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

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