Spectral inequalities for Schr\"odinger equations with various potentials

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 depends 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 LM18. As an application, we apply the spectral inequalities to obtain quantitative observability inequalities for heat equations.