Long-time dynamics of completely integrable Schr\"odinger flows on the torus

TL;DR

This paper investigates the long-time behavior of solutions to semi-classical Schrödinger equations on the torus, revealing a threshold time-scale where measures transition from singular to absolutely continuous in position.

Contribution

It introduces a threshold time-scale for semi-classical measures on the torus, distinguishing between singular and absolutely continuous measures in long-time dynamics.

Findings

Below the threshold, semi-classical measures can be singular in position.

At and beyond the threshold, all measures are absolutely continuous in position.

The results characterize the transition in measure regularity over time.

Abstract

In this article, we are concerned with long-time behaviour of solutions to a semi-classical Schr\"odinger-type 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. 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.