{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JOWGT67AL3F4G4QF75QPJA4ZO3","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":"f2b91526eee3cf3af21c18370caa819fc892cab0caea14da774e1ba9a5bcf9d3","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2012-05-20T15:21:06Z","title_canon_sha256":"62b386b598d3179bcc918aa0c18aad23448307d3f3ca787602d6613551e3a914"},"schema_version":"1.0","source":{"id":"1205.4425","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4425","created_at":"2026-05-18T03:23:10Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4425v1","created_at":"2026-05-18T03:23:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4425","created_at":"2026-05-18T03:23:10Z"},{"alias_kind":"pith_short_12","alias_value":"JOWGT67AL3F4","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JOWGT67AL3F4G4QF","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JOWGT67A","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:f4f56f5af2c2127241eab18edb81942c8af1454b4f172a63e8fedcf62a46788b","target":"graph","created_at":"2026-05-18T03:23:10Z","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 PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\\epsilon$ ($\\epsilon$ real) exhibits a phase transition at $\\epsilon=0$. When $\\epsilon\\geq0$, the eigenvalues are all real, positive, discrete, and grow as $\\epsilon$ increases. However, when $\\epsilon<0$ there are only a finite number of real eigenvalues. As $\\epsilon$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $\\epsilon=-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Ham","authors_text":"Carl M. Bender, Daniel W. Hook","cross_cats":["math-ph","math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2012-05-20T15:21:06Z","title":"Universal spectral behavior of $x^2(ix)^\\epsilon$ potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4425","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":"sha256:2d9fb3ba919102d9bff6ac6c286770c06bfef8850a582638953c1170bb42f55a","target":"record","created_at":"2026-05-18T03:23:10Z","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":"f2b91526eee3cf3af21c18370caa819fc892cab0caea14da774e1ba9a5bcf9d3","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2012-05-20T15:21:06Z","title_canon_sha256":"62b386b598d3179bcc918aa0c18aad23448307d3f3ca787602d6613551e3a914"},"schema_version":"1.0","source":{"id":"1205.4425","kind":"arxiv","version":1}},"canonical_sha256":"4bac69fbe05ecbc37205ff60f4839976ff325de61950c64e219ae221d3f4ccf3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4bac69fbe05ecbc37205ff60f4839976ff325de61950c64e219ae221d3f4ccf3","first_computed_at":"2026-05-18T03:23:10.824908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:23:10.824908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EJOs0XSyxbeIRpUeBt8EagSSq4zA/Vp/vB8P1jhSHjsv6WFMvl07OOYRq/um3ylsM3OIjuLC0csxacz5hIYFAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:23:10.825508Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4425","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d9fb3ba919102d9bff6ac6c286770c06bfef8850a582638953c1170bb42f55a","sha256:f4f56f5af2c2127241eab18edb81942c8af1454b4f172a63e8fedcf62a46788b"],"state_sha256":"12082c0af6ab57d5ae57e44b3cbe25c7bead56e3848d5b5b22163a2943d6385e"}