Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures

TL;DR

This paper investigates the long-time behavior of semi-classical Schrödinger equations on the torus, revealing a threshold time-scale that determines the regularity of semi-classical measures and establishing observability estimates using two-microlocal analysis.

Contribution

It introduces a two-microlocal framework to analyze semi-classical measures, identifying a threshold time-scale for measure regularity and deriving observability estimates for integrable systems.

Findings

Existence of a threshold time-scale for measure regularity.

Below threshold, measures can be singular; beyond, they are absolutely continuous.

Established semiclassical observability estimates for integrable systems.

Abstract

We look at the long-time behaviour of solutions to a semi-classical Schr\"odinger equation on the torus. We consider time scales which go to infinity when the semi-classical parameter goes to zero and we associate with each time-scale the set of semi-classical measures associated with all possible choices of initial data. On each classical invariant torus, the structure of semi-classical measures is described in terms of two-microlocal measures, obeying explicit propagation laws. We apply this construction in two directions. We first analyse the regularity of semi-classical measures, and we emphasize the existence of a threshold : for time-scales below this threshold, the set of semi-classical measures contains measures which are singular with respect to Lebesgue measure in the "position" variable, while at (and beyond) the threshold, all the semi-classical measures are absolutely continuous in the "position" variable, reflecting the dispersive properties of the equation. Second, the techniques of two- microlocal analysis introduced in the paper are used to prove semiclassical observability estimates. The results apply as well to general quantum completely integrable systems.