Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $S\subset \RR^d$ ensuring that the $L^2$-seminorm associated to $S$ is equivalent to the full $L^2$-norm on $\RR^d$ when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of $S$. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from $S$. In all our results, the set $S$ may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.