Quantum limits of the Martinet sub-Laplacian

TL;DR

This paper investigates the semiclassical behavior of eigenfunctions of the Martinet sub-Laplacian on a flat toroidal cylinder, revealing how they concentrate and oscillate at different scales through advanced microlocal analysis.

Contribution

It introduces adapted two-microlocal semiclassical measures to describe eigenfunction distributions and analyzes their concentration and invariance properties in various regimes.

Findings

Eigenfunctions exhibit concentration near critical points of quartic oscillators.

Effective dynamics are governed by harmonic or anharmonic oscillators depending on the regime.

Regularity properties are established at critical points of eigenvalues.

Abstract

In this article we study the semiclassical asymptotics of the Martinet sub-Laplacian on the flat toroidal cylinder $M = \mathbb{R} \times \mathbb{T}^2$. We describe the asymptotic distribution of sequences of eigenfunctions oscillating at different scales prefixed by Rothschild-Stein estimates via the introduction of adapted two-microlocal semiclassical measures. We obtain concentration and invariance properties of these measures in terms of effective dynamics governed by harmonic or an-harmonic oscillators depending on the regime, and we show additional regularity properties with respect to critical points of the eigenvalues of the Montgomery family of quartic oscillators.