Semiclassical analysis of dispersion phenomena

TL;DR

This paper provides a quantitative analysis of dispersion in semiclassical Schrödinger equations, focusing on energy localization over time and conditions affecting smoothing estimates.

Contribution

It offers new quantitative insights into energy localization and dispersion phenomena in semiclassical Schrödinger equations in non-compact settings.

Findings

Energy remains localized under certain conditions.

Dispersive effects depend on geometric and initial data.

Results can identify obstructions to smoothing estimates.

Abstract

Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schr{\"o}dinger-type equation in R d. We describe quantitatively the localisation of the energy in a long-time semiclassical limit within this non compact geometry and exhibit conditions under which the energy remains localized on compact sets. We also explain how our results can be applied in a straightforward way to describe obstructions to the validity of smoothing type estimates.