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It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\\{x: |x-x_m|<a\\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\\leq m \\leq M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\\Delta$ is defined as $N(\\Delta):=\\sum_{x_m\\in \\Delta}1$ and is given by the formula $N(\\Delta)=|V(a)|^{-1}\\int_{\\Delta}n(x)dx [1+o(1)]$ as $a\\to 0$, where $V(a)=4\\pi a^3/3$, then th"},"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":"0906.3214","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-06-17T15:26:28Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"81c1c195cee99d6dc99b9245f0c654e454ca6507b2da56ffbf8ba4de80d111d3","abstract_canon_sha256":"86b9be07b8eecb45b67d30d96276f065b0ca5d9b3b14ac5a4e5e07a25367be4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:19.364065Z","signature_b64":"WSMaFpusSAPvVHSy+aDxdzdMDkoG1IiD3R5d9z+eqicT+1ptuTw0DiV28QAswEmurbTu4CnEIwL/E5pBd8j/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b7e4bee8140c1d9d7fb87062a4528a757c4b5afda9b77cc5dada5a3ad533496","last_reissued_at":"2026-05-18T02:13:19.363301Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:19.363301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Creating desired potentials by embedding small inhomogeneities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"A.G.Ramm","submitted_at":"2009-06-17T15:26:28Z","abstract_excerpt":"The governing equation is $[\\nabla^2+k^2-q(x)]u=0$ in $\\R^3$. It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\\{x: |x-x_m|<a\\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\\leq m \\leq M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\\Delta$ is defined as $N(\\Delta):=\\sum_{x_m\\in \\Delta}1$ and is given by the formula $N(\\Delta)=|V(a)|^{-1}\\int_{\\Delta}n(x)dx [1+o(1)]$ as $a\\to 0$, where $V(a)=4\\pi a^3/3$, then th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.3214","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":"0906.3214","created_at":"2026-05-18T02:13:19.363433+00:00"},{"alias_kind":"arxiv_version","alias_value":"0906.3214v1","created_at":"2026-05-18T02:13:19.363433+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.3214","created_at":"2026-05-18T02:13:19.363433+00:00"},{"alias_kind":"pith_short_12","alias_value":"DN7EX3UBIDA5","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"DN7EX3UBIDA5TV73","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"DN7EX3UB","created_at":"2026-05-18T12:25:59.703012+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/DN7EX3UBIDA5TV73Q4DCURJIU5","json":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5.json","graph_json":"https://pith.science/api/pith-number/DN7EX3UBIDA5TV73Q4DCURJIU5/graph.json","events_json":"https://pith.science/api/pith-number/DN7EX3UBIDA5TV73Q4DCURJIU5/events.json","paper":"https://pith.science/paper/DN7EX3UB"},"agent_actions":{"view_html":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5","download_json":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5.json","view_paper":"https://pith.science/paper/DN7EX3UB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0906.3214&json=true","fetch_graph":"https://pith.science/api/pith-number/DN7EX3UBIDA5TV73Q4DCURJIU5/graph.json","fetch_events":"https://pith.science/api/pith-number/DN7EX3UBIDA5TV73Q4DCURJIU5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5/action/storage_attestation","attest_author":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5/action/author_attestation","sign_citation":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5/action/citation_signature","submit_replication":"https://pith.science/pith/DN7EX3UBIDA5TV73Q4DCURJIU5/action/replication_record"}},"created_at":"2026-05-18T02:13:19.363433+00:00","updated_at":"2026-05-18T02:13:19.363433+00:00"}