Quantitative propagation of smallness and spectral estimates for the Schrödinger operator

In this paper, we investigate the quantitative propagation of smallness properties for the Schrödinger operator on a bounded domain in ^d. We extend Logunov and Malinnikova’s results concerning the propagation of smallness for A -harmonic functions to solutions of divergence elliptic equations perturbed by real-valued Lipschitz first-order terms and a real-valued bounded zero-order term. We also prove similar results for the gradients of solutions to certain particular equations. This later result enables us to follow the recent strategy of Burq and Moyano for obtaining spectral estimates on rough sets for the Schrödinger operator. Applications to observability estimates and to the null-controllability of associated parabolic equations posed on compact manifolds or the entire Euclidean space are then considered.