Quantitative observability for the Schrödinger and Heisenberg equations: An optimal transport approach

Abstract

We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on [Formula: see text], uniform in the Planck constant [Formula: see text]. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in [Formula: see text] and all parameters involved.