Notes on Hardy's Uncertainty Principle for the Wigner distribution and Schrödinger evolutions

We consider Schrödinger 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 Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach reproduces known, sharp results for the free Schrödinger equation and the harmonic oscillator, and we also present an explicit scheme for quadratic systems based on positive definite matrices.