{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BQHTXV6SCWM5PLTXSVUUGZHCL2","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":"52056d22775f8b114e9f5f65008045219ce84d97a300979b114f25f9a0439681","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-11T09:19:29Z","title_canon_sha256":"3b8dce865c63c24ecc82db45211a6b37ab4aff6f5437458992e561ed7f73da8a"},"schema_version":"1.0","source":{"id":"1810.04918","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.04918","created_at":"2026-05-18T00:00:07Z"},{"alias_kind":"arxiv_version","alias_value":"1810.04918v3","created_at":"2026-05-18T00:00:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.04918","created_at":"2026-05-18T00:00:07Z"},{"alias_kind":"pith_short_12","alias_value":"BQHTXV6SCWM5","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BQHTXV6SCWM5PLTX","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BQHTXV6S","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:a17a4190eb283d0a646aa494b20fb02604adcdec57c10d540d0a7fd045e77bd9","target":"graph","created_at":"2026-05-18T00:00:07Z","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 the difference Schr{\\\"o}dinger equation $\\psi$(z + h) + $\\psi$(z -- h) + v(z)$\\psi$(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h $\\rightarrow$ 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = $\\pm$2. We study analytic solutions near the points z 0 satisfying v(z 0) = $\\pm$2 and v (z 0) = 0. For the finite difference equation, these points are the natural analogues of the simple turning points defined for the differential equation --$\\psi$ (z) + v(z)$\\psi$(z) = 0. In an h-ind","authors_text":"Alexander Fedotov, Fr\\'ed\\'eric Klopp (IMJ-PRG)","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-11T09:19:29Z","title":"The complex WKB method for difference equations and Airy functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04918","kind":"arxiv","version":3},"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:c6c56e0668a072675f6efe4181ce98434e82cc134ba0ad02063c9ced1a80f2b5","target":"record","created_at":"2026-05-18T00:00:07Z","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":"52056d22775f8b114e9f5f65008045219ce84d97a300979b114f25f9a0439681","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-11T09:19:29Z","title_canon_sha256":"3b8dce865c63c24ecc82db45211a6b37ab4aff6f5437458992e561ed7f73da8a"},"schema_version":"1.0","source":{"id":"1810.04918","kind":"arxiv","version":3}},"canonical_sha256":"0c0f3bd7d21599d7ae7795694364e25e97bcc70ed0496f96797f162cbdc1433d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c0f3bd7d21599d7ae7795694364e25e97bcc70ed0496f96797f162cbdc1433d","first_computed_at":"2026-05-18T00:00:07.922978Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:07.922978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LUb0Gn9lWWNFXzBHAmr+Mx0yiuVRO22TUAdE/JMpqGKYiwNbTZFij9BNzyFm3j3H7pcv7X+oNAv3xk32PFlPAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:07.923635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.04918","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6c56e0668a072675f6efe4181ce98434e82cc134ba0ad02063c9ced1a80f2b5","sha256:a17a4190eb283d0a646aa494b20fb02604adcdec57c10d540d0a7fd045e77bd9"],"state_sha256":"c32fd87e36daa1e7ecc0ba9a81bbd8302b16d8337c7dd34380e923b8275ceab1"}