Description:

Title: Observability of the Schrődinger equation with confining potential

Abstract:

We consider the Schrődinger equation with a confining potential V, in the Euclidean space. The question of observability consists in investigating the localisation properties of solutions in an open set U, over some time interval [0, T]. We will give an (almost-)characterization of the open sets U that "observe" the Schrődinger equation. The observability condition that we find is the result of some form of quantum-classical correspondence: any trajectory of the Hamiltonian flow has to spend a sufficient time in the open set U. In this setting, the Hamiltonian flow describes the trajectory of a point mass trapped in the potential V, evolving according to Newton’s second law. In particular, it is very sensitive to the profile of the potential. In two dimensions, 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.