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

TL;DR

This paper extends spectral criteria for matrix-valued Jacobi operators, generalizing known inequalities and showing constancy of certain spectral components, thereby advancing understanding of their absolutely continuous spectrum.

Contribution

It generalizes the Jitomirskaya-Last inequality and Last-Simon criterion to matrix-valued Jacobi operators and proves the constancy of their absolutely continuous spectral components.

Findings

Extended spectral criteria to matrix-valued Jacobi operators.

Proved the constancy of absolutely continuous spectral components of even multiplicity.

Generalized key inequalities from scalar to matrix-valued operators.

Abstract

We extend in this work the Jitomirskaya-Last inequality and Last-Simoncriterion for the absolutely continuous spectral component of a half-line Schr\"odinger operator to the special class of matrix-valued Jacobi operators $H:l^2(\mathbb{Z},\mathbb{C})\rightarrow l^2(\mathbb{Z},\mathbb{C})$ given by the law $[H \textbf{u}]_{n} := D_{n - 1} \textbf{u}_{n - 1} + D_{n} \textbf{u}_{n + 1} + V_{n} \textbf{u}_{n}$, 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 Schr\"odinger operators.