{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:4STEETNQLQQQSVYAAPP5FCVAGH","short_pith_number":"pith:4STEETNQ","schema_version":"1.0","canonical_sha256":"e4a6424db05c2109570003dfd28aa031d52b070b3df2049b40772c623219dc32","source":{"kind":"arxiv","id":"1803.09936","version":1},"attestation_state":"computed","paper":{"title":"Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Yong Luo, Yujin Guo, Zhi-qiang Wang","submitted_at":"2018-03-27T07:37:27Z","abstract_excerpt":"We consider minimizers of the following mass critical Hartree minimization problem:\n  \\[ e_\\lambda(N):=\\underset{\\{u\\in H^1(R^d),\\,\\|u\\|^2_2=N\\}}{\\inf} E_\\lambda(u),\\,\\ d\\ge 3, \\] where the Hartree energy functional $E_\\lambda(u)$ is defined by \\[\n  E_\\lambda(u):=\\int_{R ^d}|\\nabla u(x)|^2dx+\\lambda \\int_{R ^d}g(x)u^2(x)dx-\\frac{1}{2} \\int_{R ^d}\\int_{R ^d} \\frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\\,\\ \\lambda>0,\\] and the steep potential $g(x)$ satisfies $0=g(0)=\\inf _{R^d}g(x)\\le g(x)\\le 1$ and $1-g(x)\\in L^{\\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $\\lambda g("},"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":"1803.09936","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-03-27T07:37:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"97fe5bea41d5651b5e132afb38f33992f6cfe6be61507b3e620a04fd10db46a6","abstract_canon_sha256":"4c3588c54bed0ea357f6b4b35da9fc69f9fe3320d368bc503134f2279bf2ced6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:40.634143Z","signature_b64":"YflvQSf90TLtgavSTBH45Fp8rMjApBJk9MS8nAXsJ9tmJexOyoeBiQLTFkecX7g9giEjz+UelWvJSPiDMcjECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4a6424db05c2109570003dfd28aa031d52b070b3df2049b40772c623219dc32","last_reissued_at":"2026-05-17T23:48:40.633556Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:40.633556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Limit Behavior of Mass Critical Hartree Minimization Problems with Steep Potential Wells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Yong Luo, Yujin Guo, Zhi-qiang Wang","submitted_at":"2018-03-27T07:37:27Z","abstract_excerpt":"We consider minimizers of the following mass critical Hartree minimization problem:\n  \\[ e_\\lambda(N):=\\underset{\\{u\\in H^1(R^d),\\,\\|u\\|^2_2=N\\}}{\\inf} E_\\lambda(u),\\,\\ d\\ge 3, \\] where the Hartree energy functional $E_\\lambda(u)$ is defined by \\[\n  E_\\lambda(u):=\\int_{R ^d}|\\nabla u(x)|^2dx+\\lambda \\int_{R ^d}g(x)u^2(x)dx-\\frac{1}{2} \\int_{R ^d}\\int_{R ^d} \\frac{u^2(x)u^2(y)}{|x-y|^2}dxdy,\\,\\ \\lambda>0,\\] and the steep potential $g(x)$ satisfies $0=g(0)=\\inf _{R^d}g(x)\\le g(x)\\le 1$ and $1-g(x)\\in L^{\\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $\\lambda g("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09936","kind":"arxiv","version":1},"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":"1803.09936","created_at":"2026-05-17T23:48:40.633631+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.09936v1","created_at":"2026-05-17T23:48:40.633631+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09936","created_at":"2026-05-17T23:48:40.633631+00:00"},{"alias_kind":"pith_short_12","alias_value":"4STEETNQLQQQ","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4STEETNQLQQQSVYA","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4STEETNQ","created_at":"2026-05-18T12:32:05.422762+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/4STEETNQLQQQSVYAAPP5FCVAGH","json":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH.json","graph_json":"https://pith.science/api/pith-number/4STEETNQLQQQSVYAAPP5FCVAGH/graph.json","events_json":"https://pith.science/api/pith-number/4STEETNQLQQQSVYAAPP5FCVAGH/events.json","paper":"https://pith.science/paper/4STEETNQ"},"agent_actions":{"view_html":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH","download_json":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH.json","view_paper":"https://pith.science/paper/4STEETNQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.09936&json=true","fetch_graph":"https://pith.science/api/pith-number/4STEETNQLQQQSVYAAPP5FCVAGH/graph.json","fetch_events":"https://pith.science/api/pith-number/4STEETNQLQQQSVYAAPP5FCVAGH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH/action/storage_attestation","attest_author":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH/action/author_attestation","sign_citation":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH/action/citation_signature","submit_replication":"https://pith.science/pith/4STEETNQLQQQSVYAAPP5FCVAGH/action/replication_record"}},"created_at":"2026-05-17T23:48:40.633631+00:00","updated_at":"2026-05-17T23:48:40.633631+00:00"}