  We prove that the thickness is a necessary and sufficient geometric condition
that ensures the (rapid) stabilization or the approximate null-controllability
with uniform cost of a large class of evolution equations posed on the whole
space $\mathbb{R}^n$ and associated with operators of the form $F(\vert
D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being bounded
below and continuous. We also provide explicit feedbacks and constants
associated with these stabilization properties. Our results apply in particular
for the half heat equation associated with the function $F(t) = t$, for which
null-controllability is known to fail from thick control supports. More
generally, the notion of thickness was known to be a necessary and sufficient
condition for the null-controllability of the fractional heat equations
associated with the functions $F(t) = t^{2s}$ in the case $s>1$, and that this
null-controllability property from thick control supports does not hold when
$0<s\le 1$.
