{"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 ... (n-2)]^2}  | \\cos (\\frac{\\pi}{2} t) |  [ 1 +"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0509034","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"}