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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