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We prove that the decay of the eigenfunctions of harmonic oscillators,uniform electric or magnetic fields is not stable under 0-order complexperturbations, even if bounded, of these Hamiltonians, in the sense that we canproduce solutions to the evolutionary Schr\"odinger flows associated to theHamiltonians, with a stronger Gaussian decay at two distinct times. We thencharacterize, in a quantitative way, the sharpest possible Gaussian decay ofsolutions as a function of the oscillation frequency or the strength of thefield, depending on the Hamiltonian which is considered. This is connected tothe Hardy's Uncertainty Principle for free Schr\"odinger evolutions.

Gaussian decay of Harmonic Oscillators and related models