Observability for the Schr\"odinger Equation: an Optimal Transport Approach

TL;DR

This paper proves an observability inequality for the Schrödinger equation on Euclidean space, valid uniformly across all Planck constants in [0,1], using an approach based on optimal transport and explicit constants.

Contribution

It introduces a new uniform observability inequality for the Schrödinger equation utilizing an optimal transport framework with explicit dependence on the Planck constant.

Findings

Established a uniform observability inequality in 0 for Schrödinger equation

Derived explicit constants depending on 0 and parameters

Applied pseudometric from Golse and Paul to quantum observability

Abstract

We establish an observation inequality for the Schr\"odinger equation on $\mathbf{R}^d$, 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. 223 (2017), 57-94]. This inequality involves only effective constants which are computed explicitly in their dependence in $\hbar$ and all parameters involved.