  We make progress on a question by Vemuri on the optimal Gaussian decay of
harmonic oscillators, proving the original conjecture up to an arithmetic
progression of times. The techniques used are a suitable translation of the
problem at hand in terms of the free Schr\"odinger equation, the machinery
developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a
lemma which relates decay on average to pointwise decay.
  Such a lemma produces many more consequences in terms of equivalences of
uncertainty principles. Complementing such results, we provide endpoint results
in particular classes induced by certain Laplace transforms, both to the decay
Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the
full endpoint question.
