{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:LPT7PBLMS762JNR2VTPXJG24I6","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":"f0ddec60263939cb8a464d23eaf406e445016b2a1948faf79ae3a0a28a197ed6","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2009-01-09T09:13:14Z","title_canon_sha256":"61e8afcf3b41fd6ae60e442883cea926ff6f5f0b4c4fc36976de17a3687aa2d3"},"schema_version":"1.0","source":{"id":"0901.1190","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.1190","created_at":"2026-06-03T23:06:10Z"},{"alias_kind":"arxiv_version","alias_value":"0901.1190v1","created_at":"2026-06-03T23:06:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.1190","created_at":"2026-06-03T23:06:10Z"},{"alias_kind":"pith_short_12","alias_value":"LPT7PBLMS762","created_at":"2026-06-03T23:06:10Z"},{"alias_kind":"pith_short_16","alias_value":"LPT7PBLMS762JNR2","created_at":"2026-06-03T23:06:10Z"},{"alias_kind":"pith_short_8","alias_value":"LPT7PBLM","created_at":"2026-06-03T23:06:10Z"}],"graph_snapshots":[{"event_id":"sha256:4fe5ba275cef51a81132749b7d18d19a48d29a29b981d16c2ea639e227a2cc72","target":"graph","created_at":"2026-06-03T23:06:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0901.1190/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the linear Schr\\\"odinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time","authors_text":"Arnaud Debussche (IRMAR), Erwan Faou (INRIA - Irisa)","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2009-01-09T09:13:14Z","title":"Modified energy for split-step methods applied to the linear Schr\\\"odinger equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.1190","kind":"arxiv","version":1},"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":"sha256:91e426c9dd4bc3ca7a31644fcd2769896ce1676329dd7edc8a894d3db60378c5","target":"record","created_at":"2026-06-03T23:06:10Z","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":"f0ddec60263939cb8a464d23eaf406e445016b2a1948faf79ae3a0a28a197ed6","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2009-01-09T09:13:14Z","title_canon_sha256":"61e8afcf3b41fd6ae60e442883cea926ff6f5f0b4c4fc36976de17a3687aa2d3"},"schema_version":"1.0","source":{"id":"0901.1190","kind":"arxiv","version":1}},"canonical_sha256":"5be7f7856c97fda4b63aacdf749b5c47a98dd42f51250059eab6cb0c9a2c4264","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5be7f7856c97fda4b63aacdf749b5c47a98dd42f51250059eab6cb0c9a2c4264","first_computed_at":"2026-06-03T23:06:10.312008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T23:06:10.312008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E0EuBzE4y7vyOEtaLPByF1EpUgqsz7RMHp/QM+l+7aIadZ5TyNOXGy/bffXCY8vMa2/i7ShANaCupeEcdfVTBQ==","signature_status":"signed_v1","signed_at":"2026-06-03T23:06:10.312436Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.1190","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91e426c9dd4bc3ca7a31644fcd2769896ce1676329dd7edc8a894d3db60378c5","sha256:4fe5ba275cef51a81132749b7d18d19a48d29a29b981d16c2ea639e227a2cc72"],"state_sha256":"a7d5922674741a17ca574086f3fe45ffce43b238a07abe2f32aa69d573f2295f"}