Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials

We prove a spectral inequality (a specific type of uncertainty relation) for Schrödinger operators with confinement potentials, in particular of Shubin-type. 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, quantitative weighted L² L 2 -estimates and an H¹ H 1 -concentration estimate, all of them for functions in a spectral subspace of the operator.