{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:5PPC3QCE4EQFLBOYETWLP4WOH6","short_pith_number":"pith:5PPC3QCE","schema_version":"1.0","canonical_sha256":"ebde2dc044e1205585d824ecb7f2ce3f9b042e1f2a8145ae5bff0248d4a3cee9","source":{"kind":"arxiv","id":"2309.06044","version":2},"attestation_state":"computed","paper":{"title":"Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"\\c{S}eng\\\"ul Kuru, Lorena Acevedo, Yi\\u{g}it Can Acar","submitted_at":"2023-09-12T08:28:34Z","abstract_excerpt":"In this paper, we search the factorizations of the shape invariant Hamiltonians with Scarf II potential. We find two classes; one of them is the standard real factorization which leads us to a real hierarchy of potentials and their energy levels; the other one is complex and it leads us naturally to a hierarchy of complex Hamiltonians. We will show some properties of these complex Hamiltonians: they are not parity-time (or PT) symmetric, but their spectrum is real and isospectral to the Scarf II real Hamiltonian hierarchy. The algebras for real and complex shift operators (also called potentia"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2309.06044","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2023-09-12T08:28:34Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"46a5513c1fb1d4a1fa001dd86440eb6a71c3dafdaacf31c6f9e6ef53906dd748","abstract_canon_sha256":"da825776ea7dc25440e235cc842d800dc8d25a0944e9cc4cfbcf8f056efbd558"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:30:53.284250Z","signature_b64":"bdHmZPN8SDZY4wx+8ghLbToXPqJ2M8U/Y9K1w2P+YfksBmlTKzMBfrcdjdEWa9FuKHFaBh0CeFlVAxCpnAeODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebde2dc044e1205585d824ecb7f2ce3f9b042e1f2a8145ae5bff0248d4a3cee9","last_reissued_at":"2026-07-05T07:30:53.283771Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:30:53.283771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"\\c{S}eng\\\"ul Kuru, Lorena Acevedo, Yi\\u{g}it Can Acar","submitted_at":"2023-09-12T08:28:34Z","abstract_excerpt":"In this paper, we search the factorizations of the shape invariant Hamiltonians with Scarf II potential. We find two classes; one of them is the standard real factorization which leads us to a real hierarchy of potentials and their energy levels; the other one is complex and it leads us naturally to a hierarchy of complex Hamiltonians. We will show some properties of these complex Hamiltonians: they are not parity-time (or PT) symmetric, but their spectrum is real and isospectral to the Scarf II real Hamiltonian hierarchy. The algebras for real and complex shift operators (also called potentia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.06044","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2309.06044/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2309.06044","created_at":"2026-07-05T07:30:53.283830+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.06044v2","created_at":"2026-07-05T07:30:53.283830+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.06044","created_at":"2026-07-05T07:30:53.283830+00:00"},{"alias_kind":"pith_short_12","alias_value":"5PPC3QCE4EQF","created_at":"2026-07-05T07:30:53.283830+00:00"},{"alias_kind":"pith_short_16","alias_value":"5PPC3QCE4EQFLBOY","created_at":"2026-07-05T07:30:53.283830+00:00"},{"alias_kind":"pith_short_8","alias_value":"5PPC3QCE","created_at":"2026-07-05T07:30:53.283830+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.01326","citing_title":"Flyby-induced displacement: analytic solution","ref_index":39,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6","json":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6.json","graph_json":"https://pith.science/api/pith-number/5PPC3QCE4EQFLBOYETWLP4WOH6/graph.json","events_json":"https://pith.science/api/pith-number/5PPC3QCE4EQFLBOYETWLP4WOH6/events.json","paper":"https://pith.science/paper/5PPC3QCE"},"agent_actions":{"view_html":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6","download_json":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6.json","view_paper":"https://pith.science/paper/5PPC3QCE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.06044&json=true","fetch_graph":"https://pith.science/api/pith-number/5PPC3QCE4EQFLBOYETWLP4WOH6/graph.json","fetch_events":"https://pith.science/api/pith-number/5PPC3QCE4EQFLBOYETWLP4WOH6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6/action/storage_attestation","attest_author":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6/action/author_attestation","sign_citation":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6/action/citation_signature","submit_replication":"https://pith.science/pith/5PPC3QCE4EQFLBOYETWLP4WOH6/action/replication_record"}},"created_at":"2026-07-05T07:30:53.283830+00:00","updated_at":"2026-07-05T07:30:53.283830+00:00"}