Static and Dynamical, Fractional Uncertainty Principles

TL;DR

This paper investigates the dispersion and uncertainty principles for low-regularity solutions to the Schrödinger equation using fractional weights, revealing concentration phenomena, fluctuations, and multifractality at rational times.

Contribution

It provides a new proof of the fractional uncertainty principle and explores the evolution of Dirac comb initial data, uncovering multifractal behavior and Lévy-like fluctuations.

Findings

Lower bounds for mass concentration derived from fractional uncertainty principles.

Fluctuations at rational times resemble Lévy processes.

Evolution exhibits multifractality and complex fluctuation patterns.

Abstract

We study the process of dispersion of low-regularity solutions to the Schr\"odinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in $\mathbb{R}$. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a L\'evy process. Furthermore, the evolution exhibits multifractality.