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In this paper we use microscopic arguments to derive a nonlinearSchr\"{o}dinger equation for trapped Bose-condensed gases. This is madepossible by considering the equations of motion of various anomalous averages.The resulting equation explicitly includes the effect of repeated binaryinteractions (in particular ladders) between the atoms. Moreover, under theconditions that dressing of the intermediate states of a collision can beignored, this equation is shown to reduce to the conventional Gross-Pitaevskiiequation in the pseudopotential limit. Extending the treatment, we show firsthow the occupation of excited (bare particle) states affects the collisions,and thus obtain the many-body T-matrix approximation in a trap. In addition, wediscuss how the bare particle many-body T-matrix gets dressed by mean fieldsdue to condensed and excited atoms. We conclude that the most commonly usedversion of the Gross-Pitaevskii equation can only be put on a microscopic basisfor a restrictive range of conditions. For partial condensation, we need totake account of interactions between condensed and excited atoms, which, in aconsistent formulation, should also be expressed in terms of the many-bodyT-matrix. This can be achieved by considering fluctuations around thecondensate mean field beyond those included in the conventional finitetemperature mean field, i.e. Hartree-Fock-Bogoliubov (HFB), theory.Comment: Resolved some problems with printing of figure

Microscopic treatment of binary interactions in the nonequilibrium dynamics of partially Bose-condensed trapped gases