Criteria for the Absolutely Continuous Spectral Components of matrix-valued Jacobi operators

Abstract

We extend in this work the Jitomirskaya-Last inequality and Last-Simoncriterion for the absolutely continuous spectral component of a half-line Schrödinger operator to the special class of matrix-valued Jacobi operators $H:l^2(\mathbb{Z},\mathbb{C})\to l^2(\mathbb{Z},\mathbb{C})$ given by the law $[H\text{u}{]}_n:=D_{n-1}{\text{u}}_{n-1}+D_n{\text{u}}_{n+1}+V_n{\text{u}}_n$, where $(D_n{)}_n$ and $(V_n{)}_n$ are bilateral sequences of $l\times l$ self-adjoint matrices such that $0<{\inf }_{n\in \mathbb{Z}}{s}_l[D_n]\le {\sup }_{n\in \mathbb{Z}}{s}_1[D_n]<\infty$ (here, $s_k[A]$ stands for the $k$-th singular value of $A$). 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-Simon originally proven for scalar Schrödinger operators.