{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:HIUAVSGI73KEA6VTOK42OYQH6O","short_pith_number":"pith:HIUAVSGI","schema_version":"1.0","canonical_sha256":"3a280ac8c8fed4407ab372b9a76207f3a81a503522432ae85d8d6ba67981cb5b","source":{"kind":"arxiv","id":"math/0411143","version":1},"attestation_state":"computed","paper":{"title":"Schr\\\"odinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues","license":"","headline":"","cross_cats":["hep-th","math-ph","math.MP","quant-ph"],"primary_cat":"math.SP","authors_text":"Kwang C. Shin","submitted_at":"2004-11-07T08:37:59Z","abstract_excerpt":"For integers $m\\geq 3$ and $1\\leq\\ell\\leq m-1$, we study the eigenvalue problem $-u^{\\prime\\prime}(z)+[(-1)^{\\ell}(iz)^m-P(iz)]u(z)=\\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\\arg z=-\\frac{\\pi}{2}\\pm \\frac{(\\ell+1)\\pi}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $\\lambda_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials fr"},"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/0411143","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.SP","submitted_at":"2004-11-07T08:37:59Z","cross_cats_sorted":["hep-th","math-ph","math.MP","quant-ph"],"title_canon_sha256":"47b2577a9e9899fd8faed3f0f99f6841200c56d7eb8f5c3fbce1f6de2c2226c2","abstract_canon_sha256":"b8e5a33a63dcf0fcbaed884402270a9e7235d61c51c63364d53017dade88f064"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:20.330688Z","signature_b64":"gOx+fe0iyWRMX11hixfH2GcVCDFa77GfDB0DDvV/gAONmnFZHSzBlqS1/NF4qyEkJsPtzOkldF+HSJE9xwbKDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a280ac8c8fed4407ab372b9a76207f3a81a503522432ae85d8d6ba67981cb5b","last_reissued_at":"2026-07-04T14:39:20.330294Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:20.330294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Schr\\\"odinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues","license":"","headline":"","cross_cats":["hep-th","math-ph","math.MP","quant-ph"],"primary_cat":"math.SP","authors_text":"Kwang C. Shin","submitted_at":"2004-11-07T08:37:59Z","abstract_excerpt":"For integers $m\\geq 3$ and $1\\leq\\ell\\leq m-1$, we study the eigenvalue problem $-u^{\\prime\\prime}(z)+[(-1)^{\\ell}(iz)^m-P(iz)]u(z)=\\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\\arg z=-\\frac{\\pi}{2}\\pm \\frac{(\\ell+1)\\pi}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $\\lambda_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0411143","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0411143/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/0411143","created_at":"2026-07-04T14:39:20.330367+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0411143v1","created_at":"2026-07-04T14:39:20.330367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0411143","created_at":"2026-07-04T14:39:20.330367+00:00"},{"alias_kind":"pith_short_12","alias_value":"HIUAVSGI73KE","created_at":"2026-07-04T14:39:20.330367+00:00"},{"alias_kind":"pith_short_16","alias_value":"HIUAVSGI73KEA6VT","created_at":"2026-07-04T14:39:20.330367+00:00"},{"alias_kind":"pith_short_8","alias_value":"HIUAVSGI","created_at":"2026-07-04T14:39:20.330367+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":83,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O","json":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O.json","graph_json":"https://pith.science/api/pith-number/HIUAVSGI73KEA6VTOK42OYQH6O/graph.json","events_json":"https://pith.science/api/pith-number/HIUAVSGI73KEA6VTOK42OYQH6O/events.json","paper":"https://pith.science/paper/HIUAVSGI"},"agent_actions":{"view_html":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O","download_json":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O.json","view_paper":"https://pith.science/paper/HIUAVSGI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0411143&json=true","fetch_graph":"https://pith.science/api/pith-number/HIUAVSGI73KEA6VTOK42OYQH6O/graph.json","fetch_events":"https://pith.science/api/pith-number/HIUAVSGI73KEA6VTOK42OYQH6O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O/action/storage_attestation","attest_author":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O/action/author_attestation","sign_citation":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O/action/citation_signature","submit_replication":"https://pith.science/pith/HIUAVSGI73KEA6VTOK42OYQH6O/action/replication_record"}},"created_at":"2026-07-04T14:39:20.330367+00:00","updated_at":"2026-07-04T14:39:20.330367+00:00"}