{"paper":{"title":"Maximum Decay Rate for 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":"2012-07-09T13:13:19Z","abstract_excerpt":"In this paper, we consider global solutions for the following nonlinear Schr\\\"odinger equation $iu_t+\\Delta u+\\lambda|u|^\\alpha u=0,$ in $\\R^N,$ with $\\lambda\\in\\R$ and $0\\le\\alpha<\\frac{4}{N-2}$ $(0\\le\\alpha<\\infty$ if $N=1).$ We show that no nontrivial solution can decay faster than the solutions of the free Schr\\\"odinger equation, provided that $u(0)$ lies in the weighted Sobolev space $H^1(\\R^N)\\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\\R^N),$ or in $L^2(\\R^N),$ according to the different cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2032","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"}