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We prove that the scattering operator $\\mathscr{S}= \\mathscr{S}(V,J)$ associated with ${\\bm \\phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)\\rho(x)$, $(t,x) \\in \\mathbb{R} \\times \\mathbb{R}^3$, we represent $\\rho$ (resp. $j$) in terms of $j$ (resp. $\\rho$) and $\\mathscr{S}$."},"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":"1101.0310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-01-01T00:03:17Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"4e22ed41c1ce82f14b32ea500c93a40fb6e5a720775a52c638275b85212170be","abstract_canon_sha256":"55aab88fe40acb1c8624b166f5256a2038d74d64c42435e7c337493e1626d574"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:10.597188Z","signature_b64":"J0jtMsY3FxYZH7TZjk4U55COfgV2Xtdjyr8hr0mdr6sZ0RED6L3zRMDbkfoiTfJX7EVaTBkNWeda5Cr9A/aoBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03a563faa60f7c3a060af1f207c457b26b9e4b15a5397e0493681a9ba8607ffe","last_reissued_at":"2026-05-18T04:32:10.596770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:10.596770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An inverse scattering problem for the Klein-Gordon equation with a classical source in quantum field theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Akito Suzuki, Hironobu Sasaki","submitted_at":"2011-01-01T00:03:17Z","abstract_excerpt":"An inverse scattering problem for a quantized scalar field ${\\bm \\phi}$ obeying a linear Klein-Gordon equation $(\\square + m^2 + V) {\\bm \\phi} = J \\mbox{in $\\mathbb{R} \\times \\mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $\\mathscr{S}= \\mathscr{S}(V,J)$ associated with ${\\bm \\phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)\\rho(x)$, $(t,x) \\in \\mathbb{R} \\times \\mathbb{R}^3$, we represent $\\rho$ (resp. $j$) in terms of $j$ (resp. $\\rho$) and $\\mathscr{S}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.0310","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":"1101.0310","created_at":"2026-05-18T04:32:10.596836+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.0310v1","created_at":"2026-05-18T04:32:10.596836+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.0310","created_at":"2026-05-18T04:32:10.596836+00:00"},{"alias_kind":"pith_short_12","alias_value":"AOSWH6VGB56D","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"AOSWH6VGB56DUBQK","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"AOSWH6VG","created_at":"2026-05-18T12:26:24.575870+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/AOSWH6VGB56DUBQK6HZAPRCXWJ","json":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ.json","graph_json":"https://pith.science/api/pith-number/AOSWH6VGB56DUBQK6HZAPRCXWJ/graph.json","events_json":"https://pith.science/api/pith-number/AOSWH6VGB56DUBQK6HZAPRCXWJ/events.json","paper":"https://pith.science/paper/AOSWH6VG"},"agent_actions":{"view_html":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ","download_json":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ.json","view_paper":"https://pith.science/paper/AOSWH6VG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.0310&json=true","fetch_graph":"https://pith.science/api/pith-number/AOSWH6VGB56DUBQK6HZAPRCXWJ/graph.json","fetch_events":"https://pith.science/api/pith-number/AOSWH6VGB56DUBQK6HZAPRCXWJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ/action/storage_attestation","attest_author":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ/action/author_attestation","sign_citation":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ/action/citation_signature","submit_replication":"https://pith.science/pith/AOSWH6VGB56DUBQK6HZAPRCXWJ/action/replication_record"}},"created_at":"2026-05-18T04:32:10.596836+00:00","updated_at":"2026-05-18T04:32:10.596836+00:00"}