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

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.

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.

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We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean L 2-space. Our result then provides a bound for the L 2-norm of such functions in terms of their L 2-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernsteintype inequalities for the respective functions, complemented with a suitable covering of the underlying domain.