Sharp spectral estimates for periodic matrix-valued Jacobi operators

TL;DR

This paper derives bounds on the spectrum size of periodic matrix-valued Jacobi operators, linking spectral measure to the trace of off-diagonal elements, and estimates spectral band widths.

Contribution

It provides new spectral measure estimates for periodic matrix-valued Jacobi operators, including bounds on the spectrum's Lebesgue measure and spectral band widths.

Findings

Spectrum measure is bounded by four times the minimum trace of off-diagonal matrices.

Spectral band widths are estimated.

Results connect spectral properties with matrix element traces.

Abstract

For the periodic matrix-valued Jacobi operator $J$ we obtain the estimate of the Lebesgue measure of the spectrum $|\s(J)|\le4 \min_n\Tr(a_na_n^*)^\frac12$, where $a_n$ are off-diagonal elements of $J$. Moreover estimates of width of spectral bands are obtained.