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Shin","submitted_at":"2004-07-01T16:30:21Z","abstract_excerpt":"We study the eigenvalue problem $-u^{\\prime\\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\\arg z=-\\frac{\\pi}{2}\\pm \\frac{2\\pi}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\\geq 3$. We provide an asymptotic expansion of the eigenvalues $\\lambda_n$ as $n\\to+\\infty$, and prove that for each {\\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions."},"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":"math/0407018","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.SP","submitted_at":"2004-07-01T16:30:21Z","cross_cats_sorted":["hep-th","math-ph","math.MP","quant-ph"],"title_canon_sha256":"87d39c1cbe4eef86608c910c7d83f27393698eb9e9f669685e5da19a8f0a62d8","abstract_canon_sha256":"9da98b61119c392ead370603689b32774120ad269d4a45e4aef517c397dfb6db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T16:49:14.966405Z","signature_b64":"HpmIncow3x3NSs3ryDgTlwAJp3vg4hd3uqNXnYbd1geejUAMRhdudbfML27M/LlwwaHMDRYbSP8BDVC5UCbqDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c415192495f690dec08070df056145fd350d1553b7170e4320215b885edd817","last_reissued_at":"2026-07-04T16:49:14.966049Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T16:49:14.966049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalues of PT-symmetric oscillators with polynomial potentials","license":"","headline":"","cross_cats":["hep-th","math-ph","math.MP","quant-ph"],"primary_cat":"math.SP","authors_text":"Kwang C. 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We provide an asymptotic expansion of the eigenvalues $\\lambda_n$ as $n\\to+\\infty$, and prove that for each {\\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0407018","kind":"arxiv","version":3},"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/math/0407018/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":"math/0407018","created_at":"2026-07-04T16:49:14.966110+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0407018v3","created_at":"2026-07-04T16:49:14.966110+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0407018","created_at":"2026-07-04T16:49:14.966110+00:00"},{"alias_kind":"pith_short_12","alias_value":"DRAVDESJL5UQ","created_at":"2026-07-04T16:49:14.966110+00:00"},{"alias_kind":"pith_short_16","alias_value":"DRAVDESJL5UQ33AI","created_at":"2026-07-04T16:49:14.966110+00:00"},{"alias_kind":"pith_short_8","alias_value":"DRAVDESJ","created_at":"2026-07-04T16:49:14.966110+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.23125","citing_title":"Path integral analysis of Schr\\\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states","ref_index":68,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7","json":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7.json","graph_json":"https://pith.science/api/pith-number/DRAVDESJL5UQ33AIA4G7AVQUL7/graph.json","events_json":"https://pith.science/api/pith-number/DRAVDESJL5UQ33AIA4G7AVQUL7/events.json","paper":"https://pith.science/paper/DRAVDESJ"},"agent_actions":{"view_html":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7","download_json":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7.json","view_paper":"https://pith.science/paper/DRAVDESJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0407018&json=true","fetch_graph":"https://pith.science/api/pith-number/DRAVDESJL5UQ33AIA4G7AVQUL7/graph.json","fetch_events":"https://pith.science/api/pith-number/DRAVDESJL5UQ33AIA4G7AVQUL7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7/action/storage_attestation","attest_author":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7/action/author_attestation","sign_citation":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7/action/citation_signature","submit_replication":"https://pith.science/pith/DRAVDESJL5UQ33AIA4G7AVQUL7/action/replication_record"}},"created_at":"2026-07-04T16:49:14.966110+00:00","updated_at":"2026-07-04T16:49:14.966110+00:00"}