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For $M(R) = \\inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \\gtrsim \\exp(-C R^{\\be_0}(\\log R)^{A(R)})$, where $\\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\\la$, and the dimension, we construct exampl"},"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":"1212.4085","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-17T17:59:03Z","cross_cats_sorted":[],"title_canon_sha256":"3faa1a085fa44c2025f8e83d6817936010a43e472db41a28bba9eb16519d2886","abstract_canon_sha256":"343ba401d2bfe997a44355f287338b98172351f218144ef7be942c05651060a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:30.883574Z","signature_b64":"P5YIr4UvIyN8AmomESFk74Gtv4EUZaGvdmzTipubX36jNu9p+HIqwZ82Z6v9csCJaGOWp2PcVPh9I8GEo46+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0abfa6f07cc315bec2478577fc9a5fc971912352043cc15837335fb37e8c9866","last_reissued_at":"2026-05-18T02:54:30.883059Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:30.883059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp constructions of eigenfunctions of the magnetic Schr\\\"odinger operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Blair Davey","submitted_at":"2012-12-17T17:59:03Z","abstract_excerpt":"We prove sharpness of quantitative unique continuation results for solutions of $-\\Delta u + W\\cdot \\nabla u + V u = \\la u$, where $\\la \\in \\C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \\lesssim <x>^{-N}$ and $|W(x)| \\lesssim <x>^{-P}$. For $M(R) = \\inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \\gtrsim \\exp(-C R^{\\be_0}(\\log R)^{A(R)})$, where $\\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\\la$, and the dimension, we construct exampl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4085","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":"1212.4085","created_at":"2026-05-18T02:54:30.883146+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.4085v2","created_at":"2026-05-18T02:54:30.883146+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.4085","created_at":"2026-05-18T02:54:30.883146+00:00"},{"alias_kind":"pith_short_12","alias_value":"BK72N4D4YMK3","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"BK72N4D4YMK35QSH","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"BK72N4D4","created_at":"2026-05-18T12:27:01.376967+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/BK72N4D4YMK35QSHQV37ZGS7ZF","json":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF.json","graph_json":"https://pith.science/api/pith-number/BK72N4D4YMK35QSHQV37ZGS7ZF/graph.json","events_json":"https://pith.science/api/pith-number/BK72N4D4YMK35QSHQV37ZGS7ZF/events.json","paper":"https://pith.science/paper/BK72N4D4"},"agent_actions":{"view_html":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF","download_json":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF.json","view_paper":"https://pith.science/paper/BK72N4D4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.4085&json=true","fetch_graph":"https://pith.science/api/pith-number/BK72N4D4YMK35QSHQV37ZGS7ZF/graph.json","fetch_events":"https://pith.science/api/pith-number/BK72N4D4YMK35QSHQV37ZGS7ZF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF/action/storage_attestation","attest_author":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF/action/author_attestation","sign_citation":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF/action/citation_signature","submit_replication":"https://pith.science/pith/BK72N4D4YMK35QSHQV37ZGS7ZF/action/replication_record"}},"created_at":"2026-05-18T02:54:30.883146+00:00","updated_at":"2026-05-18T02:54:30.883146+00:00"}