{"paper":{"title":"On the splitting method for the nonlinear Schr\\\"odinger equation with initial data in $H^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.AP","authors_text":"Woocheol Choi, Youngwoo Koh","submitted_at":"2016-10-19T14:22:56Z","abstract_excerpt":"In this paper, we establish a convergence result for the operator splitting scheme $Z_{\\tau}$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schr\\\"odinger equation :\n  $$\n  \\partial_t u = i \\Delta u + i\\lambda |u|^{p} u,\\qquad u (x,0) =\\phi (x),\n  $$ where $(x,t) \\in \\mathbb{R}^d \\times [0,\\infty)$, with $0< p < 4/(d-2)$ for $d\\geq3$ and $0< p<\\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\\mathcal{O}(\\tau^{1/2})$ for this scheme with initial data in the space $H^1 (\\mathbb{R}^d)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06028","kind":"arxiv","version":3},"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"}