Notes on Hardy's uncertainty principle for the Wigner distribution and Schrödinger evolutions

Abstract

For Schrödinger 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 Schrödinger 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 Schrödinger equation, the harmonic oscillator and uniform magnetic potentials, as well as new uniqueness results.