{"paper":{"title":"Energy-critical NLS with quadratic potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Monica Visan, Rowan Killip, Xiaoyi Zhang","submitted_at":"2006-11-13T16:41:59Z","abstract_excerpt":"We consider the defocusing $\\dot H^1$-critical nonlinear Schr\\\"odinger equation in all dimensions ($n\\geq 3$) with a quadratic potential $V(x)=\\pm \\tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\\nabla u_0(x), xu_0(x) \\in L^2$. In view of the potential $V$, this is the natural energy space. In the repulsive case, we also prove scattering.\n  We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611394","kind":"arxiv","version":2},"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"}