Notes on Hardy's Uncertainty Principle for the Wigner distribution and Schr\"{o}dinger evolutions

TL;DR

This paper explores Hardy's uncertainty principle in the context of Wigner distributions for Schr"odinger equations with quadratic Hamiltonians, establishing uniqueness results and reproducing sharp known cases.

Contribution

It introduces a new approach using Hardy's uncertainty principle to prove uniqueness for Schr"odinger equations with quadratic Hamiltonians, including explicit schemes for quadratic systems.

Findings

Reproduces sharp results for free Schr"odinger and harmonic oscillator.

Establishes a uniqueness theorem based on decay properties at two times.

Provides an explicit scheme for quadratic systems using positive definite matrices.

Abstract

We consider Schr\"{o}dinger equations with real quadratic Hamiltonians, for which the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we prove a uniqueness result for such Schr\"{o}dinger equations, where the solution cannot have strong decay at two distinct times. This approach reproduces known, sharp results for the free Schr\"{o}dinger equation and the harmonic oscillator, and we also present an explicit scheme for quadratic systems based on positive definite matrices.