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Finally, we estimate $|\\:\\|u(t)\\|_X-\\|e^{it\\Delta}u_\\pm\\|_X\\:|$ under some less restrictive "},"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":"1207.2034","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-09T13:14:28Z","cross_cats_sorted":[],"title_canon_sha256":"a2ecf55f9c0412b44c0fe5ac45bc455ca0a790f0c8cb2edcbd46a20f3dba1ee3","abstract_canon_sha256":"374c23c585e6d3cd5361ca08051abb84dadbac198967f819e166e0aa987748a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:26.230273Z","signature_b64":"r4NTM/ttZus5GxuCiktovJ5loedqZXlvzjYEL6SHKEE7eVfQw3RD2uxfqL2p4I57miC672tRvNDUSPRSdhgpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5398e7d01650312236ae0a0eae08e0b726beddc62cfb378cb028bf634144397","last_reissued_at":"2026-05-18T03:51:26.229547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:26.229547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence to Scattering States in the Nonlinear Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"LJLL), Pascal B\\'egout (IMT","submitted_at":"2012-07-09T13:14:28Z","abstract_excerpt":"In this paper, we consider global solutions of the following nonlinear Schr\\\"odinger equation $iu_t+\\Delta u+\\lambda|u|^\\alpha u = 0,$ in $\\R^N,$ with $\\lambda\\in\\R,$ $\\alpha\\in(0,\\frac{4}{N-2})$ $(\\alpha\\in(0,\\infty)$ if $N=1)$ and \\linebreak $u(0)\\in X\\equiv H^1(\\R^N)\\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-it\\Delta}u(t)-u_ \\pm\\to0$ in $X$ as $t\\to\\pm\\infty$ then $u(t)-e^{it\\Delta}u_\\pm\\to0$ in $X$ as $t\\to\\pm\\infty.$ We also study the converse. Finally, we estimate $|\\:\\|u(t)\\|_X-\\|e^{it\\Delta}u_\\pm\\|_X\\:|$ under some less restrictive "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2034","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":"1207.2034","created_at":"2026-05-18T03:51:26.229668+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.2034v1","created_at":"2026-05-18T03:51:26.229668+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2034","created_at":"2026-05-18T03:51:26.229668+00:00"},{"alias_kind":"pith_short_12","alias_value":"2U4Y47IBMUBR","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"2U4Y47IBMUBREI3K","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"2U4Y47IB","created_at":"2026-05-18T12:26:50.516681+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/2U4Y47IBMUBREI3K4CQOVYEOBN","json":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN.json","graph_json":"https://pith.science/api/pith-number/2U4Y47IBMUBREI3K4CQOVYEOBN/graph.json","events_json":"https://pith.science/api/pith-number/2U4Y47IBMUBREI3K4CQOVYEOBN/events.json","paper":"https://pith.science/paper/2U4Y47IB"},"agent_actions":{"view_html":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN","download_json":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN.json","view_paper":"https://pith.science/paper/2U4Y47IB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.2034&json=true","fetch_graph":"https://pith.science/api/pith-number/2U4Y47IBMUBREI3K4CQOVYEOBN/graph.json","fetch_events":"https://pith.science/api/pith-number/2U4Y47IBMUBREI3K4CQOVYEOBN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN/action/storage_attestation","attest_author":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN/action/author_attestation","sign_citation":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN/action/citation_signature","submit_replication":"https://pith.science/pith/2U4Y47IBMUBREI3K4CQOVYEOBN/action/replication_record"}},"created_at":"2026-05-18T03:51:26.229668+00:00","updated_at":"2026-05-18T03:51:26.229668+00:00"}