A note on sharp spectral estimates for periodic Jacobi matrices

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 $$\sum _{i=1}^{p-1}|{\gamma }_i|\ge \max (\max (4(a_1...a_p{)}^{\frac{1}{p}},2\max a_n)-4\min a_n,\max b_n-\min b_n).$$ 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.

References in corpus (1)

• Spectral estimates for periodic Jacobi matrices