  We prove new bounds on the control cost for the abstract heat equation,
assuming a spectral inequality or uncertainty relation for spectral projectors.
In particular, we specify quantitatively how upper bounds on the control cost
depend on the constants in the spectral inequality. This is then applied to the
heat flow on bounded and unbounded domains modeled by a Schr\"odinger
semigroup. This means that the heat evolution generator is allowed to contain a
potential term. The observability/control set is assumed to obey an
equidistribution or a thickness condition, depending on the context.
Complementary lower bounds and examples show that our control cost estimates
are sharp in certain asymptotic regimes. One of these is dubbed homogenization
regime and corresponds to the situation that the control set becomes more and
more evenly distributed throughout the domain while its density remains
constant.
