We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].
Abstract We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting from an interacting N-particle system of bosons. We consider the interaction potential to be given either by
Wβ(x)=N−1+2βW(Nβx) Wβ(x)=N-1+2βW(Nβx) , for any
β>0 β>0 , or to be given by
VN(x)=e2NV(eNx) VN(x)=e2NV(eNx) , for some spherical symmetric, nonnegative and compactly supported
W,V∈L∞(R2,R) W,V∈L∞(R2,R) . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential
VN VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.
Consider a system of
N bosons in three dimensions interacting via a repulsive short range pair potential
N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let
HN denote the Hamiltonian of the system and let
ψN,t be the solution to the Schrodinger equation. Suppose that the initial data
ψN,0 satisfies the energy condition
≤CN>. and that the one-particle density matrix converges to a projection as
N→∞. Then, we prove that the
k-particle density matrices of
ψN,t factorize in the limit
N→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential
V. In \cite{ESY}, we proved the same statement under the condition that the interaction potential
V is sufficiently small; in the present work we extend this result to large potentials.
Abstract We derive, in the Gross–Pitaevskii (GP) limit, the time-dependent two-component GP equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose–Einstein condensate. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number N . This improves upon the works of Michelangeli and Olgiati 2014 J. Nonlinear Math. Phys. 24 466–4; Olgiati 2017 Effective Non-Linear Dynamics of Binary Condensates and Open Problems pp 239–56) by providing a sharper, N -dependent, time-independent convergence rate, and also implies a uniform-in-time bound for the one-component case. Furthermore, our approach extends the framework of Benedikter et al (2015 Commun. Pure Appl. Math., 68 1399–482) to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime.
Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.