{"paper":{"title":"A note on sharp spectral estimates for periodic Jacobi matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Anton A. 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 $a_n$, $b_n$. Moreove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06948","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}