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

. We prove a spectral inequality for Shubin-type operators and more general Schrödinger operators with confinement potentials. The sensor sets are allowed to decay exponentially, where the precise allowed decay rate depends on the potential. The proof uses an interpolation inequality derived by Carleman estimates and quantitative weighted L 2 -estimates for functions in the spectral subspace of the operator.

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