{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:UWD2JCV2ERJTLLTOVRKVONPOSH","short_pith_number":"pith:UWD2JCV2","schema_version":"1.0","canonical_sha256":"a587a48aba245335ae6eac555735ee91d35f300b1ab87702be3737de462de3b8","source":{"kind":"arxiv","id":"1611.10030","version":1},"attestation_state":"computed","paper":{"title":"The spectra of surface Maryland model for all phases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2016-11-30T07:23:19Z","abstract_excerpt":"We study the discrete Schr\\\"{o}dinger operators $H_{\\lambda,\\alpha,\\theta}$ on $\\ell^2(\\mathbb{Z}^{d+1})$ with surface potential of the form $V(n,x)=\\lambda \\delta(x)\\tan\\pi(\\alpha\\cdot n+\\theta)$, and $H_{\\lambda,\\alpha,\\theta}^{+}$ on $\\ell^2(\\mathbb{Z}^{d}\\times \\mathbb{Z}_+)$ with the boundary condition $ \\psi_{(n,-1)}=\\lambda \\tan\\pi(\\alpha\\cdot n+\\theta)\\psi_{(n,0)} $, where $\\alpha\\in \\mathbb{R}^d$ is rationally independent. We show that the spectra of $H_{\\lambda,\\alpha,\\theta}$ and $H_{\\lambda,\\alpha,\\theta}^{+}$ are $(-\\infty,\\infty)$ for all parameters. We can also determine the abs"},"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":"1611.10030","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-11-30T07:23:19Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"dec4f3a674a9e2b58ef0de72fbaa9c409def7d4e46dcb8282bbd82de8f3da340","abstract_canon_sha256":"e1763a7decc2bfc00b370faef6a6b2730ab29ff98e5ac1b7fb7f18c4839e4b5e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:11.345594Z","signature_b64":"eU3QyCOZmzV1kA8SdaF1l0J1WdxGtHG5+x6GjVfc9asr0AJB7Rhvf3is0b5tYQW4VahAm//iQxaBzeH3Bi4sAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a587a48aba245335ae6eac555735ee91d35f300b1ab87702be3737de462de3b8","last_reissued_at":"2026-05-18T00:56:11.345118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:11.345118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spectra of surface Maryland model for all phases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2016-11-30T07:23:19Z","abstract_excerpt":"We study the discrete Schr\\\"{o}dinger operators $H_{\\lambda,\\alpha,\\theta}$ on $\\ell^2(\\mathbb{Z}^{d+1})$ with surface potential of the form $V(n,x)=\\lambda \\delta(x)\\tan\\pi(\\alpha\\cdot n+\\theta)$, and $H_{\\lambda,\\alpha,\\theta}^{+}$ on $\\ell^2(\\mathbb{Z}^{d}\\times \\mathbb{Z}_+)$ with the boundary condition $ \\psi_{(n,-1)}=\\lambda \\tan\\pi(\\alpha\\cdot n+\\theta)\\psi_{(n,0)} $, where $\\alpha\\in \\mathbb{R}^d$ is rationally independent. We show that the spectra of $H_{\\lambda,\\alpha,\\theta}$ and $H_{\\lambda,\\alpha,\\theta}^{+}$ are $(-\\infty,\\infty)$ for all parameters. We can also determine the abs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10030","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":"1611.10030","created_at":"2026-05-18T00:56:11.345193+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.10030v1","created_at":"2026-05-18T00:56:11.345193+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.10030","created_at":"2026-05-18T00:56:11.345193+00:00"},{"alias_kind":"pith_short_12","alias_value":"UWD2JCV2ERJT","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UWD2JCV2ERJTLLTO","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UWD2JCV2","created_at":"2026-05-18T12:30:46.583412+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/UWD2JCV2ERJTLLTOVRKVONPOSH","json":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH.json","graph_json":"https://pith.science/api/pith-number/UWD2JCV2ERJTLLTOVRKVONPOSH/graph.json","events_json":"https://pith.science/api/pith-number/UWD2JCV2ERJTLLTOVRKVONPOSH/events.json","paper":"https://pith.science/paper/UWD2JCV2"},"agent_actions":{"view_html":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH","download_json":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH.json","view_paper":"https://pith.science/paper/UWD2JCV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.10030&json=true","fetch_graph":"https://pith.science/api/pith-number/UWD2JCV2ERJTLLTOVRKVONPOSH/graph.json","fetch_events":"https://pith.science/api/pith-number/UWD2JCV2ERJTLLTOVRKVONPOSH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH/action/storage_attestation","attest_author":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH/action/author_attestation","sign_citation":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH/action/citation_signature","submit_replication":"https://pith.science/pith/UWD2JCV2ERJTLLTOVRKVONPOSH/action/replication_record"}},"created_at":"2026-05-18T00:56:11.345193+00:00","updated_at":"2026-05-18T00:56:11.345193+00:00"}