{"paper":{"title":"${\\ell}$-oscillators from second-order invariant PDEs of the centrally extended Conformal Galilei Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"F. Toppan, N. Aizawa, Z. Kuznetsova","submitted_at":"2014-12-31T12:33:53Z","abstract_excerpt":"We construct, for any given ${\\ell}=\\frac{1}{2}+{\\mathbb{N}}_0$, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. \\par At the given ${\\ell}$, two invariant equations in one time and ${\\ell}+\\frac{1}{2}$ space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schr\\\"odinger equation (recovered for ${\\ell}=\\frac{1}{2}$) in $1+1$ dimension. The second equation (the \"$\\ell$-oscillator\") possesses a discrete, positive spectrum. It generalizes the $1+1$-dimensional harmonic oscillator (recovere"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00121","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"}