Asymptotic analysis of time-fractional quantum diffusion

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrodinger equation in $\mathbb{R}^d$. We define the time-fractional derivative by the Caputo derivative and we consider the initial-value problem for the free evolution of wave packets in $\mathbb{R}^d$ governed by the time-fractional Schrodinger equation $i^\beta \partial_t^\alpha u = - \Delta u, ~~~~u(t=0) = u_0$, parameterized by two indices $\alpha, \beta \in (0,1]$. We show distinctly different long-time evolution of the mean square displacement according to the relation between $\alpha$ and $\beta$. In particular, asymptotically ballistic motion occurs only for $\alpha=\beta$.