{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:CZMBZIQL4YNOYWHGO6CEWDKTNI","short_pith_number":"pith:CZMBZIQL","schema_version":"1.0","canonical_sha256":"16581ca20be61aec58e677844b0d536a3c96ebed7edf8683bacbf5ae2e11f0cb","source":{"kind":"arxiv","id":"1706.02549","version":1},"attestation_state":"computed","paper":{"title":"Sharp Criteria of Scattering for the Fractional NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo, Shihui Zhu","submitted_at":"2017-06-07T01:11:05Z","abstract_excerpt":"In this paper, the sharp threshold of scattering for the fractional nonlinear Schr\\\"{o}dinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\\frac{4s}{N}<p<1+\\frac{4s}{N-2s}$, and $$ M[u_{0}]^{\\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\\frac{s-s_c}{s_c}}E[Q], \\ M[u_{0}]^{\\frac{s-s_c}{s_c}}\\| u_{0}\\|^2_{\\dot H^s}<M[Q]^{\\frac{s-s_c}{s_c}}\\| Q\\|^2_{\\dot H^s}$$ then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that if $1+\\frac{4s}{N}<p<1+\\frac{4s}{N-2s}$ and $$ M[u_{0}]^{\\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\\frac{s-s_c}{s_c}}E[Q], \\ M[u_{0}]^{\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.02549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-07T01:11:05Z","cross_cats_sorted":[],"title_canon_sha256":"065b23b30ec9478a095a69219c92db7a24dc5d650d235533ccdf3acac4a5a6e2","abstract_canon_sha256":"1b561171d79b33ae195bc4eda5a727e66a84a6b2ef7205455e1694b2335c587e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:45.179416Z","signature_b64":"63Rbbxy7JHbIUuw4qL8q+t8cMJv6cYPmxKvsn2t0TU7NPmYqJ9gnHdWKCg1JVQmOrokjexNtxXlKaD6QHP1SCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16581ca20be61aec58e677844b0d536a3c96ebed7edf8683bacbf5ae2e11f0cb","last_reissued_at":"2026-05-18T00:42:45.178764Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:45.178764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Criteria of Scattering for the Fractional NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo, Shihui Zhu","submitted_at":"2017-06-07T01:11:05Z","abstract_excerpt":"In this paper, the sharp threshold of scattering for the fractional nonlinear Schr\\\"{o}dinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\\frac{4s}{N}<p<1+\\frac{4s}{N-2s}$, and $$ M[u_{0}]^{\\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\\frac{s-s_c}{s_c}}E[Q], \\ M[u_{0}]^{\\frac{s-s_c}{s_c}}\\| u_{0}\\|^2_{\\dot H^s}<M[Q]^{\\frac{s-s_c}{s_c}}\\| Q\\|^2_{\\dot H^s}$$ then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that if $1+\\frac{4s}{N}<p<1+\\frac{4s}{N-2s}$ and $$ M[u_{0}]^{\\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\\frac{s-s_c}{s_c}}E[Q], \\ M[u_{0}]^{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02549","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":"1706.02549","created_at":"2026-05-18T00:42:45.178861+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.02549v1","created_at":"2026-05-18T00:42:45.178861+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.02549","created_at":"2026-05-18T00:42:45.178861+00:00"},{"alias_kind":"pith_short_12","alias_value":"CZMBZIQL4YNO","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CZMBZIQL4YNOYWHG","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CZMBZIQL","created_at":"2026-05-18T12:31:10.602751+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/CZMBZIQL4YNOYWHGO6CEWDKTNI","json":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI.json","graph_json":"https://pith.science/api/pith-number/CZMBZIQL4YNOYWHGO6CEWDKTNI/graph.json","events_json":"https://pith.science/api/pith-number/CZMBZIQL4YNOYWHGO6CEWDKTNI/events.json","paper":"https://pith.science/paper/CZMBZIQL"},"agent_actions":{"view_html":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI","download_json":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI.json","view_paper":"https://pith.science/paper/CZMBZIQL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.02549&json=true","fetch_graph":"https://pith.science/api/pith-number/CZMBZIQL4YNOYWHGO6CEWDKTNI/graph.json","fetch_events":"https://pith.science/api/pith-number/CZMBZIQL4YNOYWHGO6CEWDKTNI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI/action/storage_attestation","attest_author":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI/action/author_attestation","sign_citation":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI/action/citation_signature","submit_replication":"https://pith.science/pith/CZMBZIQL4YNOYWHGO6CEWDKTNI/action/replication_record"}},"created_at":"2026-05-18T00:42:45.178861+00:00","updated_at":"2026-05-18T00:42:45.178861+00:00"}