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We prove a spectral inequality for Schr\"odinger equations with power growthpotentials, which particularly confirms a conjecture in \cite{DSV}. Thisspectral inequality depends on the decaying density of the sensor sets, and thegrowth rate of potentials. The proof relies on three-ball inequalities derivedfrom modified versions of quantitative global and local Carleman estimates thattake advantage of the gradient information of the potentials.

Spectral inequality for Schr\"odinger equations with power growth potentials