Spectral inequality for Schrödinger equations with power growth potentials

We prove a spectral inequality for Schrödinger equations with power growth potentials, which particularly confirms a conjecture in \cite{DSV}. This spectral inequality depends on the decaying density of the sensor sets, and the growth rate of potentials. The proof relies on three-ball inequalities derived from modified versions of quantitative global and local Carleman estimates that take advantage of the gradient information of the potentials.