  We study the spectral inequalities of Schr\"odinger operator in the whole
space for different potentials, which can be power growth or continuously
vanishing at infinity. The spectral inequalities quantitatively depend on the
density of the sensor sets with positive measure, growth rate of the potentials
and spectrum (or eigenvalues). One important component in the poof is the
adaptation of propagation of smallness argument for gradients in \cite{LM18}.
As an application, we apply the spectral inequalities to obtain quantitative
observability inequalities for heat equations.
