  We consider the time slicing approximations of Feynman path integrals,
constructed via piecewice classical paths. A detailed study of the convergence
in the norm operator topology, in the space $\mathcal{B}(L^2(\mathbb{R}^d))$ of
bounded operators on $L^2$, and even in finer operator topologies, was carried
on by D. Fujiwara in the case of smooth potentials with an at most quadratic
growth. In the present paper we show that the result about the convergence in
$\mathcal{B}(L^2(\mathbb{R}^d))$ remains valid if the potential is only assumed
to have second space derivatives in the Sobolev space $H^{d+1}(\mathbb{R}^d)$
(locally and uniformly), uniformly in time. The proof is non-perturbative in
nature, but relies on a precise short time analysis of the Hamiltonian flow at
this Sobolev regularity and on the continuity in $L^2$ of certain oscillatory
integral operators with non-smooth phase and amplitude.
