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.

Highlights Non-adaptive pooling strategies can increase testing capacity for COVID-19 without sacrifices in detection time. The number of pools needed only grows mildly with pool size and decreasing false positive probability. We calculate explicit bounds on the number of pools required and construct such sets of pools. This can lead to more efficient use of resources and fast detection in realistic scenarios.

Pooling of samples can increase lab capacity when using Polymerase chain reaction (PCR) to detect infections such as COVID-19. However, pool testing is typically performed via an adaptive testing strategy which requires a feedback loop in the lab and at least two PCR runs to confirm positive results. This can cost precious time. We discuss a non-adaptive testing method where each sample is distributed in a prescribed manner over several pools, and which yields reliable results after one round of testing. More precisely, assuming knowledge about the overall infection incidence rate, we calculate explicit error bounds on the number of false positives which scale very favourably with pool size and sample multiplicity. This allows for hugely streamlined PCR testing and cuts in detection times for a large-scale testing scenario. A viable consequence of this method could be real-time screening of entire communities, frontline healthcare workers and international flight passengers, for example, using the PCR machines currently in operation.

We prove and apply two theorems: first, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain; second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz bounds on the function parametrizing locally a chosen edge of the essential spectrum of a Schrödinger operator in dependence of a coupling constant. Analogous estimates for eigenvalues, possibly in gaps of the essential spectrum, are exhibited as well.