The parametrization of the Marchenko-Ostrovsky mapping in terms of the Dirichlet eigenvalues

TL;DR

This paper investigates the inverse spectral problem for periodic Jacobi matrices using the Marchenko-Ostrovsky mapping, focusing on Dirichlet eigenvalues and their properties, including linear independence of gradients.

Contribution

It provides a parametrization of the Marchenko-Ostrovsky mapping in terms of Dirichlet eigenvalues and demonstrates the linear independence of their gradients and norming constants.

Findings

Parametrization of the Marchenko-Ostrovsky mapping using Dirichlet eigenvalues.

Proof of linear independence of gradients of Dirichlet eigenvalues and norming constants.

Enhanced understanding of inverse spectral problems for periodic Jacobi matrices.

Abstract

We consider the inverse spectral problem for periodic Jacobi matrices in terms of the vertical slits on the quasi-momentum domain plus the Dirichlet eigenvalues, i.e., the Marchenko-Ostrovsky mapping. Moreover, we show that the gradients of the Dirichlet eigenvalues and of the so-called norming constants are linear independent.