{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VZDUVIPJBNJJTZPO7ZJ24PLJFA","short_pith_number":"pith:VZDUVIPJ","schema_version":"1.0","canonical_sha256":"ae474aa1e90b5299e5eefe53ae3d6928358ff1b9309b7fb47ed834b17fd5b905","source":{"kind":"arxiv","id":"1101.2145","version":2},"attestation_state":"computed","paper":{"title":"Scattering theory for Klein-Gordon equations with non-positive energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Christian G\\'erard (LM-Orsay)","submitted_at":"2011-01-11T15:45:57Z","abstract_excerpt":"We study the scattering theory for charged Klein-Gordon equations: \\[\\{{array}{l} (\\p_{t}- \\i v(x))^{2}\\phi(t,x) \\epsilon^{2}(x, D_{x})\\phi(t,x)=0,[2mm] \\phi(0, x)= f_{0}, [2mm] \\i^{-1} \\p_{t}\\phi(0, x)= f_{1}, {array}. \\] where: \\[\\epsilon^{2}(x, D_{x})= \\sum_{1\\leq j, k\\leq n}(\\p_{x_{j}} \\i b_{j}(x))A^{jk}(x)(\\p_{x_{k}} \\i b_{k}(x))+ m^{2}(x),\\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \\[ h[f, f]:= \\int_{\\rr^"},"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":"1101.2145","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-01-11T15:45:57Z","cross_cats_sorted":["math.AP","math.MP","math.SP"],"title_canon_sha256":"81f485b7fb9bf372f529611355581b22c82f2ed68a569d79b0d0c72b0c60dd31","abstract_canon_sha256":"b37ce8014844618a7f8fb0b15addf60c429cc9fe308828efc859809dc3422a01"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:42.977146Z","signature_b64":"ubpX47f+e/VdZ0PAf0D4U7v9rF/jB5kxYhjDt+5XwJoZ0qqpz1lOYN89MnNqjfhTV8oOrQ2Uc7paEysJZvjVDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae474aa1e90b5299e5eefe53ae3d6928358ff1b9309b7fb47ed834b17fd5b905","last_reissued_at":"2026-05-18T02:03:42.976775Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:42.976775Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering theory for Klein-Gordon equations with non-positive energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Christian G\\'erard (LM-Orsay)","submitted_at":"2011-01-11T15:45:57Z","abstract_excerpt":"We study the scattering theory for charged Klein-Gordon equations: \\[\\{{array}{l} (\\p_{t}- \\i v(x))^{2}\\phi(t,x) \\epsilon^{2}(x, D_{x})\\phi(t,x)=0,[2mm] \\phi(0, x)= f_{0}, [2mm] \\i^{-1} \\p_{t}\\phi(0, x)= f_{1}, {array}. \\] where: \\[\\epsilon^{2}(x, D_{x})= \\sum_{1\\leq j, k\\leq n}(\\p_{x_{j}} \\i b_{j}(x))A^{jk}(x)(\\p_{x_{k}} \\i b_{k}(x))+ m^{2}(x),\\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \\[ h[f, f]:= \\int_{\\rr^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2145","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":"1101.2145","created_at":"2026-05-18T02:03:42.976831+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.2145v2","created_at":"2026-05-18T02:03:42.976831+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.2145","created_at":"2026-05-18T02:03:42.976831+00:00"},{"alias_kind":"pith_short_12","alias_value":"VZDUVIPJBNJJ","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VZDUVIPJBNJJTZPO","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VZDUVIPJ","created_at":"2026-05-18T12:26:44.992195+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/VZDUVIPJBNJJTZPO7ZJ24PLJFA","json":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA.json","graph_json":"https://pith.science/api/pith-number/VZDUVIPJBNJJTZPO7ZJ24PLJFA/graph.json","events_json":"https://pith.science/api/pith-number/VZDUVIPJBNJJTZPO7ZJ24PLJFA/events.json","paper":"https://pith.science/paper/VZDUVIPJ"},"agent_actions":{"view_html":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA","download_json":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA.json","view_paper":"https://pith.science/paper/VZDUVIPJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.2145&json=true","fetch_graph":"https://pith.science/api/pith-number/VZDUVIPJBNJJTZPO7ZJ24PLJFA/graph.json","fetch_events":"https://pith.science/api/pith-number/VZDUVIPJBNJJTZPO7ZJ24PLJFA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA/action/storage_attestation","attest_author":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA/action/author_attestation","sign_citation":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA/action/citation_signature","submit_replication":"https://pith.science/pith/VZDUVIPJBNJJTZPO7ZJ24PLJFA/action/replication_record"}},"created_at":"2026-05-18T02:03:42.976831+00:00","updated_at":"2026-05-18T02:03:42.976831+00:00"}