Observability for the Schrödinger Equation: an Optimal Transportation Approach

We establish an observation inequality for the Schr\"odinger equation on $\bR^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94]. This inequality involves only effective constants which are computed explicitly in their dependence in $\hbar$ and all parameters involved.