Derivation of the time dependent Gross–Pitaevskii equation with external fields

Abstract

Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective.Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross-Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction.Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross-Pitaevskii scaling.

Quantitative Derivation of the Gross‐Pitaevskii Equation

“…In fact, the following result was established in [6,7,8,9,10], and, in a slightly different form, in [21]. Consider a family ψ N ∈ L 2 s (R 3N ) with bounded energy per particle…”

“…For N -particle initial data ψ N ∈ L 2 (R 3N ) obtained projecting down a Fock-state of the form W ( √ N ϕ)Ψ onto the N -particle sector, bounds of the form (21) were established in [2], extending the ideas developed in [22], for arbitrary Ψ ∈ F with finite number of particles and energy (note that this class of N -particle states include factorized wave functions of the form ϕ ⊗N , which are obtained choosing ψ = Ω). Using techniques similar to those proposed in [21], bounds for the rate of convergence towards the nonlinear Hartree dynamics were also obtained in [17], allowing also for potential with strong singularities. A different point of view on the mean field limit was given in [12], where the convergence towards the Hartree dynamics was interpreted as a Egorov-type theorem.…”

“…It seems natural to ask whether the coherent state approach introduced in [22] can be used to obtain a rigorous derivation of the Gross-Pitaevskii equation (11), providing at the same time bounds on the rate of the convergence of the form (21). A major difficulty in this program is immediately clear.…”

Randomization and the Gross–Pitaevskii Hierarchy

“…The derivation of a hierarchy similar to (1) coming from the limit of N -body Schrödinger dynamics was obtained in [1,2,11,[15][16][17]45,48,[51][52][53][54][55][56][57]74,[84][85][86][87]124,162] in various different contexts. The first rate of convergence result was obtained by Rodnianski and Schlein [145] and subsequent rate of convergence results have been obtained in [6,17,38,49,50,75,85,[93][94][95]110,115,121,124,137,138]. The Cauchy problem associated to a hierarchy as in (1) has been studied in its own right in [41][42][43][44]46,47,59].…”

Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

“…Note that when β = 1 (the Gross-Pitaevskii regime), the strong correlations between particles require a subtle correction: the nonlinear term w N * |u(t)| 2 in Hartree equation (3) has to be replaced by 8πa|u(t)| 2 with a being the scattering length of w. This has been justified rigorously in the context of the Bose-Einstein condensation (11); see [34,33,43] for the ground state problem and [18,17,8,45] for the dynamical problem. The norm approximation is completely open.…”