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We prove that given an arbitrary function $f(\\alpha')\\in L^2(S^2)$, an arbitrary fixed $\\alpha_0\\in S^2$, an arbitrary fixed $k>0$, and an arbitrary small $\\ve>0$, there exists a potential $q(x)\\in L^2(D)$, where $D\\subset R^3$ is a bounded domain such that \\bee\n \\|A_q(\\alpha',\\alpha_0"},"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":"math-ph/0606055","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2006-06-21T15:19:20Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"f10aa31a30e4baed04a517ab8e8c7a9e8252697f9f7c683ec2f635219b0576e8","abstract_canon_sha256":"aa182dc5d3daffdb8301edd81a36f65521c92ef18fc7aea7b9d764badc573565"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:30.025684Z","signature_b64":"QmMAzHByBsXphR9t5Tn0nSW+o5Kucv0jcqDuzzSHvYDiR5cdhAiqw+E6wVId+mJ/f+PWRxjpOImYMr3stVa7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39f1fd276f36a16ab7ed910d2b8c8ae0d1becf68ca0349bcee52198cc921bac6","last_reissued_at":"2026-05-18T01:05:30.025183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:30.025183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Inverse scattering problem with fixed energy and fixed incident direction","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"A.G.Ramm","submitted_at":"2006-06-21T15:19:20Z","abstract_excerpt":"Let $A_q(\\alpha',\\alpha,k)$ be the scattering amplitude, corresponding to a local potential $q(x)$, $x\\in\\R^3$, $q(x)=0$ for $|x|>a$, where $a>0$ is a fixed number, $\\alpha',\\alpha\\in S^2$ are unit vectors, $S^2$ is the unit sphere in $\\R^3$, $\\alpha$ is the direction of the incident wave, $k^2>0$ is the energy. We prove that given an arbitrary function $f(\\alpha')\\in L^2(S^2)$, an arbitrary fixed $\\alpha_0\\in S^2$, an arbitrary fixed $k>0$, and an arbitrary small $\\ve>0$, there exists a potential $q(x)\\in L^2(D)$, where $D\\subset R^3$ is a bounded domain such that \\bee\n \\|A_q(\\alpha',\\alpha_0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0606055","kind":"arxiv","version":3},"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":"math-ph/0606055","created_at":"2026-05-18T01:05:30.025257+00:00"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0606055v3","created_at":"2026-05-18T01:05:30.025257+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0606055","created_at":"2026-05-18T01:05:30.025257+00:00"},{"alias_kind":"pith_short_12","alias_value":"HHY72J3PG2QW","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"HHY72J3PG2QWVN7N","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"HHY72J3P","created_at":"2026-05-18T12:25:53.939244+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/HHY72J3PG2QWVN7NSEGSXDEK4D","json":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D.json","graph_json":"https://pith.science/api/pith-number/HHY72J3PG2QWVN7NSEGSXDEK4D/graph.json","events_json":"https://pith.science/api/pith-number/HHY72J3PG2QWVN7NSEGSXDEK4D/events.json","paper":"https://pith.science/paper/HHY72J3P"},"agent_actions":{"view_html":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D","download_json":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D.json","view_paper":"https://pith.science/paper/HHY72J3P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math-ph/0606055&json=true","fetch_graph":"https://pith.science/api/pith-number/HHY72J3PG2QWVN7NSEGSXDEK4D/graph.json","fetch_events":"https://pith.science/api/pith-number/HHY72J3PG2QWVN7NSEGSXDEK4D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D/action/storage_attestation","attest_author":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D/action/author_attestation","sign_citation":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D/action/citation_signature","submit_replication":"https://pith.science/pith/HHY72J3PG2QWVN7NSEGSXDEK4D/action/replication_record"}},"created_at":"2026-05-18T01:05:30.025257+00:00","updated_at":"2026-05-18T01:05:30.025257+00:00"}