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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.06001","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-24T04:32:06Z","cross_cats_sorted":[],"title_canon_sha256":"21ca701a5d3685d8ead047d1c153324c2650bfc31a57df52e025c9013bc93613","abstract_canon_sha256":"e951f2a0ffeb9daae53163a0d3bbfd3b7a7e3af3e582d100a298596e2d928dd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:47.092377Z","signature_b64":"NFETQuDOPOxcqRBaScf8wu9eAiGGhlS7GVrXjaY2TWV4xB6O7Ir8qOY2Tq0GD6Hlo0MCYxafHwk8z/5e3TdxDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9681696f12885358cf83c83b079b9451f585a55ccbd94ca67df2f342b30066d","last_reissued_at":"2026-05-18T02:28:47.092005Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:47.092005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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