{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:VDNJ3YMS3E6B5OCTMVLCZDM6BZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7a98005d637cba3c8e5345422442b4f8454d0b10c3317ebd42577c4e39dcd5a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-03T17:29:37Z","title_canon_sha256":"b9fa250b908e752a7f6eb12d711d58e5c9d715329db722cbc3e0ea1a4caa66cb"},"schema_version":"1.0","source":{"id":"1411.0576","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.0576","created_at":"2026-05-18T02:17:54Z"},{"alias_kind":"arxiv_version","alias_value":"1411.0576v2","created_at":"2026-05-18T02:17:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0576","created_at":"2026-05-18T02:17:54Z"},{"alias_kind":"pith_short_12","alias_value":"VDNJ3YMS3E6B","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"VDNJ3YMS3E6B5OCT","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"VDNJ3YMS","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:fd332a98c81958f5c50d6d20b0b99013009d24f4d7646a81bce36156d6f95d09","target":"graph","created_at":"2026-05-18T02:17:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider here solutions of the nonlinear fractional Schr\\\"odinger equation $$\\epsilon^{2s}(-\\Delta)^s u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $\\epsilon$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $\\epsilon$ is small.","authors_text":"Enrico Valdinoci, Fethi Mahmoudi, Mouhamed Moustapha Fall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-03T17:29:37Z","title":"Ground states and concentration phenomena for the fractional Schr\\\"odinger equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0576","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5b429392ac742cda9f4926c222dbe97aa45979bfe2db1fd84412f74d5fa968f6","target":"record","created_at":"2026-05-18T02:17:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7a98005d637cba3c8e5345422442b4f8454d0b10c3317ebd42577c4e39dcd5a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-03T17:29:37Z","title_canon_sha256":"b9fa250b908e752a7f6eb12d711d58e5c9d715329db722cbc3e0ea1a4caa66cb"},"schema_version":"1.0","source":{"id":"1411.0576","kind":"arxiv","version":2}},"canonical_sha256":"a8da9de192d93c1eb85365562c8d9e0e747c5233dbc6aea7300e649942ec8352","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8da9de192d93c1eb85365562c8d9e0e747c5233dbc6aea7300e649942ec8352","first_computed_at":"2026-05-18T02:17:54.390645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:54.390645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SwRuNimPPjq3KiGbavf6rpdvcVKdVIDIwXyrfrbjFW612xKlMfGhKapfyhoHWskd97R19xef57YtnJpO1QIuDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:54.391131Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.0576","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b429392ac742cda9f4926c222dbe97aa45979bfe2db1fd84412f74d5fa968f6","sha256:fd332a98c81958f5c50d6d20b0b99013009d24f4d7646a81bce36156d6f95d09"],"state_sha256":"6b4466d56b6b93a0cabd5fbab0fad2c1bffa2883a4e588a04fb48e438527a777"}