Validity and significance of time-dependent Hartree approximation for a one-dimensional system of bosons with attractiveδ-function interactions

The Hartree (mean-field) approximation to the description of the scattering of two heavy bound states is studied in a model consisting of bosons with attractive $\delta$-function interactions. The approximation is derived from matrix elements of the Heisenberg equations of motion of the system and requires careful enumeration of all necessary approximations. The arguments made are verified by comparison of exact and approximate scattering amplitudes. The derivation also yields the physical significance of the amplitude which satisfies the Hartree approximation; it is a Fourier sum over amplitudes for different channels, which can in prinicple be recovered individually. The approach is not restricted to the model studied.