  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.
