Criteria for the absolutely continuous spectral components of matrix-valued Jacobi operators

Abstract

We extend in this work the Jitomirskaya–Last inequality [Power-law subordinacy and singular spectra i. Half-line operators, Acta Math. 183 (1999) 171–189] and Last and Simon [Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999) 329–367] criterion for the absolutely continuous spectral component of a half-line Schrödinger operator to the special class of matrix-valued Jacobi operators [Formula: see text] given by the law [Formula: see text], where [Formula: see text] are bilateral sequences of [Formula: see text] self-adjoint matrices such that [Formula: see text] (here, [Formula: see text] stands for the [Formula: see text]th singular value of [Formula: see text]). Moreover, we also show that the absolutely continuous components of even multiplicity of minimal dynamically defined matrix-valued Jacobi operators are constant, extending another result from Last and Simon [Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999) 329–367] originally proven for scalar Schrödinger operators.