Solvable model for pair excitation in trapped Boson gas at zero temperature

In Bose-Einstein condensation (BEC), particles occupy a single-particle quantum state, Phi, macroscopically. At zero temperature, the wavefunction for Phi is usually described via a nonlinear Schrodinger equation (NSE). Our goal is to study time-dependent nonlocal effects beyond the NSE in trapped atomic gases. We adopt the view that atoms are excited from Phi in pairs: the scattering from Phi to other states at positions x and y is described by the pair-excitation function, K-0(x, y, t) (Wu 1961 J. Math. Phys. 2 105). This function satisfies a nonlinear, dispersive integrodifferential equation coupled with the NSE. We solve these equations under a slowly varying external potential by assuming that Phi is stationary. For zero initial excitation (K-0 = 0 at t = 0) and sufficiently large t, we evaluate K-0 asymptotically for any distance vertical bar x-y vertical bar. Implications of these results are discussed, particularly the connection of non-equilibrium properties to the coalescence of critical points in the Fourier space.