Maximal Speed of Quantum Propagation

TL;DR

This paper proves a simple bound on the maximal speed of quantum propagation for Schroedinger equations with Kato potentials, showing the probability of finding the system outside a growing ball decays as an inverse power of time.

Contribution

It provides a straightforward proof of the maximal speed bound for Schroedinger equations with general Kato potentials, including explicit constants.

Findings

Probability outside the ball decays as an inverse power of time

Explicit expression for the proportionality constant in terms of maximal energy

No decay at infinity or smoothness required for the time-independent interaction

Abstract

For Schroedinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside the ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness.