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The following estimate is true $$\n  \\sum_{i=1}^{p-1}|\\gamma_i|\\geq\\max(\\max(4(a_1...a_p)^{\\frac1p},2\\max a_n)-4\\min a_n,\\max b_n-\\min b_n). $$ 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 $a_n$, $b_n$. 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Kutsenko","submitted_at":"2018-10-16T12:18:21Z","abstract_excerpt":"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 $$\n  \\sum_{i=1}^{p-1}|\\gamma_i|\\geq\\max(\\max(4(a_1...a_p)^{\\frac1p},2\\max a_n)-4\\min a_n,\\max b_n-\\min b_n). $$ 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 $a_n$, $b_n$. 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