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
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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.
