{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ZBRADM5EYAVPYN6YVAME7H7FLC","short_pith_number":"pith:ZBRADM5E","canonical_record":{"source":{"id":"0905.4694","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-05-28T16:12:19Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d73a320cf4b887d3b0c0a479da0a1abc97ca9a26d882ed0501785cdc8d72ed2f","abstract_canon_sha256":"683bc0f316fb4cb1020bf0bd757928b9a0b251e19081884d6363d6dd3c3d26bd"},"schema_version":"1.0"},"canonical_sha256":"c86201b3a4c02afc37d8a8184f9fe5588dc98dcfbe56fac884738f936aaa63b4","source":{"kind":"arxiv","id":"0905.4694","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.4694","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"0905.4694v1","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.4694","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"ZBRADM5EYAVP","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZBRADM5EYAVPYN6Y","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZBRADM5E","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ZBRADM5EYAVPYN6YVAME7H7FLC","target":"record","payload":{"canonical_record":{"source":{"id":"0905.4694","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-05-28T16:12:19Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d73a320cf4b887d3b0c0a479da0a1abc97ca9a26d882ed0501785cdc8d72ed2f","abstract_canon_sha256":"683bc0f316fb4cb1020bf0bd757928b9a0b251e19081884d6363d6dd3c3d26bd"},"schema_version":"1.0"},"canonical_sha256":"c86201b3a4c02afc37d8a8184f9fe5588dc98dcfbe56fac884738f936aaa63b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:31.044188Z","signature_b64":"7JxqBuTgoGNgal9UHn9yv5AsHi3k00r5zM/meE+diNdzJ5BKVSSE7SB9wnjVvEnC+xk9wT+vm0kz+fyxp5hzAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c86201b3a4c02afc37d8a8184f9fe5588dc98dcfbe56fac884738f936aaa63b4","last_reissued_at":"2026-05-18T02:13:31.043357Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:31.043357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0905.4694","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZUfbR6ZfdUkSGfY6eRgBc/qcbidAtuCP96oHj5hhroatEi4xcXAflyrJPvN6pGiovTuWDmgEnWroE/PbUDbXBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T05:32:51.533071Z"},"content_sha256":"dfd1026dc0ac6f1f66c9db6272d11337ec0ed1f4455855b6e841f4e9075d3461","schema_version":"1.0","event_id":"sha256:dfd1026dc0ac6f1f66c9db6272d11337ec0ed1f4455855b6e841f4e9075d3461"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ZBRADM5EYAVPYN6YVAME7H7FLC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conduction bands in classical periodic potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Carl M. Bender, Tanwa Arpornthip","submitted_at":"2009-05-28T16:12:19Z","abstract_excerpt":"The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time in which to perform the measurement. This paper considers the possibility that $\\Delta E$, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behavior of a classical particle in a one-dimensional periodic potential $V(x)=-\\cos(x)$ is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitativel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.4694","kind":"arxiv","version":1},"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":"2026-05-18T02:13:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KWFAFKle0cx014RYbL2bTe1Wx08QmMvSaxtkL2l1Q3JqPNfbGwNA9PIdzmbJt6K+Qo6+iCqMwZJk7WnMvplQAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T05:32:51.533514Z"},"content_sha256":"a64a48b46b79904e7c9e2c197daa43fc84c3f3d1eb4d0124a55a29a1d3d4d7ec","schema_version":"1.0","event_id":"sha256:a64a48b46b79904e7c9e2c197daa43fc84c3f3d1eb4d0124a55a29a1d3d4d7ec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/bundle.json","state_url":"https://pith.science/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/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":"2026-05-20T05:32:51Z","links":{"resolver":"https://pith.science/pith/ZBRADM5EYAVPYN6YVAME7H7FLC","bundle":"https://pith.science/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/bundle.json","state":"https://pith.science/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZBRADM5EYAVPYN6YVAME7H7FLC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZBRADM5EYAVPYN6YVAME7H7FLC","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":"683bc0f316fb4cb1020bf0bd757928b9a0b251e19081884d6363d6dd3c3d26bd","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-05-28T16:12:19Z","title_canon_sha256":"d73a320cf4b887d3b0c0a479da0a1abc97ca9a26d882ed0501785cdc8d72ed2f"},"schema_version":"1.0","source":{"id":"0905.4694","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.4694","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"arxiv_version","alias_value":"0905.4694v1","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.4694","created_at":"2026-05-18T02:13:31Z"},{"alias_kind":"pith_short_12","alias_value":"ZBRADM5EYAVP","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZBRADM5EYAVPYN6Y","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZBRADM5E","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:a64a48b46b79904e7c9e2c197daa43fc84c3f3d1eb4d0124a55a29a1d3d4d7ec","target":"graph","created_at":"2026-05-18T02:13:31Z","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 energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time in which to perform the measurement. This paper considers the possibility that $\\Delta E$, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behavior of a classical particle in a one-dimensional periodic potential $V(x)=-\\cos(x)$ is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitativel","authors_text":"Carl M. Bender, Tanwa Arpornthip","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-05-28T16:12:19Z","title":"Conduction bands in classical periodic potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.4694","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:dfd1026dc0ac6f1f66c9db6272d11337ec0ed1f4455855b6e841f4e9075d3461","target":"record","created_at":"2026-05-18T02:13:31Z","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":"683bc0f316fb4cb1020bf0bd757928b9a0b251e19081884d6363d6dd3c3d26bd","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-05-28T16:12:19Z","title_canon_sha256":"d73a320cf4b887d3b0c0a479da0a1abc97ca9a26d882ed0501785cdc8d72ed2f"},"schema_version":"1.0","source":{"id":"0905.4694","kind":"arxiv","version":1}},"canonical_sha256":"c86201b3a4c02afc37d8a8184f9fe5588dc98dcfbe56fac884738f936aaa63b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c86201b3a4c02afc37d8a8184f9fe5588dc98dcfbe56fac884738f936aaa63b4","first_computed_at":"2026-05-18T02:13:31.043357Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:13:31.043357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7JxqBuTgoGNgal9UHn9yv5AsHi3k00r5zM/meE+diNdzJ5BKVSSE7SB9wnjVvEnC+xk9wT+vm0kz+fyxp5hzAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:13:31.044188Z","signed_message":"canonical_sha256_bytes"},"source_id":"0905.4694","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dfd1026dc0ac6f1f66c9db6272d11337ec0ed1f4455855b6e841f4e9075d3461","sha256:a64a48b46b79904e7c9e2c197daa43fc84c3f3d1eb4d0124a55a29a1d3d4d7ec"],"state_sha256":"76d316c7f4543013ed0cbaee16a003a12443c07ac696a6e43a243550bbc898c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ojkWH2W7JLSjdeMU27d7zh/F1xtFxF1N588SoX4ESGR8yIGSayYrJAoYZacY+9jFhDW8ekIJ5v2spJOH5/0HCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T05:32:51.536085Z","bundle_sha256":"6682ce6185a3c38941f9eb6615156b4982825e726ca3f643b9f2e77ef442c6ad"}}