  We present a microscopic derivation of the time-dependent Gross-Pitaevskii
equation starting from an interacting N-particle system of Bosons. We prove
convergence of the reduced density matrix corresponding to the exact time
evolution to the projector onto the solution of the respective Gross-Pitaevskii
equation. Our work extends a previous result by one of us (P.P.[44]) to
interaction potentials which need not to be nonnegative, but may have a
sufficiently small negative part. One key estimate in our proof is an operator
inequality which was first proven by Jun Yin, see [49].
