Second-order corrections to mean field evolution for weakly interacting Bosons. I

Inspired by the works of Rodnianski and Schlein and Wu, we derive a new nonlinear Schr\"odinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $v(x)= \epsilon \chi(x) |x|^{-1}$, where $\epsilon$ is sufficiently small and $\chi \in C_0^{\infty}$, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (part II) of this paper.