Spectral summability for the quartic oscillator with applications to the Engel group

Hajer Bahouri (1) , Davide Barilari (2) , Isabelle Gallagher (3, 4, 5) , Matthieu Léautaud (6)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 Dipartimento di Matematica "Tullio Levi-Civita"
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 UPCité - Université Paris Cité
5 UPD7 - Université Paris Diderot - Paris 7
6 LMO - Laboratoire de Mathématiques d'Orsay

In this article, we investigate spectral properties of the sublaplacian $-\Delta_{G}$ on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying $F(-\Delta_{G})u=u\star k_{F}$, for suitable scalar functions $F$, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.

Mots clés: functional embeddings; sub-Riemannian geometry; Carnot groups; Quartic oscillator; spectral analysis; semiclassical analysis; Engel group

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Théorie spectrale [math.SP]

Hajer Bahouri, Davide Barilari, Isabelle Gallagher, Matthieu Léautaud. Spectral summability for the quartic oscillator with applications to the Engel group. 2022. ⟨hal-03699288⟩