Quantum-classical correspondence and obstruction to dispersion on the Engel group

In this paper, we develop the semiclassical analysis of the lowest dimensional simply connected nilpotent Lie group of step 3, called the Engel group and denoted by ${\mathbb E}$. We are interested in the propagation of the semiclassical measures associated to solutions of the Schr\"odinger equation for the canonical subLaplacian $\Delta_{\mathbb E}$ and at different time-scales $\tau$. In particular, for $\tau=1$ we recover a quantum-classical correspondence where we observe the fundamental role of abnormal extremal lifts of the Engel group and are able to discuss the speed of the propagation of the singularities. Furthermore, in order to understand the dispersive nature of the subLaplacian, we are led to develop a second-microlocal analysis on particular cones in our phase space. This is done by relating the quasi-contact structure of the group ${\mathbb E}$ to its semiclassical analysis via harmonic analysis. As a consequence, we are able to prove obstruction to local smoothness-type estimates and Strichartz estimates for the Engel subLaplacian.