{"paper":{"title":"On the modified scattering of $3$-d Hartree type fractional Schr\\\"odinger equations with Coulomb potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changhun Yang, Gyeongha Hwang, Yonggeun Cho","submitted_at":"2017-10-23T23:55:33Z","abstract_excerpt":"In this paper we study 3-d Hartree type fractional Schr\\\"odin-ger equations: \\begin{equation} i\\partial_{t}u-|\\nabla|^{\\alpha}u = \\lambda\\left(|x|^{-\\gamma} *| u|^{2} \\right)u,\\;\\;1 < \\alpha < 2,\\;\\;0 < \\gamma < 3,\\;\\; \\lambda \\in \\mathbb R \\setminus \\{0\\}. \\end{equation} In \\cite{cho} it is known that no scattering occurs in $L^2$ for the long range ($0 < \\gamma \\le 1$). In \\cite{c0, chooz2, cho1} the short-range scattering ($1 < \\gamma < 3$) was treated for the scattering in $H^s$. In this paper we consider the critical case ($\\gamma = 1$) and prove a modified scattering in $L^\\infty$ on the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08552","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"}