Calculus of Variations and Geometric Measure Theory

R. Adami - U. Boscain - V. Franceschi - D. Prandi

Point interactions for 3D sub-Laplacians

created by franceschi on 14 Feb 2019
modified on 22 Oct 2020


Accepted Paper

Inserted: 14 feb 2019
Last Updated: 22 oct 2020

Journal: Annales IHP C - Analyse Nonlinéaire
Year: 2019


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 $\operatorname{dim}N\le 3$. 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.

Keywords: Heisenberg group, sub-Riemannian geometry, sub-Laplacian, Essential self-adjointness, point interactions, rotation of molecules