Rigorous Derivation of the Gross-Pitaevskii Equation

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schrödinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.

Figure 1

Diagrammatic expansion for ${\omega }_{m,t}^{(k)}$ as a sum of Feynman graphs with $m$ vertices, $2(k+m)$ leaves, and $2k$ roots.