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We prove a spectral inequality for Shubin-type operators and more generalSchr\"odinger operators with confinement potentials. The sensor sets areallowed to decay exponentially, where the precise allowed decay rate depends onthe potential. The proof uses an interpolation inequality derived by Carlemanestimates and quantitative weighted $L^2$-estimates for functions in thespectral subspace of the operator.

Spectral inequality with sensor sets of decaying density for Schr\"odinger operators with power growth potentials