Rate of Convergence Towards Hartree Dynamics

Abstract

We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the effective nonlinear Hartree dynamics is at most of the order 1/N, for any fixed time. The N-dependence of the bound is optimal.Comment: 26 page

Quantitative Derivation of the Gross‐Pitaevskii Equation

“…If we assume the initial state to exhibit complete condensation and we accept that condensation is preserved by the time evolution, we should expect γ N,t , however, we also want to take into account the correlations between the two particles. Describing the correlations through the solution of the zero-energy scattering equation (2), we are led to the ansatz…”

“…where f denotes the solution of the zero-energy scattering equation (2). For technical reason, which will become clear later on, it is more convenient for us to work with the solution of the modified Gross-Pitaevskii equation (50), rather than directly with the solution of (32).…”

“…Let f denote the solution of the zero-energy scattering equation (2), with boundary condition f (x) → 1 as |x| → ∞. Then, by Lemma 3.2 below, 0 ≤ f ≤ 1 and therefore (32) and, respectively, of the modified Gross-Pitaevskii equation (50), with initial data ϕ.…”

“…The kernel k ∈ L 2 (R 3 × R 3 ) has to be chosen so that T (k) creates the correct correlations among the particles. Since correlations are, in good approximation, two-body effects, we can describe them through the solution f of the zero-energy scattering equation (2). We write…”

“…Let ϕ ∈ H 4 (R 3 ), and ϕ (N ) t ∈ H 4 (R 3 ) be the solution of (50), with initial data ϕ. Let w(N x) = 1 − f (N x), where f is the solution of the zero-energy scattering equation (2). Let the kernel k t be defined as in (55), so thaṫ…”