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y^{\\prime \\prime} + v(x)y, \\quad 0 \\leq x \\leq \\pi, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \\quad a, b \\neq 0, r,s \\in \\mathbb{N}, r\\neq s. $$\n  It is shown that the system of root functions does not contain a basis in $L^2 ([0,\\pi], \\mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \\geq 3. $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3907","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":"1210.3907","created_at":"2026-05-18T03:43:16.778410+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.3907v1","created_at":"2026-05-18T03:43:16.778410+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3907","created_at":"2026-05-18T03:43:16.778410+00:00"},{"alias_kind":"pith_short_12","alias_value":"6WZESP6OIF6W","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_16","alias_value":"6WZESP6OIF6W7WFK","created_at":"2026-05-18T12:26:56.085431+00:00"},{"alias_kind":"pith_short_8","alias_value":"6WZESP6O","created_at":"2026-05-18T12:26:56.085431+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/6WZESP6OIF6W7WFKHYYZ6JSAVH","json":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH.json","graph_json":"https://pith.science/api/pith-number/6WZESP6OIF6W7WFKHYYZ6JSAVH/graph.json","events_json":"https://pith.science/api/pith-number/6WZESP6OIF6W7WFKHYYZ6JSAVH/events.json","paper":"https://pith.science/paper/6WZESP6O"},"agent_actions":{"view_html":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH","download_json":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH.json","view_paper":"https://pith.science/paper/6WZESP6O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.3907&json=true","fetch_graph":"https://pith.science/api/pith-number/6WZESP6OIF6W7WFKHYYZ6JSAVH/graph.json","fetch_events":"https://pith.science/api/pith-number/6WZESP6OIF6W7WFKHYYZ6JSAVH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH/action/storage_attestation","attest_author":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH/action/author_attestation","sign_citation":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH/action/citation_signature","submit_replication":"https://pith.science/pith/6WZESP6OIF6W7WFKHYYZ6JSAVH/action/replication_record"}},"created_at":"2026-05-18T03:43:16.778410+00:00","updated_at":"2026-05-18T03:43:16.778410+00:00"}