Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate

L. Erdös, B. Schlein, H. Yau, Communications on Pure and Applied Mathematics 59 (2006) 1659–1741.

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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.
Publishing Year
Date Published
2006-12-01
Journal Title
Communications on Pure and Applied Mathematics
Volume
59
Issue
12
Page
1659 - 1741
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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
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. https://doi.org/10.1002/cpa.20123
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 59, no. 12 (2006): 1659–1741. https://doi.org/10.1002/cpa.20123.
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, pp. 1659–1741, 2006.
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.
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.

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