Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate
László Erdös
Schlein, Benjamin
Yau, Horng-Tzer
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.
Wiley-Blackwell
2006
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doc-type:article
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http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/2747
Erdös L, Schlein B, Yau H. Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. <i>Communications on Pure and Applied Mathematics</i>. 2006;59(12):1659-1741. doi:<a href="https://doi.org/10.1002/cpa.20123">10.1002/cpa.20123</a>
info:eu-repo/semantics/altIdentifier/doi/10.1002/cpa.20123
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