{"paper":{"title":"Local existence, global existence, and scattering for the nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ivan Naumkin, Thierry Cazenave","submitted_at":"2016-03-10T10:09:01Z","abstract_excerpt":"In this paper, we construct for every $\\alpha >0$ and $\\lambda \\in {\\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schr\\\"odinger equation \\begin{equation*} \\begin{cases} iu_t + \\Delta u + \\lambda |u|^\\alpha u= 0 \\\\ u(0,x) = u_0 \\end{cases} \\end{equation*} on ${\\mathbb R}^N $. Moreover, we construct for every $\\alpha >\\frac {2} {N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\\to \\infty $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03204","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"}