We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ (−∞,E] (H L ) of a Schrödinger operator H L on a cube of side L ∈ N, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameterdependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain. Results Scale-free unique continuation and eigenvalue liftingLet d ∈ N. For L > 0 we denote by Λ L = (−L/2, L/2) d ⊂ R d the cube with side length L, and by ∆ L the Laplace operator on L 2 (Λ L ) with Dirichlet, Neumann or periodic boundary conditions. Moreover, for a measurable and bounded V : R d → R we denote by V L : Λ L → R its restriction to Λ L given by V L (x) = V (x) for x ∈ Λ L , and bythe corresponding Schrödinger operator. Note that H L has purely discrete spectrum. For x ∈ R d and r > 0 we denote by B(x, r) the ball with center x and radius r with respect to Euclidean norm. If the ball is centered at zero we write B(r) = B(0, r).Definition 2.1. Let G > 0 and δ > 0. We say that a sequenceCorresponding to a (G, δ)-equidistributed sequence we define for L ∈ GN the set

Abstract. A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite singlesite potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators we establish a finite-volume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy 'Lifshitz tail estimates' on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

Abstract. One of the fundamental results in the theory of localization for discrete Schrödinger operators with random potentials is the exponential decay of Green's function and the absence of continuous spectrum. In this paper we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.

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