---
_id: '2747'
abstract:
- lang: eng
text: 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.
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Benjamin
full_name: Schlein, Benjamin
last_name: Schlein
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
citation:
ama: Erdös L, Schlein B, Yau H. Derivation of the Gross-Pitaevskii hierarchy for
the dynamics of Bose-Einstein condensate. *Communications on Pure and Applied
Mathematics*. 2006;59(12):1659-1741. doi:10.1002/cpa.20123
apa: Erdös, L., Schlein, B., & Yau, H. (2006). Derivation of the Gross-Pitaevskii
hierarchy for the dynamics of Bose-Einstein condensate. *Communications on Pure
and Applied Mathematics*. Wiley-Blackwell. https://doi.org/10.1002/cpa.20123
chicago: Erdös, László, Benjamin Schlein, and Horng Yau. “Derivation of the Gross-Pitaevskii
Hierarchy for the Dynamics of Bose-Einstein Condensate.” *Communications on
Pure and Applied Mathematics*. Wiley-Blackwell, 2006. https://doi.org/10.1002/cpa.20123.
ieee: L. Erdös, B. Schlein, and H. Yau, “Derivation of the Gross-Pitaevskii hierarchy
for the dynamics of Bose-Einstein condensate,” *Communications on Pure and Applied
Mathematics*, vol. 59, no. 12. Wiley-Blackwell, pp. 1659–1741, 2006.
ista: Erdös L, Schlein B, Yau H. 2006. Derivation of the Gross-Pitaevskii hierarchy
for the dynamics of Bose-Einstein condensate. Communications on Pure and Applied
Mathematics. 59(12), 1659–1741.
mla: Erdös, László, et al. “Derivation of the Gross-Pitaevskii Hierarchy for the
Dynamics of Bose-Einstein Condensate.” *Communications on Pure and Applied Mathematics*,
vol. 59, no. 12, Wiley-Blackwell, 2006, pp. 1659–741, doi:10.1002/cpa.20123.
short: L. Erdös, B. Schlein, H. Yau, Communications on Pure and Applied Mathematics
59 (2006) 1659–1741.
date_created: 2018-12-11T11:59:23Z
date_published: 2006-12-01T00:00:00Z
date_updated: 2021-01-12T06:59:26Z
day: '01'
doi: 10.1002/cpa.20123
extern: 1
intvolume: ' 59'
issue: '12'
month: '12'
page: 1659 - 1741
publication: Communications on Pure and Applied Mathematics
publication_status: published
publisher: Wiley-Blackwell
publist_id: '4145'
quality_controlled: 0
status: public
title: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein
condensate
type: journal_article
volume: 59
year: '2006'
...