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In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{2-\\frac{n}{2}}$ for $|t|>1$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{1-\\frac{n}{2}},\\qquad\\textrm{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued "},"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":"1409.6323","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T20:02:21Z","cross_cats_sorted":[],"title_canon_sha256":"be8608a64713ae040b37f27f592c9c85fa3b80aaf3ef3daccada4b1e88570f7d","abstract_canon_sha256":"d5fc1f87855dbfc1d80ac6c3bf367d470a2d79932845b84ec2d71ebf0438c444"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:21.343186Z","signature_b64":"ryIlhjB97zeZ64/pRUp1pFf2LKqWI0HbDcqJv9o7uZIgHJl/6jhzOYJ/bRJjYRZCJllRq+q3Ezvfcq5wCb1jCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6925bc629eb824c270fe14c8325d139c6dfae64fb60c14b00294a3b8097278a","last_reissued_at":"2026-05-18T01:07:21.342523Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:21.342523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dispersive Estimates for higher dimensional Schr\\\"odinger Operators with threshold eigenvalues I: The odd dimensional case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Goldberg, William R. Green","submitted_at":"2014-09-22T20:02:21Z","abstract_excerpt":"We investigate $L^1(\\mathbb R^n)\\to L^\\infty(\\mathbb R^n)$ dispersive estimates for the Schr\\\"odinger operator $H=-\\Delta+V$ when there is an eigenvalue at zero energy and $n\\geq 5$ is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\\|F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{2-\\frac{n}{2}}$ for $|t|>1$ such that $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{1-\\frac{n}{2}},\\qquad\\textrm{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6323","kind":"arxiv","version":3},"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":"1409.6323","created_at":"2026-05-18T01:07:21.342621+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.6323v3","created_at":"2026-05-18T01:07:21.342621+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.6323","created_at":"2026-05-18T01:07:21.342621+00:00"},{"alias_kind":"pith_short_12","alias_value":"62JFXRRJ5OBE","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"62JFXRRJ5OBEYJYP","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"62JFXRRJ","created_at":"2026-05-18T12:28:16.859392+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/62JFXRRJ5OBEYJYP4FGIGJORHH","json":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH.json","graph_json":"https://pith.science/api/pith-number/62JFXRRJ5OBEYJYP4FGIGJORHH/graph.json","events_json":"https://pith.science/api/pith-number/62JFXRRJ5OBEYJYP4FGIGJORHH/events.json","paper":"https://pith.science/paper/62JFXRRJ"},"agent_actions":{"view_html":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH","download_json":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH.json","view_paper":"https://pith.science/paper/62JFXRRJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.6323&json=true","fetch_graph":"https://pith.science/api/pith-number/62JFXRRJ5OBEYJYP4FGIGJORHH/graph.json","fetch_events":"https://pith.science/api/pith-number/62JFXRRJ5OBEYJYP4FGIGJORHH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH/action/storage_attestation","attest_author":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH/action/author_attestation","sign_citation":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH/action/citation_signature","submit_replication":"https://pith.science/pith/62JFXRRJ5OBEYJYP4FGIGJORHH/action/replication_record"}},"created_at":"2026-05-18T01:07:21.342621+00:00","updated_at":"2026-05-18T01:07:21.342621+00:00"}