A note on sharp spectral estimates for periodic Jacobi matrices

TL;DR

This paper derives sharp spectral estimates for the gaps in the spectrum of periodic Jacobi matrices, showing the bounds are optimal for various oscillation regimes and improving previous results.

Contribution

It provides new sharp bounds on spectral gaps of periodic Jacobi matrices and demonstrates their optimality across different oscillation conditions.

Findings

The spectral gap sum estimate is sharp for minimal period Jacobi matrices.

The estimate applies to both strongly and weakly oscillated coefficients.

It improves upon recent existing spectral estimates.

Abstract

The spectrum of three-diagonal self-adjoint $p$-periodic Jacobi matrix with positive off-diagonal elements $a_n$ an real diagonal elements $b_n$ consist of intervals separated by $p-1$ gaps $\gamma_i$, where some of the gaps can be degenerated. The following estimate is true $$ \sum_{i=1}^{p-1}|\gamma_i|\geq\max(\max(4(a_1...a_p)^{\frac1p},2\max a_n)-4\min a_n,\max b_n-\min b_n). $$ We show that for any $p\in\mathbb{N}$ there are Jacobi matrices of minimal period $p$ for which the spectral estimate is sharp. The estimate is sharp for both: strongly and weakly oscillated $a_n$, $b_n$. Moreover, it improves some recent spectral estimates.