{"paper":{"title":"Reducibility of quantum harmonic oscillator on $ R^d$ with differential and quasi-periodic in time potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Zhenguo Liang, Zhiguo Wang","submitted_at":"2017-04-22T04:15:22Z","abstract_excerpt":"We improve the results by Gr\\'ebert and Paturel in \\cite{GP} and prove that a linear Schr\\\"odinger equation on $R^d$ with harmonic potential $|x|^2$ and small $t$-quasiperiodic potential as $$ {\\rm i}u_t - \\Delta u+|x|^2u+\\varepsilon V(\\omega t,x)u=0, \\ (t,x)\\in R\\times R^d $$ reduces to an autonomous system for most values of the frequency vector $\\omega\\in R^n$. The new point is that the potential $V(\\theta,\\cdot )$ is only in ${\\mathcal{C}^{\\beta}}(T^n, \\mathcal{H}^{s}(R^d))$ with $\\beta$ large enough. As a consequence any solution of such a linear PDE is almost periodic in time and remains"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06744","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"}