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

TL;DR

This paper explores spectral properties of the sublaplacian on the Engel group, introducing a new Fourier analysis approach, and establishing spectral summability of the quartic oscillator using semiclassical methods.

Contribution

It develops a novel Fourier analysis framework for the Engel group and proves spectral summability of the quartic oscillator spectrum using semiclassical techniques.

Findings

Established fine estimates for convolution kernels on the Engel group.

Proved classical functional embeddings via Fourier analysis.

Demonstrated spectral summability of the quartic oscillator spectrum.

Abstract

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.