Hypoelliptic heat kernel on 3-step nilpotent Lie groups

TL;DR

This paper explicitly links the hypoelliptic heat kernel on certain 3-step nilpotent Lie groups to the quartic oscillator, using noncommutative Fourier analysis to transform the heat equation.

Contribution

It provides a novel explicit connection between hypoelliptic heat kernels on specific nilpotent Lie groups and the quartic oscillator, advancing analytical techniques in sub-Riemannian geometry.

Findings

Derived explicit formulas for heat kernels on the Engel and Cartan groups.

Connected hypoelliptic heat equations to one-dimensional quartic oscillator problems.

Applied noncommutative Fourier analysis to simplify complex PDEs.

Abstract

In this paper we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis that permits to transform the hypoelliptic heat equation in a one dimensional heat equation with a quartic potential.