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We prove an explicit weighted estimate for the semiclassical Schrödinger operatorP = - h^{2} \partial^{2}_{x} + V(x;h)onL^{2}(\R), withV(x;h)a finite signed measure, and whereh >0is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues forP, and a limiting absorption resolvent estimate with sharph-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support. 

We consider, for h , E > 0, resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E. Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | ) − ρ ) for some ρ > 1. Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ). We find that, for any ρ ˜ > 1, there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ]. The h-dependence of the bound improves if V S decays at a faster rate toward infinity. 

We prove that a large class of N × N Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when $$W < 

We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (Commun Math Phys 259(1):185–221, 2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We furthermore provide a proof of the stability of the Z2 index in the mobility gap regime. These two-dimensional results serve as a model for the study of higher dimensional Z2 indices. 

Abstract We consider the initial‐value problem for a one‐dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges. The key ingredient of the proof is a high‐frequency resolvent estimate for an associated Helmholtz operator with a BV potential. 

Societal-scale data is playing an increasingly prominent role in social science research; examples from research on geopolitical events include questions on how emergency events impact the diffusion of information or how new policies change patterns of social interaction. Such research often draws critical inferences from observing how an exogenous event changes meaningful metrics like network degree or network entropy. However, as we show in this work, standard estimation methodologies make systematically incorrect inferences when the event also changes the sparsity of the data. To address this issue, we provide a general framework for inferring changes in social metrics when dealing with non-stationary sparsity. We propose a plug-in correction that can be applied to any estimator, including several recently proposed procedures. Using both simulated and real data, we demonstrate that the correction significantly improves the accuracy of the estimated change under a variety of plausible data generating processes. In particular, using a large dataset of calls from Afghanistan, we show that whereas traditional methods substantially overestimate the impact of a violent event on social diversity, the plug-in correction reveals the true response to be much more modest.