Pair Excitations and the Mean Field Approximation of Interacting Bosons, I

In our previous work (Grillakis et al. in Commun Math Phys 294:273–301, 2010; Adv Math 228:1788–1815, 2011) we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In Grillakis et al. (Adv Math 228:1788–1815, 2011) we carried out the analysis assuming that the interactions are independent of the number of particles N. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain a priori estimates independent of the number of particles and use these to compare the exact with the approximate dynamics.