We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Correlation Structures, Many-Body Scattering Processes, and the Derivation of the Gross-Pitaevskii Hierarchy.
- Authors
Xuwen Chen; Justin Holmer
- Abstract
We consider the dynamics of N bosons in three dimensions. We assume that the pair interaction is given by N3β−1V(Nβ⋅). By studying an associated many-body wave operator, we introduce a Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the N-body wave function, for β∈(0,1], we derive the Gross-Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the N→∞ limit, all k-body scattering processes vanish if k⩾3 and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for β∈(0,1) and weakly for β=1. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve β<1, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon [48] for β∈(0,1).
- Subjects
STATISTICAL correlation; BOSONS; GROSS-Pitaevskii equations; SPACE-time mathematical models; SCHRODINGER equation; MATHEMATICAL models
- Publication
IMRN: International Mathematics Research Notices, 2016, Vol 2016, Issue 10, p3051
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnv228