Notes on Hardy?s uncertainty principle for the Wigner distribution and Schr?dinger evolutions

For Schrodinger equations with real quadratic Hamiltonians, it is known that 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 present a general uniqueness result for such Schrodinger equations, where the solution cannot have strong decay at two distinct times. This approach gives new proofs to known, sharp Hardy type estimates for the free Schrodinger equation, the harmonic oscillator and uniform magnetic potentials, as well as new uniqueness results.(c) 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).