{"paper":{"title":"Dispersive estimates for matrix Schr\\\"{o}dinger operators in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M. Burak Erdo\\u{g}an, William R. Green","submitted_at":"2012-11-16T21:01:11Z","abstract_excerpt":"We consider the non-selfadjoint operator [\\cH = [{array}{cc} -\\Delta + \\mu-V_1 & -V_2 V_2 & \\Delta - \\mu + V_1 {array}]] where $\\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\\R^2)\\times L^1(\\R^2)\\to L^\\infty(\\R^2)\\times L^\\infty(\\R^2)$ dispersive decay estimates for the evolution $e^{it\\cH}P_{ac}$. We also obtain the following weighted estimate $$ \\|w^{-1} e^{it\\cH}P_{ac}f\\|_{L^\\infty(\\R^2)\\times L^\\infty(\\R^2)}\\les \\f1{|t|\\log^2(|t|)} \\|w f\\|_{L^1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4036","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"}