Marchenko-Ostrovski mappings for periodic Jacobi matrices

TL;DR

This paper studies 1D periodic Jacobi matrices, solving the inverse spectral problem using quasimomentum domain slits and providing estimates for these slits based on the matrices.

Contribution

It introduces a novel inverse problem solution for periodic Jacobi matrices using geometric features in the quasimomentum domain.

Findings

Explicit characterization of the inverse spectral problem

Two-sided estimates for vertical slits in the quasimomentum domain

Enhanced understanding of spectral gaps and their relation to matrix parameters

Abstract

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain . Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.