Point interactions for 3D sub-Laplacians

Riccardo Adami (1) , Ugo Boscain (2, 3, 4) , Valentina Franceschi (5) , Dario Prandi (2, 6, 7)

1 Polito - Politecnico di Torino = Polytechnic of Turin
2 CNRS - Centre National de la Recherche Scientifique
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 CaGE - Control And GEometry
5 Dipartimento di Matematica "Tullio Levi-Civita"
6 Université Paris-Saclay
7 L2S - Laboratoire des signaux et systèmes

In this paper we show that, for a sub-Laplacian $\Delta$ 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 $\Delta$ 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 Schr\"odinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

Mots clés: Sub-riemannian geometry; Heisenberg group; Essential self-adjointness; Point interactions; Sub-Laplacian; Rotation of molecules

Domaines: Equations aux dérivées partielles [math.AP]

Riccardo Adami, Ugo Boscain, Valentina Franceschi, Dario Prandi. Point interactions for 3D sub-Laplacians. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2020, ⟨10.1016/j.anihpc.2020.10.007⟩. ⟨hal-02020844⟩