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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 $$\\|e^{itH}P_{ac}-F_t\\|_{L^1\\to L^\\infty} \\lesssim |t|^{1-\\frac{n}{2}},\\,\\,\\,\\,\\,\\text{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operat"},"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":"1409.6328","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T20:07:00Z","cross_cats_sorted":[],"title_canon_sha256":"a5a75684b6faff91e1865ed656f6bde0bae2a7e7831e2326564ac9e4e56f0cdd","abstract_canon_sha256":"bd4884c09346ed6e7d5a718598373f1c43ba701a6c1bc429b0419bd3c40c612d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:56.972335Z","signature_b64":"9OVVH30VJon0i9Yo1PLEYi6rmxcHx9G4pgJ9WUFadzjt0fX6rEgWRA04c+10SLzGsnWRX6yJJYTWfni10TOgCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40eef2f030b3ac2a68bfdadc91aff0a113f6ac66cb02091367dee3bfd3693997","last_reissued_at":"2026-05-18T00:05:56.971925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:56.971925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dispersive Estimates for higher dimensional Schr\\\"odinger Operators with threshold eigenvalues II: The even 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":"2014-09-22T20:07:00Z","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 in even dimensions $n\\geq 6$. 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