{"volume":59,"doi":"10.1002/cpa.20123","type":"journal_article","publication_status":"published","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"László Erdös","orcid":"0000-0001-5366-9603","first_name":"László"},{"last_name":"Schlein","full_name":"Schlein, Benjamin","first_name":"Benjamin"},{"first_name":"Horng","full_name":"Yau, Horng-Tzer","last_name":"Yau"}],"date_updated":"2021-01-12T06:59:26Z","page":"1659 - 1741","publication":"Communications on Pure and Applied Mathematics","date_published":"2006-12-01T00:00:00Z","publisher":"Wiley-Blackwell","issue":"12","month":"12","publist_id":"4145","day":"01","extern":1,"title":"Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate","quality_controlled":0,"abstract":[{"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.","lang":"eng"}],"year":"2006","intvolume":"        59","_id":"2747","status":"public","citation":{"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.","ieee":"L. Erdös, B. Schlein, and H. Yau, “Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate,” <i>Communications on Pure and Applied Mathematics</i>, vol. 59, no. 12. Wiley-Blackwell, pp. 1659–1741, 2006.","ama":"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>","short":"L. Erdös, B. Schlein, H. Yau, Communications on Pure and Applied Mathematics 59 (2006) 1659–1741.","apa":"Erdös, L., Schlein, B., &#38; Yau, H. (2006). Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell. <a href=\"https://doi.org/10.1002/cpa.20123\">https://doi.org/10.1002/cpa.20123</a>","chicago":"Erdös, László, Benjamin Schlein, and Horng Yau. “Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate.” <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell, 2006. <a href=\"https://doi.org/10.1002/cpa.20123\">https://doi.org/10.1002/cpa.20123</a>.","mla":"Erdös, László, et al. “Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate.” <i>Communications on Pure and Applied Mathematics</i>, vol. 59, no. 12, Wiley-Blackwell, 2006, pp. 1659–741, doi:<a href=\"https://doi.org/10.1002/cpa.20123\">10.1002/cpa.20123</a>."},"date_created":"2018-12-11T11:59:23Z"}