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For even $n$ we prove: if $t, n $ are fixed, then, for $ \\alpha \\to 0, $\n  $$ \\gamma_n = | \\frac{8\\alpha^n}{2^n [(n-1)!]^2} \\prod_{k=1}^{n/2}  (t^2 - (2k-1)^2)  |  (1 + O(\\alpha)), $$\n and if $ \\alpha, t $ are fixed, then, for $ n \\to \\infty, $\n  $$ \\gamma_n = \\frac{8 |\\alpha/2|^n}{[2 \\cdot 4 ... (n-2)]^2}  | \\cos (\\frac{\\pi}{2} t) |  [ 1 +"},"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":"math-ph/0509034","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2005-09-16T12:45:32Z","cross_cats_sorted":["math.FA","math.MP"],"title_canon_sha256":"e79b59cfd0d67c35f127a36291c2c2eee910dae36da6c4086de2a7f42641461f","abstract_canon_sha256":"364aa6f231b29ee987010e38a4f0219eff62c66d22d05600bc5c350db247f312"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:30.089538Z","signature_b64":"+j+PaubNlbR2ieHkuDkj7on9hxvauEkkvi0dBtG2qcY8VxrIxuwhG3LJ4dibqzPH4P4NwWhTQHm9xj2z1zbZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bd8209e31a46caa5e5f78e3b001e888d1ff3374c30e636522b35fbd07d99838","last_reissued_at":"2026-05-18T01:05:30.088901Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:30.088901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of instability zones of the Hill operator with a two term potential","license":"","headline":"","cross_cats":["math.FA","math.MP"],"primary_cat":"math-ph","authors_text":"Boris Mityagin, Plamen Djakov","submitted_at":"2005-09-16T12:45:32Z","abstract_excerpt":"Let $\\gamma_n $ denote the length of the $n$-th zone of instability of the Hill operator $Ly= -y^{\\prime \\prime} - [4t\\alpha \\cos2x + 2 \\alpha^2 \\cos 4x ] y,$ where $\\alpha \\neq 0, $ and either both $\\alpha, t $ are real, or both are pure imaginary numbers. For even $n$ we prove: if $t, n $ are fixed, then, for $ \\alpha \\to 0, $\n  $$ \\gamma_n = | \\frac{8\\alpha^n}{2^n [(n-1)!]^2} \\prod_{k=1}^{n/2}  (t^2 - (2k-1)^2)  |  (1 + O(\\alpha)), $$\n and if $ \\alpha, t $ are fixed, then, for $ n \\to \\infty, $\n  $$ \\gamma_n = \\frac{8 |\\alpha/2|^n}{[2 \\cdot 4 ... 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