Point interactions for 3D sub-Laplacians

Abstract

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 $C_0^{\infty }(M\setminus \{q_0\left\}\right)$ is essentially self-adjoint in $L^2\left(M\right)$. 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 $C_0^{\infty }(N\setminus \{q_0\left\}\right)$ is never essentially self-adjoint in $L^2\left(N\right)$, 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.