In this talk, I will address some questions concerning spectral properties of the sublaplacian $-\Delta_{G}$ on Carnot groups. The attention will focus on the Engel group, which is the main example of a Carnot group of step~3.
Thanks to Fourier analysis on the Engel group in terms of a frequency set, we give fine estimates on the convolution kernel satisfying $F(-\Delta_{G})u=u\star k_{F}$, for suitable scalar functions $F$, proving an interesting summation formula for the spectrum of the sublaplacian.
This analysis requires a summability property on the spectrum of the quartic oscillator, which is of independent interest. If time permits we will discuss possible questions and generalization of this result to more general Carnot groups.
This is a joint work with H.Bahouri, I.Gallagher and M.Léautaud