Semiclassical Analysis of Dispersion Phenomena

Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schrodinger-type equation in \(\mathbf{R}^d\). 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.