Beyond mean field: condensate coupled with pair excitations

We prove existence results for a system of partial differential equations describing the approximate condensate wavefunction and pair-excitation kernel of a dilute (T=0) Bose gas in the stationary setting, in the presence of a trapping potential and repulsive pairwise atomic interactions. Notably, the Hartree-type equation for the condensate in this system contains contributions from non-condensate particles, and the pair excitation kernel satisfies a nonlinear operator equation. The equations studied here are inspired by the work of Griffin, who derived this system in the study of finite temperature condensates. The techniques employed include a variational principle, which exploits the connection between unitary Bogoliubov rotations and a nonlinear operator equation for the pair excitation kernel. An iterative procedure for constructing solutions is also included.