On the measure of the spectrum of direct integrals

TL;DR

This paper provides new estimates for the Lebesgue measure of the spectrum of direct integrals of matrix-valued functions, with applications to periodic operators like Jacobi matrices and Schrödinger operators.

Contribution

It introduces sharp spectral bounds for 1D and 2D periodic operators using measure estimates of their spectra.

Findings

New spectral bounds for 1D periodic Jacobi matrices

Sharp estimates for 2D discrete periodic Schrödinger operators

Applicable to a broad class of discrete periodic operators

Abstract

We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example: these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.