbation. Then we show an energy parameter belongs to the spectrum of a Jacobi operator, possibly singular, if and only if the associated Jacobi cocycle does not admit dominated splitting. This generalizes the results obtained by the second author [J. Spectr. Theory 10 (2020), pp. 1471–1517] in the scenario of Schrödinger operators. Finally, we consider dynamically defined Jacobi operators whose base dynamics is only assumed to be topologically transitive. We show an energy parameter belongs to the spectrum of the operator defined by the base point with a dense orbit if and only if the dynamically defined Jacobi cocycle does not admit dominated splitting. This includes the original Johnson’s theorem obtained by R. Johnson [J. Differential Equations 61 (1986), pp. 54–78] for Schrödinger operators and the main theorem obtained by C. Marx [Nonlinearity 27 (2014), pp. 3059–3072] for Jacobi operators as special cases.