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We prove an explicit Sommerfeld radiation condition\n\\notag \\int_{\\Rd} |\\D u - in_{\\infty}^{1/2}\\frac{x}{|x|}u|^{2} \\frac{dx}{1+|x)} < + \\infty\nfor solutions obtained from the limiting absorption principle and we also give a new energy estimate\n"},"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":"1201.0494","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-02T16:55:19Z","cross_cats_sorted":[],"title_canon_sha256":"0aa952a2176622fe083f294fba8bcdf2f2efe07de50bd483abc9652a5e29eece","abstract_canon_sha256":"8a7516885f53e61eec45c1602b3ad743a82ac157ffef2837ab06881128199c8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:47.167409Z","signature_b64":"EW8mgxReZ04RSayeCsmK58oPn7zTksg4lrNPGOskZ1Tkz7tNK4zE0JEVleEZobbuYfRrzYCtk8cHmlN3vf5LAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4c55c2db3e6a6a65773d1141e1a7981ca05451426d6e425e7742c888004e356","last_reissued_at":"2026-05-18T03:51:47.166771Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:47.166771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Miren Zubeldia","submitted_at":"2012-01-02T16:55:19Z","abstract_excerpt":"We study the electromagnetic Helmholtz equation \n\\notag (\\nabla + ib(x))^{2}u(x) + n(x)u(x) = f(x), \\quad x\\in\\Rd\nwith the magnetic vector potential $b(x)$ and $n(x)$ a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependency like $n(x) \\to n_{\\infty}(\\frac{x}{|x|})$ as $|x|\\to\\infty$. We prove an explicit Sommerfeld radiation condition\n\\notag \\int_{\\Rd} |\\D u - in_{\\infty}^{1/2}\\frac{x}{|x|}u|^{2} \\frac{dx}{1+|x)} < + \\infty\nfor solutions obtained from the limiting absorption principle and we also give a new energy estimate\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0494","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":"1201.0494","created_at":"2026-05-18T03:51:47.166842+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.0494v2","created_at":"2026-05-18T03:51:47.166842+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0494","created_at":"2026-05-18T03:51:47.166842+00:00"},{"alias_kind":"pith_short_12","alias_value":"UTCVYLNT42TK","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"UTCVYLNT42TKMV3T","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"UTCVYLNT","created_at":"2026-05-18T12:27:25.539911+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/UTCVYLNT42TKMV3T2EKB4GTZQH","json":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH.json","graph_json":"https://pith.science/api/pith-number/UTCVYLNT42TKMV3T2EKB4GTZQH/graph.json","events_json":"https://pith.science/api/pith-number/UTCVYLNT42TKMV3T2EKB4GTZQH/events.json","paper":"https://pith.science/paper/UTCVYLNT"},"agent_actions":{"view_html":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH","download_json":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH.json","view_paper":"https://pith.science/paper/UTCVYLNT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.0494&json=true","fetch_graph":"https://pith.science/api/pith-number/UTCVYLNT42TKMV3T2EKB4GTZQH/graph.json","fetch_events":"https://pith.science/api/pith-number/UTCVYLNT42TKMV3T2EKB4GTZQH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH/action/storage_attestation","attest_author":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH/action/author_attestation","sign_citation":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH/action/citation_signature","submit_replication":"https://pith.science/pith/UTCVYLNT42TKMV3T2EKB4GTZQH/action/replication_record"}},"created_at":"2026-05-18T03:51:47.166842+00:00","updated_at":"2026-05-18T03:51:47.166842+00:00"}