{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:S7VW25DIVSOT27TKWTCVKDEG35","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":"2b1e6fbe3c98bc62e4040f664094b38b9e8c2ff73b0f65c0e440c72608a77c45","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-05-19T17:29:43Z","title_canon_sha256":"3511f30bce091d4acf56f879961fdca1446ab9f28a8864e080e4dce9b0207944"},"schema_version":"1.0","source":{"id":"1205.4351","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4351","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4351v4","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4351","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"S7VW25DIVSOT","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"S7VW25DIVSOT27TK","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"S7VW25DI","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:9276179ed3a7ef7626d106c5325995b002bf6ba3647bae620f7c58a79045e2eb","target":"graph","created_at":"2026-05-18T02:38:37Z","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 study spectral theory for bounded Borel subsets of $\\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator $\\Ds = \\frac1{2\\pi i}\\frac{d}{dx}$ with domain consisting of $C^\\infty$ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding selfadjoint extensions of $\\Ds$ and for the associated unitary groups of local translations. While motivated by scatter","authors_text":"Dorin Ervin Dutkay, Palle E. T. Jorgensen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-05-19T17:29:43Z","title":"Unitary groups and spectral sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4351","kind":"arxiv","version":4},"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":"sha256:0e89fb3701d1761a27e9a0a2e2d13764b2571ccf21000fdcfe25f580c297a93a","target":"record","created_at":"2026-05-18T02:38:37Z","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":"2b1e6fbe3c98bc62e4040f664094b38b9e8c2ff73b0f65c0e440c72608a77c45","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-05-19T17:29:43Z","title_canon_sha256":"3511f30bce091d4acf56f879961fdca1446ab9f28a8864e080e4dce9b0207944"},"schema_version":"1.0","source":{"id":"1205.4351","kind":"arxiv","version":4}},"canonical_sha256":"97eb6d7468ac9d3d7e6ab4c5550c86df78d33af3b360d1b642539d3864714e86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97eb6d7468ac9d3d7e6ab4c5550c86df78d33af3b360d1b642539d3864714e86","first_computed_at":"2026-05-18T02:38:37.596762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:37.596762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"swgCwlkPOgEvbhNFLFuS43DpZwQbvhWFVvQzpYh0nnFQPIoe+0qQU84tWV1v07srgWszPXPy0FFFgTyZt9Y4Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:37.597262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4351","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e89fb3701d1761a27e9a0a2e2d13764b2571ccf21000fdcfe25f580c297a93a","sha256:9276179ed3a7ef7626d106c5325995b002bf6ba3647bae620f7c58a79045e2eb"],"state_sha256":"77f668935f7e92b8d2a6b798553d8b7be368bc20278cb3551bcdffad9a35dd92"}