In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a contact point $q_0\in M$ exists. When $M$ is complete w.r.t. the associated sub-Riemannian structure, this means that $\Delta$ acting on $C^\infty_0(M\setminus{q_0})$ is essentially self-adjoint. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold $N$, whose associated Laplace-Beltrami operator is never essentially self-adjoint on $C^\infty_0(N\setminus{q_0})$, if $\dim N\le 3$. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

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