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If $V$ has sufficient pointwise decay, the wave operators $W_{\\pm}=s-\\lim_{t\\to \\pm\\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\\mathbb R^4)$ for all $1\\leq p\\leq \\infty$ if zero is not an eigenvalue or resonance, and on $\\frac43<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p(\\mathbb R^4)$ for $1\\leq p\\leq \\frac43$ by direct examination of the integral kernel of the leading terms"},"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":"1606.06691","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-21T18:14:13Z","cross_cats_sorted":[],"title_canon_sha256":"45d4be544c4b2ea035e07d24e78354c581d590b481becead869b3b01ad95ed0d","abstract_canon_sha256":"d754769bab6beafaba995857d46fb2ba8072a92d1696c77f358d90d8778f4032"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:56.846174Z","signature_b64":"i1P4/lj3ejZWOy8N+UFicgLkzRQHqNnU6Elsy6KMA50XP3v1UpoDbBgu3WtCjYan4ET/FKig3IW2fQunoYJyDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36b55d6304b6d59954b49a65803857beab36e7cc3ddfe62865ee7703061a0c85","last_reissued_at":"2026-05-18T00:05:56.845706Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:56.845706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $L^p$ boundedness of wave operators for four-dimensional Schr\\\"odinger Operators with a threshold eigenvalue","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":"2016-06-21T18:14:13Z","abstract_excerpt":"Let $H=-\\Delta+V$ be a Schr\\\"odinger operator on $L^2(\\mathbb R^4)$ with real-valued potential $V$, and let $H_0=-\\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\\pm}=s-\\lim_{t\\to \\pm\\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\\mathbb R^4)$ for all $1\\leq p\\leq \\infty$ if zero is not an eigenvalue or resonance, and on $\\frac43<p<4$ if zero is an eigenvalue but not a resonance. 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