Convergence in $L^p$ for Feynman path integrals

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.