{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BILAK2ILX3QL6HCREQVBFJQ2NP","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":"37394c666b52be6f96adf5d56b0b57c373617a83a70a89680de08ee0d30a7e64","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-07-11T22:32:43Z","title_canon_sha256":"f3fd6549a3be23aa3b9e97058245696aae208f503e1716b6ba7d991952c7b0f8"},"schema_version":"1.0","source":{"id":"1707.03482","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.03482","created_at":"2026-05-18T00:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1707.03482v1","created_at":"2026-05-18T00:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03482","created_at":"2026-05-18T00:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"BILAK2ILX3QL","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BILAK2ILX3QL6HCR","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BILAK2IL","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:0574f31650266ddaf50f3147654f5ee931ab72a7236f191cbd641950c6dba992","target":"graph","created_at":"2026-05-18T00:15:21Z","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":"In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schr\\\"odinger operators on $\\mathbb{Z}^d$ lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even.","authors_text":"Rui Han, Svetlana Jitomirskaya","cross_cats":["math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-07-11T22:32:43Z","title":"Discrete Bethe-Sommerfeld Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03482","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":"sha256:6aae0a61acf12f850b54b0c770b01ae4bdcafcfaad96dba960581340f3a6f74d","target":"record","created_at":"2026-05-18T00:15:21Z","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":"37394c666b52be6f96adf5d56b0b57c373617a83a70a89680de08ee0d30a7e64","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-07-11T22:32:43Z","title_canon_sha256":"f3fd6549a3be23aa3b9e97058245696aae208f503e1716b6ba7d991952c7b0f8"},"schema_version":"1.0","source":{"id":"1707.03482","kind":"arxiv","version":1}},"canonical_sha256":"0a1605690bbee0bf1c51242a12a61a6bc2c0073e1d95603162c24cae29adb39b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a1605690bbee0bf1c51242a12a61a6bc2c0073e1d95603162c24cae29adb39b","first_computed_at":"2026-05-18T00:15:21.687237Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:21.687237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JXo+DWv8jkGq9ZhqJC4ODj9hXHPUNf5KIM0W2dZrO9UN2G16eeKpLxgc3gCSP9CBoptdh3DMKBFiUyK66HHFBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:21.687796Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.03482","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6aae0a61acf12f850b54b0c770b01ae4bdcafcfaad96dba960581340f3a6f74d","sha256:0574f31650266ddaf50f3147654f5ee931ab72a7236f191cbd641950c6dba992"],"state_sha256":"435d881a87e66d88eeaa9dc7aee650b2c7b07241fae6f305877ce43ef7c51310"}