{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:2GMHPNM3I7GTTEMBW7HVZH4RJI","short_pith_number":"pith:2GMHPNM3","schema_version":"1.0","canonical_sha256":"d19877b59b47cd399181b7cf5c9f914a19bb7c821c3935ed54b181963a3527ef","source":{"kind":"arxiv","id":"1509.01748","version":2},"attestation_state":"computed","paper":{"title":"Decoupling of Deficiency Indices and Applications to Schr\\\"odinger-Type Operators with Possibly Strongly Singular Potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Gerald Teschl, Irina Nenciu, Marius Mitrea","submitted_at":"2015-09-05T23:43:55Z","abstract_excerpt":"We investigate closed, symmetric $L^2(\\mathbb{R}^n)$-realizations $H$ of Schr\\\"odinger-type operators $(- \\Delta +V)\\upharpoonright_{C_0^{\\infty}(\\mathbb{R}^n \\setminus \\Sigma)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $\\Sigma_j$, $j \\in J$, of $n$-dimensional Lebesgue measure zero, with $J \\subseteq \\mathbb{N}$ an index set and $\\Sigma = \\bigcup_{j \\in J} \\Sigma_j$. We show that the defect, $\\mathrm{def}(H)$, of $H$ can be computed in terms of the individual defects, $\\mathrm{def}(H_j)$, of closed, symmetric $L^2(\\mathbb{R}^n)$-re"},"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":"1509.01748","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-05T23:43:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"3689607984c1240c838ca36672d033c8175660896ff7d1018f9022dcfa9ab40c","abstract_canon_sha256":"50014228581c286b72ba4d0c8fef00779a93681704ca156e24583c69b82aa990"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:26.410948Z","signature_b64":"+eQjEDMpRDGbYHNJ4VmlS2Qgh1h4cqaC9bP0NgnVIR7QRJgEAS59uHO68N/67Q1oVfQpMmsUpEt6uLNZHIleDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d19877b59b47cd399181b7cf5c9f914a19bb7c821c3935ed54b181963a3527ef","last_reissued_at":"2026-05-18T01:08:26.410025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:26.410025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decoupling of Deficiency Indices and Applications to Schr\\\"odinger-Type Operators with Possibly Strongly Singular Potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Fritz Gesztesy, Gerald Teschl, Irina Nenciu, Marius Mitrea","submitted_at":"2015-09-05T23:43:55Z","abstract_excerpt":"We investigate closed, symmetric $L^2(\\mathbb{R}^n)$-realizations $H$ of Schr\\\"odinger-type operators $(- \\Delta +V)\\upharpoonright_{C_0^{\\infty}(\\mathbb{R}^n \\setminus \\Sigma)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $\\Sigma_j$, $j \\in J$, of $n$-dimensional Lebesgue measure zero, with $J \\subseteq \\mathbb{N}$ an index set and $\\Sigma = \\bigcup_{j \\in J} \\Sigma_j$. We show that the defect, $\\mathrm{def}(H)$, of $H$ can be computed in terms of the individual defects, $\\mathrm{def}(H_j)$, of closed, symmetric $L^2(\\mathbb{R}^n)$-re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01748","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":"1509.01748","created_at":"2026-05-18T01:08:26.410192+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.01748v2","created_at":"2026-05-18T01:08:26.410192+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01748","created_at":"2026-05-18T01:08:26.410192+00:00"},{"alias_kind":"pith_short_12","alias_value":"2GMHPNM3I7GT","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"2GMHPNM3I7GTTEMB","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"2GMHPNM3","created_at":"2026-05-18T12:28:59.999130+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/2GMHPNM3I7GTTEMBW7HVZH4RJI","json":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI.json","graph_json":"https://pith.science/api/pith-number/2GMHPNM3I7GTTEMBW7HVZH4RJI/graph.json","events_json":"https://pith.science/api/pith-number/2GMHPNM3I7GTTEMBW7HVZH4RJI/events.json","paper":"https://pith.science/paper/2GMHPNM3"},"agent_actions":{"view_html":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI","download_json":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI.json","view_paper":"https://pith.science/paper/2GMHPNM3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.01748&json=true","fetch_graph":"https://pith.science/api/pith-number/2GMHPNM3I7GTTEMBW7HVZH4RJI/graph.json","fetch_events":"https://pith.science/api/pith-number/2GMHPNM3I7GTTEMBW7HVZH4RJI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI/action/storage_attestation","attest_author":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI/action/author_attestation","sign_citation":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI/action/citation_signature","submit_replication":"https://pith.science/pith/2GMHPNM3I7GTTEMBW7HVZH4RJI/action/replication_record"}},"created_at":"2026-05-18T01:08:26.410192+00:00","updated_at":"2026-05-18T01:08:26.410192+00:00"}