  We study the influence on diffusion in one dimension of a potential energy
perturbation varying as a power in space and time. We concentrate on the case
of a parabolic perturbation in space decaying as $t^{-\omega}$ which shows a
rich variety of scaling behaviours. When $\omega=1$, the perturbation is truly
marginal and leads to anomalous (super)diffusion with a dynamical exponent
varying continuously with the perturbation amplitude below some negative
threshold value. For slower decay, $\omega<1$, the perturbation becomes
relevant and the system is either subdiffusive for an attractive potential or
displays a stretched-exponential behaviour for a repulsive one. Exact results
are obtained for the mean value and the variance of the position as well as for
the surviving probability.
