{"paper":{"title":"Some Results on the Scattering Theory for Nonlinear Schr\\\"{o}dinger Equations in Weighted $L^{2}$ Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Wei Dai","submitted_at":"2011-08-16T05:51:07Z","abstract_excerpt":"We investigate the scattering theory for the nonlinear Schr\\\"{o}dinger equation $i \\partial_{t}u+ \\Delta u+\\lambda|u|^\\alpha u=0$ in $\\Sigma=H^{1}(\\mathbb{R}^{d})\\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\\pm}$ exist in $\\Sigma$ when $\\alpha_{d}<\\alpha<\\frac{4}{d-2}$, $d\\geq3$, $\\lambda\\in \\mathbb{R}$ with certain smallness assumption on the initial data $u_{0}$, and when $\\alpha(d)\\leq \\alpha< \\frac{4}{d-2}$($\\alpha\\in [\\alpha(d), \\infty)$, if $d=1,2$), $\\lambda>0$ under suitable conditions on $u_{0}$, where $\\alpha_{d}$, $\\alpha(d)$ are the positive root of the polynomial $d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3158","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"}