{"paper":{"title":"On reducibility of Quantum Harmonic Oscillator on $\\mathbb{R}^d$ with quasiperiodic in time potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Beno\\^it Gr\\'ebert (LMJL), Eric Paturel (LMJL)","submitted_at":"2016-03-24T07:34:03Z","abstract_excerpt":"We prove that a linear d-dimensional Schr{\\\"o}dinger equation on $\\mathbb{R}^d$  with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\\partial\\_t u -- \\Delta u + |x|^2 u + \\epsilon V (t\\omega, x)u = 0, x \\in \\mathbb{R}^d$  reduces to an autonomous system for most values of the frequency vector $\\omega \\in \\mathbb{R}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07455","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"}