Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate

Abstract

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^{2}V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let ${\psi }_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data ${\psi }_{N,0}$ satisfies the energy condition \[ < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k \] for $k=1,2,...$. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to \infty$. We prove that the $k$-particle density matrices of ${\psi }_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of ${\psi }_{N,0}$ is assumed in a stronger sense.

Latex file, 66 pages; new version, with an appendix to include a new class of inital states

References in corpus (2)

• Mean Field Dynamics of Boson Stars
• Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

Cited by in corpus (4)

• Mean Field Dynamics of Boson Stars
• Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
• Derivation of the two dimensional nonlinear Schrodinger equation from many body quantum dynamics
• Ground State Energy of Dilute Bose Gas in Small Negative Potential Case