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We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for any sequences $(\\s_n)_{1}^\\iy, \\s_n\\in \\{0,1\\}$, and $(\\vk_n)_1^\\iy\\in \\ell^2, \\vk_n\\ge 0$, there exists a potential $p$ s"},"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":"1107.2692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-07-13T22:46:48Z","cross_cats_sorted":[],"title_canon_sha256":"a2a2d8438f4917e8c151ea9d33f608d736d3ab59846fdfb9b25eafb04ceec6dc","abstract_canon_sha256":"24180b82f2391f0d3976d0a0855d6826ef7e975ec00efc3192b5215be38e192f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:15.822154Z","signature_b64":"qGxXxLh2yq88QfkV2ELCPDF2cHe3CwYdu9Pa2+Ej4lqh0Tp0O03HgkZhL4HmFE640cndyAPh1jcg0V6nRBj0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0809e24c2b373e7fd1597bffaab59f283d29ae3c62dc7b3acb064661cc6e6d94","last_reissued_at":"2026-05-18T04:18:15.821765Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:15.821765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the resonances and eigenvalues for a 1D half-crystal with localised impurity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Evgeny L. 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Conversely, for any potential $q$ and for any sequences $(\\s_n)_{1}^\\iy, \\s_n\\in \\{0,1\\}$, and $(\\vk_n)_1^\\iy\\in \\ell^2, \\vk_n\\ge 0$, there exists a potential $p$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2692","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":"1107.2692","created_at":"2026-05-18T04:18:15.821825+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.2692v1","created_at":"2026-05-18T04:18:15.821825+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2692","created_at":"2026-05-18T04:18:15.821825+00:00"},{"alias_kind":"pith_short_12","alias_value":"BAE6ETBLG47H","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BAE6ETBLG47H7UKZ","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BAE6ETBL","created_at":"2026-05-18T12:26:24.575870+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/BAE6ETBLG47H7UKZPP72VNM7FA","json":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA.json","graph_json":"https://pith.science/api/pith-number/BAE6ETBLG47H7UKZPP72VNM7FA/graph.json","events_json":"https://pith.science/api/pith-number/BAE6ETBLG47H7UKZPP72VNM7FA/events.json","paper":"https://pith.science/paper/BAE6ETBL"},"agent_actions":{"view_html":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA","download_json":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA.json","view_paper":"https://pith.science/paper/BAE6ETBL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.2692&json=true","fetch_graph":"https://pith.science/api/pith-number/BAE6ETBLG47H7UKZPP72VNM7FA/graph.json","fetch_events":"https://pith.science/api/pith-number/BAE6ETBLG47H7UKZPP72VNM7FA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA/action/storage_attestation","attest_author":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA/action/author_attestation","sign_citation":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA/action/citation_signature","submit_replication":"https://pith.science/pith/BAE6ETBLG47H7UKZPP72VNM7FA/action/replication_record"}},"created_at":"2026-05-18T04:18:15.821825+00:00","updated_at":"2026-05-18T04:18:15.821825+00:00"}