{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:HXKS4BXP2VKBTEUPRVSD2QTB7C","short_pith_number":"pith:HXKS4BXP","canonical_record":{"source":{"id":"math-ph/0403050","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2004-03-26T21:15:35Z","cross_cats_sorted":["cond-mat.other","hep-th","math.MP"],"title_canon_sha256":"65640784a9eec5e75eb7726106df6f761056ad86052a7c102c8a727ac21bdd5b","abstract_canon_sha256":"348f134180b3e698b831f6981fc06fcc60e590e682320cd7ad418776e68b8b82"},"schema_version":"1.0"},"canonical_sha256":"3dd52e06efd55419928f8d643d4261f880df54a68b0348623166d0369bfdbf9e","source":{"kind":"arxiv","id":"math-ph/0403050","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0403050","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0403050v1","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0403050","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"HXKS4BXP2VKB","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"HXKS4BXP2VKBTEUP","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"HXKS4BXP","created_at":"2026-07-04T15:20:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:HXKS4BXP2VKBTEUPRVSD2QTB7C","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0403050","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2004-03-26T21:15:35Z","cross_cats_sorted":["cond-mat.other","hep-th","math.MP"],"title_canon_sha256":"65640784a9eec5e75eb7726106df6f761056ad86052a7c102c8a727ac21bdd5b","abstract_canon_sha256":"348f134180b3e698b831f6981fc06fcc60e590e682320cd7ad418776e68b8b82"},"schema_version":"1.0"},"canonical_sha256":"3dd52e06efd55419928f8d643d4261f880df54a68b0348623166d0369bfdbf9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:20:32.841862Z","signature_b64":"nN9tMZEFpVA2tUcg5K5dshu50CBihxCqAM9/a3n2+FsiVCE/vwaz5SJ0WqxyDP9JYpiTvdqgBC9CTu7Mzj4KDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dd52e06efd55419928f8d643d4261f880df54a68b0348623166d0369bfdbf9e","last_reissued_at":"2026-07-04T15:20:32.841506Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:20:32.841506Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0403050","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":"2026-07-04T15:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zgBKxEpjeZuHiFGw8e5BgvRCU44rf5ZLjWo89L/smV1/Sb3f71qB+EzgMjF8P4qKqimDD4S2k9Akb05+W1YTAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:05:54.093455Z"},"content_sha256":"7ce04449db3bd0e2850752a61fe0f3b89d527c17e9642e304fcc6477656f5b36","schema_version":"1.0","event_id":"sha256:7ce04449db3bd0e2850752a61fe0f3b89d527c17e9642e304fcc6477656f5b36"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:HXKS4BXP2VKBTEUPRVSD2QTB7C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Functional determinants for general Sturm-Liouville problems","license":"","headline":"","cross_cats":["cond-mat.other","hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"Alan J. McKane, Klaus Kirsten","submitted_at":"2004-03-26T21:15:35Z","abstract_excerpt":"Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0403050","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math-ph/0403050/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ae7kzHESPDYphylug6T1Rj7iamKXIZDB635DjYnakZFJ0+cTwGwzpDmnMSftyulNUERqMBVn5iOrYYnLtNXpBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T15:05:54.094018Z"},"content_sha256":"bc9dd2cf00655d3f36b03cd339f1072672903074316f96460843eff3f8012193","schema_version":"1.0","event_id":"sha256:bc9dd2cf00655d3f36b03cd339f1072672903074316f96460843eff3f8012193"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/bundle.json","state_url":"https://pith.science/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/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":"2026-08-03T15:05:54Z","links":{"resolver":"https://pith.science/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C","bundle":"https://pith.science/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/bundle.json","state":"https://pith.science/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HXKS4BXP2VKBTEUPRVSD2QTB7C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:HXKS4BXP2VKBTEUPRVSD2QTB7C","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":"348f134180b3e698b831f6981fc06fcc60e590e682320cd7ad418776e68b8b82","cross_cats_sorted":["cond-mat.other","hep-th","math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2004-03-26T21:15:35Z","title_canon_sha256":"65640784a9eec5e75eb7726106df6f761056ad86052a7c102c8a727ac21bdd5b"},"schema_version":"1.0","source":{"id":"math-ph/0403050","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0403050","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0403050v1","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0403050","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"HXKS4BXP2VKB","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"HXKS4BXP2VKBTEUP","created_at":"2026-07-04T15:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"HXKS4BXP","created_at":"2026-07-04T15:20:32Z"}],"graph_snapshots":[{"event_id":"sha256:bc9dd2cf00655d3f36b03cd339f1072672903074316f96460843eff3f8012193","target":"graph","created_at":"2026-07-04T15:20:32Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math-ph/0403050/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses t","authors_text":"Alan J. McKane, Klaus Kirsten","cross_cats":["cond-mat.other","hep-th","math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2004-03-26T21:15:35Z","title":"Functional determinants for general Sturm-Liouville problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0403050","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":"sha256:7ce04449db3bd0e2850752a61fe0f3b89d527c17e9642e304fcc6477656f5b36","target":"record","created_at":"2026-07-04T15:20:32Z","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":"348f134180b3e698b831f6981fc06fcc60e590e682320cd7ad418776e68b8b82","cross_cats_sorted":["cond-mat.other","hep-th","math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2004-03-26T21:15:35Z","title_canon_sha256":"65640784a9eec5e75eb7726106df6f761056ad86052a7c102c8a727ac21bdd5b"},"schema_version":"1.0","source":{"id":"math-ph/0403050","kind":"arxiv","version":1}},"canonical_sha256":"3dd52e06efd55419928f8d643d4261f880df54a68b0348623166d0369bfdbf9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3dd52e06efd55419928f8d643d4261f880df54a68b0348623166d0369bfdbf9e","first_computed_at":"2026-07-04T15:20:32.841506Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:20:32.841506Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nN9tMZEFpVA2tUcg5K5dshu50CBihxCqAM9/a3n2+FsiVCE/vwaz5SJ0WqxyDP9JYpiTvdqgBC9CTu7Mzj4KDw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:20:32.841862Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0403050","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ce04449db3bd0e2850752a61fe0f3b89d527c17e9642e304fcc6477656f5b36","sha256:bc9dd2cf00655d3f36b03cd339f1072672903074316f96460843eff3f8012193"],"state_sha256":"4fca4a473652e42c44611d4589042548af90974518d12b9813888c5cab14ad11"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qqht1hmX20YX8wQSa9Rc5i80d3WxDVAJO9lIAn68N3Ah5DNZozWxuthJawqpSGDXliQ2dfEwoiGhx4Dp6CDiBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T15:05:54.097640Z","bundle_sha256":"0b2d61ccacbe23ad56e35ba5c9f4fa6d79b01cfdd2cfe0abd58b9496f9004672"}}