  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.
