{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AHV3C2CVR552DEVYJU2FDVWUUA","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":"ddb334fccea1f7a102607351eae9f4b7e8ec0aa40621035c191e74d6d90b7cc3","cross_cats_sorted":["math.MP","math.SP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-18T16:09:40Z","title_canon_sha256":"2e07e309103803be74475e3213b6be35e055ef1d55ea40a497a3bda07ea6c4bb"},"schema_version":"1.0","source":{"id":"1107.3489","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.3489","created_at":"2026-05-18T03:31:54Z"},{"alias_kind":"arxiv_version","alias_value":"1107.3489v6","created_at":"2026-05-18T03:31:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3489","created_at":"2026-05-18T03:31:54Z"},{"alias_kind":"pith_short_12","alias_value":"AHV3C2CVR552","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AHV3C2CVR552DEVY","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AHV3C2CV","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:c1055f5e29b33383a34b84bf600c87efcaa97b7fd89111606e6ab43cd558f5ab","target":"graph","created_at":"2026-05-18T03:31:54Z","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":"The paper addresses the the number of nodal domains for eigenfunctions of Schr\\\"{o}dinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a \"nodal deficiency\" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains.\n  It was suggested ","authors_text":"Gregory Berkolaiko, Peter Kuchment, Uzy Smilansky","cross_cats":["math.MP","math.SP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-18T16:09:40Z","title":"Critical partitions and nodal deficiency of billiard eigenfunctions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3489","kind":"arxiv","version":6},"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":"sha256:3e09c50a9e3b069b34a6fa4b7fe3498eed898428a2ad9f22da581ccc5211a4a4","target":"record","created_at":"2026-05-18T03:31:54Z","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":"ddb334fccea1f7a102607351eae9f4b7e8ec0aa40621035c191e74d6d90b7cc3","cross_cats_sorted":["math.MP","math.SP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-18T16:09:40Z","title_canon_sha256":"2e07e309103803be74475e3213b6be35e055ef1d55ea40a497a3bda07ea6c4bb"},"schema_version":"1.0","source":{"id":"1107.3489","kind":"arxiv","version":6}},"canonical_sha256":"01ebb168558f7ba192b84d3451d6d4a0199b1d116cd87318441856727ad718c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"01ebb168558f7ba192b84d3451d6d4a0199b1d116cd87318441856727ad718c2","first_computed_at":"2026-05-18T03:31:54.474121Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:31:54.474121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FlSyFzMDbSCXWNhLMTDOCuvwi/cwEjWvmZfRYHGYPErrvocq+uXaIilOWfPpXPbS0hmSTzbb7/WeiYdA8boRBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:31:54.474668Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.3489","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e09c50a9e3b069b34a6fa4b7fe3498eed898428a2ad9f22da581ccc5211a4a4","sha256:c1055f5e29b33383a34b84bf600c87efcaa97b7fd89111606e6ab43cd558f5ab"],"state_sha256":"2bd9f1e5b6c9fc3e4d7dd504cc526e8493f5c615a3523c56356f9d7a50f9a284"}