{"paper":{"title":"Generalized Hermite Polynomials and Spectral Degeneracies of a Singular Sextic Oscillator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Davide Guzzetti, Dmitrii Rachenkov","submitted_at":"2026-06-29T14:25:08Z","abstract_excerpt":"We study a quasi-exactly solvable singular sextic oscillator and its algebraic spectrum. For a distinguished range of parameters, we prove that the discriminant of the characteristic polynomial of the matrix determining the algebraic spectrum admits a natural factorization into three factors. One of these factors is the square of a generalized Hermite polynomial $H_{mn}$, whose zeros are poles of a rational solution of the fourth Painlev\\'e equation. Hence, the spectral degeneracies (level crossing points) corresponding to a component of the discriminant locus are in exact correspondence with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30357","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/2606.30357/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"}