{"paper":{"title":"Universal spectral behavior of $x^2(ix)^\\epsilon$ potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","quant-ph"],"primary_cat":"hep-th","authors_text":"Carl M. Bender, Daniel W. Hook","submitted_at":"2012-05-20T15:21:06Z","abstract_excerpt":"The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\\epsilon$ ($\\epsilon$ real) exhibits a phase transition at $\\epsilon=0$. When $\\epsilon\\geq0$, the eigenvalues are all real, positive, discrete, and grow as $\\epsilon$ increases. However, when $\\epsilon<0$ there are only a finite number of real eigenvalues. As $\\epsilon$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $\\epsilon=-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Ham"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4425","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":""},"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"}