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 \(L^2\) $L^2$ -estimates and an \(H^1\) $H^1$ -concentration estimate, all of them for functions in a spectral subspace of the operator.