# Talk by Antoine Prouff

On October 25th, 2023, **Antoine Prouff** (Paris Saclay University) gave a talk about **"Observability of the Schrödinger equation with confining potential"** as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

We consider the Schrödinger equation with a subquadratic confining potential V, in the Euclidean space. Observability inequalities allow to quantify the localisation properties of solutions in an open set U, over some time interval [0, T]. We will give a characterization of the open sets U and times T for which such an inequality holds (up to slightly enlarging U). The observability condition that we find is the result of some form of quantum-classical correspondence: any trajectory of the underlying classical mechanics has to spend a sufficient time in the open set U. This dynamical condition can change dramatically according to the shape of the potential. To illustrate this, we shall take a closer look at the example of harmonic

oscillators, where the potential V is quadratic, and see why the arithmetic properties of the oscillator’s characteristic frequencies matter in this problem.