The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds

We study the eigenvalues and eigenfunctions of the Folland-Stein operator ${\mathcal{L}}_{\alpha }$ on some examples of 3-dimensional Heisenberg Bieberbach manifolds, that is, compact quotients $\mathrm{\Gamma }\mathrm{\backslash }\mathbb{H}$ of the Heisenberg group $\mathbb{H}$ by a discrete torsion-free subgroup $\mathrm{\Gamma }$ of $\mathbb{H}⋊U(1)$.