{"paper":{"title":"On the reality of the eigenvalues for a class of PT-symmetric oscillators","license":"","headline":"","cross_cats":["hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"K. C. Shin","submitted_at":"2002-01-07T01:19:33Z","abstract_excerpt":"We study the eigenvalue problem -u\"(z)-[(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{2\\pi}{m+2}, where P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z is a real polynomial and m\\geq 2. We prove that if for some 1\\leq j\\leq\\frac{m}{2}, we have (j-k)a_k\\geq 0 for all 1\\leq k\\leq m-1, then the eigenvalues are all positive real. We then sharpen this to a slightly larger class of polynomial potentials.\n  In particular, this implies that the eigenvalues are all positive real for the potentials \\al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0201013","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/math-ph/0201013/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"}