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

TL;DR

This paper proves that a pseudo-relativistic quantum particle's probability of reaching a certain region outside its light cone decays exponentially, establishing a maximal velocity bound in a time-dependent potential.

Contribution

It introduces a maximal velocity bound for pseudo-relativistic particles in external potentials, extending understanding of relativistic quantum dynamics with time dependence.

Findings

Probability outside light cone decays exponentially with distance

Maximal velocity bound established for pseudo-relativistic particles

Results apply to particles in time-dependent external potentials

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 $X$.