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We prove that if zero is a regular point of the spectrum of $H=-\\Delta+V$, then $$ \\|w^{-1} e^{itH}P_{ac}f\\|_{L^\\infty(\\R^2)}\\les \\f1{|t|\\log^2(|t|)} \\|w f\\|_{L^1(\\R^2)},  |t| >2, $$ with $w(x)=\\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights."},"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":"1202.0050","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T23:32:59Z","cross_cats_sorted":[],"title_canon_sha256":"a509ebfd838cb50dd62038ef75f63522df62b50c206d61bfdd6811b090450438","abstract_canon_sha256":"f7eed57c0ca51a7625a00befa1cb34e710c16f1c41debdbc93231875eef8b998"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:11.777208Z","signature_b64":"qBLt/OBownUBTp0pSf2avYCMB3m4SO6hpcns7UI1EOS8Ymc0SBBN+OFSFpjPYJHiSJWGBsmF6jYX8dA+0jQIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7338ae920af0f33b5215a753f32c5d1ec1ca5bf45ef2860991837a0cb0b355a","last_reissued_at":"2026-05-18T03:19:11.776504Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:11.776504Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A weighted dispersive estimate for 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":"2012-01-31T23:32:59Z","abstract_excerpt":"Let $H=-\\Delta+V$, where $V$ is a real valued potential on $\\R^2$ satisfying $|V(x)|\\les \\la x\\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H=-\\Delta+V$, then $$ \\|w^{-1} e^{itH}P_{ac}f\\|_{L^\\infty(\\R^2)}\\les \\f1{|t|\\log^2(|t|)} \\|w f\\|_{L^1(\\R^2)},  |t| >2, $$ with $w(x)=\\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0050","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":"1202.0050","created_at":"2026-05-18T03:19:11.776625+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.0050v1","created_at":"2026-05-18T03:19:11.776625+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0050","created_at":"2026-05-18T03:19:11.776625+00:00"},{"alias_kind":"pith_short_12","alias_value":"U4ZYV2JAV4HT","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"U4ZYV2JAV4HTHNJB","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"U4ZYV2JA","created_at":"2026-05-18T12:27:23.164592+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/U4ZYV2JAV4HTHNJBLJ2T6MWF2H","json":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H.json","graph_json":"https://pith.science/api/pith-number/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/graph.json","events_json":"https://pith.science/api/pith-number/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/events.json","paper":"https://pith.science/paper/U4ZYV2JA"},"agent_actions":{"view_html":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H","download_json":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H.json","view_paper":"https://pith.science/paper/U4ZYV2JA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.0050&json=true","fetch_graph":"https://pith.science/api/pith-number/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/graph.json","fetch_events":"https://pith.science/api/pith-number/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/action/storage_attestation","attest_author":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/action/author_attestation","sign_citation":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/action/citation_signature","submit_replication":"https://pith.science/pith/U4ZYV2JAV4HTHNJBLJ2T6MWF2H/action/replication_record"}},"created_at":"2026-05-18T03:19:11.776625+00:00","updated_at":"2026-05-18T03:19:11.776625+00:00"}