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In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim 1/\\log t$ for $t>2$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim t^{-1},\\,\\,\\,\\,\\,\\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero ener"},"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":"1310.6302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T17:30:21Z","cross_cats_sorted":[],"title_canon_sha256":"3ea96a367db5adef697e8375d11aa9106b3beb80c6b596ec1b408e8e297bf5cc","abstract_canon_sha256":"6ddb27fbcb6ef185e220c5cf5b8b7ef3d85c1b941fb30b4c60bfc0214d0084a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:06.108509Z","signature_b64":"TWHT7tEzA/Qz/tAjRBuKFdQaTgz9j8T64q+F+mw9W7Da0jHwVMFHgR4FdcVTySCEubywR72Np1l4fUBZa8KPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"072249d3adfb644278d9a70ea9581566c2261c0acdf36de94b5a69e44f3b3525","last_reissued_at":"2026-05-18T02:42:06.107895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:06.107895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dispersive estimates for four dimensional Schr\\\"{o}dinger and wave equations with obstructions at zero energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M. Burak Erdogan, Michael Goldberg, William R. Green","submitted_at":"2013-10-23T17:30:21Z","abstract_excerpt":"We investigate $L^1(\\mathbb R^4)\\to L^\\infty(\\mathbb R^4)$ dispersive estimates for the Schr\\\"odinger operator $H=-\\Delta+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim 1/\\log t$ for $t>2$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim t^{-1},\\,\\,\\,\\,\\,\\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero ener"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6302","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":"1310.6302","created_at":"2026-05-18T02:42:06.107973+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.6302v1","created_at":"2026-05-18T02:42:06.107973+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6302","created_at":"2026-05-18T02:42:06.107973+00:00"},{"alias_kind":"pith_short_12","alias_value":"A4RETU5N7NSE","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"A4RETU5N7NSEE6GZ","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"A4RETU5N","created_at":"2026-05-18T12:27:38.830355+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/A4RETU5N7NSEE6GZU4HKSWAVM3","json":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3.json","graph_json":"https://pith.science/api/pith-number/A4RETU5N7NSEE6GZU4HKSWAVM3/graph.json","events_json":"https://pith.science/api/pith-number/A4RETU5N7NSEE6GZU4HKSWAVM3/events.json","paper":"https://pith.science/paper/A4RETU5N"},"agent_actions":{"view_html":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3","download_json":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3.json","view_paper":"https://pith.science/paper/A4RETU5N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.6302&json=true","fetch_graph":"https://pith.science/api/pith-number/A4RETU5N7NSEE6GZU4HKSWAVM3/graph.json","fetch_events":"https://pith.science/api/pith-number/A4RETU5N7NSEE6GZU4HKSWAVM3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3/action/storage_attestation","attest_author":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3/action/author_attestation","sign_citation":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3/action/citation_signature","submit_replication":"https://pith.science/pith/A4RETU5N7NSEE6GZU4HKSWAVM3/action/replication_record"}},"created_at":"2026-05-18T02:42:06.107973+00:00","updated_at":"2026-05-18T02:42:06.107973+00:00"}