  In this paper we establish an observability inequality for the heat equation
with bounded potentials on the whole space. Roughly speaking, such a kind of
inequality says that the total energy of solutions can be controlled by the
energy localized in a subdomain, which is equidistributed over the whole space.
The proof of this inequality is mainly adapted from the parabolic frequency
function method, which plays an important role in proving the unique
continuation property for solutions of parabolic equations. As an immediate
application, we show that the null controllability holds for the heat equation
with bounded potentials on the whole space.
