On stability in the Borg--Hochstadt theorem for periodic Jacobi matrices

TL;DR

This paper provides a quantitative extension of the Borg--Hochstadt theorem for periodic Jacobi matrices, establishing bounds relating spectral gaps to oscillations in matrix entries.

Contribution

It introduces two-sided bounds linking the oscillations of matrix diagonals with the size of spectral gaps, advancing understanding of spectral stability.

Findings

Established bounds between diagonal oscillations and spectral gaps.

Quantified stability conditions for periodic Jacobi matrices.

Extended classical theorem with explicit quantitative relations.

Abstract

A result of Borg--Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving the two-sided bounds between oscillations of the matrix entries along the diagonals and the length of the maximal gap in the spectrum.