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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 afinite box $\Lambda$ with either Dirichlet or periodic boundary condition. Weprove 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 appropriatepotentials $W\ge 0$ and intervals $I$. As an application, we obtain optimalWegner estimates at all energies for a class of non-ergodic random Schr\"odinger operators with alloy{-type random potentials (`crooked' AndersonHamiltonians). We also prove optimal Wegner estimates at the bottom of thespectrum with the expected dependence on the disorder (the Wegner estimateimproves as the disorder increases), a new result even for the usual (ergodic)Anderson Hamiltonian. These estimates are applied to prove localization at highdisorder for Anderson Hamiltonians in a fixed interval at the bottom of thespectrum.

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