Mathematical theory and numerical methods for Bose-Einstein condensation

Abstract

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.Comment: 135 pages and 10 figure

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

“…If ν = 0, then we will speak about the time-dependent Linear Schrödinger Equation (LSE). In the Physics literature, the first equation of system (1) is also called the Gross-Pitaevskii Equation (GPE) [2,11,16], when considering Bose-Einstein Condensates (BEC) (see e.g. [38,39]).…”

New approach to derivation of quantum kinetic equations with initial correlations

“…, j n )), n ≥ 1. As a result of this equality we establish the validity of statement (16) for expansion (8) …”

“…We note that in case of initial states specified by sequences (23) the average values (mean values) of marginal observables (8) are determined by the following positive continuous linear functional…”

“…In this paper we consider the problem of the rigorous description of the kinetic evolution in the presence of initial correlations of quantum particles. Such initial states are typical for the condensed states of quantum gases [5][6][7][8] in contrast to the gaseous state. For example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition specified by correlations of the condensed state [5].…”

Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity

“…• Energy conservation: if Ω = 0 and f (|ψ| 2 ) = β |ψ| 2 , the energy conservation [19] is given by: ∀n ∈ N * , E 0,β (ψ n ) = E 0,β (ψ 0 ), where…”