{"paper":{"title":"Convergence to Scattering States in the Nonlinear Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"LJLL), Pascal B\\'egout (IMT","submitted_at":"2012-07-09T13:14:28Z","abstract_excerpt":"In this paper, we consider global solutions of the following nonlinear Schr\\\"odinger equation $iu_t+\\Delta u+\\lambda|u|^\\alpha u = 0,$ in $\\R^N,$ with $\\lambda\\in\\R,$ $\\alpha\\in(0,\\frac{4}{N-2})$ $(\\alpha\\in(0,\\infty)$ if $N=1)$ and \\linebreak $u(0)\\in X\\equiv H^1(\\R^N)\\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-it\\Delta}u(t)-u_ \\pm\\to0$ in $X$ as $t\\to\\pm\\infty$ then $u(t)-e^{it\\Delta}u_\\pm\\to0$ in $X$ as $t\\to\\pm\\infty.$ We also study the converse. Finally, we estimate $|\\:\\|u(t)\\|_X-\\|e^{it\\Delta}u_\\pm\\|_X\\:|$ under some less restrictive "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2034","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"}