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Here $H_{0}$ is the free Dirac operator and $\\zeta\\left( t\\right) $ is such that $\\zeta\\left( t\\right) =1$ for $0\\leq t\\leq1$ and $\\zeta\\left( t\\right) =0$ for $t>1.$ This approach allows us to obtain the time delay operator $\\delta \\mathcal{T}\\left( f\\right) $ for initial states $f$ in $\\mathcal{H} _{2}^{3/2+\\varepsilon"},"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":"1506.08079","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-06-26T13:57:13Z","cross_cats_sorted":["math.MP","quant-ph"],"title_canon_sha256":"699ef06ae8b544348ed979d2bc0f908a4b194a4c19cabb96c3a2671b09c98f45","abstract_canon_sha256":"457f6b16f578e71a507a4db67885b2f57627a6c564e0acc34647ac4dda68fbf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:04:43.762603Z","signature_b64":"NHaEZWO2Crqa7xWbnHI+Ggmqh9BxCiii/kKKr2baJJLXJzQ/gBC5qdBtnAqliV/iPsnGy3fHcx70b2Q6WUXBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a23b0c79a340ca5c1cc30014504e96597c005354de3511b5c6428ffeba43f8cb","last_reissued_at":"2026-07-05T01:04:43.762215Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:04:43.762215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Time delay for the Dirac equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Ivan Naumkin, Ricardo Weder","submitted_at":"2015-06-26T13:57:13Z","abstract_excerpt":"We consider time delay for the Dirac equation. 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