{"paper":{"title":"Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Thomas Strohmer, Tim Wertz","submitted_at":"2015-01-24T04:32:06Z","abstract_excerpt":"The almost Mathieu operator is the discrete Schr\\\"odinger operator $H_{\\alpha,\\beta,\\theta}$ on $\\ell^2(\\mathbb{Z})$ defined via $(H_{\\alpha,\\beta,\\theta}f)(k) = f(k + 1) + f(k - 1) + \\beta \\cos(2\\pi \\alpha k + \\theta) f(k)$. We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator. We furthermore show that the (properly rescaled) $m$-th Hermite function $\\phi_m$ is an approximate eigenvector of this operator, and that it satisfies the same properties that characterize the true eigenvector associated to the $m$-th largest ei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06001","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"}