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We find the ionization probability in the limit $t\\to\\infty$ for all $\\lambda$ and $\\omega$. The long pulse limit is very singular, and, for $\\omega=0$, the survival probability is $const \\lambda^{1/3}$, much larger than $O(\\lambda)$, the one in the abrupt transition counterpart, $V(x,t)=\\delta(x)\\mathbf{1}_{\\{t\\ge 1/\\lambda\\}}$ where $\\"},"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":"0901.0724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-01-06T21:41:43Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"013222e642aff1fe7ff46bfe6b9de78920a71c5af52e7f51f218ae8387c9400c","abstract_canon_sha256":"cefea8199bc87c1850c62908a1b23f9fd0c7031d5311f2bf132e52d2c3a9b1ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:03.190524Z","signature_b64":"wD4EBXySI1do+mUb5SyktNc4LMs38JfMa5E8bnhHW2uNWiJ0jRk15W633SRKoVMMkdTjghNwd7rp9LzPJ4SnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c64ebfd9b665390892d3025b4b82f0206a34d8825bc2eb25d21a1d7c02fc63b","last_reissued_at":"2026-05-18T02:15:03.190094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:03.190094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ionization in damped time-harmonic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"M. Huang, O. Costin, Z. Qiu","submitted_at":"2009-01-06T21:41:43Z","abstract_excerpt":"We study the asymptotic behavior of the wave function in a simple one dimensional model of ionization by pulses, in which the time-dependent potential is of the form $V(x,t)=-2\\delta(x)(1-e^{-\\lambda t} \\cos\\omega t)$, where $\\delta$ is the Dirac distribution. We find the ionization probability in the limit $t\\to\\infty$ for all $\\lambda$ and $\\omega$. The long pulse limit is very singular, and, for $\\omega=0$, the survival probability is $const \\lambda^{1/3}$, much larger than $O(\\lambda)$, the one in the abrupt transition counterpart, $V(x,t)=\\delta(x)\\mathbf{1}_{\\{t\\ge 1/\\lambda\\}}$ where $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0724","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0901.0724","created_at":"2026-05-18T02:15:03.190156+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.0724v1","created_at":"2026-05-18T02:15:03.190156+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0724","created_at":"2026-05-18T02:15:03.190156+00:00"},{"alias_kind":"pith_short_12","alias_value":"FRSOX7M3MZJZ","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"FRSOX7M3MZJZBCJN","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"FRSOX7M3","created_at":"2026-05-18T12:25:59.703012+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI","json":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI.json","graph_json":"https://pith.science/api/pith-number/FRSOX7M3MZJZBCJNGAS3JOBPAI/graph.json","events_json":"https://pith.science/api/pith-number/FRSOX7M3MZJZBCJNGAS3JOBPAI/events.json","paper":"https://pith.science/paper/FRSOX7M3"},"agent_actions":{"view_html":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI","download_json":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI.json","view_paper":"https://pith.science/paper/FRSOX7M3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.0724&json=true","fetch_graph":"https://pith.science/api/pith-number/FRSOX7M3MZJZBCJNGAS3JOBPAI/graph.json","fetch_events":"https://pith.science/api/pith-number/FRSOX7M3MZJZBCJNGAS3JOBPAI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI/action/storage_attestation","attest_author":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI/action/author_attestation","sign_citation":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI/action/citation_signature","submit_replication":"https://pith.science/pith/FRSOX7M3MZJZBCJNGAS3JOBPAI/action/replication_record"}},"created_at":"2026-05-18T02:15:03.190156+00:00","updated_at":"2026-05-18T02:15:03.190156+00:00"}