{"paper":{"title":"A Perturbation of the Dunkl Harmonic Oscillator on the Line","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.SP","authors_text":"Carlos Franco, Jes\\'us A. \\'Alvarez L\\'opez, Manuel Calaza","submitted_at":"2014-12-15T16:10:08Z","abstract_excerpt":"Let $J_\\sigma$ be the Dunkl harmonic oscillator on ${\\mathbb{R}}$ ($\\sigma>-\\frac{1}{2}$). For $0<u<1$ and $\\xi>0$, it is proved that, if $\\sigma>u-\\frac{1}{2}$, then the operator $U=J_\\sigma+\\xi|x|^{-2u}$, with appropriate domain, is essentially self-adjoint in $L^2({\\mathbb{R}},|x|^{2\\sigma} dx)$, the Schwartz space ${\\mathcal{S}}$ is a core of $\\overline U^{1/2}$, and $\\overline U$ has a discrete spectrum, which is estimated in terms of the spectrum of $\\overline{J_\\sigma}$. A generalization $J_{\\sigma,\\tau}$ of $J_\\sigma$ is also considered by taking dif\\/ferent parameters $\\sigma$ and $\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4655","kind":"arxiv","version":9},"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"}