{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:FXDBU4P76U6YNCD2YURS3F2BI2","short_pith_number":"pith:FXDBU4P7","canonical_record":{"source":{"id":"1307.7320","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-27T23:11:40Z","cross_cats_sorted":[],"title_canon_sha256":"2b5007cf0d63d862771024b383c7f9149a3e1a898bdc470291460da5b27f712c","abstract_canon_sha256":"7bdfa12e08db1a6ee3de57e7cb7214784ddf56651a9368662bdc6a7fcd75166a"},"schema_version":"1.0"},"canonical_sha256":"2dc61a71fff53d86887ac5232d9741469e1387834e763a9377daa4a84df614c9","source":{"kind":"arxiv","id":"1307.7320","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.7320","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1307.7320v2","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.7320","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"FXDBU4P76U6Y","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FXDBU4P76U6YNCD2","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FXDBU4P7","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:FXDBU4P76U6YNCD2YURS3F2BI2","target":"record","payload":{"canonical_record":{"source":{"id":"1307.7320","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-27T23:11:40Z","cross_cats_sorted":[],"title_canon_sha256":"2b5007cf0d63d862771024b383c7f9149a3e1a898bdc470291460da5b27f712c","abstract_canon_sha256":"7bdfa12e08db1a6ee3de57e7cb7214784ddf56651a9368662bdc6a7fcd75166a"},"schema_version":"1.0"},"canonical_sha256":"2dc61a71fff53d86887ac5232d9741469e1387834e763a9377daa4a84df614c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:36.511943Z","signature_b64":"61aRw6obCZiM+SRp3Bd/CAzCxTEw/WgC4uSiCL3VSGOvuKE6TjHlaSArn3RYsgz7hNeySm6bOzSQxSA2LPS0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2dc61a71fff53d86887ac5232d9741469e1387834e763a9377daa4a84df614c9","last_reissued_at":"2026-05-18T03:08:36.511481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:36.511481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.7320","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":"2026-05-18T03:08:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ONw+QNVm9V65xDeIJfXg2sM0gXb9YHzeZ9nvwbc3W6efqHprN5A91HfdIGpEeJVmujO+xBkewS+eMgpjWyXyBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:30:51.683535Z"},"content_sha256":"27ccfc02149f526c3e567a238c59f5a879be5ec6d50a60443c2f37cf7c72c133","schema_version":"1.0","event_id":"sha256:27ccfc02149f526c3e567a238c59f5a879be5ec6d50a60443c2f37cf7c72c133"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:FXDBU4P76U6YNCD2YURS3F2BI2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Schr\\\"odinger equations with periodic potentials and with nonperiodic nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fei Fang","submitted_at":"2013-07-27T23:11:40Z","abstract_excerpt":"We consider the Schr\\\"{o}dinger equation $-\\Delta u +V(x)u=f(x, u)$, where $V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the continuous spectrum of $A:=-\\Delta +V(x)$. We use M. Willem and W. M. Zou's linking theorem and M. Schechter's method to establish an existence result for this problem in weak superlinear cases. In a sense, we enrich a recent result of M. Willem and W.M. Zou [M. Willem and W.M. Zou, On a Schr\\\"{o}dinger Equation with Periodic Potential and Spectrum Point Zero,Indiana Univ. Math. J. 2003]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.7320","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":"2026-05-18T03:08:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p06kRJOFW6eWDiQU70K8v3TpKzop6CRPIfS5zNPhjTbg629D6jFUFfSfiN8d+A4R0r1xb2rDMEdl85vO1EPwBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:30:51.683876Z"},"content_sha256":"e6bda3467c75566e49a49e39515ca6159895626cd6c3668d2e883adf0d50b4cd","schema_version":"1.0","event_id":"sha256:e6bda3467c75566e49a49e39515ca6159895626cd6c3668d2e883adf0d50b4cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FXDBU4P76U6YNCD2YURS3F2BI2/bundle.json","state_url":"https://pith.science/pith/FXDBU4P76U6YNCD2YURS3F2BI2