{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:ADDJ75G3LVV262WXI5UUG7DEMB","short_pith_number":"pith:ADDJ75G3","schema_version":"1.0","canonical_sha256":"00c69ff4db5d6baf6ad74769437c646066766ca2c9ac7156e5876845040d9a1e","source":{"kind":"arxiv","id":"1505.04904","version":1},"attestation_state":"computed","paper":{"title":"Homogeneity of the spectrum for quasi-periodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"David Damanik, Michael Goldstein, Mircea Voda, Wilhelm Schlag","submitted_at":"2015-05-19T08:25:10Z","abstract_excerpt":"We consider the one-dimensional discrete Schr\\\"odinger operator $$ \\bigl[H(x,\\omega)\\varphi\\bigr](n)\\equiv -\\varphi(n-1)-\\varphi(n+1) + V(x + n\\omega)\\varphi(n)\\ , $$ $n \\in \\mathbb{Z}$, $x,\\omega \\in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,\\omega)>0$ for all $E$. Let $\\mathcal{S}_\\omega$ be the spectrum of $H(x,\\omega)$. For all $\\omega$ obeying the Diophantine condition $\\omega \\in \\mathbb{T}_{c,a}$, we show the following: if $\\mathcal{S}_\\omega \\cap (E',E\")\\neq \\emptyset$, then $\\mathcal{S}_\\omega \\cap (E',E\")$ is homogeneous in the sense of Carleson (see [Car83]). Furtherm"},"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":"1505.04904","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-05-19T08:25:10Z","cross_cats_sorted":[],"title_canon_sha256":"5872f0cae72116acaa66a36c52c3b43059d7ac071848892e96cc73609c57039b","abstract_canon_sha256":"79f04a987ebd87befe5098d9447bb92915c5f6528f13935f74908188a9e2ca16"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:50.549596Z","signature_b64":"B8nw57jn3sOb/zVYuiL63A3lLZ++cpbqYzuunrdT8y/6cfmocWxPBCNot8wXfAr7MutGwnH5Iz3krY3VoNxNBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00c69ff4db5d6baf6ad74769437c646066766ca2c9ac7156e5876845040d9a1e","last_reissued_at":"2026-05-18T00:04:50.548873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:50.548873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homogeneity of the spectrum for quasi-periodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"David Damanik, Michael Goldstein, Mircea Voda, Wilhelm Schlag","submitted_at":"2015-05-19T08:25:10Z","abstract_excerpt":"We consider the one-dimensional discrete Schr\\\"odinger operator $$ \\bigl[H(x,\\omega)\\varphi\\bigr](n)\\equiv -\\varphi(n-1)-\\varphi(n+1) + V(x + n\\omega)\\varphi(n)\\ , $$ $n \\in \\mathbb{Z}$, $x,\\omega \\in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,\\omega)>0$ for all $E$. Let $\\mathcal{S}_\\omega$ be the spectrum of $H(x,\\omega)$. For all $\\omega$ obeying the Diophantine condition $\\omega \\in \\mathbb{T}_{c,a}$, we show the following: if $\\mathcal{S}_\\omega \\cap (E',E\")\\neq \\emptyset$, then $\\mathcal{S}_\\omega \\cap (E',E\")$ is homogeneous in the sense of Carleson (see [Car83]). Furtherm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04904","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":"1505.04904","created_at":"2026-05-18T00:04:50.548984+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.04904v1","created_at":"2026-05-18T00:04:50.548984+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04904","created_at":"2026-05-18T00:04:50.548984+00:00"},{"alias_kind":"pith_short_12","alias_value":"ADDJ75G3LVV2","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"ADDJ75G3LVV262WX","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"ADDJ75G3","created_at":"2026-05-18T12:29:10.953037+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/ADDJ75G3LVV262WXI5UUG7DEMB","json":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB.json","graph_json":"https://pith.science/api/pith-number/ADDJ75G3LVV262WXI5UUG7DEMB/graph.json","events_json":"https://pith.science/api/pith-number/ADDJ75G3LVV262WXI5UUG7DEMB/events.json","paper":"https://pith.science/paper/ADDJ75G3"},"agent_actions":{"view_html":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB","download_json":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB.json","view_paper":"https://pith.science/paper/ADDJ75G3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.04904&json=true","fetch_graph":"https://pith.science/api/pith-number/ADDJ75G3LVV262WXI5UUG7DEMB/graph.json","fetch_events":"https://pith.science/api/pith-number/ADDJ75G3LVV262WXI5UUG7DEMB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB/action/storage_attestation","attest_author":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB/action/author_attestation","sign_citation":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB/action/citation_signature","submit_replication":"https://pith.science/pith/ADDJ75G3LVV262WXI5UUG7DEMB/action/replication_record"}},"created_at":"2026-05-18T00:04:50.548984+00:00","updated_at":"2026-05-18T00:04:50.548984+00:00"}