Exponential Decay outside of the Light Cone for the Pseudo-Relativistic Non-Autonomous Schr\"odinger Equation

Abstract

We establish a maximal velocity bound for a pseudo-relativistic quantum particle in an external time-dependent potential. Our estimate shows that the probability for the particle, starting in a convex set $X\subset\mathbb{R}^d$ at $t=0$, to reach a convex set $Y\subset\mathbb{R}^d$ at a time $t>0$, is bounded by $e^{-2\delta}$ where $\delta$ is the distance from $Y$ to the section at time $t$ of the light cone generated by $X$.