{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:EYBA2VR6CNFE6TNTY6UGBR6GJD","short_pith_number":"pith:EYBA2VR6","schema_version":"1.0","canonical_sha256":"26020d563e134a4f4db3c7a860c7c648efa1b9f481c66165ba90727fd6ec77ff","source":{"kind":"arxiv","id":"1211.4036","version":1},"attestation_state":"computed","paper":{"title":"Dispersive estimates for matrix Schr\\\"{o}dinger operators in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M. Burak Erdo\\u{g}an, William R. Green","submitted_at":"2012-11-16T21:01:11Z","abstract_excerpt":"We consider the non-selfadjoint operator [\\cH = [{array}{cc} -\\Delta + \\mu-V_1 & -V_2 V_2 & \\Delta - \\mu + V_1 {array}]] where $\\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\\R^2)\\times L^1(\\R^2)\\to L^\\infty(\\R^2)\\times L^\\infty(\\R^2)$ dispersive decay estimates for the evolution $e^{it\\cH}P_{ac}$. We also obtain the following weighted estimate $$ \\|w^{-1} e^{it\\cH}P_{ac}f\\|_{L^\\infty(\\R^2)\\times L^\\infty(\\R^2)}\\les \\f1{|t|\\log^2(|t|)} \\|w f\\|_{L^1"},"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":"1211.4036","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-16T21:01:11Z","cross_cats_sorted":[],"title_canon_sha256":"5a39b8bfd9094923edd9ac725db4d96e9a2cfd6c831e3e38a0eb83f851bfd3ba","abstract_canon_sha256":"9983e13bcbf4e19962cc6ba4b5f6e9ec51e41d7a727805ce678b5087e5e92435"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:04.699972Z","signature_b64":"c1/PkJ2cVUKz3q9u4j9D+F6KGKnRa0ZQeRo6+cfpMnljWSc0C5BD9xTCGZclAiXDq+NVGqljo7bk0Vc49dGVAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26020d563e134a4f4db3c7a860c7c648efa1b9f481c66165ba90727fd6ec77ff","last_reissued_at":"2026-05-18T03:19:04.699168Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:04.699168Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dispersive estimates for matrix Schr\\\"{o}dinger operators in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M. Burak Erdo\\u{g}an, William R. Green","submitted_at":"2012-11-16T21:01:11Z","abstract_excerpt":"We consider the non-selfadjoint operator [\\cH = [{array}{cc} -\\Delta + \\mu-V_1 & -V_2 V_2 & \\Delta - \\mu + V_1 {array}]] where $\\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\\R^2)\\times L^1(\\R^2)\\to L^\\infty(\\R^2)\\times L^\\infty(\\R^2)$ dispersive decay estimates for the evolution $e^{it\\cH}P_{ac}$. We also obtain the following weighted estimate $$ \\|w^{-1} e^{it\\cH}P_{ac}f\\|_{L^\\infty(\\R^2)\\times L^\\infty(\\R^2)}\\les \\f1{|t|\\log^2(|t|)} \\|w f\\|_{L^1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4036","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":"1211.4036","created_at":"2026-05-18T03:19:04.699338+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.4036v1","created_at":"2026-05-18T03:19:04.699338+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4036","created_at":"2026-05-18T03:19:04.699338+00:00"},{"alias_kind":"pith_short_12","alias_value":"EYBA2VR6CNFE","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"EYBA2VR6CNFE6TNT","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"EYBA2VR6","created_at":"2026-05-18T12:27:04.183437+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/EYBA2VR6CNFE6TNTY6UGBR6GJD","json":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD.json","graph_json":"https://pith.science/api/pith-number/EYBA2VR6CNFE6TNTY6UGBR6GJD/graph.json","events_json":"https://pith.science/api/pith-number/EYBA2VR6CNFE6TNTY6UGBR6GJD/events.json","paper":"https://pith.science/paper/EYBA2VR6"},"agent_actions":{"view_html":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD","download_json":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD.json","view_paper":"https://pith.science/paper/EYBA2VR6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.4036&json=true","fetch_graph":"https://pith.science/api/pith-number/EYBA2VR6CNFE6TNTY6UGBR6GJD/graph.json","fetch_events":"https://pith.science/api/pith-number/EYBA2VR6CNFE6TNTY6UGBR6GJD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD/action/storage_attestation","attest_author":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD/action/author_attestation","sign_citation":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD/action/citation_signature","submit_replication":"https://pith.science/pith/EYBA2VR6CNFE6TNTY6UGBR6GJD/action/replication_record"}},"created_at":"2026-05-18T03:19:04.699338+00:00","updated_at":"2026-05-18T03:19:04.699338+00:00"}