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n\\in \\mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $\\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\\lambda_n^-, \\, \\lambda_n^+ $ (counted with multiplicity).\n  We give estimates for the asymptotics of the spectral gaps $\\gamma_n = \\lambda_n"},"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":"1309.1751","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-09-06T19:41:43Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"dd54478398b1936d56d172b58289557acf6d979b1f2422d6057b409e6b948cea","abstract_canon_sha256":"a7a0386bba91e61f932cf5480fe0f03b15512205254e0e573cac8c7d1d59de9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:58.718096Z","signature_b64":"xal4OWRh7gWCRQZsS/9CfZsKATqCWzzfoDOoaL7VKvjFU6NKuCS4iOPvZD4nNud+01zdu/H/uYTPZa3VjaxOAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"630c066f31b2b9703f8f97cd770be577d1cd09d0bcf48ce6a1f2ba644b385cf8","last_reissued_at":"2026-05-18T03:13:58.717590Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:58.717590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Boris Mityagin, Plamen Djakov","submitted_at":"2013-09-06T19:41:43Z","abstract_excerpt":"Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,\\pi].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \\; 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