{"paper":{"title":"On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Nata\\v{s}a Pavlovi\\'c, Thomas Chen","submitted_at":"2008-12-18T18:16:36Z","abstract_excerpt":"We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\\R^d$, $d\\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\\cH_\\xi^\\alpha$ of sequences of marginal density matrices. The regularity is accounted for by $\\alpha>\\frac12& if $d=1$, $\\alpha>\\frac d2-\\frac{1}{2(p-1)} $ if $d\\geq2$ and $(d,p)\\neq(3,2)$, and $\\alpha\\geq1$ if $(d,p)=(3,2)$, where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\\xi>0$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.3463","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"}