{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DQ7QZD6L2TMCD5Z3S6LNCQTVFF","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":"93a80eb7fe65f47af383387ef1ded6c243d7689602a1a2efde4e324f8958a5cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-07-07T09:09:35Z","title_canon_sha256":"f9f1d94e6375c1ca35e9b0b963b95479507036e19d04f6eae2774e84404c6625"},"schema_version":"1.0","source":{"id":"1307.1841","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.1841","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"1307.1841v2","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1841","created_at":"2026-05-18T03:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"DQ7QZD6L2TMC","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DQ7QZD6L2TMCD5Z3","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DQ7QZD6L","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:0a743a4df8e32c6b845322417730749a479ed93c9f06946f03605de0d444aafe","target":"graph","created_at":"2026-05-18T03:04:06Z","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 Schr\\\"odinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schr\\\"odinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific gr","authors_text":"Evgeny Korotyaev, Natalia Saburova","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-07-07T09:09:35Z","title":"Schr\\\"odinger operators on periodic discrete graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1841","kind":"arxiv","version":2},"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":"sha256:9eb9132243872b10928b1206261e8b40c220c85e96cf11326ad22c0cbd109671","target":"record","created_at":"2026-05-18T03:04:06Z","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":"93a80eb7fe65f47af383387ef1ded6c243d7689602a1a2efde4e324f8958a5cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-07-07T09:09:35Z","title_canon_sha256":"f9f1d94e6375c1ca35e9b0b963b95479507036e19d04f6eae2774e84404c6625"},"schema_version":"1.0","source":{"id":"1307.1841","kind":"arxiv","version":2}},"canonical_sha256":"1c3f0c8fcbd4d821f73b9796d14275295a1ae3e70d034d935441d70926d94d75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1c3f0c8fcbd4d821f73b9796d14275295a1ae3e70d034d935441d70926d94d75","first_computed_at":"2026-05-18T03:04:06.440114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:06.440114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"stwGCfBSLZYYUUAqSyYmx4ha4DIuWSqn57FL9osmleYz9TsnMLZDwHTbdlVOy9CG1M6qZO7Khlc5e4zU7SRGBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:06.440772Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.1841","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9eb9132243872b10928b1206261e8b40c220c85e96cf11326ad22c0cbd109671","sha256:0a743a4df8e32c6b845322417730749a479ed93c9f06946f03605de0d444aafe"],"state_sha256":"f03f2f17eec886bf7919adab6755976ccb79503c1336fc4b7e5a99739b575572"}