Transport of a quantum particle in a time-dependent white-noise potential

We show that a quantum particle in $mathbb{R}^d$, for $d geq 1$, subject to a white-noise potential, moves super-ballistically in the sense that the mean square displacement $int |x|^2 langle rho(x,x,t) rangle ~dx$ grows like $t^{3}$ in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension (see refs. Fischer, Leschke, Muller and Javannar, Kumar}. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice $mathbb{Z}^d$, for $d geq 1$, the mean square displacement is diffusive growing like $t^{1}$. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice $mathbb{Z}^d$ with a time-dependent Markovian potential (see ref. Kang, Schenker).