Dependence of High-Frequency Waves with Respect to Potentials

In this article, we consider the wave equation in a bounded domain Ω of R with a potential q. Our goal then is to show that the high-frequency part of the corresponding solutions weakly depends on the potential. We will in particular focus on two instances of interest arising in data assimilation and control theory, respectively corresponding to the problem of recovering an initial data from a measurement and to the problem of computing a control. In these two cases, we derive an explicit bound on the error of the high-frequency part of the solution induced by a W (Ω)-error on the potential for s ∈ (0, 1] and p ∈ (max{d, 2},∞]. In order to do that and to express it in a quantified form, we introduce spectral truncations. Our main tool is a commutator estimate.