BOSE-EINSTEIN CONDENSATION BEYOND MEAN FIELD: MANY-BODY BOUND STATE OF PERIODIC MICROSTRUCTURE

We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrodinger-type equation, the Gross-Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105-123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro-partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, Phi. For a scattering length with periodic microstructure of subscale epsilon, we formally describe via classical homogenization the lowest many-body bound state in terms of Phi and K up to second order in epsilon. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.