{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:APZJXV6D5MA4MNGWI4B4HGK3GY","short_pith_number":"pith:APZJXV6D","canonical_record":{"source":{"id":"1112.4041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-17T09:16:05Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"86dbe75d662c2dd7289115d23ff84524fb384b1bdebbf8592aed9965e7354cc2","abstract_canon_sha256":"3e708953ae36ba183b0d15c3b1d16d0da6d27800e1eafbe642e64d1cf42d1bd6"},"schema_version":"1.0"},"canonical_sha256":"03f29bd7c3eb01c634d64703c3995b360e066f4a82bd5a7c43dc125fa5534d3e","source":{"kind":"arxiv","id":"1112.4041","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4041","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4041v1","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4041","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"pith_short_12","alias_value":"APZJXV6D5MA4","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"APZJXV6D5MA4MNGW","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"APZJXV6D","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:APZJXV6D5MA4MNGWI4B4HGK3GY","target":"record","payload":{"canonical_record":{"source":{"id":"1112.4041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-17T09:16:05Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"86dbe75d662c2dd7289115d23ff84524fb384b1bdebbf8592aed9965e7354cc2","abstract_canon_sha256":"3e708953ae36ba183b0d15c3b1d16d0da6d27800e1eafbe642e64d1cf42d1bd6"},"schema_version":"1.0"},"canonical_sha256":"03f29bd7c3eb01c634d64703c3995b360e066f4a82bd5a7c43dc125fa5534d3e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:44.038996Z","signature_b64":"P+SHiVqwsa6g4o0IQI0S4ftm24NOmp6MfLgRnxCKIUtq5Vkfl8dSIChA05L76Mj84T+1Sy6bDeJq69xADR7gCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03f29bd7c3eb01c634d64703c3995b360e066f4a82bd5a7c43dc125fa5534d3e","last_reissued_at":"2026-05-18T03:02:44.038419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:44.038419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.4041","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AT+4KYXC9HoONKvqTleK7NQSFCH1Hof/TKl/gl65SpF/AKgj+2exilhckyx454Hq5AMBxELGNRglGk5QjIwzDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:19:12.700710Z"},"content_sha256":"b190a8d7b9a7591da02ece2dd85487a81077f233d9f0af204a85ff11575af588","schema_version":"1.0","event_id":"sha256:b190a8d7b9a7591da02ece2dd85487a81077f233d9f0af204a85ff11575af588"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:APZJXV6D5MA4MNGWI4B4HGK3GY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singular Schrodinger operators in one dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.SP","authors_text":"E. B. Davies","submitted_at":"2011-12-17T09:16:05Z","abstract_excerpt":"We consider a class of singular Schr\\\"odinger operators $H$ that act in $L^2(0,\\infty)$, each of which is constructed from a positive function $\\phi$ on $(0,\\infty)$. Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of $\\phi$. For a large class of $H$ that have discrete spectrum, we prove that the eigenvalue asymptotics of $H$ does not depend on rapid oscillations of $\\phi$ or of the potential. Similar comments apply to our treatment of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4041","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"brdkCBabW2o4QAuLLs1SUKg+m6gzGwJJ+LymeZpQDvreGMvEpaT4C/QgyBvCGRzzpd75T+hTjwKcgpP8X1XlDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:19:12.701050Z"},"content_sha256":"343127874d1e2d56a437e459480eae5ea9ee3aecaf199915be3b1cd00456e8f6","schema_version":"1.0","event_id":"sha256:343127874d1e2d56a437e459480eae5ea9ee3aecaf199915be3b1cd00456e8f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/bundle.json","state_url":"https://pith.science/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T09:19:12Z","links":{"resolver":"https://pith.science/pith/APZJXV6D5MA4MNGWI4B4HGK3GY","bundle":"https://pith.science/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/bundle.json","state":"https://pith.science/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/APZJXV6D5MA4MNGWI4B4HGK3GY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:APZJXV6D5MA4MNGWI4B4HGK3GY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3e708953ae36ba183b0d15c3b1d16d0da6d27800e1eafbe642e64d1cf42d1bd6","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-17T09:16:05Z","title_canon_sha256":"86dbe75d662c2dd7289115d23ff84524fb384b1bdebbf8592aed9965e7354cc2"},"schema_version":"1.0","source":{"id":"1112.4041","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4041","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4041v1","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4041","created_at":"2026-05-18T03:02:44Z"},{"alias_kind":"pith_short_12","alias_value":"APZJXV6D5MA4","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"APZJXV6D5MA4MNGW","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"APZJXV6D","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:343127874d1e2d56a437e459480eae5ea9ee3aecaf199915be3b1cd00456e8f6","target":"graph","created_at":"2026-05-18T03:02:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider a class of singular Schr\\\"odinger operators $H$ that act in $L^2(0,\\infty)$, each of which is constructed from a positive function $\\phi$ on $(0,\\infty)$. Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of $\\phi$. For a large class of $H$ that have discrete spectrum, we prove that the eigenvalue asymptotics of $H$ does not depend on rapid oscillations of $\\phi$ or of the potential. Similar comments apply to our treatment of the","authors_text":"E. B. Davies","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-17T09:16:05Z","title":"Singular Schrodinger operators in one dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4041","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b190a8d7b9a7591da02ece2dd85487a81077f233d9f0af204a85ff11575af588","target":"record","created_at":"2026-05-18T03:02:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3e708953ae36ba183b0d15c3b1d16d0da6d27800e1eafbe642e64d1cf42d1bd6","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-17T09:16:05Z","title_canon_sha256":"86dbe75d662c2dd7289115d23ff84524fb384b1bdebbf8592aed9965e7354cc2"},"schema_version":"1.0","source":{"id":"1112.4041","kind":"arxiv","version":1}},"canonical_sha256":"03f29bd7c3eb01c634d64703c3995b360e066f4a82bd5a7c43dc125fa5534d3e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03f29bd7c3eb01c634d64703c3995b360e066f4a82bd5a7c43dc125fa5534d3e","first_computed_at":"2026-05-18T03:02:44.038419Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:44.038419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P+SHiVqwsa6g4o0IQI0S4ftm24NOmp6MfLgRnxCKIUtq5Vkfl8dSIChA05L76Mj84T+1Sy6bDeJq69xADR7gCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:44.038996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.4041","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b190a8d7b9a7591da02ece2dd85487a81077f233d9f0af204a85ff11575af588","sha256:343127874d1e2d56a437e459480eae5ea9ee3aecaf199915be3b1cd00456e8f6"],"state_sha256":"81c93435fe3ca074fa06daa4409648fb4c488e9c49a7e274ac75872478d6062a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D29RPh+q5oQ+FlQO20AA21VM9sFKNL/yhlUUvAZ9TeUvfCdH71Z87Lw/L5zoGGNHeR1lwaxIaUVnD+17cy9IAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T09:19:12.702890Z","bundle_sha256":"cadaef8d9faac15ce10e2b75c7eb403d23e14bace02916bfc4a08068f64cf0d1"}}