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We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\\nu=\\nu_+\\vee\\nu_-$ where $\\nu_\\pm(x)=\\pm\\int^{\\pm 1}_{\\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\\nu\\not\\in L_2(0,1)$ and a unique submarkovian extension if and only if $\\nu\\"},"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":"0909.0567","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-09-03T04:01:19Z","cross_cats_sorted":[],"title_canon_sha256":"6c1d43c74cfbd17b1207233dd4602e70cdf242c0b96e47b625777e7e8ef8fc2f","abstract_canon_sha256":"07ed404924a5ef4b4f1e8771b4426ddb0b35df64090e54b443b664e90933d4e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:33.138846Z","signature_b64":"n1gWYBWW+4Y+CyUU2wgEa3cEkusUEnTUcCITfNik+3JIBonX4A9rG7j/suAmmqWSBvDxCmLPOMLmkiWK4QoOCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39868ea6b2cdeb54cb42746529ebcffbfd56cd459bacf3f662c353af0af75849","last_reissued_at":"2026-05-18T03:03:33.138013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:33.138013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Degenerate elliptic operators in one dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2009-09-03T04:01:19Z","abstract_excerpt":"Let $H$ be the symmetric second-order differential operator on $L_2(\\Ri)$ with domain $C_c^\\infty(\\Ri)$ and action $H\\varphi=-(c \\varphi')'$ where $ c\\in W^{1,2}_{\\rm loc}(\\Ri)$ is a real function which is strictly positive on $\\Ri\\backslash\\{0\\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\\nu=\\nu_+\\vee\\nu_-$ where $\\nu_\\pm(x)=\\pm\\int^{\\pm 1}_{\\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\\nu\\not\\in L_2(0,1)$ and a unique submarkovian extension if and only if $\\nu\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0567","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":"0909.0567","created_at":"2026-05-18T03:03:33.138155+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.0567v1","created_at":"2026-05-18T03:03:33.138155+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0567","created_at":"2026-05-18T03:03:33.138155+00:00"},{"alias_kind":"pith_short_12","alias_value":"HGDI5JVSZXVV","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"HGDI5JVSZXVVJS2C","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"HGDI5JVS","created_at":"2026-05-18T12:25:59.703012+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/HGDI5JVSZXVVJS2CORSST26P7P","json":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P.json","graph_json":"https://pith.science/api/pith-number/HGDI5JVSZXVVJS2CORSST26P7P/graph.json","events_json":"https://pith.science/api/pith-number/HGDI5JVSZXVVJS2CORSST26P7P/events.json","paper":"https://pith.science/paper/HGDI5JVS"},"agent_actions":{"view_html":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P","download_json":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P.json","view_paper":"https://pith.science/paper/HGDI5JVS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.0567&json=true","fetch_graph":"https://pith.science/api/pith-number/HGDI5JVSZXVVJS2CORSST26P7P/graph.json","fetch_events":"https://pith.science/api/pith-number/HGDI5JVSZXVVJS2CORSST26P7P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P/action/storage_attestation","attest_author":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P/action/author_attestation","sign_citation":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P/action/citation_signature","submit_replication":"https://pith.science/pith/HGDI5JVSZXVVJS2CORSST26P7P/action/replication_record"}},"created_at":"2026-05-18T03:03:33.138155+00:00","updated_at":"2026-05-18T03:03:33.138155+00:00"}