Asymptotic analysis of time-fractional quantum diffusion

Abstract

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 $\alpha =\beta$.