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

2022/06/21 by Hajer Bahouri, Davide Barilari, Bahouri, Hajer +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · #Protein Tyrosine Phosphatases

paper · pdf · doi:10.48550/arxiv.2206.10396

Abstract

In this article, we investigate spectral properties of the sublaplacian -Δ_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(-Δ_G)u=u⋆ 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.

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