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

We study the large-time asymptotics of the edge current for a family of time-fractional Schrödinger equations with a constant, transverse magnetic field on a half-plane $(x,y)∈Rx+×Ry$⁠. The time-fractional Schrödinger equation is parameterized by two constants (α, β) in (0, 1], where α is the fractional order of the time derivative, and β is the power of i in the Schrödinger equation. We prove that for fixed α, there is a transition in the transport properties as β varies in (0, 1]: For 0 < β < α, the edge current grows exponentially in time, for α = β, the edge current is asymptotically constant, and for β > α, the edge current decays in time. We prove that the mean square displacement in the $y∈R$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin [Phys. Rev. E 62, 3135 (2000)] that the latter two cases, α = β and α < β, are the physically relevant ones.