Point interactions for 3D sub-Laplacians
Ugo Boscain, Valentina Franceschi, Dario Prandi, Riccardo Adami · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2020
In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M , no point interaction centered at a point q_{0} \in M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C_{0}^{\infty }(M \setminus \{q_{0}\}) is essentially self-adjoint in L^{2}(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 C_{0}^{\infty }(N \setminus \{q_{0}\}) is never essentially self-adjoint in L^{2}(N) , if \mathrm{\dim }N \leq 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.