Several weeks ago I wrote about the Fourier uncertainty principle which gives a lower bound on the product of the variance of a function f and the variance of its Fourier transform. This post expands on the earlier post by quoting some results from a recent paper .
The earlier post said that the inequality in the Fourier uncertainty principle is exact when f is proportional to a Gaussian probability density. G. H. Hardy proved this result in 1933 in the form of the following theorem.
Let f be a square-integrable function on the real line and assume f and its Fourier transform satisfy the following bounds
for some constant C. Then if ab > 1/4, then f = 0. And if ab = 1/4, f(x) = c exp(-ax²) for some constant c.
Let’s translate this into probability terms by setting
Now Hardy’s theorem says that if f is bounded by a multiple of a Gaussian density with variance σ² and its Fourier transform is bounded by a multiple of a Gaussian density with variance τ², then the product of the two variances is no greater than 1. And if the product of the variances equals 1, then f is a multiple of a Gaussian density with variance σ².
Theorem 3 in  says that if u(t, x) is a solution to the free Schrödinger’s equation
then u at different points in time satisfies a theorem similar to Hardy’s theorem. In fact, the authors show that this theorem is equivalent to Hardy’s theorem.
Specifically, if u is a sufficiently smooth solution and
then αβ > (4T)-2 implies u(t, x) = 0, and αβ = (4T)-2 implies
 Aingeru Fernández-Bertolin and Eugenia Malinnikova. Dynamical versions of Hardy’s uncertainty principle: A survey. Bulletin of the American Mathematical Society. DOI: https://doi.org/10.1090/bull/1729