Point interactions for 3D sub-Laplacians

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint in L2(M). 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 acting on C0∞(N∖{q0}) is never essentially self-adjoint in L2(N), if dim⁡N≤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.