  We consider the Carleson's problem regarding small time almost everywhere
convergence to initial data for the Schr\"odinger equation, both linear and
nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the
Schr\"odinger flow, that the (sharp) result proved by Dahlberg and Kenig for
initial data in Sobolev spaces still holds when one considers the full
Schr\"odinger equation with a certain class of potentials. As for
$s<\frac{1}{4}$, the failure of $L^p$-boundedness of the (localized) maximal
operator is investigated.
