### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 050, 14 pages      arXiv:1809.07136      https://doi.org/10.3842/SIGMA.2019.050

### On Direct Integral Expansion for Periodic Block-Operator Jacobi Matrices and Applications

Leonid Golinskii a and Anton Kutsenko b
a) B. Verkin Institute for Low Temperature Physics and Engineering, 47 Science Ave., Kharkiv 61103, Ukraine
b) Jacobs University, Campus Ring 1, 28759 Bremen, Germany

Received December 03, 2018, in final form June 23, 2019; Published online July 02, 2019

Abstract
We construct a functional model (direct integral expansion) and study the spectra of certain periodic block-operator Jacobi matrices, in particular, of general 2D partial difference operators of the second order. We obtain the upper bound, optimal in a sense, for the Lebesgue measure of their spectra. The examples of the operators for which there are several gaps in the spectrum are given.

Key words: functional model; block Jacobi matrices; partial difference operators; periodicity; spectrum.

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