{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WZDWWN4L5JUAZCWMWRQDHJ75XC","short_pith_number":"pith:WZDWWN4L","canonical_record":{"source":{"id":"1903.02224","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-03-06T07:55:12Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"b345c4869566bca9545fcd867baf25ac0569d93de0c6794fd258dd6de4a315d3","abstract_canon_sha256":"83c930667e80be71d8c5a19f21abb6dc1d0e48fab2d719ed89fe716dd992cdc7"},"schema_version":"1.0"},"canonical_sha256":"b6476b378bea680c8accb46033a7fdb8bfab84e4210979a76372034ab722d081","source":{"kind":"arxiv","id":"1903.02224","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.02224","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.02224v2","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.02224","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"pith_short_12","alias_value":"WZDWWN4L5JUA","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"WZDWWN4L5JUAZCWM","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"WZDWWN4L","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WZDWWN4L5JUAZCWMWRQDHJ75XC","target":"record","payload":{"canonical_record":{"source":{"id":"1903.02224","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-03-06T07:55:12Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"b345c4869566bca9545fcd867baf25ac0569d93de0c6794fd258dd6de4a315d3","abstract_canon_sha256":"83c930667e80be71d8c5a19f21abb6dc1d0e48fab2d719ed89fe716dd992cdc7"},"schema_version":"1.0"},"canonical_sha256":"b6476b378bea680c8accb46033a7fdb8bfab84e4210979a76372034ab722d081","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:01.701378Z","signature_b64":"e8I1BVskahRy2SKHgtX+tm7Q4ExrpSWlshDr7yyCFc13XEBrZoaJM/hq4755jUhD7UvigCKSORAFJ5pTpw2gCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6476b378bea680c8accb46033a7fdb8bfab84e4210979a76372034ab722d081","last_reissued_at":"2026-05-17T23:47:01.700655Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:01.700655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.02224","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:47:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XR78MQvb34Z8YVs7XeE1GgQQ9px7OJn+iaM+npE1rvylB/CA2XfsNEbAZ6xx2sNcNGmwCVdcL9B27ppwheLcDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:56:45.763244Z"},"content_sha256":"2c026bf7c1472699581e5697507e8cb7879fc12465d7a0ef5e1fa7d0859fd723","schema_version":"1.0","event_id":"sha256:2c026bf7c1472699581e5697507e8cb7879fc12465d7a0ef5e1fa7d0859fd723"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WZDWWN4L5JUAZCWMWRQDHJ75XC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"WKB asymptotics of meromorphic solutions of difference equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Alexander Fedotov (Department of Mathematical Physics), Fr\\'ed\\'eric Klopp (IMJ-PRG, SU)","submitted_at":"2019-03-06T07:55:12Z","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 and $h$ is a small positive parameter. If $v$ is an analytic function, then, for $h$ sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for  differential equations.  In the present paper, we assume that $v$ has a simple pole and, in its neighborhood, we study the asymptotics  of meromorphic solutions to the difference Schr{\\''o}dinger equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.02224","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:47:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TCDfOoOMk676Z42A6a2fGfqs92SMqjYwhLyQ8tXYyr3U53C+XG49osWXDxZ0XN/KVeWD4kbADN26RUF6kUd/AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:56:45.763608Z"},"content_sha256":"452c2709d633d92e162fe2d55b1619d5737afa9b2bf3d8e4f785004d529e15de","schema_version":"1.0","event_id":"sha256:452c2709d633d92e162fe2d55b1619d5737afa9b2bf3d8e4f785004d529e15de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/bundle.json","state_url":"https://pith.science/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T19:56:45Z","links":{"resolver":"https://pith.science/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC","bundle":"https://pith.science/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/bundle.json","state":"https://pith.science/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WZDWWN4L5JUAZCWMWRQDHJ75XC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WZDWWN4L5JUAZCWMWRQDHJ75XC","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":"83c930667e80be71d8c5a19f21abb6dc1d0e48fab2d719ed89fe716dd992cdc7","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-03-06T07:55:12Z","title_canon_sha256":"b345c4869566bca9545fcd867baf25ac0569d93de0c6794fd258dd6de4a315d3"},"schema_version":"1.0","source":{"id":"1903.02224","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.02224","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"arxiv_version","alias_value":"1903.02224v2","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.02224","created_at":"2026-05-17T23:47:01Z"},{"alias_kind":"pith_short_12","alias_value":"WZDWWN4L5JUA","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"WZDWWN4L5JUAZCWM","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"WZDWWN4L","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:452c2709d633d92e162fe2d55b1619d5737afa9b2bf3d8e4f785004d529e15de","target":"graph","created_at":"2026-05-17T23:47:01Z","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 and $h$ is a small positive parameter. If $v$ is an analytic function, then, for $h$ sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for  differential equations.  In the present paper, we assume that $v$ has a simple pole and, in its neighborhood, we study the asymptotics  of meromorphic solutions to the difference Schr{\\''o}dinger equation.","authors_text":"Alexander Fedotov (Department of Mathematical Physics), Fr\\'ed\\'eric Klopp (IMJ-PRG, SU)","cross_cats":["math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-03-06T07:55:12Z","title":"WKB asymptotics of meromorphic solutions of difference equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.02224","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":"sha256:2c026bf7c1472699581e5697507e8cb7879fc12465d7a0ef5e1fa7d0859fd723","target":"record","created_at":"2026-05-17T23:47:01Z","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":"83c930667e80be71d8c5a19f21abb6dc1d0e48fab2d719ed89fe716dd992cdc7","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2019-03-06T07:55:12Z","title_canon_sha256":"b345c4869566bca9545fcd867baf25ac0569d93de0c6794fd258dd6de4a315d3"},"schema_version":"1.0","source":{"id":"1903.02224","kind":"arxiv","version":2}},"canonical_sha256":"b6476b378bea680c8accb46033a7fdb8bfab84e4210979a76372034ab722d081","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6476b378bea680c8accb46033a7fdb8bfab84e4210979a76372034ab722d081","first_computed_at":"2026-05-17T23:47:01.700655Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:01.700655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e8I1BVskahRy2SKHgtX+tm7Q4ExrpSWlshDr7yyCFc13XEBrZoaJM/hq4755jUhD7UvigCKSORAFJ5pTpw2gCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:01.701378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.02224","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c026bf7c1472699581e5697507e8cb7879fc12465d7a0ef5e1fa7d0859fd723","sha256:452c2709d633d92e162fe2d55b1619d5737afa9b2bf3d8e4f785004d529e15de"],"state_sha256":"c83236c58e999bdcf4a97c3c472b0cb5a75ccc481b7d5b400f04ed64b3f88f50"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hWvVhYwJI0ex2WiASPq3Sfdg2x3jAVzWJahMeaV743Vd4yeqFHf83gz+GyDTFuK/nVLXlMcww6D7dNBMG+RmAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T19:56:45.765655Z","bundle_sha256":"0f6c7ad45412331e043218633912e93b8ed8beb5f49bc62ad4c387f4362c7474"}}