  We consider a class of Schrodinger equations with time-dependent smooth
magnetic and electric potentials having a growth at infinity at most linear and
quadratic, respectively. We study the convergence in $L^p$ with loss of
derivatives, $1<p<\infty$, of the time slicing approximations of the
corresponding Feynman path integral. The results are completely sharp and hold
for long time, where no smoothing effect is available. The techniques are based
on the decomposition and reconstruction of functions and operators with respect
to certain wave packets in phase space.