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FXDBU4P76U6YNCD2YURS3F2BI2/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":"2026-06-03T14:30:51Z","links":{"resolver":"https://pith.science/pith/FXDBU4P76U6YNCD2YURS3F2BI2","bundle":"https://pith.science/pith/FXDBU4P76U6YNCD2YURS3F2BI2/bundle.json","state":"https://pith.science/pith/FXDBU4P76U6YNCD2YURS3F2BI2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FXDBU4P76U6YNCD2YURS3F2BI2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FXDBU4P76U6YNCD2YURS3F2BI2","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":"7bdfa12e08db1a6ee3de57e7cb7214784ddf56651a9368662bdc6a7fcd75166a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-27T23:11:40Z","title_canon_sha256":"2b5007cf0d63d862771024b383c7f9149a3e1a898bdc470291460da5b27f712c"},"schema_version":"1.0","source":{"id":"1307.7320","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.7320","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"arxiv_version","alias_value":"1307.7320v2","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.7320","created_at":"2026-05-18T03:08:36Z"},{"alias_kind":"pith_short_12","alias_value":"FXDBU4P76U6Y","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FXDBU4P76U6YNCD2","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FXDBU4P7","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:e6bda3467c75566e49a49e39515ca6159895626cd6c3668d2e883adf0d50b4cd","target":"graph","created_at":"2026-05-18T03:08:36Z","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 Schr\\\"{o}dinger equation $-\\Delta u +V(x)u=f(x, u)$, where $V$ is periodic and $f$ is non-periodic, 0 is a boundary point of the continuous spectrum of $A:=-\\Delta +V(x)$. We use M. Willem and W. M. Zou's linking theorem and M. Schechter's method to establish an existence result for this problem in weak superlinear cases. In a sense, we enrich a recent result of M. Willem and W.M. Zou [M. Willem and W.M. Zou, On a Schr\\\"{o}dinger Equation with Periodic Potential and Spectrum Point Zero,Indiana Univ. Math. J. 2003].","authors_text":"Fei Fang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-27T23:11:40Z","title":"On Schr\\\"odinger equations with periodic potentials and with nonperiodic nonlinearities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.7320","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":"sha256:27ccfc02149f526c3e567a238c59f5a879be5ec6d50a60443c2f37cf7c72c133","target":"record","created_at":"2026-05-18T03:08:36Z","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":"7bdfa12e08db1a6ee3de57e7cb7214784ddf56651a9368662bdc6a7fcd75166a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-27T23:11:40Z","title_canon_sha256":"2b5007cf0d63d862771024b383c7f9149a3e1a898bdc470291460da5b27f712c"},"schema_version":"1.0","source":{"id":"1307.7320","kind":"arxiv","version":2}},"canonical_sha256":"2dc61a71fff53d86887ac5232d9741469e1387834e763a9377daa4a84df614c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2dc61a71fff53d86887ac5232d9741469e1387834e763a9377daa4a84df614c9","first_computed_at":"2026-05-18T03:08:36.511481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:36.511481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"61aRw6obCZiM+SRp3Bd/CAzCxTEw/WgC4uSiCL3VSGOvuKE6TjHlaSArn3RYsgz7hNeySm6bOzSQxSA2LPS0Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:36.511943Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.7320","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27ccfc02149f526c3e567a238c59f5a879be5ec6d50a60443c2f37cf7c72c133","sha256:e6bda3467c75566e49a49e39515ca6159895626cd6c3668d2e883adf0d50b4cd"],"state_sha256":"cbe068fefa3d0b6c2403719650aa7c0cd5a6f9d3de7394b250bd772a19a18d53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vQdrXp8+F4jdjOrN7Vdc/1dOiY6ljqNs7/6ununTJHg9U/mOvzdI8hxlaXQob/0DnH5Crnotx6rY426ig37bBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T14:30:51.685769Z","bundle_sha256":"9183f0fdc579e09a667f01d86cce1bcb1222894727e66bf55daf7e031d858585"}}