Unique continuation principle for spectral projections of Schr\" odinger operators and optimal Wegner estimates for non-ergodic random Schr\" odinger operators

We prove a unique continuation principle for spectral projections of Schr\" odinger operators. We consider a Schr\" odinger operator $H= -\Delta + V$ on $\mathrm{L}^2(\mathbb{R}^d)$, and let $H_{\Lambda}$ denote its restriction to a finite box $\Lambda$ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type $\chi_I (H_\Lambda) W \chi_I (H_\Lambda) \ge \kappa\, \chi_I (H_\Lambda) $ with $\kappa >0$ for appropriate potentials $W\ge 0$ and intervals $I$. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schr\" odinger operators with alloy{-type random potentials (`crooked' Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.