A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

Abstract

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the k-th marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see [MS], [Lu]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H 1 -regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional a N introduced by Pickl in [Pi].

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

“…Instead of describing the condensate as the vacuum of a Fock space of fluctuations, this approach remains in the N -body setting and uses projection operators to factor out the condensate. This strategy was successfully applied to prove effective dynamics for N -boson systems in various situations, e.g., [4,8,17,30,31,34,40,41].…”

“…which implies that assumption A3 is satisfied. Let us conclude the discussion of our assumptions with a remark on the relation between A3 and the so-called Wick property of quasi-free states 4 For a quasi-free state χ on a Fock space F, it is known (e.g. [44,Lemma 5]) that for every a ≥ 1, there exists a constant C a > 0 such that…”

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

“…In case β = 0, the approximation of the form (10) has been studied in [38,37,3,39]. Note that (10) is stronger than the standard definition of the Bose-Einstein condensation…”

Central limit theorem for Bose gases interacting through singular potentials

“…For β = 0 (mean-field regime), the solution of (1.2) can be approximated by products of the Hartree equation i∂ t ϕ t = −∆ϕ t + V * |ϕ t | 2 ϕ t with initial data ϕ 0 ∈ L 2 (R 3 ). See for example [1,2,3,4,5,9,15,19,20,21,23,24,31]. For 0 < β ≤ 1, on the other hand, the solution ψ N,t of (1.2) can be approximated by the non-linear Schrödinger equation…”