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We present a deta"},"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":"1803.10395","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-28T03:07:49Z","cross_cats_sorted":[],"title_canon_sha256":"febe8a9a8e45f45ce96a1d6703714b1bf21838c625aa0e64fe69be78b68c8d0a","abstract_canon_sha256":"f1c6f6e28c9657ed654b84604d014a5b48a4019ad273ab37b7782968b43063ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:03.798747Z","signature_b64":"IbJw/gncuE1MeypM+qawE3r2mU/Vhn7ye2N/2pV7Yiw14rmY8R3n/I/hKTkvd/ITStlPFz0lLqNMRatZYsD8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c9a2e615b8a4c8df328fd35ad087189d2b6ecd23c83ddf0b1f1559d79dd38e6","last_reissued_at":"2026-05-18T00:12:03.797929Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:03.797929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mass Concentration and Local Uniqueness of Ground States for $L^2$-subcritical Nonlinear Schr\\\"{o}dinger Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Shuai Li, Xincai Zhu","submitted_at":"2018-03-28T03:07:49Z","abstract_excerpt":"We consider ground states of $L^2$-subcritical nonlinear Schr\\\"{o}dinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem\n  $$ e(\\rho):=\\inf\\{E_{\\rho}(u):u\\in \\mathcal{H}(\\mathbb{R}^d),\\|u\\|_2^2=1\\}.$$ The energy functional $E_{\\rho}(u)$ is defined by $$ E_{\\rho}(u):=\\frac{1}{2}\\int_{\\mathbb{R}^d}|\\nabla u|^2dx +\\frac{1}{2}\\int_{\\mathbb{R}^d}V(x)|u|^2dx-\\frac{\\rho^{p-1}}{p+1}\\int_{\\mathbb{R}^d}|u|^{p+1}dx,$$ where $d\\geq1$, $\\rho>0$, $p\\in\\big(1, 1+\\frac{4}{d}\\big)$ and $0\\leq V(x)\\to\\infty$ as $|x| \\to\\infty$. We present a deta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10395","kind":"arxiv","version":2},"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":"1803.10395","created_at":"2026-05-18T00:12:03.798053+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.10395v2","created_at":"2026-05-18T00:12:03.798053+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.10395","created_at":"2026-05-18T00:12:03.798053+00:00"},{"alias_kind":"pith_short_12","alias_value":"BSNC4YK3RJGI","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"BSNC4YK3RJGI34ZI","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"BSNC4YK3","created_at":"2026-05-18T12:32:16.446611+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/BSNC4YK3RJGI34ZI7U222CDRRH","json":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH.json","graph_json":"https://pith.science/api/pith-number/BSNC4YK3RJGI34ZI7U222CDRRH/graph.json","events_json":"https://pith.science/api/pith-number/BSNC4YK3RJGI34ZI7U222CDRRH/events.json","paper":"https://pith.science/paper/BSNC4YK3"},"agent_actions":{"view_html":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH","download_json":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH.json","view_paper":"https://pith.science/paper/BSNC4YK3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.10395&json=true","fetch_graph":"https://pith.science/api/pith-number/BSNC4YK3RJGI34ZI7U222CDRRH/graph.json","fetch_events":"https://pith.science/api/pith-number/BSNC4YK3RJGI34ZI7U222CDRRH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH/action/storage_attestation","attest_author":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH/action/author_attestation","sign_citation":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH/action/citation_signature","submit_replication":"https://pith.science/pith/BSNC4YK3RJGI34ZI7U222CDRRH/action/replication_record"}},"created_at":"2026-05-18T00:12:03.798053+00:00","updated_at":"2026-05-18T00:12:03.798053+00:00"}