{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:CW3W6YWBLSS6RAWRTU4YUQ26MG","short_pith_number":"pith:CW3W6YWB","schema_version":"1.0","canonical_sha256":"15b76f62c15ca5e882d19d398a435e61893f1d140f4aff543f55ba16177bd9a4","source":{"kind":"arxiv","id":"1709.05614","version":2},"attestation_state":"computed","paper":{"title":"Continuous quasiperiodic Schr\\\"odinger operators with Gordon type potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2017-09-17T07:02:12Z","abstract_excerpt":"Let us concern the quasi-periodic Schr\\\"odinger operator in the continuous case, \\begin{equation*}\n  (Hy)(x)=-y^{\\prime\\prime}(x)+V(x,\\omega x)y(x), \\end{equation*} where $V:(\\R/\\Z)^2\\to \\R$ is piecewisely $\\gamma$-H\\\"older continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $\\beta(\\omega)$ as \\begin{equation*}\n  \\beta(\\omega)= \\limsup_{k\\to \\infty}\\frac{-\\ln ||k\\omega||}{k}. \\end{equation*} We prove that $H$ admits no eigenvalue in regime $\\{E\\in\\R:L(E)<\\gamma\\beta(\\omega)\\}$."},"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":"1709.05614","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-09-17T07:02:12Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"8d4cfa04d36fa79b65ed4f25ee6200d7dcf08ff0cb44290744b5729f6d066de2","abstract_canon_sha256":"6ad5a304cd7976d5a1a5d4722a74bff876e691080b81d69b616c3c94be4b3ccd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:06.196863Z","signature_b64":"i/rTj7pt6b25jcYoynX0XJLApMQJi3Cn4bsRaeLp5D6AVpUKuQVwZhnZ1RFWJSWZ+mjvHQOsVE79SBVhZ0xzDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15b76f62c15ca5e882d19d398a435e61893f1d140f4aff543f55ba16177bd9a4","last_reissued_at":"2026-05-17T23:49:06.196137Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:06.196137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Continuous quasiperiodic Schr\\\"odinger operators with Gordon type potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2017-09-17T07:02:12Z","abstract_excerpt":"Let us concern the quasi-periodic Schr\\\"odinger operator in the continuous case, \\begin{equation*}\n  (Hy)(x)=-y^{\\prime\\prime}(x)+V(x,\\omega x)y(x), \\end{equation*} where $V:(\\R/\\Z)^2\\to \\R$ is piecewisely $\\gamma$-H\\\"older continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $\\beta(\\omega)$ as \\begin{equation*}\n  \\beta(\\omega)= \\limsup_{k\\to \\infty}\\frac{-\\ln ||k\\omega||}{k}. \\end{equation*} We prove that $H$ admits no eigenvalue in regime $\\{E\\in\\R:L(E)<\\gamma\\beta(\\omega)\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05614","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":"1709.05614","created_at":"2026-05-17T23:49:06.196257+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.05614v2","created_at":"2026-05-17T23:49:06.196257+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.05614","created_at":"2026-05-17T23:49:06.196257+00:00"},{"alias_kind":"pith_short_12","alias_value":"CW3W6YWBLSS6","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CW3W6YWBLSS6RAWR","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CW3W6YWB","created_at":"2026-05-18T12:31:10.602751+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/CW3W6YWBLSS6RAWRTU4YUQ26MG","json":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG.json","graph_json":"https://pith.science/api/pith-number/CW3W6YWBLSS6RAWRTU4YUQ26MG/graph.json","events_json":"https://pith.science/api/pith-number/CW3W6YWBLSS6RAWRTU4YUQ26MG/events.json","paper":"https://pith.science/paper/CW3W6YWB"},"agent_actions":{"view_html":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG","download_json":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG.json","view_paper":"https://pith.science/paper/CW3W6YWB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.05614&json=true","fetch_graph":"https://pith.science/api/pith-number/CW3W6YWBLSS6RAWRTU4YUQ26MG/graph.json","fetch_events":"https://pith.science/api/pith-number/CW3W6YWBLSS6RAWRTU4YUQ26MG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG/action/storage_attestation","attest_author":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG/action/author_attestation","sign_citation":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG/action/citation_signature","submit_replication":"https://pith.science/pith/CW3W6YWBLSS6RAWRTU4YUQ26MG/action/replication_record"}},"created_at":"2026-05-17T23:49:06.196257+00:00","updated_at":"2026-05-17T23:49:06.196257+00:00"}