Sharp Hardy uncertainty principle and gaussian profiles of covariant Schr\"odinger evolutions

TL;DR

This paper establishes a precise Hardy uncertainty principle for Schrödinger equations with electromagnetic potentials, demonstrating how solutions can exhibit critical Gaussian decay at two different times, based on convexity properties.

Contribution

It introduces a sharp Hardy uncertainty principle for Schrödinger equations with electromagnetic potentials and provides an example of solutions with critical Gaussian decay.

Findings

Proves a sharp Hardy uncertainty principle for Schrödinger equations with electromagnetic potentials.

Shows existence of solutions with critical Gaussian decay at two different times.

Utilizes logarithmic convexity properties of Schrödinger evolutions.

Abstract

We prove a sharp version of the Hardy uncertainty principle for Schr\"odinger equations with external bounded electromagnetic potentials, based on logarithmic convexity properties of Schr\"odinger evolutions. We provide, in addition, an example of a real electromagnetic potential which produces the existence of solutions with critical gaussian decay, at two distinct times.