Hardy's Uncertainty Principle, Convexity and Schr\"odinger Evolutions

TL;DR

This paper establishes new convexity properties of solutions to Schr"odinger and heat equations, leading to generalized Hardy-type uncertainty principles and uniqueness results for these evolutions.

Contribution

It proves logarithmic convexity of decay measures for Schr"odinger solutions, extending Hardy's uncertainty principle and applying similar results to heat equations.

Findings

Logarithmic convexity of decay measures for Schr"odinger solutions

Generalized Hardy's uncertainty principle for Schr"odinger evolutions

Corresponding results for heat evolutions

Abstract

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schr\"odinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.