2004/10/02 by László Erdős, Erdos, Laszlo, Benjamin Schlein +3 · 2 citations
Physics and Astronomy · #35Q55 #81Q05 #81V70 #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectroscopy and Quantum Chemical Studies #Strong Light-Matter Interactions
paper · pdf · doi:10.48550/arxiv.math-ph/0410005
openalex publication_date 2004/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a system of N bosons on the three dimensional unit torus interacting via a pair potential N2V(N(xi-xj)), where \bx=(x1, ..., xN) denotes the positions of the particles. Suppose that the initial data ψN,0 satisfies the condition lt; ψN,0, HN2 ψN,0 gt; ≤ C N2 where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN,0= WN ϕN,0 where WN is an approximate ground state to HN and ϕN,0 is regular. Let ψN,t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices \⊗k ut, k≥ 1 \ solves the GP hierarchy. We prove that as N→ ∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the N boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least n particles from coming close together, for any fixed n.