Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field

Abstract

We study the large-time asymptotics of the edge current for a family of time-fractional Schrodinger equations with a constant, transverse magnetic field on a half-plane $(x,y)\in {\mathbb{R}}_x^{+}\times {\mathbb{R}}_y$. The TFSE is parameterized by two constants $(\alpha ,\beta )$ in $(0,1]$, where $\alpha$ is the fractional order of the time derivative, and $\beta$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $\alpha$, there is a transition in the transport properties as $\beta$ varies in $(0,1]$: For $0<\beta <\alpha$, the edge current grows exponentially in time, for $\alpha =\beta$, the edge current is asymptotically constant, and for $\beta >\alpha$, the edge current decays in time. We prove that the mean square displacement in the $y\in \mathbb{R}$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, $\alpha =\beta$ and $\alpha <\beta$, are the physically relevant ones.