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res:
bibo_abstract:
- Consider a system of N bosons on the three-dimensional unit torus interacting
via a pair potential N 2V(N(x i - x j)) where x = (x i, . . ., x N) denotes the
positions of the particles. Suppose that the initial data ψ N,0 satisfies the
condition 〈ψ N,0, H 2 Nψ N,0) ≤ C N 2 where H N is the Hamiltonian of the Bose
system. This condition is satisfied if ψ N,0 = W Nφ N,t where W N is an approximate
ground state to H N and φ N,0 is regular. Let ψ N,t denote the solution to the
Schrödinger equation with Hamiltonian H N. Gross and Pitaevskii proposed to model
the dynamics of such a system by a nonlinear Schrödinger equation, the Gross-Pitaevskii
(GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so
that if u t solves the GP equation, then the family of k-particle density matrices
⊗ k |u t?〉 〈 t | solves the GP hierarchy. We prove that as N → ∞ the limit points
of the k-particle density matrices of ψ N,t are solutions of the GP hierarchy.
Our analysis requires that the N-boson dynamics be described by a modified Hamiltonian
that cuts off the pair interactions whenever at least three particles come into
a region with diameter much smaller than the typical interparticle distance. Our
proof can be extended to a modified Hamiltonian that only forbids at least n particles
from coming close together for any fixed n.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: László
foaf_name: László Erdös
foaf_surname: Erdös
foaf_workInfoHomepage: http://www.librecat.org/personId=4DBD5372-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0001-5366-9603
- foaf_Person:
foaf_givenName: Benjamin
foaf_name: Schlein, Benjamin
foaf_surname: Schlein
- foaf_Person:
foaf_givenName: Horng
foaf_name: Yau, Horng-Tzer
foaf_surname: Yau
bibo_doi: 10.1002/cpa.20123
bibo_issue: '12'
bibo_volume: 59
dct_date: 2006^xs_gYear
dct_publisher: Wiley-Blackwell@
dct_title: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein
condensate@
...