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We first establish an interaction Morawetz estimate that is positive definite when $\\| u_{0} \\|_{L^{2}(\\mathbf{R}^{d})} < \\| Q \\|_{L^{2}(\\mathbf{R}^{d})}$, and has the appropriate scaling. Next, we will prove a frequency localized interaction Morawetz estimate similar to the estimat"},"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":"1104.1114","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-06T14:40:10Z","cross_cats_sorted":[],"title_canon_sha256":"f951593f3d5b7f7fa7ba22ff3378187c2b4662c21f93e41b7c85e461d1f124b4","abstract_canon_sha256":"1c47e7b584d14d87e9413878983403afade543fb9e66b0fa0231c4fe913523c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:47.569776Z","signature_b64":"9UwEd0moVp0t2u8XdWs7idGa8pvEd3g9+v1dkxnZ1KAyDhrwzzAo2G4HNlPrvewaTJtUAufpm7D9ipwSq9wUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a064cf2d33523dda2dd2a2c5106924490f945b714c7b5ee2ecbc227778b8547","last_reissued_at":"2026-05-18T04:23:47.569059Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:47.569059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global well-posedness and scattering for the mass critical nonlinear Schr{\\\"o}dinger equation with mass below the mass of the ground state","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benjamin Dodson","submitted_at":"2011-04-06T14:40:10Z","abstract_excerpt":"In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{\\\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \\in L^{2}(\\mathbf{R}^{d})$, $\\| u_{0} \\|_{L^{2}(\\mathbf{R}^{d})} < \\| Q \\|_{L^{2}(\\mathbf{R}^{d})}$, where $Q$ is the ground state, and $d \\geq 1$. 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