{"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"}